Derived Manifolds from Functors of Points [1 ed.] 9783832591649, 9783832534059

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22

Augsburger Schriften zur Mathematik, Physik und Informatik

Derived Manifolds from Functors of Points

Franz Vogler

λογος

Franz Vogler

Derived Manifolds from Functors of Points

λογος

Augsburger Schriften zur Mathematik, Physik und Informatik Band 22 herausgegeben von: Professor Dr. F. Pukelsheim Professor Dr. W. Reif Professor Dr. D. Vollhardt

• Erstgutachter: Prof. Dr. Marc A. Nieper-Wißkirchen • Zweitgutachter: Prof. Dr. Marco Hien • Tag der m¨ undlichen Pr¨ ufung: 26. Februar 2013

Bibliografische Information der Deutschen Nationalbibliothek Die Deutsche Nationalbibliothek verzeichnet diese Publikation in der Deutschen Nationalbibliografie; detaillierte bibliografische Daten sind im Internet u ¨ber http://dnb.d-nb.de abrufbar. c

Copyright Logos Verlag Berlin GmbH 2013 Alle Rechte vorbehalten.

ISBN 978-3-8325-3405-9 ISSN 1611-4256 Logos Verlag Berlin GmbH Comeniushof, Gubener Str. 47, 10243 Berlin Tel.: +49 030 42 85 10 90 Fax: +49 030 42 85 10 92 INTERNET: http://www.logos-verlag.de

Begriffe ohne Anschauung sind leer, Anschauung ohne Begriffe ist blind. [Kant 1787]

Contents 1 Introduction 1.1 Motivation and Accomplishments 1.2 Organisation of the Chapters . . 1.3 Background and Notation . . . . 1.4 Acknowledgements . . . . . . . .

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2 Analysis of Model Structures 2.1 Objectwise Adjoints . . . . . . . . . . . . . . 2.2 The Global Model Structures . . . . . . . . . 2.3 Simplicial Enrichment of Model Categories . . 2.4 Absolute derived Functors . . . . . . . . . . . 2.5 Homotopy Limits as absolute derived Functors 2.6 The left Bousfield Localization . . . . . . . . . 2.7 Sites and their Morphisms . . . . . . . . . . . 2.8 Homotopy Sheaves . . . . . . . . . . . . . . . 3 Smooth Functors as C∞ -Schemes 3.1 Commutative Algebra with C∞ -Rings 3.2 C∞ -rings and Topology . . . . . . . . 3.3 C∞ -Schemes from Smooth Functors . 3.4 The Big Structure Sheaf . . . . . . . 3.5 C∞ -ringed Spaces as C∞ -Schemes . . 3.6 From Smooth Functors to C∞ -Ringed

. . . . . . . . . . . . . . . . . . . . Spaces

4 Derived Manifolds as Functors 4.1 The Layout for Smooth Rings . . . . . . . 4.2 Topology for Smooth Rings . . . . . . . . 4.3 Derived Manifolds from Smooth-Simplicial Functors . . . . . . . . . . . . . . . . . . . 4.4 The Global Structure Sheaf . . . . . . . . 4.5 Derived Manifolds as Ringed Spaces . . . . 4.6 The Functor approach to LRS . . . . . . .

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124 129 135 139

Chapter 1 Introduction 1.1

Motivation and Accomplishments

The work on ”Derived Manifolds from Functors of Points” touches four areas of mathematics: Homotopical Algebra invented by D. Quillen, Synthetic Differential Geometry developed by E. Dubuc, A. Kock and I. Moerdijk, Homotopical Algebraic Geometry in the sense of B. To¨en and G. Vezzosy, and smooth Derived Manifolds after D. Spivak as a part of J. Lurie’s Derived Algebraic Geometry programme. The purpose of this dissertation is to give a categorial approach to the construction of geometrical objects, especially to find a functorial construction of derived manifolds in the sense of Spivak. To accomplish this one has to pass from ringed spaces to functors as local models. The theory of manifolds and the theory of schemes are both based on the concept of local models. A manifold is a local euclidian space, i.e. something that looks locally like some Rn , and a scheme is an object that is given locally by the spectrum of some commutative ring with unit. To specify this one has to discuss rings, ringed spaces, locally ringed spaces, and affine schemes as well as one has to consider manifolds as ringed spaces. Considering manifolds as locally ringed spaces has the advantage to facilitate generalizations. All this is studied by graduate students in their first courses on Differential Geometry and Algebraic Geometry. However, there is another way to look at schemes as it is done by the authors of [DeG 70, Ch. I, § 1] and [Eis 00, Ch. VI]. From their point of view affine schemes are ◦ defined to be functors in the essential image of the Yoneda functor h(·) : Aff → SetAff where Aff denotes the dual of the category Ringfp of finitely presented commutative rings with unit. One can turn Aff into a site by defining a set of morphisms (A → A[a−1 i ] | i ∈ I) to be an open co-cover of A ∈ Ringfp if and only if the ideal J := (ai | i ∈ I) is the unit ideal in A as it is discussed in [Mac 92, Ch. III, § 4]. More generally, according to [DeG 70, Ch. I, § 1, 3.11] a scheme X is a functor X ∈ SetRingfp that is a sheaf for the Grothendieck site Aff that can be covered by a family (Xi )i∈I of affines. Since the Grothendieck site Aff is subcanonical, affine schemes are indeed schemes by definition. Using the Grothendieck topology on Aff one can define open subfunctors as well as it was done in [Eis 00, Ch. VI, § 1, Def. VI-5]: A subfunctor α : G → F in SetRingfp is an open subfunctor if, for each 3

4

Chapter 1: Introduction

morphism ψ : X → F from the functor X represented by an affine scheme, the base change of α along ψ yields a morphism ψ −1 G → X isomorphic to the injection from the functor represented by some open subscheme. This way open subfunctors of affine schemes are precisely those represented by open maps, i.e. by arrows of some open co-cover. These considerations were used in [To¨e 07] to generalize the notion of schemes. As mentioned above, it is easier to generalize manifolds considering manifolds as locally ringed spaces. For instance, the notion of C∞ -schemes as generalization of manifolds appeared first in [Dub(a) 81]. In his work [Dub(a) 81] E. Dubuc aimed at incorporating Differential Geometry into the field of applicability of the techniques and tools of Algebraic Geometry. Even more important, he constructed a model of Synthetic Differential Geometry sufficiently general and with enough good properties such that synthetic reasoning is valid. The theory of C∞ -schemes has not been developed very far in Synthetic Differential Geometry because after at least one model was found where synthetic arguements are happening the job was done as far as Synthetic Differential Geometry is concerned. After a while D. Joyce, whose interest in this area comes from Symplectic Geometry, layed down in [Joy 09] the foundations of Algebraic Geometry over C∞ -rings, in which he replaces commutative rings in classical Algebraic Geometry by C∞ -rings to develop the theory of C∞ -schemes much further than it was done by the Synthetic Differential geometers. C∞ -rings appeared in a very different context as part of the definition of derived manifolds. D. Spivak extended in [Spi 07] parts of J. Lurie’s Derived Algebraic Geometry programme [Lur 11] to Differential Geometry. The construction given by D. Spivak is complex and technical, using simplicial methods and Homotopical Algebra in the sense of D. Quillen [Qui 67], therefore his derived manifolds form a simplicial model category, i.e. a kind of ∞-category. Both concepts of generalizing manifolds, the one of C∞ -schemes from Synthetic Differential Geometry and the one of derived manifolds from the Derived Algebraic Geometry programme, have one thing in common: They use the language of ringed spaces from Algebraic Geometry to develop their theory. In the light of the functorial approach to classical schemes we mentioned above, the first question one may ask at this point is whether there is a way to obtain both the theory of C∞ -schemes as well as the theory of derived manifolds from functors. The answer is: Yes, there is a functorial approach to the theory of C∞ -schemes as well as to the theory of derived manifolds as will be shown in this thesis. To accomplish this we derive much of our inspiration from [To¨e 07]. In their work [To¨e 07] B. To¨en and M. Vaqui´e start with the functorial point of view to generalize schemes focusing on the quality characteristics that the Grothendieck topology of the affine building stones has to satisfy. As a next step they incorporate their approach to the homotopical ∞ setting. We follow this and define a functor F ∈ SetC Ringfp from the category of finitely presented C∞ -rings to sets to be a C∞ -scheme if it is a sheaf for the Grothendieck topology on C∞ Ring◦fp we introduce following [Dub(b) 81] and if in addition F can be covered by affines. Since the Grothendieck topology from [Dub(b) 81] does not enjoy the properties required by [To¨e 07], we have to check step by step how far the ideas can be transferred. It turns out that the set of open subfunctors SubC∞ (F ) associated with a C∞ -scheme F forms a frame. Hence its dual is a locale, i.e. nothing different than the category of opens

1.2 Organisation of the Chapters

5

Op(X) associated with a topological space X. Beyond the results of [To¨e 07] we provide an ∞ explicit construction of how to assign locally ringed spaces to C∞ -schemes F ∈ SetC Ringfp in a functorial way. It turns out that the C∞ -schemes one obtains by the ringed spaces approach from [Dub(a) 81], [Joy 09], [Quˆe 87], built up by finitely presented C∞ -rings, are in the essential image of the functorial assignment. More precisely, up to isomorphism there is no difference between the functorial and the classical ringed space approach. Using the experience from C∞ -schemes we carry over the functorial theory to the homotopical setting by substitution of the Yoneda functor with the simplicial mapping space functor. In contrast to [To¨e 07] we develop the theory on the level of model categories and not on the associated homotopy categories. We define affine derived manifolds to be simplicial presheaves on a certain subcategory of the simplicial model category of smooth rings that belong to the essential image of the simplicial mapping space functor. Derived manifolds in this context are defined to be simplicial homotopy sheaves that possess a cover by affines. The proof that every affine derived manifold is a derived manifold becomes intricate. Nevertheless, this can be accomplished by manipulating the model structure in an appropriate way. For every derived manifold we define open subfunctors and the set of all open subfunctors forms also a locale. As in the case of C∞ -schemes we associate locally ringed spaces to derived manifolds. It will be proved that this assignment gives a simplicial functor. At the end we prove that the derived manifolds coming from simplicial homotopy sheaves and those from ringed spaces that both possess a finite cover by affines are equivalent as ∞-categories.

1.2

Organisation of the Chapters

The first Chapter gives the introduction to the thesis. Chapter two serves as a recollection of fundamental facts about model categories which we will use throughout the thesis. We are going to introduce the direct and inverse image functor for presheaves as special Kan extensions and show that they behave well with respect to global and local model structures on simplicial model categories. The concept of homotopy limits will be explained putting two points of view together. One of G. Maltsiniotis [Mal 07] and one of W. Dwyer [Dwy 95], the first one introducing homotopy limits in the setting of arbitrary derived categories and the second approaching to homotopy limits on model categories via derived functors. In Chapter three we start with an overview over the theory of C∞ -rings which will be used to build C∞ -schemes. By means of a Grothendieck topology on the dual Aff ∞ of ∞◦ a full subcategory of the C∞ -rings, we introduce C∞ -schemes as functors F ∈ SetAff that are sheaves with respect to the topology and that possess a cover by affines, i.e. by ∞◦ subfunctors that one obtains as image of the embedding Aff ∞ → SetAff . To every ∞◦ C∞ -scheme F ∈ SetAff we will assign a locally C∞ -ringed space in a functorial way. It turns out that the locally C∞ -ringed spaces in the essential image of the assignment are precisely those one obtains by the classical approach to C∞ -schemes via ringed spaces when restricted to finitely presented C∞ -rings. To that end the category of locally C∞ -ringed

6

Chapter 1: Introduction

spaces that are locally covered by finitely presented C∞ -schemes is equivalent to those ∞◦ C∞ -schemes coming from functors in SetAff . The last chapter is the heart of the thesis. We put our results from chapter one and two together to develop a functorial theory towards derived manifolds. Replacing sets by simplicial sets, C∞ -rings by smooth rings and colimits by homotopy colimits we get the layout in the homotopical setting. With the results from chapter two we can turn the dual of an appropriate subcategory of smooth rings into a site and define derived manifolds as simplicial set valued homotopy sheaves on this site that possess a cover by affine subfunctors. Finally, we show that the subcategory of derived manifolds with finite cover is equivalent to the full subcategory of derived manifolds in ringed spaces that are covered by a finite set of affines in the sense of ∞-categories.

1.3

Background and Notation

The reader is assumed to be familiar with category theory and to have some basic background on model categories as described in [Hov 99] in the first three chapters. In contrast to [Hov 99] we follow Quillen’s original work [Qui 67] on Homotopical Algebra and assume model categories not to possess functorial fibrant and cofibrant replacements. The symbol [n] refers to the linear ordered category with n + 1 objects. The set of all linear ordered categories can be organized itself into a category we will denote by ∆ and whose terminal object is [0] with one object and one morphism. Given any category C, the notation C ◦ will always denote its dual. A diagram in a category C is a functor F : I → C, where I is a small category. We denote the category of functors from I to C and natural transformations between them by C I . If F : C → D is a functor, then it induces a functor FI : C I → DI and we generally say that F induces FI object-wise. The notation F a G will always denote an adjoint pair, in which F is the left adjoint and G is the right adjoint. For example, sSet is the category of all functors ∆◦ → Set and we will use the term level-wise instead of object-wise. The n-th simplicial level of a simplicial set X ∈ sSet will be notated Xn . A simplicial set S ∈ sSet will be called discrete or constant if its non-degenerate simplices are concentrated in degree 0. In other words, S is in the essential image of the functor c∗ : Set → sSet which takes an object S : [0] → Set to the composition ∆◦ → [0] → Set where the first arrow is the terminal map. Note that c∗ possesses a left adjoint denoted by π0 . Because of the fully faithful right adjoint c∗ : Set → sSet we sometimes will not differentiate between discrete simplicial sets and sets. In the case one has two functors F and G from C to D together with a natural transformation α : F → G and two other functors H and K from D to E with a morphism β : H → K we will always denote by β ? α the horizontal composition1 , i.e. the natural transformation β ? α : H ◦ F → K ◦ G which is determined on objects X via (β ? α)X = βGX ◦ H(αX ) = K(αX ) ◦ βF X . For the vertical composition, i.e. the composition η ◦ τ : F → H of two natural transformations η : F → G and τ : G → H where F, G, H : C → D are three functors, 1

In [Bor 94] it is called the Godement product.

1.4 Acknowledgements

7

we simply suppress the composition sign and write ητ . We assume a lot of familiarity with the properties of limits and colimits in categories. The symbol ∆I denotes the constant diagramm functor I → C I . Let F ∈ C I be an arbitrary I-diagram. Recall that a colimit for F is an object C such that there is a bijection Hom(C, X) = Hom(F, ∆I (X)) of sets. This can also be read as follows. A colimit of F is a pair (C, t) where C is an object of C and t : F → ∆I (X) is a natural transformation such that for every other object Y of C and every natural transformation s : C → ∆I (Y ) there exists an unique map r : C → Y in C such that the equation ∆I (r) ◦ t = s holds (cf. [Mac 91, Ch. III, § 3]). In this case we write − colim −−−→F and say ”the left adjoint of the constant diagram ∆I is defined at F ”. Dually, we write ← lim −−F in the case that the right adjoint of the constant diagram ∆I is defined at F . The symbol ← holim −−−−F stands for the homotopy limit and −hocol −−−→F for the homotopy colimit of F . When computing homotopy colimits and homotopy limits we always assume the result to be cofibrantly-fibrantly replaced. When dealing with topological spaces we will occasionally write U ⊂◦ X to indicate that a subset U ⊂ X of a topological space X is open and Op(X) will always indicate the ordered category of opens associated to the space X. We will write SC for the site whose Grothendieck topology is induced by some Grothedieck pretopology on a category C (cf. [Moe 91, Ch. III] for a detailed discussion of Grothendieck topologies and pretopologies). The category of set-valued presheaves will sometimes be written SC∧ , and SC∼ refers to the full reflective subcategory of sheaves. In this thesis manifolds are always assumed to be smooth and finite dimensional and we denote their category by Man. The terminal object in Man as well as in the category Top of topological spaces will be notated as R0 .

1.4

Acknowledgements

I would like to thank my advisor Prof. Dr. Marc A. Nieper-Wißkirchen for giving me the opportunity to do research in this area at his chair and for conducting me with encouraging and helpful discussions.

Chapter 2 Analysis of Model Structures In this section we collect our basic tools we are going to work with throughout the thesis. First of all we introduce a sequence of adjoint functors. After that we define the global model structures that will allow us to carry over the model structure of a given category to its category of diagrams. In paragraph three we recall the definition of a simplicial model category and we point out that the adjoint functors from paragraph one respect both the global model structures and the simplicial structure of simplicial model categories. Moreover, we recall the definition of the category of elements and extend it to the simplicial setting. After that we present the notion of derived functors on localized categories in the context of absoluteness and we apply it to Quillen functors. As a special case we will treat homotopy limits and homotopy colimits in this context. The last sections of this chapter focus upon local model structures. It is common use to say that a local model structure on a category of presheaves is one whose weak equivalences are not defined objectwise but by the help of covers on some site. We use this local weak equivalences to pass from presheaves to sheaves in the homotopical setting as it was done in [DHI 04]. To do so we shortly list some facts about the left Bousfield localization of model categories and we show how to detect classical sheaves in the local model structure. Then we show that the adjoint functors of paragraph one respect the local model structure and in the end we show that the simplicial mapping space presheaf is local with respect to this model structure, i.e. is a homotopy sheaf.

2.1

Objectwise Adjoints

In order to compare model categories one has to consider morphisms between them. Such morphisms are given by a pair of adjoint functors that respect the model structure in a certain way. Since we are going to deal with global model structures in the next chapter, we start with a special kind of adjoint functors we are going to discuss now. Definition 2.1.1 Let f : C → D be a functor between small categories and K be a complete (respective cocomplete) category, i.e. arbitrary small limits (respective small 9

10

Chapter 2: Analysis of Model Structures

colimits) exist. The restriction of functors f∗ : K

D◦

→K

C◦

given by P 7→ P ◦ f induces a functor f∗ : K

C◦

→K

D◦

defined on objects via f∗ P (d) :=

lim ← −−

P (c)

(f (c)→d)∈(f ↓d)

and f! : K

C◦

→K

D◦

via f! P (d) :=

−colim −−−→

P (c)

(d→f (c))∈(d↓f )

respectively. Theorem 2.1.2 Let K be a complete and cocomplete category and f : C → D a functor between small categories. Then there is a sequence f! a f ∗ a f∗ of adjoint functors. Proof:

We prove that f! is left adjoint to f ∗ . We have to show that there is a bijection HomK D◦ (f! P, F ) = HomK C◦ (P, f ∗ F )

of sets. Taken a natural transformation from the right hand side we can associate objectwise exactly one morphism f! P (d) → F (d) because of the universal property of the colimit. That these maps define indeed a natural transformation from f! P to F is an immediate consequence of the functorality of the colimit. The adjointness f ∗ a f∗ follows in a similar fashion. q.e.d. Remark. Note that if D has binary products the category (d ↓ f ) is filtered for every ◦ ◦ d ∈ D and since the functor f! : K C → K D is defined by filtered colimits, f! commutes with finite limits by [Mac 91, Ch. IX, § 2, Thm. 1]. Thus, if we replace K by the topos Set of sets (or any other topos), the pairs (f! , f ∗ ) and (f ∗ , f∗ ) define geometric morphisms. In the case that f possesses a left adjoint g one can enlarge the sequence of adjoint functors to g! a f! a f ∗ a f∗ where g ∗ ∼ = f! and g∗ ∼ = f ∗ (cf. [SGA4 72, Expos´e I, § 5, Prop. 5.5]).

11

2.1 Objectwise Adjoints Proposition 2.1.3 In the situation from above both the counit f ∗ f ∗ → 1K C ◦ of the adjunction f ∗ a f∗ and the unit 1K C ◦ → f ∗ f ! of the adjunction f! a f ∗ are isomorphisms. Proof:

Let P ∈ K

C◦

and c ∈ C be any object. Then we compute

f ∗ f∗ P (c) = (f∗ P ) (f (c)) = ← lim −− P (a) = P (c), (f ↓f (c))

because (c, f (c) → f (c)) is terminal in (f ↓ f (c)). Similarly we have f ∗ f! P (c) = (f! P ) (f (c)) = − colim −−−→P (a) = P (c), (f (c)↓f )

since now (c, f (c) → f (c)) is the initial object in (f (c) ↓ f ). q.e.d. Recall that the left Kan extension (Lanf P, α) of a functor P : C ◦ → K along f : C → D is a tuple consisting of a functor Lanf P : D◦ → K and a morphism α : P → Lanf P ◦ f which is universal from the left (which explains the name left Kan extension), i.e. for every other couple (G, β) where G : D◦ → K is a functor and β : P → G ◦ f is a morphism of functors there is a unique natural transformation γ : Lanf P → G such that β = (γ ? 1P )α (here 1P denotes the identical natural transformation on P ). Of course a left Kan extension, if it exists, is unique up to unique isomorphism. The notion of right Kan extension (Ranf P, ω) is defined in the dual way. With this in mind we can conclude the following. Corollary 2.1.4 With the notation from above we have (f! P, 1P ) = (Lanf P, α) and (f∗ P, 1P ) = (Ranf P, ω) . In other words, the left adjoint f! of the restriction of functors f ∗ and its right adjoint f∗ respectively form a left Kan extension of P along f and a right Kan extension of P along f respectively.

12 Proof:

Chapter 2: Analysis of Model Structures We have α : P → Lanf P ◦ f = f ∗ Lanf P = f ∗ f! P = P

where the last equality follows from Proposition 2.1.3. Thus α := 1P is initial from the left. The same reasoning proves the statement about f∗ . q.e.d. Remark.

In the case of the restriction of covariant functors K : C → K we have f! K(d) = Lanf K(d) =

−colim −−−→

K(c)

lim ← −−

K(c).

(f (c)→d)∈(f ↓c)

and f∗ K(d) = Ranf K(d) =

(d→f (c))∈(d↓f )

This case is treated in [Mac 91, Ch. X, §3, Th. 1] for the right Kan extension under the headline the Kan extension as a point-wise limit. We have chosen the notation f! and f∗ for the adjoint functors of f ∗ in honor of J. L. Verdier such that our notation agrees with SGA (cf. [SGA4 72, Expos´e III]). Later on after introducing sites we will switch notation in order to emphasize the geometric point of view.

2.2

The Global Model Structures

In this paragraph we recall the notion of global model structures on functor categories. After recollecting some properties of the two global model structures, the injective and the projective one, we point out that the adjoint functors from the preceeding paragraph respect the global model structures. First of all we begin with the definition of morphisms between model categories. Definition 2.2.1 Let F : M → N and G : N → M be two functors between model categories that are an adjoint pair with F as the left adjoint. We call F a left Quillen functor if it preserves cofibrations and we call G a right Quillen functor if it preserves fibrations. The pair (F, G) with F a left and G a right Quillen functor is called a Quillen adjunction. A Quillen adjunction is an (elementary) Quillen equivalence if and only if the adjoint pair F a G induces an equivalence of the associated homotopy categories. Following [Hov 99, Ch. 1, § 1.4] one can organize all model categories in a bicategory. Objects are model categories, 1-morphisms are Quillen pairs, and 2-morphisms are natural transformations between the left Quillen parts. Thus, we will simply refer to a morphism of

2.2 The Global Model Structures

13

model categories (F, G) : N → M in the case where (F, G) is an Quillen adjunction. Here the arrow is in the direction of the right adjoint as in the case of geometric morphisms, where the arrow is drawn in the direction of the direct image parts. Definition 2.2.2 We will say that M is Quillen equivalent to N iff there is a chain M → K1 ← K2 → · · · ← N of elementary Quillen equivalences between M and N . Lemma 2.2.3 The condition that an adjoint pair F a G forms a Quillen adjunction is equivalent to either one of the following two conditions: (1) F preserves cofibrations and acyclic cofibrations. (2) G preserves fibrations and acyclic fibrations. Proof: The proof is an easy exercise playing around with the lifting properties and using the adjointness of the functors involved. q.e.d. Example 2.2.4 Let Top be the model category of topological spaces and sSet the model category of simplicial sets. Then the geometric realization | · |gr : sSet → Top and the singular functor Sing : Top → sSet give a morphism (| · |gr , Sing) : Top → sSet of model categories. This is even a Quillen equivalence and hence an equivalence of model categories (cf. [Qui 67, Ch. I, § 4.8, Examples, 2.]). The following lemma is taken from [Hov 99]. Lemma 2.2.5 (The Ken Brown–lemma) Let M be a model category and (C, W ) be a localizer (see Definition 2.4.2 below) whose localizing class W satisfies the 2-out-of-3–axiom1 . Suppose F : M → C is a functor that takes acyclic cofibrations between cofibrant objects to W . Then F takes all weak equivalences between cofibrant objects to morphisms in W . Proof:

Given a weak equivalence f : A → B the principal idea is to factor (f, idB ) : A q B → B

into a a cofibration followed by an acyclic fibration. The details are carried out in [Hov 99, Ch. 1, § 1.1, Lemma 1.1.12]. q.e.d. The dual statement about acyclic fibrations between fibrant objects is true as well. As a consequence we have 1

If f and g are morphisms in C such that g ◦ f is defined and two of f , g, g ◦ f are in W , then so is the third.

14

Chapter 2: Analysis of Model Structures

Corollary 2.2.6 Left Quillen functors preserve weak equivalences between cofibrant objects and right Quillen functors preserve weak equivalences between fibrant objects. There are two global model structures, the projective and the injective one, which are frequently used and of highest importance for our purposes. Definition 2.2.7 Let M be a model category and I be a small category. The projective model structure on MI is given, if it exists, in the following way. The weak equivalences and fibrations in MI are determined object-wise and the cofibrations are defined to be those maps which have the left lifting property with respect to the maps which are both weak equivalences and fibrations in MI . A model category M is called I-projective if MI can be equipped with the projective model structure. A model category M is projective if it is I-projective for every small category I. The other global model structure is defined in the dual way, i.e. for the injective model structure weak equivalences and cofibrations on MI are defined object-wise and the fibrations are determined by the right lifting property with respect to acyclic cofibrations. The category of simplicial sets will be of particular interest throughout the thesis. Hence we turn our attention towards the following Example 2.2.8 The category sSet of simplicial sets is both projective and injective. A proof for the projectivity of sSet is given in [Bou 73, Ch. XI, § 8, Proof of 8.1]. Injectivity is discussed in [Hel 88]. Theorem 2.2.9 Assume I = {a ← b → c} is a pushout index category and J = {a → b ← c} is a pullback index category. Then every model category M is I-projective and J-injective. Proof: We give an explicit description on how to equip MI with the projective and MJ with the injective model structure using results of [Dwy 95]. Therefore let f: X →Y be a morphism in MI . Now define f∗ X(a) := − colim −−−→ (Y (b) ← X(b) → X(a)) , f∗ X(c) := − colim −−−→ (Y (b) ← X(b) → X(c)) , and f∗ X(b) := Y (b). Note that the last definition fits into this picture as the cobase change along the identity gives the identity. There are maps ia : f∗ X(a) → Y (a), ic : f∗ X(c) → Y (c) and ib = id : f∗ X(b) → Y (b) obtained by the universal property of colimits.

15

2.2 The Global Model Structures

Now define f : X → Y to be a weak equivalence or fibration iff fa , fb and fc are weak equivalences or fibrations in M and let f be a cofibration if and only if ia , ib and ic are cofibrations in M. In [Dwy 95, § 10.4, Prop. 10.6] it is proved that this determines a model structure on MI . This is the projective model structure since weak equivalences and fibrations are determined object-wise. Thus, the cofibrations are fixed by the lifting property. Dually, for the injective model structure on MJ define for a morphism f : X → Y in MJ the objects fb−1 Y (a) := ← lim −− (X(b) → Y (b) ← Y (a)) , fb−1 Y (b) := X(b) and fb−1 Y (c) := ← lim −− (X(b) → Y (b) ← Y (c)) . Let pa : X(a) → fb−1 Y (a), pa : X(b) → fb−1 Y (b) and pa : X(c) → fb−1 Y (c) be the induced maps. Now define f to be a cofibration or weak equivalence iff fa , fb and fc are such in the model structure of M and let f be a fibration iff the induced maps pa , pb , pc are fibrations in M. These choices provide a model structure on MJ which coincides with the injective one since fibrations are already determined by the left lifting property with respect to acyclic cofibrations. q.e.d. The injective model structure on sSetI has the following advantage: Lemma 2.2.10 For the injective model structure on sSetI every object is cofibrant. Proof: This is an immediate consequence from the fact that the object ∅ ∈ sSetI defined via i 7→ ∅ is initial and cofibrations in sSetI are determined object-wise. q.e.d. Lemma 2.2.11 Let M be I-projective. A cofibration in the projective model structure on MI is also an object-wise cofibration. Proof: This can be found in [Hir 03, Part II, § 11.6, Prop. 11.6.3] or checked using the lifting properies. q.e.d. Recall from the introduction that an object is simplicially constant or simply discrete if it is in the image of the constant simplicial set functor c∗ : Set → sSet ,

X 7→ ([n] 7→ X).

The constant simplicial set functor possesses a left adjoint π0 , the simplicial path component functor. We will always identify ordinary sets with their image under c∗ in the simplicial context.

16

Chapter 2: Analysis of Model Structures

Lemma 2.2.12 For the injective model structure on sSetI any object that is constant in the simplicial direction is fibrant. Proof:

Let X → Y be an acyclic cofibration in the commutative diagram X −−−→   qy

F   y

Y −−−→ ∆[0], where F is simplicially constant and ∆[0] the terminal object in sSet. We have to show that there is a lift l : Y → F in the above square making both resulting triangles commute. Since we are working in the injective model structure we see that for every object i of I the map qi in X (i) −−−→ F (i)     qi y y Y (i) −−−→ ∆[0], remains an acyclic cofibration. Our main tool to show the existence of li is the fact that the simplicial path component functor π0 from simplicial sets to the category of sets is left adjoint to the constant simplicial set functor c∗ . Without loss of generality we may assume the objects on the left hand side in the above diagram to be fibrant. Thus, we see in particular that π0 (X (i) → Y (i)) is an isomorphism which provides a map π0 (Y (i)) → F (i). Using the adjointness π0 a c∗ with the constant simplicial set functor c∗ as the right adjoint, of which F is in the image, we get our map li : Y (i) → F and the set of all li gives the desired lift. q.e.d. Lemma 2.2.13 Being fibrant in the injective model structure implies being object-wise fibrant. Proof: This is an easy consequence of the fact that cofibrations and weak equivalences are determined object-wise. q.e.d. Recall that a presheaf is said to be representable if and only if it is in the essential image of the Yoneda functor (cf. Example 2.3.24). Lemma 2.2.14 Every representable presheaf is cofibrant for the projective model structure on sSetI . Lemma 2.2.15 Simplicially constant objects are fibrant with respect to the projective structure.

2.2 The Global Model Structures

17

Proof: Since cofibrations for the projective model structure are also object-wise cofibrations, the proof of the injective case carries over to the projective one. q.e.d. We will use both the injective and the projective model structure throughout the thesis. Therefore we give the following Definition 2.2.16 For a small category C and a C ◦ -injective model category M let Pre(C, M) ◦

denote the category MC of presheaves on C with values in M equipped with the injective model structure. Dually, in the case where M is C ◦ -projective let Pro(C, M) ◦

be the presheaf category MC with the projective model structure. The following proposition says that the two global model structures are Quillen equivalent. Proposition 2.2.17 Let M be a model category and C be a small category such that M is both C ◦ -injective and C ◦ -projective. Then the identity functor 1MC◦ gives a morphism (1MC◦ , 1MC◦ ) : Pre(C, M) → Pro(C, M) of model categories which is in fact a Quillen equivalence. Proof: By Lemma 2.2.11 the identity functor 1MC◦ : Pro(C, M) → Pre(C, M) preserves cofibrations. Since the class of weak equivalences are the same for both global model structures, the identity functor preserves acyclic cofibrations as well. Due to Lemma 2.2.3 the couple (1MC◦ , 1MC◦ ) is a morphism of model categories. The claim about the Quillen equivalence is immediate from the fact that the weak equivalences are determined object-wise in both model categories. q.e.d. Proposition 2.2.18 Let f : C → D be a functor between small categories. Then the sequence f! a f ∗ a f∗ of adjoints gives two morphisms (1) (f! , f ∗ ) : Pro(D, M) → Pro(C, M) and

18

Chapter 2: Analysis of Model Structures

(2) (f ∗ , f∗ ) : Pre(C, M) → Pre(D, M) of model categories. Proof: The restriction of functors f ∗ preserves fibrations and acyclic fibrations for the projective model structure as well as cofibrations and acyclic cofibrations for the injective model structure. Using Lemma 2.2.3 we are done. q.e.d. Proposition 2.2.19 Let M be a model category whose class of cofibrations is closed under filtered colimits and f : C → D be a functor between small categories. Assume D to have binary products, i.e. f! is defined via a filtered colimit. Then (f! , f ∗ ) : Pre(D, M) → Pre(C, M) is a morphism of model categories. Proof: Let φ : F → G be a cofibration in Pre(C, M). We have to check on objects that f! (φ) : f! F → f! G is a cofibration in Pre(D, M). But this is clear by assumption since f! (φ)(U ) = − colim colim colim −−−→φV : − −−−→F (V ) → − −−−→G(V ) (U ↓f )

(U ↓f )

(U ↓f )

is a filtered colimit for every U ∈ D. Assume now that F → G is an acyclic cofibration fitting into a commutative square f! F −−−→   y

X   y

f! G −−−→ Y in Pre(D, M) with X → Y a fibration. Due to Lemma 2.2.13 the map X(U ) → Y (U ) is a fibration for every U ∈ D in M and yields a commutative square F (V ) −−−→ X(U )     y y G(V ) −−−→ Y (U ) for every (V, V → f (U )) ∈ (U ↓ f ) which has a lift G(V ) → X(U ) dividing the commutative square into two commuting triangles. The universal property of − colim −−−→G(V ) gives (U ↓f )

a map f! G(U ) → X(U ). Using the functoriality of the colimit one verifies that these morphisms glue and give a natural transformation. Hence we have found a lift f! G → X in the original square and f! F → f! G is a cofibration. q.e.d.

2.3 Simplicial Enrichment of Model Categories

19

Corollary 2.2.20 Let f : C → D be a morphism between small categories. Then (f! , f ∗ ) : Pre(D, sSet) → Pre(C, sSet) is a morphism of model categories. Proof: Cofibrations in sSet are precisely the monomorphisms. Recall that a morphism X → Y is a monomorphism if and only if X → X ×Y X is an isomorphism. Since filtered colimits commute with finite limits, f! preserves monomorphisms, i.e. f! (X) → f! (X) ×f! (Y ) f! (X) is again an isomorphism and hence the previous proposition applies. q.e.d.

2.3

Simplicial Enrichment of Model Categories

Now we are going to deal with simplicial model categories and simplicial functors. We prove that the adjoints of the very first paragraph respect the simplicial structure and therefore, taking paragraph two into account, form morphisms of simplicial model categories. Beyond that we recall the Grothendieck construction of set-valued functors which leads to the category of elements. The next step is to generalize the category of elements by performing the Grothendieck construction on simplicial-set-valued functors. We use this to construct a left adjoint to the mapping space functor. As a consequence, we have that every simplicial presheaf can be displayed as as a colimit of representables. A simplicial category C is a category enriched over simplicial sets, i.e. for every pair X and Y of objects in C there is an object Map(X, Y ) ∈ sSet, the so called simplicial function complex or simplicial mapping space, with its own natural composition rule and a natural isomorphism Map(X, Y )0 = HomC (X, Y ) of sets which respects the composition rule (cf. [Hir 03, Ch. 9, § 9.1, Definition 9.1.2]). A simplicial model category M is a model category M enriched over simplicial sets satisfying the following two additional axioms: For every finite2 simplicial set K ∈ sSet there are objects X ⊗ K (the tensor object) and X K (the cotensor object) in M satisfying the two natural isomorphisms MapM (X ⊗ K , Y ) = MapsSet (K , Map(X, Y )) = MapM (X, Y K ). This data is subject to some natural compatibility conditions which can be found in [Qui 67, Ch. II, § 2] or [Hir 03, Ch. 9, § 9.1, Def. 9.1.6]. One last axiom relates the model structure 2

A simplicial set is called finite iff it has only finitely many non-degenerated simplices.

20

Chapter 2: Analysis of Model Structures

of M to the simplicial structure: If i : A → B is a cofibration in M and p : X → Y is a fibration in M, then the map of simplicial sets MapM (B, X) → MapM (A, X) ×MapM (A,Y ) MapM (B, Y )

(2.3.1)

is a fibration that is an acyclic fibration if either i or p is a weak equivalence. Example 2.3.1 The category sSet of simplicial sets is the prototype of a simplicial model category. Indeed, take X ⊗ K to be the the cartesian product X × K of simplicial sets, define MapsSet (X, Y )n := HomsSet (X ⊗ ∆[n], Y ) and choose X K to be MapsSet (K, X). It can easily be checked that this data gives sSet the structure of a simplicial category. Consult [Qui 67, Ch. II, § 3, Thm. 3] to see that it is a simplicial model category. Remark. One can discuss simplicial model categories from a functorial point of view. That means the simplicial structure within a simplicial model category can be seen as the additional data of three bifunctors Map : M◦ × M → sSet , (X, Y ) 7→ MapM (X, Y ), ⊗ : M × sSetf → M , (X, S ) 7→ X ⊗ S , exp : M × sSet◦f → M , (Y, S ) 7→ Y S satisfying the natural isomorphisms from above, where sSetf denotes the subcategory of finite simplicial sets. From this angle the induced functors X ⊗ ( · ) : sSetf → M and MapM (X, · ) : M → sSet, MapM ( · , Y ) : M◦ → sSet and Y ( · ) : sSetf → M◦ , as well as ( · ) ⊗ S : M → M and ( · )S : M → M give pairs of adjoint functors X ⊗ ( · ) a MapM (X, · ) , Y ( · ) a MapM ( · , Y ) and ( · ) ⊗ S a ( · )S .

The existence of tensor objects allows an explicit description of the n-simplices of the simplicial function complex given in the lemma below. Lemma 2.3.2 Let M be a simplicial model category. For all objects X and Y of M and all n ∈ N0 there is a natural isomorphism MapM (X, Y )n = HomM (X ⊗ ∆[n], Y ) between the set of n-simplices of the simplicial function complex and the set of maps from X ⊗ ∆[n] to Y .

2.3 Simplicial Enrichment of Model Categories Proof:

21

The proof is just a simple application of Yoneda’s lemma: MapM (X, Y )n = HomsSet (∆[n], MapM (X, Y )) = MapsSet (∆[n], MapM (X, Y ))0 = MapM (X ⊗ ∆[n], Y )0 = HomM (X ⊗ ∆[n], Y )

for all n ∈ N0 . q.e.d. Example 2.3.3 The first example of this section can be generalized in the following way. Let C be a small category, M be a simplicial model category both projective and injective ◦ and choose either the projective or injective model structure on MC . Following [DHI 04, ◦ § 2] or [Lur 09, A.3.3.4] one can turn the category MC of M-valued presheaves into a simplicial model category where F ⊗ S and F S for a finite simplicial set S and a presheaf F are defined objectwise, i.e. (F ⊗ S )(C) := F (C) ⊗ S

and (F S )(C) := F (C)S

for all objects C of C. By Lemma 2.3.2 it is clear how to define the simplicial mapping space, namely MapMC◦ (X, Y )n := HomMC◦ (X ⊗ ∆[n], Y ). The axioms of a simplicial model category are self dual. To be precise, we have the following Proposition 2.3.4 Let M be a simplicial model category. Let D : M → M◦ denote the dualisation functor. Then M◦ is a simplicial model category with tensor object D(Y ) ⊗ K := D(Y K ) and cotensor object (D(X))K := D(X ⊗ K ). Proof:

Because of Lemma 2.3.2 and the fact that HomM (X ⊗ ∆[n], Y ) = HomM◦ (D(Y ), D(X ⊗ ∆[n])),

the assertion on tensors and cotensors can easily be checked. Let i : A → B a cofibration in M and p : X → Y a fibration. By the definition of the dual model structure the arrow D(i) : D(B) → D(A) is a fibration and D(p) : D(Y ) → D(X) is a cofibration in M◦ . Therefore the induced map MapM◦ (D(Y ), D(B)) → MapM◦ (D(X), D(B)) ×MapM◦ (D(X),D(B)) MapM◦ (D(Y ), D(A)) is an fibration which is an acyclic one if either i of p is. q.e.d.

22

Chapter 2: Analysis of Model Structures

Lemma 2.3.5 For a closed simplicial model category M, a I-diagram X ∈ MI and a simplicial set S ∈ sSet there is a natural isomorphism colim colim − −−−→(X ⊗ S ) = (− −−−→X) ⊗ S of tensor objects. Proof:

General nonsense gives for every object Y of M HomM (−colim lim −−−→(X ⊗ S ), Y ) = ← −−HomM (X ⊗ S , Y ) S =← lim −−HomM (X, Y )

= HomM ((−colim −−−→X) ⊗ S , Y ). q.e.d. In the sequel we will need a slight generalization of the well known Yoneda lemma. We will refer to it as the simplicial Yoneda lemma. ◦

Lemma 2.3.6 Let F ∈ sSetC is a simplicial presheaf. Looking at simplicial mapping spaces we find an isomorphism Map(Hom( · , X), F ) ∼ = F (X) of simplicial sets. Proof: yields

Using the simplicial structure and applying the classical Yoneda lemma twice MapsSetC◦ (Hom( · , X), F )n = HomsSetC◦ (Hom( · , X) ⊗ ∆[n], F ) = HomsSetC◦ (Hom( · , X), F ∆[n] )
= HomsSet ∆[0], (F ∆[n] )(X)
= HomsSet ∆[0], (F (X))∆[n] = HomsSet (∆[n], F (X)) = F (X)n

for all n ∈ N0 . q.e.d. Definition 2.3.7 A simplicial functor F : M → N is a morphism of simplicial categories, i.e. the functor F consists of a function F : Ob(M) → Ob(N ) and simplicial set maps F : MapM (X, Y ) → MapN (F (X), F (Y )) which respects identities and composition at all simplicial levels.

2.3 Simplicial Enrichment of Model Categories

23

Let M be a simplicial category. The category sSetM of simplicial functors taking values in the category of simplicial sets is of particular interest. Example 2.3.8 The corepresentable simplicial functor MapM (X, · ) : M → sSet is the simplicial functor which assigns to each Y ∈ M the simplicial set MapM (X, Y ) and to each morphism g ∈ MapM (Y, Z)n the composite simplicial set map MapM (X, Y ) × ∆[n] → MapM (X, Y ) × MapM (Y, Z) → MapM (X, Z). It will turn out to be central for our work that the Yoneda lemma carries over to this situation. We will refer to it as the simplicial enriched Yoneda lemma which we have taken from [Goe 99]. Lemma 2.3.9 Suppose A ∈ M is an object of a simplicial category M, and that X : M → sSet is a simplicial functor. Then there is a natural isomorphism MapsSetM (MapM (A, · ), X) ∼ = X(A) of simplicial sets. Proof: The idea is to carry over the classical proof to natural transformations of simplicial functors. A detailed proof can be found in [Goe 99, Ch. IX, § 1, Lemma 1.2]. q.e.d.

Proposition 2.3.10 Let M be a projective and injective simplicial model category and f : C → D be a functor between small categories. Then the restriction of functors ◦

f ∗ : MD → MC

◦

preserves both tensor and cotensor objects for the simplicial structure, i.e. one has f ∗ (F ⊗ S ) = (f ∗ F) ⊗ S and
f ∗ Y S = (f ∗ Y )S ◦

for all simplicial sets S ∈ sSet and F, Y ∈ MD . Moreover, we have f! (F ⊗ S ) = (f! F) ⊗ S and
f∗ Y S = (f∗ Y )S .

24

Chapter 2: Analysis of Model Structures

Proof: We first show the isomorphism of the first equation. Following Example 2.3.3 we compute f ∗ (F ⊗ S ) (U ) = (F ⊗ S ) (f (U )) = F(f (U )) ⊗ S = (f ∗ F) (U ) ⊗ S = (f ∗ F ⊗ S ) (U ) ◦

for all U ∈ D, S ∈ sSet and F ∈ MC . In the same way we conclude by Example 2.3.3 that
f ∗ Y S = (f ∗ Y )S . For the third isomorphism use the adjunction f! a f ∗ and the preservation of cotensors by f ∗ or use that by Definition 2.1.1 the functor f! is defined by a colimit. Thus, by Lemma 2.3.5, the functor f! preserves tensor objects. From the adjointness f ∗ a f∗ one deduces the preservation of cotensors by f∗ . Hence we are done. q.e.d. Corollary 2.3.11 Let M be a simplicial model category and f : C → D be a functor. Then the sequence of adjoint functors from Theorem 2.1.2 f! a f ∗ a f∗ induced by the restriction of functors ◦

f ∗ : MD → MC

◦

is simplicially enriched, i.e. we have isomorphisms MapMD◦ (f! X, Y ) = MapMC◦ (X, f ∗ Y ) and MapMC◦ (f ∗ Y, Z) = MapMD◦ (Y, f∗ Z) of the corresponding simplicial mapping spaces. Proof: Using Lemma 2.3.2 and Theorem 2.1.2 it suffices to show that f! and f ∗ preserve tensor objects. But this is true by Proposition 2.3.10. q.e.d. Corollary 2.3.12 Let M be a simplicial model category and f : C → D be a functor between small categories. Then there are two morphisms

2.3 Simplicial Enrichment of Model Categories

25

(1) (f! , f ∗ ) : Pro(D, M) → Pro(C, M) and (2) (f ∗ , f∗ ) : Pre(C, M) → Pre(D, M) of simplicial model categories. Proof:

This is just a combination of Proposition 2.2.18 and Corollary 2.3.11. q.e.d.

Now we introduce the category of elements that is a special kind of a fibered category. Recall the definition of fibered categories from [Vis 05, Ch. 3, 3.1, Def. 3.1]: Definition 2.3.13 Let I be a category. A category over I is a category C equipped with a functor pC : C → I. A fibered category over I is a category C over I, such that given a morphism f : i → j in I and an object c ∈ C with pC (c) = j, there is a cartesian arrow ϕ : x → c in C such that pC (ϕ) = f . Example 2.3.14 Let pC : C → I be a fibered category. The category (pC ↓ i) for i ∈ I fixed denotes the full subcategory of C whose objects are mapped to i under pC , the fiber over i in C. Recall the Grothendieck construction which leads to the category of elements (cf. [Mac 92, Ch. I, § 5]): Given a functor P : I → Set the category of elements Z P I

is the category whose objects are pairs (i, x) with i ∈ I and x ∈ P (i) and where an arrow (i, x) → (j, y) is defined to be an arrow f : i → j in I such that P (f )(x) = y. In particular, the category of elements has a projection functor Z πP : P →I I

mapping (i, x) to i which turns the category of elements into a category fibered over I. Its fibers (πP ↓ i) are simply given by the set P (i). This justifies the integral sign to notate the category of elements since we are ”summing up all fibers”.

26

Chapter 2: Analysis of Model Structures

Example 2.3.15 Let C be a small category and X be an object of C. Then we have Z hX = (C ↓ X), C

where hX = HomC ( · , X) is the image under the Yoneda functor. More general, given a functor F : C → D and an object X ∈ D one has Z (F ↓ X) = HomD (F ( · ), X). C

Remark. The terminology ”Grothendieck construction” is commonly used to include the case where the category Set of sets is replaced by the category CAT of all small categories. From this angle (and the assumption that one believes in the axiom of choice as we do) one can interpret the Grothendieck construction for any fixed category C as a lax 2-functor Z : Pseu(C) → Fib(C) between the 2-category Pseu(C) of pseudo-functors on C and the category Fib(C) of categories fibered over C which provides an equivalence of 2-categories (cf. [Vis 05, Part 1, Ch. 2, § 3.1]). Proposition 2.3.16 Let P : C ◦ → Set be a presheaf. There exists an equivalence ◦ (SetC ↓ P ) ' Set(

P)

R C

◦

of categories. Proof:

First of all we define the functor ◦ Φ : (SetC ↓ P ) → Set(

R C

P)

◦

.

◦

−1 Let (η : F → P ) be an object of (SetC ↓ P ). Define Φ(η)(X, p) to be ηX ({p}). Since η is a natural transformation it can readily be checked that Φ(η) behaves functorially with R ◦ P ( respect to (X, p) and thus gives an object of Set C ) . Given a morphism Λ : η → ξ in ◦ (SetC ↓ P ) it induces a map −1 −1 Φ(Λ)(X,p) := ΛX |η−1 ({p}) : ηX ({p}) → ξX ({p}) X

and naturality carries over and gives a morphism Φ(Λ) : Φ(η) → Φ(ξ) in Set( Now let us define the functor Ψ : Set(

R C

P)

◦

◦

→ (SetC ↓ P ).

◦

R C

P)

.

27

2.3 Simplicial Enrichment of Model Categories For G ∈ Set(

◦

R C

P)

define a

Ψ(G)(X) :=

G(X, p) → P (X)

p∈P (X)

to be the natural map induced by G(X, p) → P (X), y 7→ p. Using the universal property of coproducts one checks functoriality in X and naturality in G. Thus one sees that Ψ is well defined. Now one computes (Φ ◦ Ψ(G)) (X, p) = Ψ(G)−1 ({p}) ∼ = G(X, p) X

and

 (Ψ ◦ Φ(η)) (X) = 

 −1 ηX ({p}) → P (X) ∼ = (F (X) → P (X)) .

a

p∈P (X)

q.e.d. Assume SC is a site, i.e. its underlying category C has Ra Grothendieck pretopology. Let (fi : (Ui , ai ) → (U, a) | i ∈ I) be an open cover of (U, a) ∈ C F if and only if (πX (fi ) : Ui → U |Ri ∈ I) ∈ covC (U ) is an open cover of RU in C. This gives a Grothendieck pretopology on C F turning the category of elements C F into a site. We will write SRC F for the corresponding site. Proposition 2.3.17 Suppose C can be equipped with some Grothendieck pretopology and let SC denote the resulting site. Let F be a sheaf on the site SC . Then there is an equivalence   (Sh(SC ) ↓ F) ' Sh SRC F of the corresponding set-valued sheaves. Proof: Now let (η : G → F) ∈ (Sh(SC ) ↓ F). To prove Φ(η) ∈ Sh( (fi : (Xi , yi ) → (X, z) | i ∈ I) ∈ covRC F (X, y) and look at Y Y −1 −1 −1 ηX ({z}) → ηX ({y }) ⇒ ηX ({(yi , yj )}) i i i,j i∈I

R C

F), let

i,j∈I

which maps injectively into F(X) →

Y i∈I

F(Xi ) ⇒

Y

F(Xij ).

i,j∈I

Thus Φ(η) defines indeed a sheaf on the site SRC F . The other way round follows from the fact that Ψ is a right adjoint and as such preserves limits. Hence we are done. q.e.d.

28

Chapter 2: Analysis of Model Structures

Corollary 2.3.18 Let SC be a subcanonical site. Then SRC F is subcanonical. Corollary 2.3.19 There are equivalences (C ↓ X)∧ ' (C ∧ ↓ hX ) of presheaf- and (C ↓ X)∼ ' (C ∼ ↓ hX ) sheaf-categories. Proof:

This is just the previous proposition together with Example 2.3.15. q.e.d.

Now we are going to carry over the Grothendieck construction to the simplicial setting since we want to prove the fact that simplicial presheaves can be displayed as a colimit of representable simplicial functors (cf. Example 2.3.8). Therefore let M be a small simplicial category and F : M◦ → sSet be a simplicial functor. We define Z F M

to be the category whose objects are triples (x, [n], M ) with objects [n] ∈ ∆ and M ∈ M such that x ∈ F(M )n and whose morphisms are tuples (ϕ, f ) : (x, [n], M ) → (y, [m], N ) where f ∈ HomM (M, N ) and ϕ ∈ Hom∆ ([n], [m]) such that F(f )ϕ ∈ Hom(F(N )m , F(M )n ) maps y to x. Note that the category has an obvious projection functor Z πF : F → M × ∆. M

Example 2.3.20 Let X ∈ sSet

M◦

where M is a simplicial category. Then we have Z (M ↓ X) = MapM ( · , X), M

where (M ↓ X) is the category whose objects are natural transformations MapM ( · , A) ⊗ ∆[n] → X of simplicial functors and whose morphisms are all commuting triangles MapM ( · , A) ⊗ ∆[n] −−−→ X     id y y MapM ( · , B) ⊗ ∆[m] −−−→ X.

29

2.3 Simplicial Enrichment of Model Categories

The following proposition is a generalization of the ”brute force adjunction” in [Moe 91, Ch. I, § 5, Thm. 2]. Theorem 2.3.21 Let C and E be simplicially enriched categories with C a small category and E a cocomplete category. Assume G : C → E to be a simplicial functor. Then the simplicial functor ◦ R : E → sSetC defined by Z 7→ R(Z) := (X 7→ MapsSet (G(X), Z)) possesses a left adjoint ◦

L : sSetC → E given via L(F) := − colim −−−→ [(G ⊗ ∆[ · ]) ◦ πF ((x, [n], C))] . R

F

C

Moreover, the adjunction L a R is simplicially enriched. In particular, L is the left Kan extension of R along G. Proof:

An element τ ∈ HomsSetC◦ (F, R(Z)) is just a family (τX : F(X) → MapE (G(X), Z))X∈C

of morphisms of simplicial sets indexed by objects of C. Moreover, since τ is assumed to be a natural transformation between simplicial functors, for every morphism f : Y ⊗∆[n] → X in C the square τX F(X) −−− → Map(G(X), Z)     F (f )y G(f )∗ y τY ×id[n]∗

F(Y ) × ∆[n] −−−−−→ Map(G(Y ), Z) × ∆[n] commutes. But such a natural transformation τ may also be seen as a family (τY (x) : G(Y ) × ∆[n] → Z)(x,[n],Y )∈R F C

of morphisms in E. From this point of view the commutativity of the first square means that the diagram τX (x)

(G ⊗ ∆[ · ]) ◦ πF ((x, [n], X)) = G(X) × ∆[n] −−−→   G(f )⊗ϕ∗ y τY (y)

Z  id y Z

(G ⊗ ∆[ · ]) ◦ πF ((y, [m], Y )) = G(Y ) × ∆[m] −−−→ Z

30

Chapter 2: Analysis of Model Structures

commutes for every morphism (ϕ, f ) : (x, [n], X) → (y, [m], Y ). As in the proof of Theorem 2.1.2 we have precisely one morphism colim − −R−−→ [(G ⊗ ∆) ◦ πF ((x, [n], C))] → Z F

C

in E which is clearly functorial in F. Since L is defined via a filtered colimit it preserves tensor objects by Lemma 2.3.5. Thus, the very same reasoning as in Corollary 2.3.11 proves that the adjunction L a R is in actual fact a simplicial one. q.e.d. With Example 2.3.8 in mind and Theorem 2.3.21 we have the following Corollary 2.3.22 Every simplicial presheaf is colimit of representables. To be precise, let M be a simplicial category and F : M◦ → sSet be a simplicial functor. Then there is an isomorphism F ∼ colim = − −−−→ (MapM ( · , A) ⊗ ∆[n]) R

MapM ( · ,F )

M

of simplicial functors. Corollary 2.3.23 Retreating to vertices, i.e. looking at the 0-th simplicial level one gets the following result: If G : C → E is a functor from a small category C to a co-complete category E, the functor R : E → SetC

◦

defined by R(E) = HomE (G( · ), E) has a left adjoint

◦

L : SetC → E given on objects via L(P ) = − colim −−−→G ◦ πP (x, C). R

P

C

This is precisely the statement of [Moe 91, Ch. I, § 5, Thm. 2] which can be seen as another version of of the particular left Kan extension discussed at the beginning of this chapter. ◦

Example 2.3.24 In the situation of the corollary above let E := SetC and G : C → SetC be the Yoneda functor, i.e. G(X) := hX = HomSet ( · , X). Then we have R(P )(X) = HomSetC◦ (hX , P ) ∼ = P (X)

◦

31

2.4 Absolute derived Functors

due to the Yoneda lemma. Thus, R is isomorphic to the identity functor and hence one has P ∼ colim =− −−−→G ◦ πP R

P

C

by the uniqueness of adjoints. In particular, every presheaf can be displayed as a colimit of representables.

2.4

Absolute derived Functors

Quillen functors allow extensions to functors between the associated homotopy categories. These will be left and right derived versions of the left and right Quillen part. Before discussing this, we explain what is meant by localizing categories and deriving functors in a more general setting than that of model categories. Therefore we introduce the notion of localizers and absolute derived functors the author has learned from G. Maltsiniotis [Mal 07]. His famous absolute adjunction theorem applies to model categories and can be used to give a fairly easy proof of Quillen’s adjunction theorem for derived functors. In contrast to common literature, i.e. especially [Hir 03] and [Hov 99], Quillen’s adjunction theorem can therefore be proven without the additional assumption that the cofibrant and fibrant replacements behave functorially. The version here is even stronger than Quillen’s original in [Qui 67] since we do not require the left adjoint to preserve cofibrations and the right adjoint to preserve fibrations. Definition 2.4.1 Let C be a category and W be a class of morphisms in C. A localization of C by W is a tuple (C[W −1 ], γ), where C[W −1 ] is a category together with a functor γ : C → C[W −1 ], the localization functor, that carries morphisms from W to isomorphisms in C[W −1 ] and is universal amoung all functors with this property in the following sense: For every other category D and every functor F : C → D which maps morphisms from W to isomorphisms in D, there is a functor F : C[W −1 ] → D unique up to isomorphism and a natural isomorphism F ◦ γ ∼ = F of functors. Despite the bad taste from the point of view of category theory one often merely requires for the universal property that F is unique. Then the definition can be reformulated for small categories. Let C be a small category and NatW (C, D) the set of all functors with the property that their image of W is contained in the class of isomorphisms in D. The universal property of the localization of categories says the map Nat(C[W −1 ], D) → NatW (C, D) is a bijection of sets. The following concept of localizers is taken from [Mal 07] in order to avoid set-theoretical difficulties.

32

Chapter 2: Analysis of Model Structures

Definition 2.4.2 A localizer is a pair (C, W ), where C is a category and W is a class of morphisms in C such that the localized category C[W −1 ] exists and is locally small. A morphism of localizers F : (C, W ) → (C 0 , W 0 ) is a functor F : C → C 0 such that F (W ) belongs to W 0 . Example 2.4.3 Let A be an abelian category, Kom(A) its associated complex category and qis the class of quasi-isomorphisms. The derived category D(A) = Kom(A)[qis−1 ] is locally small (cf. [Gel 96, Ch. II, §4]) and (Kom(A), qis) therefore a localizer. Example 2.4.4 Let (C, W ) be a localizer and I be a small category. Define WI := {α ∈ Mor(C I ) | ∀i∈I αi ∈ W } to be the localizing class of C I . The pair (C I , WI ) is a localizer. Thus, the constant diagram functor ∆I : (C, W ) → (C I , WI ) is a morphism of localizers. More generally, let F : (C, W ) → (D, V ) be a morphism of localizers and I be an index category. Then the induced functor FI : C I → D I is a morphism (C I , WI ) → (DI , VI ) of localizers. Remark. With the notion of localizers in mind the universal property of the localization can be understood as a 2-functor LOC → CAT from the category of small localizers whose arrows are morphisms of localizers to the category of all small categories by sending a localizer (C, W ) to its localization (C[W −1 ], γ) and a morphism of localizers F : (C, W ) → (C 0 , W 0 ) to F : (C[W −1 ], γ) → (C 0 [W 0−1 ]; γ 0 ). By Definition 2.4.1 F is merely unique up to isomorphism, i.e. we have to involve the axiom of choice. The existence of the natural isomorphism in Definition 2.4.1 makes sure that the described assignment yields indeed a pseudo-functor in the sense of [Vis 05, Ch. 3, § 3.1, Def. 3.10].

Example 2.4.5 Let (C, W ) be a localizer and u : I → J be a morphism between index categories, i.e. small categories. Then the restriction of functors (cf. Definition 2.1.1) u∗ : C J → C I gives a morphism (C J , WJ ) → (C I , WI ) of localizers.

2.4 Absolute derived Functors

33

Remark. Let M be a model category and let W denote the class of weak equivalences. Then the derived category Ho(M) := M[W −1 ] is locally small (cf. [Qui 67, Ch.I, §1]), so the pair (M, W ) is a localizer. We will always write Ho(M) for the localization of a model category with respect to its weak equivalences and call it the homotopy category associated to M. When we speak of homotopy categories we always refer to derived categories whose localizer is given by a model category.

Definition 2.4.6 A localizer (C, W ) is called saturated if any morphism γ(f ) in C[W −1 ] is an isomorphism if and only if f belongs to the localizing class W of C. Example 2.4.7 Let M be a model category. Then by [Qui 67, Ch. I, § 5.5., Proposition 1] its localizer (M, W ) is saturated. In order to define an extension of functors F : C → D where the domain is a localizer (C, W ) but F does not carry morphisms in W to isomorphisms in D we follow Quillen [Qui 67, Ch.I, §4] and introduce the notion of derived and total derived functors. G. Maltsiniotis put in [Mal 07] derived functors of model categories in the context of absoluteness as we will do. Definition 2.4.8 Let (C, W ) be a localizer and F : C → D a functor. (1) A left derived functor of F is a pair (LF, l), where LF : C[W −1 ] → D is a functor and l : LF ◦ γ → F a natural transformation satisfying the following universal property. For every functor G : C[W −1 ] → D together with a natural transformation σ : G◦γ → F there is a unique α : G → LF such that σ = l(α ? 1γ ). (2) The pair (LF, l) is an absolute left derived functor of F if for every functor H : D → E the couple (H ◦ LF, 1H ? l) is a left derived functor of H ◦ F . Example 2.4.9 Every absolute left derived functor is in particular a left derived functor. To prove this choose H to be the identity functor in the definition above. The notion of right derived functors and absolute right derived functors is defined in the dual way. Remark. The notion of absoluteness is not that exotic as it might appear at first glance. For example, in [Bor 94, Ch. 2, § 2.10] the notion of an absolute colimit is introduced. The following example is taken from [Bor 94, Ch. 2, § 2.10, 2.10.3.d]: Let R be a ring with unit and M be a left R-module. The unit and the multiplication of R yield morphisms e : Z → R and m : R⊗Z R → R as well as the scalar multiplication on M yields µ : R⊗Z M → M . Then (M, µ) is the absolute coequalizer R ⊗Z R ⊗Z M ⇒ R ⊗Z M → M in the category of abelian groups of the pair (1R ⊗Z µ, m ⊗Z 1M ) in the diagram above.

34

Chapter 2: Analysis of Model Structures

Example 2.4.10 Let (C, W ) be a localizer and F : C → D be a functor to some category D with the property that F maps arrows in W to isomorphisms. Define W 0 to be the class of isomorphisms in D. This way (D, W 0 ) becomes a trivial localizer and F is a morphism of localizers. Due to the universal property of (C[W −1 ], γ) we find that (F , 1F ) is both absolute left and right derived functor of F . The universal property of a left derived functor means exactly that the functor LF is a right Kan extension of F along the localization functor γ : C → C[W −1 ]. Hence left derived functors are unique up to unique isomorphism and we simply refer to it as the left derived functor. The same consideration holds true for right derived functors. We can say that left derived functors are universal from the left or initial from the left in the sense that they are the best possible approximation of a given functor which is to be extended to the derived category of the target category. Note that morphisms of localizers enjoy the property that there is an (up to isomorphism) unique extension of it between the associated localized categories as seen in Example 2.4.10. We now introduce a concept which tells us how to extend functors which are no morphism of localizers. Definition 2.4.11 Let (C, W ) and (C 0 , W 0 ) be two localizers with localization functors γ and γ 0 . Let F : C → C 0 be an arbitrary functor. (1) A total left derived functor of F is a tuple (LF, l), where LF is the left derived functor L(γ 0 ◦ F ) of the functor F followed by the localization functor. (2) An absolute total left derived functor of F is a tuple (LF, l), where LF is the absolute left derived functor L(γ 0 ◦ F ) of the functor F followed by the localization functor. Example 2.4.12 Let (C, W ) be a localizer and I be a small category. The functor ∆I : C → C I is a morphism of localizers (C, W ) → (C I , WI ) and hence induces a functor ∆I : C[W −1 ] → C I [WI−1 ] unique up to isomorphism (due to the universal property of the localization γ : C → C[W −1 ] applied to γI ◦ ∆I ). We find γI ◦ ∆I ∼ = ∆I ◦ γ, i.e. C   γy

∆

−−−I→

CI  γI y

∆

C[W −1 ] −−−I→ C I [WI−1 ] commutes up to isomorphism. The pair (∆I , 1) is both the absolute left and the absolute right derived functor of ∆I .

35

2.4 Absolute derived Functors Theorem 2.4.13 (G. Maltsiniotis’ Absolute Adjunction Theorem) Let (C, W ) and (C 0 , W 0 ) two localizers and F: Co

/

C0 : G

a pair of adjoint functors such that the absolute total left derived functor of F and the absolute right derived functor of G exist. Then the pair of functors LF : C[W −1 ] : C o

/

C 0 [W 0−1 ] : RG

is a pair of adjoint functors. Proof: A detailled proof can be found in [Mal 07]. Note that the proof is purely formal such that the assumption of local smallness of the derived categories, i.e. that (C, W ) and (C 0 , W 0 ) are localizers, could be omitted. q.e.d. We are mainly interested in localizations of model structures so we are going to apply the machinery developed so far to model categories. Therefore we will apply the Ken Brown–lemma to the case where the localizer (C, W ) is given by a model category. The following proposition is due to [Qui 67, Chapter I, 4.2 Proposition 1] and we present it in a slight more general version. This version is mentioned in [Mal 07] and taken from there. Proposition 2.4.14 Let C be a category, M a model category and F : M → C a functor. Suppose that F carries weak equivalences between cofibrant objects into isomorphisms in C. Then the absolute left derived functor LF : Ho(M) → C exists and its natural transformation lX : LF (X) → F (X) is an isomorphism in C for X in M cofibrant. Proof: Quillen’s existence theorem for derived functors [Qui 67, Ch. I, 4.2, Proposition 1] or [Dwy 95, Proposition 9.3] uses the fact that F sends weak equivalences between cofibrant objects to isomorphisms. It was observed by G. Maltsiniotis in [Mal 07] that for any functor H the composition H ◦ F still sends weak equivalences between cofibrant objects to isomorphisms and hence the same construction gives a left derived functor of H ◦ F isomorphic to (H ◦ LF, 1H ? l). q.e.d.

36

Chapter 2: Analysis of Model Structures

Remark. (1) The proposition is true if F is a left Quillen functor. (2) The dual statement about the existence of RF in the case where F is a functor carrying weak equivalences (or acyclic fibrations respectively) between fibrant objects to isomorphisms in C is true.

Corollary 2.4.15 Suppose F : M → N is a functor between model categories which takes weak equivalences between cofibrant objects of M into weak equivalences in N . Then the absolute total left derived functor LF : Ho(M) → Ho(N ) exists and its natural transformation lX : LF (X) → F (X) is an isomorphism in Ho(N ) for X in M cofibrant. Corollary 2.4.16 (Quillen’s Adjunction Theorem) Let (F, G) : M0 → M be an adjoint pair of functors between model categories in such a way that F preserves weak equivalences between cofibrant objects and G preserves weak equivalences between fibrant objects. Then the absolute total left derived functor LF : Ho(M) → Ho(M0 ) of F and the absolute total right derived functor RG : Ho(M0 ) → Ho(M) of G exist, and one has LF a RG, i.e. the functor LF is left adjoint to RG. Proof: Since F is a left Quillen functor it preserves weak equivalences between cofibrant objects, hence the absolute total left derived functor LF exists due to Proposition 2.4.14. Dual reasoning gives the existence of RG and using Theorem 2.4.13 we are done. q.e.d. In the case that (F, G) : M0 → M is a morphism of model categories we have the original formulation of Quillen in [Qui 67].

37

2.4 Absolute derived Functors

Theorem 2.4.17 Let M, M0 and M00 be three model categories, F : M → M0 , F 0 : M0 → M00 two functors, and suppose that F and F 0 carry trivial cofibrations between cofibrant objects to weak equivalences. If F carries cofibrant objects of M to cofibrant objects of M0 , then the absolute total left derived functor (L(F 0 ◦ F ), β) of the composition F 0 ◦ F exists and can be described in terms of the absolute total left derived functors (LF 0 , α0 ) and (LF, α). To be precise we have a canonical isomorphism ∼ =

(LF 0 ◦ LF, (α0 ? F )(LF 0 ? α)) → (L(F 0 ◦ F ), β) of absolute total derived functors. Proof: The additional assumption on F forces the composition F 0 ◦ F to take acyclic cofibrations between cofibrant objects in M to weak equivalences in M00 . So its absolute total left derived functor (L(F 0 ◦ F ), β) exists due to Proposition 2.4.14. It remains to show that for every functor G : Ho(M) → Ho(M00 ) and every natural transformation σ : G ◦ γ → γ 00 ◦ (F 0 ◦ F ) the composition LF 0 ?α

α0 ?F

LF 0 ◦ LF ◦ γ −−−→ LF 0 ◦ (γ 0 ◦ F ) = (LF 0 ◦ γ 0 ) ◦ F −−−→ (γ 00 ◦ F 0 ) ◦ F is initial from the left. Since the author could not find a proof in literature that avoids the additional assumption of functorial cofibrant replacement we give a proof on our own inspired by Quillen’s original proof on the existence of derived functors in [Qui 67, Ch. I, 4.2, proof of Proposition 1]. We will give an explicit construction of the natural transformation τ : G → LF 0 ◦ LF which has to satisfy (α0 ? F ) ◦ (LF 0 ? α) ◦ (τ ? γ) = σ, i.e. which is universal from the left. Let qX : QX → X be a cofibrant replacement of a given object X ∈ M. We define τX : G(X) → LF 0 ◦ LF (X) to be the composition τX := LF 0 ◦ LF ◦ γ(qX ) ◦ LF 0 (αQX −1 ) ◦ αF0 QX

−1

◦ σQX ◦ Gγ(qX )−1

displayed diagramatically as Gγ(X) −→ Gγ(QX) −→ γ 00 ◦ F 0 ◦ F (QY ) −→ (LF 0 ◦ γ 0 ) ◦ F (QX) −→ LF 0 ◦ LF ◦ γ(QX) −→ LF 0 ◦ LF ◦ γ(X). Note that the definition does not depend on the choice of the cofibrant replacement since different replacements belong to the same isomorphism class in the associated homotopy category, consequently we have uniqueness in the case of existence. However, τX behaves naturally with respect to X and gives therefore the (unique) desired natural transformation G ◦ γ → LF 0 ◦ LF ◦ γ if we can prove the desired equality on objects. Since every map in the homotopy category is a finite composition of maps γ(f ) and γ(g)−1 with f a morphism

38

Chapter 2: Analysis of Model Structures

of M and g a weak equivalence, τ : G ◦ γ → LF 0 ◦ LF ◦ γ can be lifted to be a natural transformation τ : G → LF 0 ◦ LF . The commutative diagram G(pX )−1

σQX

G(X) −−−−−→ G(QX) −−−→ F 0 ◦ F (QX) −−−→ LF 0 ◦ LF (X)        α0 ◦LF 0 (αX ) G(pX )y F (pX )y idGX y y F (X) id

G(X) −−GX −→

σ

id

X G(X) −−− → F 0 ◦ F (X) −−−→

F 0 ◦ F (X)

shows that αF0 (X) ◦ LF 0 (αX ) ◦ τX = σX , hence τX does satisfy the required condition on objects, so (α0 ? F )(LF 0 ? α) has the required universal property. q.e.d. Remark. The dual assertion about the absolute total right derived functor of a composition for functors G and G0 where G takes fibrant objects to fibrant objects holds true and can be proven in a similar fashion.

2.5

Homotopy Limits as absolute derived Functors

In this section we follow ideas of [Dwy 95] and use the derivation of functors to show the existence of homotopy limits and homotopy colimits in the setting of model categories. Before doing this we define homotopy limits and homotopy colimits in a more general setting than that of model categories using localizers. In our opinion the homotopy invariance is one of the key features of homotopy limits and homotopy colimits so we approach from a more general point of view to prove the homotopy invariance of homotopy limits and homotopy colimits. Note that this is in contrast to [Bou 73] and [Hir 03] since our homotopy limits and colimits are homotopy invariant as an immediate consequence from the definition whereas theirs is not. Recall that we say ”the left adjoint of the constant diagram ∆I is defined at F ” if colim − −−−→F exists. Now we are going to extend this to localized categories. Definition 2.5.1 Let (C, W ) be a localizer. We say that F ∈ C I [WI−1 ] (see Example 2.4.4 for notation) has a homotopy colimit if and only if the left adjoint of ∆I is defined at F , i.e. if and only if there is a bijection −1 HomC[W −1 ] (−hocol −−−→F, X) = HomC I [W ] (F, ∆I (X)). I

I

The homotopy limit is defined in the dual way. It is obvious from the definition that hocol holim −−−−→ and ← −−−− behave functorially in the case of their existence. In addition, the definition chosen above eases the proof of the following proposition.

2.5 Homotopy Limits as absolute derived Functors

39

Proposition 2.5.2 Let (C, W ) be a localizer and X, Y ∈ C I be I-diagrams such that both their limit respectively homotopy limit and colimit respectively homotopy colimit is defined at X and Y . Then there are universal maps hocol colim − −−−→X −−−→ − −−−→X I

and

I

holim lim −−−−X ← −−X −−−→ ← I

I

and the homotopy invariance, i.e. given a morphism (X → Y ) ∈ WI we have isomorphisms ∼ =

hocol hocol − −−−→X −−−→ − −−−→Y I

and

I

∼ =

holim holim ← −−−−X −−−→ ← −−−−Y I

in C[W

−1

Proof:

I

]. We prove the colimit case only. Chasing around idX in HomC I (X, X) → HomC (−colim colim −−−→X, − −−−→X) = HomC I (X, ∆I (−colim −−−→X)) → HomC I [W −1 ] (X, γI ◦ ∆I (−colim −−−→X)) I

= HomC I [W −1 ] (X, ∆I ◦ γI (−colim −−−→X)) I

= HomC I [W −1 ] (−hocol colim −−−→X, − −−−→X) I

I

gives the desired universal map. Given (f : X → Y ) ∈ WI we have γI (f ) ∈ Iso(C I [WI−1 ]) and consequently −1 HomC[W −1 ] (−hocol −−−→Y, K) = HomC I [W ] (Y, ∆I (K)) I

I

∼ = HomC I [WI−1 ] (X, ∆I (K)) = HomC[W −1 ] (−hocol −−−→X, K) I

for any object K in C. Hence we are done. q.e.d. Let (C, W ) be a localizer and X ∈ C be an object. We form the slice category (C ↓ X) and define (W ↓ X) to be the class of commuting triangles α : (U, f ) → (V, g) whose

40

Chapter 2: Analysis of Model Structures

connecting morphism α belongs to W . Then the slice category ((C ↓ X), (W ↓ X)) is again a localizer. A diagram F ∈ ((C ↓ X), (W ↓ X))I induces a diagram F 0 ∈ C I by composing F with the natural projection functor (C ↓ X) → C that sends an object Y → X of the slice category to its codomain Y . Proposition 2.5.3 Let (C, W ) be a localizer such that homotopy colimits in (C I , WI ) are defined at all objects. Then (C ↓ X) has all homotopy colimits and its homotopy colimits can be calculated in terms of those of C, i.e. hocol hocol − −−−→F ' (−hocol −−−→Ai , − −−−→ai ) I

I

I

for F ∈ C I with F (i) = (Ai , ai ). Proof:

We define the bijection Hom(F, ∆I (Y, g)) ∼ hocol hocol = Hom((− −−−→Ai , − −−−→ai ), (Y, g)) I

I

by sending η ∈ Hom(F, ∆I (Y, g)) to hocol hocol hocol − −−−→ηi ∈ Hom((− −−−→Ai , − −−−→ai ), (Y, g)). I

I

I

q.e.d. Remark. The very same reasoning gives a description of ordinary colimits on the slice category in terms of those of the underlying category. We now specialize to the case of homotopy colimits in model categories. In this case the homotopy invariance can be understood as follows: Lemma 2.5.4 Let (M, W ) be a localizer given by a model category M such that homotopy colimits exist and f ∈ WI be a weak equivalence between I-diagrams. Then hocol − −−−→(f ) ∈ W I

is a weak equivalence. Proof: Every model category is saturated (cf. [Hov 99, 1.2.10 (iv)] or [Dwy 95, Proposition 5.8]) and every functor category MI associated to M with model structure where the weak equivalences are determined objectwise is saturated as well (cf. [DHKS 03, Proposition 33.9 (v)]). q.e.d. If M is a model category and X an object in M we can put a model structure on the slice category (M ↓ X) by defining a commuting triangle (U, f ) → (Y, g) to be a weak

2.5 Homotopy Limits as absolute derived Functors

41

equivalence, cofibration, fibration iff U → Y is a weak equivalence, cofibration, fibration in M (cf. [Hir 03, Theorem 7.6.5 (2)]). The description of homotopy colimits in the model category (M ↓ X) in terms of those of M now follows from Proposition 2.5.3. Corollary 2.5.5 Let M be a category which possesses all homotopy colimits. Then (M ↓ X) has all homotopy colimits and its homotopy colimits can be calculated in terms of those of M, i.e. hocol hocol − −−−→F ' (−hocol −−−→Ai , − −−−→ai ) I

I

I

for F ∈ C I with F (i) = (Ai , ai ). Proposition 2.5.6 Let (F, G) : N → M be a morphism of model categories. Then the left Quillen part F preserves homotopy colimits and the right Quillen part G preserves homotopy limits in the case of their existence. To be precise, there are weak quivalences   F − hocol X '− hocol −−−→ −−−→ (F (X)) and   G ← holim Y '← holim −−−− −−−− (G(Y )) . Proof: Using the adjunction LF a RG on the level of the associated homotopy categories, the result can easily be derived from the isomorphism ∆ ◦ RG ∼ = RG ◦ ∆ using the saturation of model categories and the saturation of the associated diagram categories. q.e.d. Generalizing the ideas developed in [Dwy 95] we have the following Proposition 2.5.7 Let (M, W ) be a localizer given by a model category and I a small category such that M is I-projective. Suppose M has colimits of type I, i.e. I colim − −−−→ : M → M I

is a functor. Then the category M admits homotopy colimits of type I, i.e. we have a functor I hocol − −−−→ : Ho(M ) → Ho(M). I

42

Chapter 2: Analysis of Model Structures

Proof: We have an isomorphism R ∆I ∼ = ∆I of functors since ∆I takes weak equivalences between fibrant objects in M to weak equivalences in MI . Due to the absolute adjunction theorem we have the adjunction L− colim −−−→ a ∆I , I

so the result follows from the uniqueness of adjoint functors. q.e.d. Corollary 2.5.8 For the projective model structure on MI there is a morphism ! colim − −−−→, ∆I

: M → MI

I

of model categories. Proof: This is just the fact that ∆I preserves fibrations and acyclic fibrations for the projective model structure together with lemma 2.2.3. q.e.d. For general reasons it remains true that the colimit of cofibrant objects gives a cofibrant object no matter how the model structure was chosen. Corollary 2.5.9 The category of simplicial sets sSet possesses all homotopy colimits and homotopy limits. Proof:

This is just example 2.2.8 together with Proposition 2.5.7. q.e.d.

We want to give a description of how to compute homotopy pushouts and homotopy pullbacks in an arbitrary model category. It turns out that finding a fibrant replacement X → RX for X ∈ MJ , where J = {a → b ← c} is a pullback category, involves replacing X(b) by a fibrant object and replacing the maps X(a) → X(b) and X(c) → X(b) by fibrations. Proposition 2.5.10 Let M be a model category. Assume I and J are given as above. An object X ∈ MI is cofibrant for the projective model structure on MI if and only if X(b) ∈ M is a cofibrant object and both morphisms X(b) → X(a) and X(b) → X(c) are cofibrations in M. Dually, an object Y ∈ MJ is fibrant if and only if Y (b) is a fibrant object in M and both morphisms Y (a) → Y (b) and Y (c) → Y (b) are fibrations in the model structure of M. Proof: The required model structures exist due to Theorem 2.2.9. Spelling out the explicit construction given in the proof of the theorem one checks the claim straight ahead. q.e.d.

2.5 Homotopy Limits as absolute derived Functors

43

Corollary 2.5.11 Every model category possesses homotopy pushouts and homotopy pullbacks. Remark. Proposition 2.4.14 and Proposition 2.5.10 tell us how to compute the homo∼ topy colimit of a given diagram X ∈ MI . First choose a cofibrant replacement QX → X of X for the projective model structure and then compute the ordinary colimit of QX and it follows that hocol colim colim − −−−→X ' L − −−−→X ' − −−−→QX. I

I

I

(Because of Proposition 2.5.2 and Lemma 2.5.4 the above formula is independent of all choices involved.) This resembles the situation in Homological Algebra where a functor which fails to be exact is going to be ”repaired” by first choosing an appropriate resolution and then applying the functor. Like in Homological Algebra it can be quite cumbersome to find a cofibrant replacement in the homotopical setting. Nevertheless, Proposition 2.5.7 makes sure that the homotopy colimits we are going to deal with do exist. Note that the dual statement of Proposition 2.5.7, i.e. the existence of homotopy limits for the injective model structure on the category of I-diagrams is true and can be proven in a very similar fashion. The following proposition is a slight generalisation of the interchange law for colimits given in [Mac 91, Ch. IX, §2]. Proposition 2.5.12 Let C be a fibered category over I and let (pC ↓ i) denote the fiber of pC over i ∈ I. Consider a functor F : C → M into a cocomplete category M. Write πi : (pC ↓ i) → C for the inclusion functor and let πi∗ : MC → M(pC ↓i) be the associated restriction of functors. Then the canonical map ∗ colim colim colim − −−−→F → − −−−→− −−−→πi F C

I

(pC ↓i)

is an isomorphism. ∗ The colimit −colim −−−→πi F exists for every i ∈ I. Every arrow f ∈ HomI (i, j) in I

Proof:

∗

(pC ↓i) (pC ↓j)

induces a functor f : M

→ M(pC ↓i) which in turn gives a canonical map ∗ ∗ colim colim − −−−→πj F → − −−−→πi F. (pC ↓j)

(pC ↓i)

In other words, there is a functor L : I ◦ → M defined by ∗ L(i) := − colim −−−→πi F. (pC ↓i)

Since − colim −−−→L is a cocone for F , there is a canonical map I

colim colim − −−−→F → − −−−→L. C

I

44

Chapter 2: Analysis of Model Structures

We will prove that this map is an isomorphism. Given a cocone κ : L → K we construct a cone λ : F → K with base F and vertex K by defining λc : F (c) → K for every c ∈ C to be the composition of F (c) → πi∗ F (c) → L(i) → K by choosing c ∈ C with p : c 7→ i. Using the existence of cartesian arrows in C one easily verifies that all λc determine a cocone λ : F → K. By the universal property of − colim −−−→F there exists a unique map colim − −−−→F → K, C

proving that the canonical map is an isomorphism as desired. q.e.d. As a special case we get the classical result about the interchange of colimits (cf. [Mac 91, Ch. IX, §2]). Corollary 2.5.13 Let F : I × J → M be a bifunctor to a cocomplete category M. Then there is a canonical isomorphism ∼ colim colim F. colim colim − −−−→− −−−→F = − −−−→−−−−→ I

J

J

I

Proof: This is just the proposition above with the symmetric product category C = I ×J considered as a fibered category over I and J. q.e.d. The following corollary is the Fubini–like theorem for homotopy limits and colimits. Corollary 2.5.14 Let M be a closed model category. If M is projective and cocomplete, then for every diagram F ∈ MI×J there is a weak equivalence hocol hocol hocol hocol − −−−→− −−−→F ' − −−−→− −−−→F. I

J

J

I

Proof: Let F and G be two diagrams I×J → M and MI×J equipped with the projective model structure. Therefore a morphism f : F → G is a weak equivalence or fibration iff f(i,j) : F (i, j) → G(i, j) is a weak equivalence or fibration in M. Every diagram F ∈ MI×J gives a functor FJ : J → MI by the exponential law MI×J ∼ = (MI )J . Define fJ : FJ → GJ to be a weak equivalence or fibration iff fJ (j) if a weak equivalence or fibration for all i, i.e. iff fJ (i)(j) = f (i, j) is a weak equivalence or fibration in M. This way we get a two tower of projective model structures on (MI )J and because of the symmetry we can do the same with (MJ )I . Since the colimit functor I J I colim − −−−→ : (M ) → M J

45

2.6 The left Bousfield Localization

preserves cofibrant objects, all involved homotopy colimits do exist due to Proposition 2.5.7 and the composition theorem (Theorem 2.4.17) for derived functors can be applied. Using the classical interchange law we find ∼ hocol hocol colim colim − −−−→− −−−→F = (L− −−−→)(L− −−−→)F I

J

I

J

' L(−colim colim −−−→)F −−−→− J

I

∼ colim = L(−colim −−−→)F −−−→− I

J

'− hocol hocol −−−→− −−−→F. J

I

q.e.d.

2.6

The left Bousfield Localization

In this section we will define the notion of sheaves in the homotopical setting following the results of [DHI 04]. To do this we have to look at local model structures, i.e. model structures one obtains by the left Bousfield localization of the global model structures at the set of hypercovers. To motivate this approach in the homotopical setting lets take a different look at sheaves in the classical sense: We follow [DHI 04] and write rX for the image of of an object X under the Yoneda functor. Let − colim −−−→U denote the coequalizer of a a,b

rUa ×rX rUb ⇒

a

rUa

a

associated to an open cover (Ui → X | i ∈ I) in the category of presheaves. One can rephrase the sheaf condition: The condition that a presheaf F is a sheaf is equivalent to saying that F turns the canonical maps −colim −−−→U → rX into isomorphisms, i.e. Hom(−colim lim −−−→U , F ) = ← −−F (U ) = F (X) = Hom(rX, F ). From this point of view sheaves are ”local objects” with respect to the collection of maps {− colim −−−→U → rX} induced by open covers (Ui → X | i ∈ I). Recall that the category of sheaves SC∼ on a site SC is universal with respect to the following properties (cf. [Moe 91, Ch. I, § 5, Cor. 4]): The category of sheaves SC∼ is a cocomplete category together with a colimit-preserving functor SC∧ → SC∼ from the category of presheaves which carries each canonical morphism −colim −−−→U → rX to an isomorphism. Hence we can consider the process of passing from presheaves to sheaves as

46

Chapter 2: Analysis of Model Structures

a kind of localization SC∧ → SC∼ at the collection of maps {− colim −−−→U → rX}. In order to pass from simplicial presheaves to sheaves in the homotopical setting we have to do some technical preparations which we recollect now. Let M be a left proper cellular model category and S be a set of morphisms in M. Then by [Hir 03, Part I, Ch. 4, § 4.1, Thm. 4.1.1] the left Bousfield localization LS M of M at S exists. That is, the class of weak equivalences in LS M equals the class of S-local equivalences of M, the class of cofibrations stay the same and the class of fibrations of LS M is the class of morphisms with the right lifting property with respect to those maps that are both cofibrations and S-local equivalences. The left Bousfield localization is a left localization, i.e. it enjoys the following universal property: LS M has a left Quillen functor j : M → LS M such that its absolute total left derived functor Lj takes images in Ho(M) of elements in S into isomorphisms in Ho(LS M) and is initial among all functors with that property, i.e. for any other model category N and left Quillen functor α : M → N sending images of elements in S in Ho(M) to isomorphisms in Ho(N ) there is a functor δ : LS M → N unique up to isomorphism such that there is a natural isomorphism α → δj. It turns out that in the case of left Bousfield localizations the universal property holds true even in a strong sense. Moreover, the left Bousfield localization LS M of a left proper cellular simplicial model category M yields a left proper cellular simplicial model category where LS M inherits the simplicial structure of M, i.e. both categories are isomorphic as simplicially enriched categories. In the case that M is a left proper simplicial model category the fibrant objects of its localization LS M are the S-local objects of M (by [Hir 03, Part I, Ch. 4, § 4.1, Thm. 4.1.1]). To be more succinct, F ∈ LS M is fibrant iff F ∈ M is fibrant and for all g : X → Y in S the morphism MapM (g, idF ) : MapM (Y, F ) → MapM (X, F ) is a weak equivalence of simplicial sets. (That is precisely the S-locality condition.) Remark. We will briefly explain the technical requirements mentioned above. Left properness refers to a good behaviour for pushouts, i.e. in a left proper model category every pushout of a weak equivalence along a cofibration is a weak equivalence again. For example, the simplicial model category sSet of simplicial sets is left proper ([Hir 03, Ch. 13, §13.1, Cor. 13.1.4]) and every model category with injective model structure is left proper (since due to Lemma 2.2.10 all objects are cofibrant and thereby [Hir 03, Ch. 13, §13.1, Cor. 13.1.3] applies). Dually, right properness is defined. Model categories that are both left proper and right proper are called proper. A cellular model category is a model category that is cofibrantly generated, i.e. a model category such that there is a set of generating cofibrations that permits the small object argument and there is a set of generating trivial cofibrations that also permits the small object argument. Cellularity requires in addition that the domains and codomains of

47

2.7 Sites and their Morphisms

the generating sets satisfy certain compactness properties. All these technical details are beyond the thesis and we recommend chaper 11 and chapter 12 in [Hir 03] for further reading. As far as we are concerned, it suffices to note that the category sSet of simplicial sets is a left proper cellular model category (cf. [Hir 03, Part I, Ch. 4, § 4.1, Prop. 4.1.4]). Beyond this, the global model structures are proper and cellular ([Lur 09, A.3.3.3 and A.3.3.5]). It is important to note that the localization LS M of a left proper cellular model category M is again left proper and cellular ([Hir 03, Part I, Ch. 4, § 4.1, Thm. 4.1.1]). Hence, further localization is possible.

2.7

Sites and their Morphisms

First of all we want to give a definition of what we mean when talking about big and small sites. In [SGA4 72, Ch. IV, § 2.5] this is explained as follows: ”... . Pour tout objet X de Top, considerons la cat´egorie (Top ↓ X) des objets de Top au-dessus de X, ... . Ce site est apppel´e le gros site associ´e a` X. ... ” Of course, this can easily be generalized by replacing the categorie Top of topological spaces by the category Man of smooth manifolds, schemes and so on. The aim of the following definition is to give ”big” and ”small” sites a meaning in an arbitrary setting. The following definition was inspired by [Lur 11, §1.2]. Definition 2.7.1 Let C be a category and C ad ⊂ C be a subcategory containing every object of C and whose morphisms are required to satisfy the following conditions: • All isomorphisms of C belong to C ad . • C ad is closed under base change along all morphisms in C. Morphisms of C which belong to C ad will be called admissible. An admissibility structure on C consists of a subcategory C ad satisfying the previous axioms. An admissibility structure C ad on C satisfies the axioms of a Grothendieck pretopology, i.e. of a distinguished class of morphisms which satisfies stability under base change, that all isomorphisms belong to them and that they are of local character, i.e. if (Xi → X) covers X and (Xij → Xi ) covers Xi for every i, then (Xij → X) covers X again. Definition 2.7.2 Let C be a category with an admissibility structure. An admissible cover is of X ∈ C is a family (αi : Ui → X | i ∈ I) of admissible morphisms that is jointly surjective, i.e. such that a Ui → X i∈I

is an epimorphism.

48

Chapter 2: Analysis of Model Structures

Example 2.7.3 Let C := Top be the category of topological spaces. Let C ad be the subcategory of Top whose morphisms consist of open injections. Since open injections in Top are stable under base change along continuous maps, C ad indeed defines an admissibility structure. A family of admissible maps (vi : Ui → X | i ∈ I) is called admissible cover of X iff it is jointly surjective, i.e. if [ vi (Ui ) = X. i∈I

The class of admissible covers defines a Grothendieck pretopology on C and we will write (vi : Ui → X | i ∈ I) ∈ CovC (X) for an admissible cover of X. Example 2.7.4 Let X ∈ Top be a topological space. Consider the category Op(X) whose objects are open sets in X and whose morphisms are given via one point sets HomOp(X) (V, U ) if V ⊂ U or empty sets if V * U . Note that base change in Op(X) is given by intersection. We immediately see that Op(X) is an admissibility structure on Op(X). To be precise, we have Op(X)ad = Op(X). The following definition has to be seen in the light of the previous examples. Definition 2.7.5 Let C be a category with admissibility structure C ad . The site SC with the Grothendieck topology generated by admissible covers is called the small site of C if C = C ad and the big site associated to C ad if C ad $ C. Example 2.7.6 Let X = (X, OX ) be a scheme. Then the ordered category Op(X) of opens is called the small Zariski site of X . Let Zar be the category whose objects are schemes and whose morphisms are morphisms of schemes. Let (Ui → X | i ∈ I) be a cover of X ∈ Zar if and only if it is a jointly surjective family of scheme-theoretic open immersions. Then the slice category (Zar ↓ X ) is called the big site of X . Example 2.7.7 Given a site SC by some Grothendieck pretopology on C one can turn (C ↓ X) into a site. Define ((Yi , gi ) → (Y, g))i∈I to be a covering iff (Yi → Y )i∈I is a covering in C. This is the comma topology (cf. [Vis 05, Part 1, Ch. 2, §2.3, Definition 2.58]). The resulting site is notated S↓X . Let f : SD → SC be a morphism of sites, i.e. a functor f † : C → D of the underlying categories which preserves the pretopology, i.e. such that for every S = (Ui → X)i∈I ∈ CovC (X) we have f † (S) := (f † (Ui ) → f † (X))i∈I ∈ CovD (f † (X)) and for every morphism ∼ = (W → V ) ∈ Mor(C) the canonical morphism f † (Ui ×V W ) → f † (U ) ×f † (V ) f † (W ) is an isomorphism. In the case of a morphism of sites f : SD → SC we tend to write f ∗ a f∗ a f ‡ for the sequence of adjoint functors of Proposition 2.1.2 obtained by the restriction of ◦ ◦ functors f∗ := (f † )∗ : K SD → K SC so as to emphasize the geometric point of view.

49

2.7 Sites and their Morphisms

Example 2.7.8 Let f : X → Y be a morphism of topological spaces, i.e. f is continuous. Let C := Op(Y ), CovC := TY (the topology of Y ) and D := Op(X), CovD := TX , then the functor f † : C → D , V 7→ f −1 (V ) makes f into a morphism f : SD → SC of sites between the associated sites. In this situation the direct image part f∗ : SD∧ → SC∧ and inverse image part f ∗ : SD∧ → SC∧ of the geometric morphism (f ∗ , f∗ ) are the pushforward and the pullback of presheaves. Example 2.7.9 Let X be a topological space and x ∈ X be a point. This can be seen as a morphism p : R0 → X sending the one-point space R0 = {∗} to x ∈ X. From this point of view we have a geometric morphism (p∗ , p∗ ) : Sh({∗}) → Sh(SX ) and we call the inverse image p∗ F of F ∈ Sh(SX ) the stalk of F at x. It is common use to write Fx instead of p∗ F for the stalk at x. Let C be a category with fibered products and f : X → Y be a morphism in C. By making a choice of pullback diagrams one can associate to f a functor f −1 : (C ↓ Y ) → (C ↓ X), the change of base functor. Note that (C ↓ X) always has a terminal object, namely idX : X → X. In the case that C has a terminal object 1 one has an isomorphism C ∼ = (C ↓ 1) of categories. If t : X → 1 is the terminal map with domain X, the associated change of base functor t−1 : C → (C ↓ X) can be chosen in a way that the description on objects gives t−1 : A 7→ (pr2 : A × X → X), where pr2 denotes the projection on the second factor. P Note that the change Qof base functor −1 f always possesses for general reasons both a left f and a right adjoint f (cf. [Moe 91, Ch. I, § 9, Thm. 4]). Proposition 2.7.10 Let S be a site whose topology is generated by some pretopology. Let f : X → Y be a morphism in S . Then the associated change of base functor f −1 : S↓Y → S↓X is a morphism of sites.

50

Chapter 2: Analysis of Model Structures

Proof: The functor f −1 has a left adjoint and thus f −1 is left exact. It remains to show that f −1 preserves open covers. Recall the slice pretopology from Example 2.7.7. Denote by f −1 U the pullback of U → Y along f : X → Y . Since open covers are stable under change of base and Ui ×U f −1 U −−−→ f −1 U −−−→ X       y y y −−−→

Ui

U

−−−→ Y

are both cartesian we know from the description of the slice pretopology and the formation of limits on the slice category (cf. the remark after Proposition 2.5.5) that (f −1 Ui = Ui ×U f −1 U → f −1 U | i ∈ I) is an open cover of f −1 U . Hence we are done. q.e.d. Example 2.7.11 Let C be a category with a Grothendieck pretopology and SC its associated site. For an object U ∈ C we get two functors i†U : C → (C ↓ U ) ,

Y 7→ (Y × U → U )

and jU† : (C ↓ U ) → C ,

(Y → U ) 7→ Y.

By Proposition 2.7.10 we thus have two morphisms of sites: iU : S↓U → SC and jU : SC → S↓U that give two sequences (iU )∗ a (iU )∗ a (iU )‡ and (jU )∗ a (jU )∗ a (jU )‡ of adjoint functors according to Theorem 2.1.2. We claim that jU† a i†U is an adjoint pair. The bijection f

HomC (jU† (V → U ), W ) o

Φ Ψ

/

f

Hom(C↓U ) (V → U, i†U (W ))

is given by the mutually inverse maps Φ and Ψ that are defined as follows: Φ(α) := (α, f ) ◦ ∆, where ∆ : V → V × V is the diagonal morphism, and Ψ(β) := pr1 ◦ β with pr1

51

2.8 Homotopy Sheaves

the projection on the first factor. According to the remark after Theorem 2.1.2 we can conclude that there are isomorphisms (iU )∗ = (jU )∗ and (iU )∗ = (jU )‡ of functors. Note that (jU )∗ is the extension by zero and (iU )∗ is the restriction of sheaves on SC to U . Indeed, suppose X is a topological space and C the category Op(X) of its open subsets. Let U ⊂◦ X be an open set. Then there is an isomorphism (C ↓ U ) = Op(U ) of categories and hence iU : Op(U ) → Op(X) induces ∧ ∧ → SOp(U i∗U : SOp(X) )

sending F to F|U := i∗U F. The inverse image part jU∗ of the corresponding geometric morphism (jU∗ , (jU )∗ ) is ∧ ∧ jU∗ : SOp(U ) → SOp(X) ,

the extension by zero. By definition we have jU∗ F(V ) =

colim − −−−→

(V

† →jU (W ))∈(V

F(W ) = † ↓jU )

−colim − −−→ ◦

F(W ).

(W ⊂ X,V ⊂W )

If V ⊂ U then V is initial in (V ↓ jU† ) and thus jU∗ F(V ) = F(V ). In the case of V * U the category (V ↓ jU† ) is empty and the colimit jU∗ F(V ) gives the initial object of M. The direct image part (jU )∗ : SX∧ → SU∧ is the restriction to U . In the case of a topological open injection U → X it is common use to write (iU )! instead of jU∗ to denote the extension by zero.

2.8

Homotopy Sheaves

The idea to pass from presheaves that take values in some model category to a homotopical version of sheaves is to perform the left Bousfield localization on some global model structure at the set of all hypercovers. The idea of a hypercover is to begin with a covering (Ui → X)i∈I of a space X, then at the next level to pick coverings (Vij → Ui )j∈J of the Ui ’s, then at the next level to pick coverings (Wijk → Vij )k∈K of the Vij ’s an so on, with the (n + 1)–th stage arising as a cover of everything from stages in degrees ≤ n. The precise way of formulating this latter condition is in terms of of the degree (n + 1)-part of the coskelet functor. The following definition is due to [SGA4 72, 7.3.1]. Again, we write rX for the presheaf represented by the object X.

52

Chapter 2: Analysis of Model Structures

Definition 2.8.1 ◦ Let C be a site. A simplicial presheaf K ∈ sSetC together with an augmentation K → rX is said to be a hypercover of X ∈ C if, for all n ≥ 0, the natural adjunction map K → coskn skn (K ) induces a covering map Kn+1 → Mn+1 K := coskn skn (K )n+1

(2.8.1)

and each Kn is semirepresentable, i.e. a coproduct of representables. We call Mn K the n–th matching object. Using coskn skn (∆[n]) = ∂∆[n] and the simplicial Yoneda lemma one finds Mn K = Hom(∂∆[n], K ). So condition 2.8.1 says that for every map of the simplicial set ∂∆[n] to K there is at least one map of the n–simplex ∆[n] inducing it. Looking at M0 K ∼ = K−1 = rX we see that K0 → rX is a covering. We will only have occasion to to deal with a special kind of hypercovers which we ˇ will introduce in the following example, the Cech hypercover associated to an open cover (Ui → Y )i∈I . Example 2.8.2 Let SC = (C, CovC ) be a site (i.e. C is the underlying category of SC and CovC the class of covering maps of the Grothendieck pretopology generating SC ) and X ∈ C be an object. (1) Let (Ui → Y )i∈I ∈ CovC (Y ) be a cover of X. Define a Un := rUin , in ∈I n+1

where Uin = U(i0 ,...,in ) := Ui0 ×Y Ui1 ×Y · · · ×Y Uin and U−1 := rX. The simplicial presheaf U together with its natural augmentation U → rX defines a hypercover of ˇ X. This is the Cech hypercover of (Ui → Y )i∈I . (2) Recall Example 2.7.7. Using this we can define hypercoverings of the slice category in the following way. We call an arrow (U , x ) → (rY, g) a hypercover of (Y, g) in (C ↓ X) iff the image U → rY of the projection on the first factor is a hypercover of Y in C.

53

2.8 Homotopy Sheaves Definition 2.8.3 Let SC = (C, CovC ) be a site. We will write HypC

ˇ ˇ for the class of all Cech hypercovers of C and (U → rX) ∈ HypC (X) for a Cech hypercover of X ∈ C. Next we show that morphisms of sites map hypercovers to hypercovers. Lemma 2.8.4 Let f : SC → SD a morphism of sites. Then the functor ◦

f ∗ : sSetSD −−−→ sSetSC

◦

given by 

 P 7→ V 7→ f ∗ P (V ) =

colim − −−−→

P (U )

(V →f † (U ))∈(V ↓f † )

(cf. Definition 2.1.1) takes hypercovers to hypercovers, i.e. induces a morphism f ∗ : HypD −−−→ HypC . Proof: Let (Ui → Y )i∈I ∈ CovD (Y ) be an open cover of Y and (U → rY ) ∈ HypD (Y ) ◦ be the associated hypercover of Y in D. We have to show that (V → r(f † (Y ))) ∈ sSetSC given by Vn := f ∗ (Un ) gives an object of HypC (f † (Y )). Using Hom (f ∗ (rV ), F ) = Hom (rV, f∗ F ) = f∗ F (V ) = F (f † (V ))

= Hom r f † V , F , ◦ with F ∈ sSetC we see that (f † )∗ (rV ) ∼ = r(f † (V )) for every V ∈ D. Moreover we have   a rUin  Vn = f ∗ 

in ∈I n+1

=

a

f ∗ rUin




in ∈I n+1

=

a


r f † (Uin ) .

in ∈I n+1

So we end up with f ∗ (U → rY ) = (Vn → rf † (Y )) ∈ HypC (f † (Y )) and we are done. q.e.d.

54

Chapter 2: Analysis of Model Structures

Proposition 2.8.5 Let M be a simplicial model category and Y ∈ M a fibrant object. Then there is a weak equivalence Map(−hocol holim −−−→F, Y ) ' ← −−−−Map(F, Y ). I

Proof:

I

Since Hom(−colim lim −−−→QF ⊗∆[n], Y ) = ← −−Hom(QF ⊗∆[n], Y ) is true for all n ∈ N0 I

I

(because of Lemma 2.3.5), general nonsense implies Map(−hocol −−−→F, Y ) ' Map(−colim −−−→QF, Y ) I

I

'← lim −−Map(QF, Y ) I

'← holim −−−−Map(F, Y ), I

where Q denotes some cofibrant replacement. We have used that Map(QF, Y ) is fibrant. q.e.d. Remark. Looking at the proof we see that in the above proposition the assumption on Y to be a fibrant object with respect to the model structure can be omitted by choosing additionally a fibrant replacement for Y . In this case our result agrees with the one from [Hir 03, Ch. 18, § 18.1, Theorem 18.1.10]. Note that our approach is more natural and does not need an explicit construction of the limits involved whereas [Hir 03] does. ˇ Recall that for a site SC every cover Ui → V gives a Cech hypercover U → rV . Definition 2.8.6 Let C be a small category and SC be a site. Define for every object V ∈ C a set n o ψV := − hocol U  → rV U → rV ∈ HypC (V ) . −−−→ Let SC be the set of all ψV where V ranges over all objects in C. Define Shv(SC , sSet) := LSC Pre(SC , sSet) to be the left Bousfield localization of the injective model structure on the category of ˇ sSet-valued presheaves on SC at the class of all Cech hypercovers. Example 2.8.7 Let U → rY ∈ Hyp(SC ) (Y) be a hypercover of Y ∈ C and F ∈ Shv(SC , sSet) be fibrant. Then there is a weak equivalence Y F (Uin ) Map(−hocol U  , F ) ' holim −−−→ ←−−−− [n]∈∆

of simplicial sets.

[n]∈∆

in ∈I n+1

55

2.8 Homotopy Sheaves

Remark. The cofibrations of Pre(SC , sSet) and Shv(SC , sSet) are equal (note that both model categories are equal as categories but they do carry different model structures). Every weak equivalence in Pre(SC , sSet) is also a weak equivalence in Shv(SC , sSet). Moreover, an object F ∈ Shv(SC , sSet) is fibrant if and only if it is fibrant for the model structure on Pre(SC , sSet) having the additional property of being SC -local (cf. [DHI 04, Cor. 6.3]), i.e. in this situation that for every hypercover U → rV associated with (Ui → V ) the map Y F (Uin ) F (V ) → ← holim −−−− is a weak equivalence as can be seen using Example 2.8.7. Thus, we do call fibrant objects in Shv(SC , sSet) homotopy sheaves and we will write F (V ) → ← holim −−−−F (U ) for conciseness. Proposition 2.8.8 Let c∗ : Sh(SC , Set) → Shv(SC , sSet) be the constant simplicial set functor from the category of set-valued sheaves over SC . Then the counit π0 ◦ c∗ → 1 of the adjunction π0 a c∗ is an isomorphism. Proof: A constant simplicial presheaf F ∈ Shv(SC , sSet) is fibrant in Pre(SC , sSet) and therefore it is fibrant in Shv(SC , sSet) if and only if the morphism ! Y Y F (V ) → ← holim F (Ui ) ⇒ F (Ui,j ) . . . −−−− i

i,j

is a weak equivalence for every object V ∈ C and every cover (Ui → V ). But the simplicial sets F (U ) are constant and so the homotopy limit is an ordinary limit. It is also true that a limit of such a cosimplicial diagramm of ordinary sets can in turn be calculated as the equalizer of the truncated two term diagramm ! Y Y F (X) → ← lim F (Ui ) ⇒ F (Ui,j ) . −− i

i,j

But this is exactly the condition for F being a sheaf. q.e.d. The proposition above reads as follows. Corollary 2.8.9 A constant simplicial presheaf F is fibrant in Shv(SC , sSet) if and only if F is a sheaf.

56

Chapter 2: Analysis of Model Structures

Proposition 2.8.10 Let f : SC → SD be a morphism of sites. Then the adjoint pair (f ∗ , f∗ ) : LSC Pro(SC , sSet) → LSD Pro(SD , sSet) is a morphism of model categories. Proof:

We already know by Proposition 2.2.18 that (f ∗ , f∗ ) : Pro(SC , sSet) → Pro(SD , sSet)

is a morphism of model categories. Since by Lemma 2.8.4 f ∗ sends hypercovers to hypercovers, the universal property of left Bousfield localization implies that f ∗ can be extended to a left Quillen functor f ∗ : LSC Pro(SC , sSet) → LSD Pro(SD , sSet) and we are done. q.e.d. Proposition 2.8.11 Let f : SD → SC be a morphism of sites. Then the adjoint pair (f ∗ , f∗ ) : Shv(SD , sSet) → Shv(SC , sSet) is a morphism of model categories. Proof:

This is an immediate consequence of from Proposition 2.2.19 and Lemma 2.8.4. q.e.d.

The proposition implies that the direct image of a homotopy sheaf is again a homotopy sheaf. Corollary 2.8.12 Let C be a category with a Grothendieck pretopology and U be an object in C. By Proposition 2.8.11 the functors from Example 2.7.11 give morphisms (i∗U , (iU )∗ ) : Shv(S↓U , sSet) → Shv(SC , sSet) and (jU∗ , (jU )∗ ) : Shv(SC , sSet) → Shv(S↓U , sSet) of model categories. Since i∗U is isomorphic to (jU )∗ , the functor (iU )∗ is both a left and right Quillen functor. As a consequence, the restriction F|U of cofibrant respectively fibrant objects F ∈ Shv(SC , sSet) is again cofibrant respectively fibrant. A question one may ask at this point is whether for a morphism of sites f : SC → SD the pullback f ∗ : Shv(SD , M) → Shv(SC , M) preserves fibrant ojects, i.e. whether the image of homotopy sheaves under f ∗ are homotopy sheaves again. Looking at conventional sheaf theory this is wrong. Therefore the pullback of sheaves is defined to be the composition of

57

2.8 Homotopy Sheaves

the pullback functor of presheaves followed by the sheafification functor. Taking Corollary 2.8.9 into account we have a commuting square f∗

Sh(SD , sSet) −−−→ Sh(SC , sSet)     c∗ y c∗ y f∗

Shv(SD , sSet) −−−→ Shv(SC , sSet). Thus, we see that f ∗ : Shv(SD ) → Shv(SC ) cannot preserve fibrancy since otherwise the classical pullback of sheaves would not need sheafification. Consequently, the pullback of homotopy sheaves f ∗ : Shv(SD )fib → Shv(SC ) should be seen as the composition of the ordinary pullback of presheaves obtained by object-wise left Kan extension followed by some fibrant replacement. Theorem 2.8.13 Let SC be a site. Then there is an equivalence LS Pro(SC , sSet) → Shv (SC , sSet) of model categories induced by the identity functor. Proof: By Proposition 2.2.17 Pro(SC , sSet) → Pre(SC , sSet) is an equivalence of model categories. Since the identity is a morphism HypC → HypC of hypercovers, it preserves the localizing class of morphisms such that the left Quillen part extends to a left Quillen functor with source LS Pro(SC , sSet). q.e.d. Remark. We will tend to write Shv(SC , sSet) instead of LS Pro(SC , sSet) which is not much abuse of notation since both categories are not only equal as categories but also equivalent as model categories. The following definition is due to [Spi 07, Ch. 2, § 2.2, Def. 2.2.2]. Definition 2.8.14 Let C be a small category and M be a simplicial model category. For ◦ each pair F, G ∈ MC of presheaves the presheaf mapping space MapM (F, G) : C ◦ → sSet is the presheaf X 7→ MapM (F|X , G|X ) of simplicial sets. Lemma 2.8.15 Let C be a small category. Assume F, G, H to be simplicial presheaves. Then one has a canonical isomorphism Map(F ⊗ H, G) ∼ = Map(H, Map(F, G)).

58 Proof:

Chapter 2: Analysis of Model Structures The assertion follows immediately from the object-wise naturality. q.e.d.

Remark. The definition of the presheaf mapping space should be seen in the light of the following situation: Let D be a small category and M be the category of simplicial ◦ presheaves sSetD with the injective model structure, i.e. M = Pre(D, sSet) as simplicial model categories. For every object X ∈ D the evaluation functor evX : M → sSet sending F to F (X) is a right Quillen functor whose left Quillen part is hX ⊗ ( · ) : sSet → M with hX the functor of points. This can be seen as follows: As discussed in paragraph 2.3 we have isomorphisms MapM (hX ⊗ S , F ) = MapsSet (S , MapM (hX , F )) = MapsSet (S , F (X)). Moreover, because of [Hir 03, Part II, Ch. 9, Prop. 9.3.9 (1)(b)] the map hX ⊗S → hX ⊗T is a cofibration (respectively an acyclic cofibration) if S → T is a cofibration (respectively an acyclic cofibration). Beyond this, let SC be a site associated to some small category C. From Lemma 2.8.15 and the fact that cofibrations and weak equivalences are determined object-wise for the injective model structure (i.e. [Hir 03, Part II, Ch. 9, Prop. 9.3.9 (1)(b)] applies again) one readily deduces that evX : Pre(SC , M) → Pre(SC , sSet) ,

F 7→ (U 7→ F(U, X))

is a right Quillen functor whose left Quillen part is hX ⊗ ( · ) : Pre(SC , sSet) → Pre(SC , M), where hX denotes the constant presheaf on SC given by the functor of points in X. The main question at issue at present is whether we can find the initial left Bousfield localization j : Pre(SC , M) → Shv(SC , M) such that the lower morphism in the diagram ev

Pre(SC , M) −−−X→ Pre(SC , sSet)     jy y ev

Shv(SC , M) −−−X→ Shv(SC , sSet). is right Quillen. By the universal property of the left Bousfield localization this can be accomplished if we localize at the image of the left Quillen part Pre(SC , M) →

59

2.8 Homotopy Sheaves

Pre(SC , sSet). Thus, we simply tensor all hypercovers with hX for each object X ∈ C, i.e. we consider o n ⊗ U ) → (h ⊗ rV ) | U → rV ∈ Hyp LX := − hocol (h   C . X −−−→ X Now let L :=

[

LX .

X∈C

Then we can define Shv(SC , M) := LL Pre(SC , M). This construction was used in a special case by D. Spivak (cf. [Spi 07, Ch. 2, §§2.2, 2.3]) to construct the simplicial model category of sheaves of smooth rings. In chapter four we will have occasion to deal with that. Let F ∈ Shv(SC , M) be fibrant. Computation gives the following chain of weak equivalences F(V, X) = Map (hX , F(V )) = Map (hX , MapsSetC◦ (rV, F)) = Map (hX ⊗ rV, F)   = Map − hocol −−−→(hX ⊗ U ), F =← holim −−−−Map (hX ⊗ U , F) =← holim −−−−F(U , X) for any hypercover U → rV of V ∈ C and any object X ∈ D. Hence we conclude F(V ) ' ← holim −−−−F(U ) for any hypercover, i.e. F ∈ Shv(SC , M) is indeed a homotopy sheaf in our sense. It is important to note that all our considerations carry over to this setting: Let f : SC → SD be a morphism of sites. Then the inverse image part f ∗ of the geometric morphism (f ∗ , f∗ ) maps hypercovers to hypercovers and since f ∗ preserves tensor objects, it preserves the localizing set L from above, and by the universal property of the left Bousfield localization we end up with a morphism (f ∗ , f∗ ) : Shv(SC , M) → Shv(SC , M) of model categories. In other words, all statements about Shv(SC , sSet) remain valid in the case of Shv(SC , M). Lemma 2.8.16 Let SC be a site and M an injective model category of simplicial presheaves, i.e. M = Pre(D, sSet) for some small category D. Assume F, G ∈ Shv(SC , M) are presheaves with G ∈ Shv(SC , M) fibrant. Then the presheaf Map(F, G) : SC◦ → sSet is a homotopy sheaf.

60

Chapter 2: Analysis of Model Structures

Proof: All objects of Shv(SC , M) are cofibrant and since G is fibrant by assumption we have that MapsSet (F, G) is fibrant in Pro(SC , sSet). It remains to show that MapsSet (F, G) is SC -local (cf. Definition 2.8.6). Now let (Ui → U | i ∈ I) be an open cover of U ∈ C. Note that the restriction G|U is fibrant again. Consequently, the mapping spaces Map(F|Ui , G|Ui ) are fibrant for the standard model structure on sSet and so are products of such, so the homotopy limit Y Map(F|U , G|U ) → ← holim Map(F|Uin , G|Uin ) −−−− in ∈I n+1

in the diagram can be calculated simply as a limit by [Bou 73, Ch. XI, §4, 4.1]. Since limits of simplicial sets are taken level-wise it suffices to perform the calculation at each simplicial level which in turn can be calculated as the equalizer of the truncated two-term diagram. Thus we have boiled down to check exactness of   Y Y  Hom(F|Ui ⊗ ∆[n], G|Ui ) ⇒ Hom(F|U(i,j) ⊗ ∆[n], G|U(i,j) ) . Map(F|U , G|U )n → ← lim −− (i,j)∈I 2

i∈I

Q Consider φ, ψ ∈ Map(F, G)n (U ) whose images in Map(F, G)n (Ui ) are equal. Hence we have φ|Ui = ψ|Ui . Let # " Y Y F(Uij ) ⊗ ∆[n] . . . ΦF := F(Ui ) ⊗ ∆[n] ⇒ i,j∈I

i∈I

be the cosimplicial diagram for short. Then one has the following diagram φ

U F(U ) ⊗ ∆[n] −−− → ψU   y

holim ← −−−−ΦF

G(U )  ∼ y=

−−−→ ← holim −−−−ΦG

in Ho(Shv(SC , M)). Since G is fibrant (in Shv(SC , M)), the arrow on the right hand side in the commutative square is a weak equivalence and hence an isomorphism in the homotopy category which means that φU = ψU in Ho(Shv(SC , M)). For every morphism V → U we conclude φV = ψV and consequently we have φ = ψ, i.e. Map(F, G) is separated in the homotopy category. Now let (φi : F|Ui ⊗ ∆[n] → G|Ui )i∈I be a family of morphisms with φi |Uij = φj |Uij . The compatibility on twofold intersections provides a map holim holim ← −−−−ΦF −−−→ ← −−−−ΦG due to the functorality of the homotopy limit and because of F(U ) ⊗ ∆[n] → ← holim holim −−−−ΦF → ← −−−−ΦG ' G(U )

61

2.8 Homotopy Sheaves we get an arrow ψU : F(U ) ⊗ ∆[n] → G(U )

in the homotopy category. This is compatible with restrictions to V → U which follows in a similar way. Hence we get a map φ : F|U ⊗ ∆[n] → G|U . Taken all together we get an isomorphism   Y Y ∼ =  Hom(F|Ui ⊗ ∆[n], G|Ui ) ⇒ Map(F|U , G|U )n → ← lim Hom(F|U(i,j) ⊗ ∆[n], G|U(i,j) ) −− (i,j)∈I 2

i∈I

in the homotopy category. Because of the saturation of model categories we get our desired weak equivalence in Shv(SC , sSet). q.e.d. As a nice consequence we have the well-known classical result. Corollary 2.8.17 Let X ∈ Top be a topological space and F be a set-valued presheaf. Then there is an endofunctor H om(F, · ) : X ∼ → X ∼ on the category X ∼ of set valued sheaves on X given by G 7→ (U 7→ Hom(F|U , G|U )) . Proof: We have F, G ∈ Shv(X, sSet) where we consider F and G to be constant in the simplicial direction. Since simplicially constant simplicial sheaves are fibrant in Shv(X, sSet) if and only if they are sheaves in the classical sense, G is fibrant in Shv(X, sSet) and according to the previous lemma Map(F, G) is fibrant. Since Map(F, G)0 ∼ = H om(F, G) is simplicially constant and fibrant, it is a sheaf in the classical setting. q.e.d. Remark.

Corollary 2.8.17 gives another proof for the well known fact that the presheaf U 7→ Hom(F|U , G|U )

is in actual fact a sheaf. The first time one encounters this fact one assumes F to be a sheaf, too. (cf. [Liu 06, Ch. 2, § 2.2, exercise 2.11] glueing of morphisms [Har 06, Ch. II, § 1, exercise 1.15] sheaf of local morphisms). But even in conventional Algebraic Geometry it is not necessary that F is a sheaf. We know that the restriction F|U of a presheaf F on X to an open subset U is given by i∗ F if i : U → X denotes the inclusion of the open subset. The adjoint pair a a ι, where a denotes the sheafification functor and ι the forgetful functor to presheaves, gives the isomorphism a (F|U ) = (aF) |U ,

62

Chapter 2: Analysis of Model Structures

because i∗ is given by a colimit. Thus, using Hom(F, ιG) = Hom(aF, G) we see that the assumption on F being a sheaf is not necessary.

Chapter 3 Smooth Functors as C∞-Schemes This chapter presents a new approach to C∞ -schemes and we show that it leads to the same result as the classical one. To do so we summarize all well known facts about C∞ -rings, the basic layout for C∞ -schemes, in the first paragraph. The second paragraph relates C∞ rings to topology. That is, we define the category of affine schemes to be the dual of the full subcategory of finitely presented C∞ -rings and turn it into a site using results from E. Dubuc [Dub(b) 81] and A. Kock [Koc 06]. Beyond this we define a topological realization functor that associates to every affine scheme a topological space. The next paragraph connects both topologies. More precisely, we define the category of C∞ -schemes as a full subcategory of the category of set-valued sheaves for the Grothendieck topology on Aff ∞ whose objects possess a cover by affines. It turns out that the set of all open subfunctors forms a locale and that for any affine C∞ -scheme this locale is isomorphic to the category of open sets obtained by its topological realization. In paragraph four we provide the big structure sheaf that will be needed to compare C∞ -schemes coming from smooth functors to those coming from ringed spaces. Paragraph five recalls the conventional approach to C∞ -schemes via ringed spaces. We show how our functor approach fits into this picture and construct an adjoint pair whose left adjoint sends C∞ -schemes coming from functors to locally C∞ -ringed spaces. Finally, we prove that the smooth functor approach gives the expected result when applying the classical construction to finitely presented C∞ -rings.

3.1

Commutative Algebra with C∞-Rings

This section serves as a recollection of facts that are of particular interest for our purposes to approach to C∞ -schemes both by functors and by ringed spaces. We will develop tools similar to classical Commutative Algebra that allow to construct C∞ -schemes in the classical way using methods similar to those in [Liu 06] and [Har 06] to build up schemes. Therefore we summarize well known results and provide proofs to those where none is available in literature. We basically follow [Koc 06] and [Moe 91] which give an introduction to C∞ -rings emphasizing the functorial point of view. For a more algebraic approach to C∞ -rings [Joy 09] can be recommended. This section might also be useful if 63

64

Chapter 3: Smooth Functors as C∞ -Schemes

one deals with C∞ -rings for the first time. We start with the definition. Definition 3.1.1 The category C∞ Ring is the category of finite product preserving functors F : Euk → Set from the category Euk of euclidian spaces, i.e. the category whose objects are Rn for some n ∈ N0 and whose morphisms are all smooth maps, to the category of sets. We call it the category of C ∞ -rings. An object F ∈ C∞ Ring is called C ∞ -ring. We will occasionally write F n instead of F (Rn ) and sometimes F instead of F (R). We continue with a fundamental example. Note that manifolds for our purposes are always assumed to be smooth and of finite dimension. Example 3.1.2 Let M ∈ Man be a manifold. We define a functor C ∞ (M ) : Euk → Set on the euclidian category by sending Rd to HomMan (M, Rd ) which is product preserving. ∞ ∞ (M ) where on the right hand side (M ) × · · · × CM To be precise we have C ∞ (M )(Rd ) = CM ∞ of differentiable functions we have the d-fold product of global sections of the sheaf CM on M . More generally, we have a contravariant functor C ∞ : Man → C∞ Ring sending a manifold M ∈ Man to C ∞ (M ) := HomMan (M, · ). The most important example of an C ∞ -ring induced by manifolds is C ∞ (Rn ) since it is the C ∞ -ring on n generators, these generators being the projections on the real line (the reason for this will become clearer soon). From the categorial point of view this example is not amazing since the Hom-functor is left exact. Therefore we give an example in order to show that C∞ -rings are more general than those coming from manifolds. It is taken from [Moe 91, Ch. I, Thm. 1.3] and can also be found in [Joy 09, § 2.5]. Example 3.1.3 The R-algebra of formal power series R[[X1 , . . . , Xn ]] can be made into a C∞ -ring using the Taylor expansion at zero and the chain rule. We will give an easier description later on.

3.1 Commutative Algebra with C∞ -Rings

65

Lemma 3.1.4 The inclusion functor j : Eukpoly → Euk, where Eukpoly is the category whose objects are Rd for d ∈ N0 and whose morphisms are polynomials with real coefficients, induces via restriction of functors a functor j ∗ : C∞ Ring → R-Alg,

A 7→ j ∗ (A)

from the category C∞ Ring of C∞ -rings to the category R-Alg of commutative R-algebras. We call it the underlying R-algebra functor. In the sequel we will tend to write ϕR instead of j ∗ (ϕ) for the underlying R-algebra morphism. The image of A ∈ C∞ Ring under j ∗ is called the underlying R-algebra of A. By abuse of notation we will often write A instead of j ∗ (A). Proof: Recall that an R-algebra is a ring (we assume all rings in this thesis to be unitary and commutative), which is equipped with a homomorphism R → A determining the scalar multiplication. Equivalently, every map p = (p1 , . . . , pn ) : Rd → Rn which is given by polynomials pi with real coefficients can be interpreted as a map A(p) : Ad → An , in such a way that projections, composition and identity are preserved. In other words, an R-algebra A is nothing but a product preserving functor A : Eukpoly → Set. q.e.d. Example 3.1.5 For C ∞ (M ) ∈ C∞ Ring from the first example we have ∞ j ∗ (C ∞ (M )) = CM (M ),

i.e. the set of global sections on M is the underlying R-algebra. In particular, we have j ∗ (C ∞ (R0 )) = R. The underlying R-algebra of a C∞ -ring enables us to extend fundamental constructions from conventional Commutative Algebra to the theory of C∞ -rings. We will explain the solutions to the three universal problems of forming the quotient by an ideal, adjoining an indeterminate element to a C∞ -ring and of inverting elements. The fact that these techniques carry over to C∞ -rings is important in so far as it will allow a construction of a spectrum functor using methods of conventional Algebraic Geometry. All the time we talk about elements or subsets of a C∞ -ring we mean elements and subsets of its underlying R-algebra. Definition 3.1.6 An ideal I of an C∞ -ring A ∈ C∞ Ring is an ideal I ⊂ A of the underlying R-algebra of A.

Chapter 3: Smooth Functors as C∞ -Schemes

66

Our definiton agrees with [Joy 09, Definition 2.7], [Koc 06, Ch. III, § 5, Proposition 5.4], [Dub(a) 81] and [Moe 91, Ch. I, § 1, Prop. 1.2]. The following proposition solves the universal problem of making zero the elements of an ideal of a given C∞ -ring. It is remarkable that the R-algebra structure can be uniquely extended to C∞ -rings. Proposition 3.1.7 Let A ∈ C∞ Ring be an C∞ -ring and I ⊂ A be an ideal. Then the R-algebra A/I carries a unique structure of a C∞ -ring such that the canonical projection p : A → A/I induces a morphism of C∞ -rings. Proof: The basic idea to prove the claim is to use Hadamard’s Lemma. Detailed proofs are available in [Joy 09, Definition 2.7], [Koc 06, Ch. III, § 5, Proposition 5.4] and [Moe 91, Ch. I, § 1, Proposition 1.2]. q.e.d.

Example 3.1.8 Proposition 3.1.7 enables us to define the ring of dual numbers in the category C∞ Ring, i.e. R[] := C ∞ (R)/(x2 ) with x ∈ C ∞ (R) the identity. Following [Moe 91, Ch. I, § 1], there is an easy explicit description of the C∞ -ring structure: Let f : Rd → R be a smooth map, then its interpretation R[](f ) : (R[])d → R[] is the map (y1 + z1 , . . . , y1 + z1 ) 7→ f (y1 , . . . , yn ) +

n X ∂f (y1 , . . . , yn ) · zk . ∂x k k=1

Proposition 3.1.9 Let ϕ : A → R be a morphism of C∞ -rings and I ⊂ A be an ideal such that I is contained in ker(ϕR ) the kernel of ϕR = j ∗ (ϕ) : A → R. Then there is a unique morphism α : A/I → R of C∞ -rings such that α ◦ p = ϕ, where p is the canonical morphism A → A/I from Proposition 3.1.7. Proof: Of course, due to the conventional fundamental theorem on homomorphisms we have a unique morphism αR : A/I → R such that αR ◦ pR = ϕR . It remains to check that this can be extended to a natural transformation. If πk : Rn → R denotes the projection on the k–th coordinate we have ϕR ◦ A(πk ) = R(πk ) ◦ ϕRn and consequently ϕRn (a1 , . . . , an ) = (ϕR (a1 ), . . . , ϕR (an )) because all functors involved

3.1 Commutative Algebra with C∞ -Rings

67

are product preserving. Hence we can define αRn : (A/I)n → Rn by αRn (a1 +I, . . . , an +I) = ϕRn (a1 , . . . , an ). For an arbitrary morphism f = (f1 , . . . , fd ) : Rn → Rd we compute (R(f ) ◦ αRn )(a1 + I, . . . , an + I) = (R(f ) ◦ ϕRn )(a1 , . . . , an ) = (ϕRd ◦ A(f ))(a1 , . . . , an ) = αRd (A(f1 )(a1 , . . . , an ) + I, . . . , A(fd )(a1 , . . . , an ) + I) = (ϕRd ◦ (A/I)(f ))((a1 , . . . , an ) + I), i.e.

α

n (A/I)n −−− → Rn     (A/I)(f )y R(f )y

α

d (A/I)d −−− → Rd

commutes. Hence it follows that α is indeed a natural transformation of product preserving functors. q.e.d. Example 3.1.10 Returning to Example 3.1.5 we can give the following description with Proposition 3.1.7 in mind. Let I ⊂ C ∞ (Rn ) be the ideal of functions which are flat at 0 ∈ Rn , i.e. whose partial derivatives vanish at 0. Since every formal power series in X1 , . . . , Xn is the Taylor series at 0 of some smooth function f ∈ C ∞ (Rn ), we have an epimorphism T0 : C ∞ (Rn ) → R[[X1 , . . . , Xn ]] whose kernel is precisely I. Thus we have R[[X1 , . . . , Xn ]] ∼ = C ∞ (Rn )/I. Definition 3.1.11 A C∞ -ring A is of finite type if and only if it is finitely generated as a C∞ -ring, i.e. if there is a n ∈ N0 and elements a1 , . . . , an ∈ A such that for every a ∈ A there is a global section f ∈ CR∞n (Rn ) in the ring of differentiable functions on Rn such that A(f )(a1 , . . . , an ) = a. The a1 , . . . , an are called generators of A. We write C∞ Ringft for the full subcategory of C∞ -rings of finite type. Remark. The definition uses the richer structure of C∞ -rings. Thus, not only addition and multiplication are admitted to generate A but all differentiable functions. The following example illustrates the universal property of the C∞ -ring C ∞ (Rn ).

Chapter 3: Smooth Functors as C∞ -Schemes

68

Example 3.1.12 Let A be a finitely generated C∞ -ring with generators a1 , . . . , an . Then we define a morphism φ : C ∞ (Rn ) → A in C∞ Ring via f 7→ φR (f ) := A(f )(a1 , . . . , an ). Thus, we have πk 7→ φR (πk ) := A(πk )(a1 , . . . , an ) = ak , where πk : Rn → R denotes the k-th coordinate projection. Hence φ is an epimorphism with I := ker φR and we can write A = C ∞ (Rn )/I due to Proposition 3.1.7. It is common use (cf. [Joy 09] and [Koc 06]) to write I → C ∞ (Rn ) → A for the presentation. We see that C ∞ (Rn ) plays the same role as R[X1 , . . . , Xn ] in the category R-Alg of commutative R-algebras with unit in conventional Commutative Algebra. For this reason we call C ∞ (Rn ) the C ∞ -ring in n generators, the projections being the generators. Definition 3.1.13 We call a C∞ -ring A finitely presented if and only if A is of finite type and fits into a presentation I → C ∞ (Rn ) → A, where I is finitely generated (as an ideal of j ∗ (A)). We write C∞ Ringfp for the full subcategory of finitely presented C∞ -rings. The following lemma gives a nice and useful explicit description of finitely presented C -rings as a colimit. ∞

Example 3.1.14 Let ϕ

I −−−→ C ∞ (Rn ) −−−→ A be a presentation of A and g1 , . . . , gk ∈ C ∞ (Rn ) generators of I. Define g : Rn → Rk to be the smooth function g(x) := (g1 (x), . . . , gk (x)). Let ψ be the morphism ψ : C ∞ (Rk ) → C ∞ (Rn ) , f 7→ ψR (f ) := g ∗ f = (x 7→ f (g1 (x), . . . , gk (x))) . Then one can readily verify that A fits into 0

C ∞ (Rk ) −−−→ C ∞ (R0 )     ψy y ϕ

C ∞ (Rn ) −−−→

A,

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which is a cocartesian diagram in C∞ Ring. On the other hand, every pushout diagram of this form gives a finite presentation of A where I = (π1∗ ψ, . . . , πk∗ ψ). A difference with conventional Algebraic Geometry is that C ∞ (Rn ) is not noetherian. Thus, finitely generated C∞ -rings need not to be finitely presented. Lemma 3.1.15 Every C∞ -ring can be displayed as a colimit of finitely generated C∞ rings. Proof:

Let E be any set such that C ∞ (RE ) → A

is a surjection on the underlying R-algebras, whereas C ∞ (RE ) is the C∞ -ring of functions RE → R which smoothly depend on finitely many variables only. Define Ψ := {D ⊂ E | D finite such that C ∞ (RD ) → A is a subring }. Then Ψ gives a directed system and ∞ D A0 := −colim −−−→C (R ) Ψ

is an object of C∞ Ring again. Now one easily checks A0 = A as in conventional Commutative Algebra since the colimit is computed pointwise. q.e.d. Recall the categorial concept of finite presentation: Given a category C, an object A ∈ C is called finitely presented iff the functor HomC (A, · ) preserves filtered colimits. The following proposition relates the categorial and the C∞ -concept of finite presentation. Proposition 3.1.16 A C ∞ -ring is finitely presented in the categorial sense if and only if it is finitely presented as a C ∞ -ring. Proof: Let A ∈ C∞ Ring be a finitely presented C∞ -ring. Due to Example 3.1.14 the ∞ n ∞ k ∞ 0 C∞ -ring A can be identified with − colim I be −∞−−→(C (R ) ← C (R ) → C (R )). Now let ∞ a directed category and X : I → C Ring a diagram. Note that −colim −−−→X is a C -ring. I

Then we have ∞ ∞ colim colim − −−−→ HomC Ring (A, X) = HomC Ring (A, − −−−→X)

I

I

because filtered colimits commute with finite limits. Assume now A ∈ C∞ Ring is finitely presented in the categorial sense. Let (Ai )i∈I be the filtered system of finitely generated subrings of A (cf. Lemma 3.1.15). Then we have colim − −−−→Ai = A. Since ∞ ∞ colim colim − −−−→HomC Ring (A, Ai ) → HomC Ring (A, − −−−→Ai )

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is surjective, there is a j ∈ I such that idA ∈ HomC∞ Ring (A, A) has a preimage in HomC∞ Ring (A, − colim −−−→Aj ). Since Aj is a finitely generated subring, it follows that A is also of finite type, i.e. A = C ∞ (Rd )/J for some d ∈ N and some ideal J ⊂ C ∞ (Rd ). It remains to prove that the ideal J of relations is finitely generated. Let (Ji )i∈I the filtered system of finitely generated subideals of J. Define Bi := C ∞ (Rd )/Ji . We have − colim Bi = C ∞ (Rd )/J = A since − − − → ∞ colim − −−−→Ji = J and dividing by ideals of C -rings commutes with inductive limits, i.e. directed colimits are left exact. Looking at ∼ ∞ ∞ colim − −−−→HomC Ring (A, Bi ) = HomC Ring (A, A) we see that there must be a morphism of C∞ -rings si : A → Bi factoring the identity idA = pri ◦ si through the projection pri : Bi → A. Let xj be the image of the j-th generator πj of C ∞ (Rd ) in A. Thus pri ◦ si (xj ) = xj and hence si (xj ) − πj ∈ J/Ji . Choosing i sufficiently large we may assume si (xj ) = πj . Let pr : C ∞ (Rd ) → A be the canonical projection. It follows that si ◦ pr : C ∞ (Rd ) → Bi is the canonical projection for i sufficiently large. For a ∈ I it is pr(a) = 0, hence Ji = J and J is finitely generated. Hence we are done. q.e.d. We are going to cope with solving the problem of universally inverting a subset S ⊂ A. To do so we recall that all small coproducts of C∞ -rings exist and give a detailed proof of this fact which is not available in literature as far as the author knows. It is remarkable that the idea of how to construct the ordinary tensor product of ordinary rings carries over to the C∞ -setting although the proof is technically more intricate. Let A, B, R ∈ C∞ Ring be of finite type, say A = C ∞ (Rn )/IA , B = C ∞ (Rm )/IB and R = C ∞ (Rd )/IR with representation maps ϕ : C ∞ (Rd ) → R, ψ : C ∞ (Rn ) → A and η : C ∞ (Rm ) → B. Let α : R → A and β : R → B be given maps. We want to give an explicit construction of the pushout M of α along β and use this to prove that category C∞ Ring possesses coproducts. Therefore note that we have canonical morphisms pr∗n : C ∞ (Rn ) → C ∞ (Rn+m ) and pr∗m : C ∞ (Rm ) → C ∞ (Rn+m ) coming from the projections prn : Rn+m → Rn and prm : Rn+m → Rm on the first n and last m coordinates. In order to obtain a pushout diagram α R −−−→ A     βy βy α

B −−−→ M we have to assure that all elements coming from R will be equalized in M , i.e. β ◦ α(r) = α ◦ β(r) for all r ∈ R. Let r1 , . . . rd ∈ R be generators coming from the projections π1 , . . . , πd ∈ C ∞ (Rd ), i.e. ri = ϕ(πi ) for i ∈ {1, . . . , d}. It suffices to make sure that

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the images of all ri in M are going to be equalized. Looking at the images α(ri ) ∈ A and β(ri ) ∈ B and taking into account that A and B are of finite type, there are finite sequences f1 , . . . , fd in C ∞ (Rn ) and g1 , . . . , gd in C ∞ (Rm ) such that α(ri ) = ψ(fi ) and β(ri ) = η(gi ). Using this we define an ideal I := (pr∗n IA , pr∗m IB , pr∗n (fi ) − pr∗m (gi ) | i ∈ {1, . . . , d}) in C ∞ (Rn+m ), the ideal I ⊂ C ∞ (Rn+m ) generated by IA , IB and the relations fi − gi . Now let M := C ∞ (Rn+m )/I be the quotient, i.e. M fits into the presentation p

I I −−−→ C ∞ (Rn+m ) −−− → M

with the canonical projection pI as presentation map. Define α : B → M via α(b) := pI ◦ pr∗m (g) for some preimage g ∈ C ∞ (Rn ) of b ∈ B. The map is well defined because of the definition of the ideal I of relations for M . In the very same way we define β : A → M and we end up with a commuting square whose right lower corner is filled by M . It remains to verify that M is initial with this property. Let Z ∈ C∞ Ring be another C∞ -ring and ξ, ζ morphisms filling α R −−−→ A     βy ξy ζ

B −−−→ Z to a commutative square. Define ω : M → Z on generators using ψ(πi ) = ai for i ∈ {1, . . . , n} and η(πl ) = bl for l ∈ {1, . . . , m} and π i := β(ai ) = pI ◦ pr∗n (πi ) and π n+l := α(bl ) = pI ◦ pr∗m (πl ). We abbreviate (π 1 , . . . , π n , π n+1 , . . . , π n+m ) by ~π . We define ωR (π i ) := ξ(ai ) for i ∈ {1, . . . , n} and ωR (π n+l ) := ξ(bl ) for l ∈ {1, . . . , m}. Let m ∈ M be an arbitrary element. Since M is finitely generated, there exists a smooth function f : Rn+m → R such that m = M (f )(~π ). Thus we define ωR (m) to be Z(f )(ξ(a1 ), . . . , ξ(an ), ζ(b1 ), . . . , ζ(bm )). It remains to check that this definition yields a morphism ω : M → Z of C∞ -rings with ξ = ω ◦ β and ζ = ω ◦ α. We write ~x for (ξ(a1 ), . . . , ξ(an ), ζ(b1 ), . . . , ζ(bm )). Then we compute Z(f ) ◦ ωRk (m1 , . . . , mk ) = Z(f )(ωR (m1 ), . . . , ωR (md )) = Z(f ) (Z(g1 )(~x), . . . , Z(gk )(~x)) = Z(f ) ◦ Z(g)(~x) = ωR ◦ M (f ◦ g)(~π ) = ωR ◦ M (f ) M (g1 )(~π ), . . . , M (gk )(~π ) = ωR ◦ M (f )(m1 , . . . , mk ). and hence it follows that ω is indeed a morphism of C∞ -rings.




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It remains to check that the resulting triangles commute: ξR (a) = ξR ◦ A(f )(a1 , . . . , an ) = Z(f ) ◦ ξRn (a1 , . . . , an ) = Z(f )(ξ(a1 ), . . . , ξ(an )) = ωR ◦ M (f )(β(a1 ), . . . , β(an )) = ωR ◦ M (f ) ◦ β Rn (a1 , . . . , an ) = ωR ◦ β R ◦ A(f )(a1 , . . . , an ) = ωR (β(a)). We sum up: M fills the diagram in the right lower corner to a commutative square and by construction M is initial via a unique map amoung all C∞ -rings having this property. Hence M is unique up to isomorphism in C∞ Ring and we will therefore write ∞

A ⊗R B := M and call this the ∞-tensor product. As a special consequence we see that binary coproducts of finitely generated C∞ -rings exist since ∞

∞

A ⊗ B := A ⊗C ∞ (R0 ) B and C ∞ (R0 ) is the initial C∞ -ring. As an immediate consequence of the preceeding consideration we have the following Corollary 3.1.17 For the C∞ -ring in n and m generators we have ∞

C ∞ (Rn ) ⊗ C ∞ (Rm ) = C ∞ (Rn+m ) and for ideals I ∈ C ∞ (Rn ) and J ∈ C ∞ (Rm ) we find ∞

C ∞ (Rn )/I ⊗ C ∞ (Rm )/J = C ∞ (Rn+m )/(I, J). We can extend our considerations following [Moe 91, p. 22] and show that all coproducts of C∞ -rings exist. Proposition 3.1.18 The category C∞ Ring possesses all coproducts. Proof: Every C∞ -ring is a directed colimit of finitely generated ones. Since forming the quotient is equivalent to a pushout by Example 3.1.14 we are done by the interchange law of colimits (cf. Corollary 2.5.13). q.e.d.

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Corollary 3.1.19 The subcategories C∞ Ringfp and C∞ Ringft are closed under the ∞tensor product. Proof: This is an immediate consequence of the explicit description of the C∞ -tensor product given above. q.e.d. Adjoining an indeterminate t to A ∈ C∞ Ring consists just in forming ∞

A{t} := A ⊗ C ∞ (R). The universal problem of inverting an element s ∈ A is solved by ∞

A[s−1 ] := A{t}/(s · t − 1) = (A ⊗ C ∞ (R))/(s · t − 1), i.e. by applying the fundamental theorem on homomorphisms of C∞ -rings to the ideal I = (st − 1) after adjoining an indeterminate. The localization of A at a subset S ⊂ A is thus given by −1 A[S −1 ] := − colim −−−→A[s ]. s∈S

This is not the ring of fractions with some power of s as a denominator. But A[S −1 ] posses the well known universal property of localization from conventional Commutative Algebra which reads in our context as follows: The localization of a C∞ -ring A at a subset S of A is a C∞ -ring B together with a morphism η : A → B of C∞ -rings such that η(s) is a unit in B for all s ∈ S and which is universal amoung such morphisms of C∞ -rings, i.e. given any other C∞ -ring C and morphism ξ : A → C such that ξ(s) ∈ B × for all s ∈ S, then there is a unique morphism φ : B → C of C∞ -rings with ξ = φ ◦ η. Thus, a localization is unique up to unique isomorphism in C∞ Ring and we will write A[S −1 ] for B. One can check, using the universal property of dividing by ideals and of colimits, that our construction from above satisfies the universal property we require. Hence, a localization always exists. The following example will be useful for our approach to derived manifolds in the next chapter. Example 3.1.20 For a finitely presented C∞ -ring A ∈ C∞ Ringfp its localization can be explicitely described as a colimit: Let I = (g1 , . . . , gk ) the ideal in C ∞ (Rn ) representing A as seen in Example 3.1.14. Define J := (g1 , . . . gk , g·, πn+1 − 1), where g ∈ C ∞ (Rn ) is some preimage of s ∈ A. For general reasons we find A[s−1 ] ∼ = C ∞ (Rn+1 )/J using the universal properties involved. From this point of view we can write the localization of A as a colimit k+1 A[s−1 ] = − colim −−−→Γψ,n+1 ,

where the diagram ψ

0

∞ n+1 Γk+1 ) ←−−− C ∞ (Rk+1 ) −−−→ C ∞ (R0 )) ψ,n+1 := (C (R

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is determined by ψ : C ∞ (Rk+1 ) → C ∞ (Rn+1 ) with ψ(f ) := ((x1 , . . . , xn+1 ) 7→ f (g1 ((x1 , . . . , xn )), . . . , gk ((x1 , . . . , xn )), g((x1 , . . . , xn )) · xn+1 − 1)) . Lemma 3.1.21 Every morphism ϕ : A → R of C∞ -rings induces a canonical morphism ϕ[a−1 ] : A[a−1 ] → R[(ϕ(a))−1 ] of the localized C∞ -rings. Proof: Composing ϕ with the canonical morphism R → R[(ϕ(a))−1 ] gives a morphism which sends a ∈ A to a unit. Because of the universal property of A[a−1 ] we get the desired morphism. q.e.d. The functor C ∞ ( · ) : Man → C∞ Ring has some remarkable properties we are going to summarize now. But first of all, we give a definition which will also be useful later on when we will relate topology to C∞ -rings. Definition 3.1.22 Let X ∈ Top be a topological space and U ⊂◦ X an open subset. Let i : R× ,→ R denote the open inclusion of the pointed line R× = R \ {0}. A continuous function f : X → R, i.e. a global section of CX0 (X), the sheaf of continuous functions on X, such that U fits into a cartesian diagram U −−−→   y

R×   iy

f

X −−−→ R is called a characteristic function for U . Lemma 3.1.23 Let M be a manifold. Every open subset U ⊂◦ M has a characteristic ∞ function, i.e. there exists a global section f ∈ CM (M ) such that U −−−→   y

R×   iy

f

M −−−→ R is cartesian in the category Man of smooth manifolds. Proof: Using the partition of unity, it suffices to consider the case M ∼ = Rd . In [Moe 91, Ch. I, § 1, Lem. 1.4] this case is already proven. q.e.d.

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∞ Proposition 3.1.24 Let U ⊂◦ M be an open submanifold of M ∈ Man and f ∈ CM (M ) −1 × a characteristic function for U , i.e. the equality f (R ) = U holds. Then we have a natural isomorphism C ∞ (U ) = C ∞ (M )[f −1 ]

of C∞ -rings. Proof: It suffices to check that C ∞ (U ) has the required universal property of C ∞ (M )[f −1 ]. We have a natural restriction r : C ∞ (M ) → C ∞ (U ) coming from the inclusion U ⊂ M which maps f to f |U . Now let ϕ : C ∞ (M ) → A be a morphism of C∞ -rings with the property that ϕR (f ) ∈ A× is a unit. In order to define a morphism ψ : C ∞ (U ) → A such that ϕ factors through r we need to extend functions defined on U to functions on M . To achieve this we use partition of unity applied to Vi0 := U and some open cover {Vj }j∈J of M \ U , where gi0 is defined to be g on Vi0 and zero for all other gj on Vj for j ∈ J. q.e.d.

Definition 3.1.25 A R-point p of A ∈ C∞ Ring is a morphism p ∈ HomC∞ Ring (A, C ∞ (R0 )) from A to the initial object C ∞ (R0 ) of C∞ Ring. A C∞ -ring is called local if and only if it has only one R-point. Lemma 3.1.26 A C∞ -ring is local if and only if its underlying R-algebra is local. Proof: Every R-point p ∈ Hom(A, C ∞ (R0 )) gives a morphism pR : j ∗ (A) → R of Ralgebras which is a surjection. Thus, we see that A/ker(pR ) is a field. Hence we have a bijection between the set of R-points of A and the maximal ideals of its underlying Ralgebra. Thus, A is local if and only if its underlying R-algebra has only one maximal ideal. By the axiom of choice this is equivalent to ∀x,y∈R (x + y = 1 ⇒ ∃z(x · z = 1) ∨ ∃z(y · z = 1)), which is the geometric definition of locality. Thus j ∗ (A) is a local R-algebra and we are done. q.e.d. Remark. In this context ”geometric” means that the formula fits into first-order logic as discussed in [Moe 91, Ch. X]. Because of the previous lemma the following definition is sensible.

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Definition 3.1.27 A morphism ϕ : A → B of local C∞ -rings is a local morphism if and only if its image j ∗ (ϕ) : A → B under the underlying R-algebra functor reflects invertibility, i.e. the commutative square A× −−−→ B ×     y y j ∗ (ϕ)

A −−−→ B is cartesian. Remark.

The condition of being local is a geometric formula: (∀a∈A )(ϕ(a) ∈ B × ⇒ a ∈ A× ).

The following proposition states the fact that in contrast to conventional Commutative Algebra there is no difference between morphisms and local morphisms in C∞ Ring. Proposition 3.1.28 Every morphism between local C∞ -rings is local. Proof: Again, using the axiom of choice one can prove that a morphism ϕ : A → R between C∞ -rings is local if and only if ϕ−1 R (mR ) = mA with mA the maximal ideal of A and mR the maximal ideal of R. This condition can readily be verified using the fact that A/mA ∼ =R∼ = R/mR . Hence we are done. q.e.d. We will use the fact that every element a ∈ A operates on the set of all R-points, i.e. there is a map a∗ : HomC∞ Ring (A, C ∞ (R0 )) → R defined by evaluation a∗ p := pR (a) of the natural transformation p. This is analogue to the situation in Linear Algebra or Functional Analysis, in which one uses this map to interpret a vector space as a subspace of its double dual. Definition 3.1.29 A C∞ -ring is point-determined if and only if for any a ∈ A the equality a∗ p = 0 for all R-points p implies a = 0 in A. Theorem 3.1.30 Let M ∈ Man be a smooth manifold. Then for all R-points p of C ∞ (M ) ∞ there exists a unique x ∈ M such that for all f ∈ C ∞ (M ) = CM (M ) the equation f ∗ p = f (x) holds.

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Proof: This can be found in [Koc 06, Ch. III, § 5, Thm. 5.7] and [Moe 91, Ch. I § 3, Prop. 3.6 (a)]. q.e.d. Corollary 3.1.31 Let M ∈ Man be a smooth manifold. Then there is a bijection HomC∞ Ring (C ∞ (M ), C ∞ (R0 )) = M between the R-points of C ∞ (M ) and points of M . Corollary 3.1.32 For every smooth manifold the C∞ -ring C ∞ (M ) is point-determined. Corollary 3.1.33 The functor C ∞ ( · ) : Man → C∞ Ring is full and faithful. But one can say even more: Corollary 3.1.34 For every A ∈ C∞ Ring there is a bijection HomC∞ Ring (A, C ∞ (R0 )) ∼ = HomR-Alg (A, R) between the R-points and the morphisms of the underlying R-algebras. Proof: This is an immediate consequence of the theorem from above since due to Lemma 3.1.15 every C∞ -ring can be displayed as a colimit of finitely generated C∞ -rings. It also can be found in [Moe 91, Ch. I, § 3, Proposition 3.6 (b)]. q.e.d. Definition 3.1.35 For an element f ∈ A we set  D(f ) := (f ∗ )−1 (R× ) = x ∈ HomC∞ Ring (A, C ∞ (R0 )) | f ∗ x 6= 0 and call it a principal open subset of HomC∞ Ring (A, C ∞ (R0 )). Principal open subsets will do their job later after we have topologized the set of all R-points of a given C∞ -ring. Corollary 3.1.36 For a ∈ A one has D(a) = HomC∞ Ring (A[a−1 ], C ∞ (R0 )). Proof: This is a consequence of the Corollary from above and the universal property of localization. q.e.d. Using the last isomorphism one can readily verify the following

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Lemma 3.1.37 The intersection of two principal opens is again principal open, i.e. D(a) ∩ D(f ) = D(a · f ). Definition 3.1.38 Let A ∈ C∞ Ring, and p be a R-point. The stalk at p, denoted Ap , is the localized C∞ -ring A[σp−1 ] where σp ⊂ A is given by σp := {a ∈ A | pR (a) 6= 0}. If a ∈ A, its image under the localization A → A[σp−1 ] is denoted ap and called the germ of a at p. For an ideal I ⊂ A we will write Ip for the ideal in Ap generated by the set {fp | f ∈ I}. Our definitions are taken from literature (cf. [Dub(a) 81, Def. 11], [Koc 06, Ch. III, § 6, Def. 6.2], [Moe 91, Ch. I, § 4, Def. 4.1, Prop 4.2]). ∞ Example 3.1.39 For a smooth manifold M define as usual CM,x to be the set of equivalence classes fx := [(U, f )] of pairs (U, f ), where U ⊂ M is an open subset containing x ∈ M and f : U → R is a smooth function. Two germs fx = [(U, f )] and fx0 = [(U 0 , f 0 )] are equal iff there exists an open neighbourhood V ⊂ U ∩ U 0 of x such that f |V and f 0 |V agree. ∞ into a C∞ -ring Cx∞ (M ) in the obvious way, i.e. we define Cx∞ (M )(Rd ) := We turn CM,x ∞ ∞ to be the d-fold cartesian product of all germs at x ∈ M . Then one can × · · · × CM,x CM,x prove Cx∞ (M ) ∼ = (C ∞ (M ))x

using parition of unity. A proof is carried out in [Koc 06, Ch. III, § 5, Prop. 5.6]. Thus, Definition 3.1.38 gives the expected result for C∞ -rings in the essential image of C ∞ ( · ). The following proposition is taken from [Moe 91, Ch. III, Lemma 3.1], but we provide a different proof. Proposition 3.1.40 Let ϕ : A → B be an epimorphism of non-trivial C∞ -rings. Assume A to be local. Then B is local and ϕ is a local morphism. Proof:

Since ϕ : A → B is an epimorphism, the induced map

ϕ∗ : HomC∞ Ring B, C ∞ (R0 ) → HomC∞ Ring A, C ∞ (R0 )

is injective. Thus B is local. The locality of ϕ now follows from Proposition 3.1.28. q.e.d. Corollary 3.1.41 Let A ∈ C∞ Ring be of finite type and x ∈ HomC∞ Ring (A, C ∞ (R0 )) be a R-point of A. Then the stalk Ax is local.

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Proof: By assumption there is an epimorphism α : C ∞ (Rd ) → A for some d ∈ N inducing an epimorphism Cp∞ (Rd ) → Ax with α∗ (x) = p. The claim follows from Proposition 3.1.40. q.e.d. Definition 3.1.42 Let A ∈ C∞ Ring be a C∞ -ring and I ⊂ A an ideal. The germ-radical of I is the ideal I ∧ in A given by  I ∧ := a ∈ A | ∀p∈Hom(A,C ∞ (R0 )) ap ∈ Ip . We always have I ⊂ I ∧ . Following [Dub(a) 81, 11. Definition] we give the following Definition 3.1.43 Let A ∈ C∞ Ring be a C∞ -ring. We say that an ideal I ⊂ A is of local character if and only if I = I ∧ . A C∞ -ring is called germ-detemined or of local character if and only if its zero ideal is of local character. We denote by C∞ Ringlc the full subcategory of germ-determined C∞ -rings of finite type. In other words, we define C∞ Ringlc to be the full subcategory of C∞ Ring whose objects are C∞ -rings of finite type represented by an ideal of local character (in [Koc 06] such ideals are called germ-determined ), i.e. every object of C∞ Ringlc is isomorphic to C ∞ (Rn )/I for some n ∈ N0 where I is an ideal of local character (these C∞ -rings are called fair in [Joy 09, § 2.4, Definition 2.4]). Example 3.1.44 The ring of dual numbers R[] = C ∞ (R)/(x2 ) is germ-determined since (x2 )∧ = (x2 ). But it is not point-determined. Proposition 3.1.45 The functor C ∞ : Man → C∞ Ring factors through the category C∞ Ringfp of finitely presented C∞ -rings. Proof: This is already proven and several proofs are available: [Dub(a) 81, p. 687 below], [Koc 06, Ch. III, § 6, Thm. 6.6] and [Moe 91, Ch. I, § 2, Thm. 2.3]. q.e.d. There is a criterion for C∞ -rings to be of local character: Lemma 3.1.46 Let M ∈ Man be a manifold and J ⊂ C ∞ (M ) be an ideal of local character. Then the ideal I := (J, h1 , . . . , hn ) obtained by adjoining finitely many generators h1 , . . . , hn ∈ C ∞ (M ) is again of local character.

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Proof: Proofs can be found in literature. For example in [Dub(a) 81, Proposition 12] and [Koc 06, Ch. III, § 6, Thm. 6.3]. q.e.d. In particular, every finitely generated ideal in C ∞ (M ) is of local character. Corollary 3.1.47 The category C∞ Ringfp of finitely presented C∞ -rings is closed under localization. ∞ In particular, for every manifold M ∈ Man and every function f ∈ CM (M ) the localization C ∞ (M )[f −1 ] yields a finitely presented C∞ -ring. The following example is taken from [Dub(b) 81, 7. Examples]. It teaches us that C∞ Ringlc is a proper subcategory of the category C∞ Ringft of C∞ -rings of finite type. Example 3.1.48 Let A := C0∞ (R) and B := C ∞ (R). Both rings are represented by an ideal of local character and hence they are germ-determinded, i.e. both C∞ -rings are objects of C∞ Ringlc . Looking at the binary coproduct we find ∞

A ⊗ B = C ∞ (R2 )/I according to Lemma 3.1.17, where  I = f : R2 → R | ∃ε>0 ∀(x,y)∈]−ε,ε[×R f (x, y) = 0 is the ideal of functions in C ∞ (R2 ) vanishing on some subset of R2 around the y-axis. Now one can show that Ip is the unit ideal for any p that does not belong to the y-axis. As a consequence, if f : R2 → R has fp = 0 for all points p of the y-axis, it follows that f ∈ I ∧ . Hence we have I ( I ∧ . In particular, we conclude that the ideal I is not of local character and as a consequence we have that the coproduct ∞

A⊗B ∈ / C∞ Ringlc of germ-determined rings in C∞ Ring is in general not germ-determined again. We want to check the behaviour with respect to localization. Since localization is defined via the ∞-tensor product one might aspect some difficulties when localizing in the category C∞ Ringlc . We turn our attention towards the following important example which is also taken from [Dub(b) 81, 7. Examples]. Example 3.1.49 Let A = C0∞ (R) and a = x0 the stalk of x ∈ C ∞ (R) at 0. The localization of A at a is given by ∞

∞ ∞ −1 A[a−1 ] = C0∞ (R)[x−1 0 ] = C0 (R) ⊗C ∞ (R) C (R)[x ].

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Define J := {f : R× → R | ∃U ⊂◦ R,0∈U f |U = 0} and m := {f ∈ C ∞ (R) | ∃U ⊂◦ R,0∈U f |U = 0}. Then one has isomorphisms C ∞ (R)[x−1 ] = C ∞ (R× ) and C0∞ (R) = C ∞ (R)/m and therefore ∞

A[a−1 ] = C ∞ (R)/m ⊗C ∞ (R) C ∞ (R× ) = C ∞ (R× )/J. Using Corollary 3.1.17 we can derive a presentation for A[a−1 ] and get ∞

A[a−1 ] = C ∞ (R)/m ⊗C ∞ (R) C ∞ (R2 )/(xy − 1) = C ∞ (R2 )/(I, xy − 1) with I ⊂ C ∞ (R2 ) the ideal from Example 3.1.48 above. Since (I, xy − 1)∧ = C ∞ (R2 ), we find that the localization of A at a performed in C∞ Ringft does not belong to C∞ Ringlc . Taken all together, we see that C∞ Ringlc is not closed under localization, i.e. localization depends on the subcategory we are working with. The germ-determined C∞ -rings of local character are the geometrically interesting ones (as we will see in the next section), so the fact that C∞ Ringlc is not closed under localization might be a serious drawback. However, it was pointed out by E. Dubuc in [Dub(b) 81] that there is a universal way to map any C∞ -ring A ∈ C∞ Ringft of finite type to a germ-determined one using I ⊂ I ∧ and Proposition 3.1.7: A = C ∞ (Rd )/I → A∧ = C ∞ (Rd )/I ∧ . We will spell that out in the following Proposition 3.1.50 There is a reflection R : C∞ Ringft → C∞ Ringlc , i.e. the forgetful functor V : C∞ Ringlc → C∞ Ringft possesses a left adjoint R a V . The reflection is determined via R(A) = A∧ . Proof: Recall that since A is of finite type it is isomorphic to C ∞ (Rn )/I for some n ∈ N0 and some ideal I in C ∞ (Rn ). Thus, since I ⊂ I ∧ there is a canonical projection A → C ∞ (Rn )/I ∧ . Suppose there is a morphism ϕ : A → B of C∞ -rings of finite type, say C ∞ (Rn )/I → C ∞ (Rd )/J. Hence ϕ(I) ⊂ J and by the fundamental theorem on homomorphisms we get a morphism ϕ∧ : C ∞ (Rn )/I ∧ → C ∞ (Rd )/J ∧ . These assignments are functorial. Thus, setting R(A) := C ∞ (Rn )/I ∧ on objects and R(ϕ) := ϕ∧ on morphisms determines a functor which is obviously left adjoint to the inclusion functor V : C∞ Ringlc → C∞ Ringft . q.e.d. Example 3.1.51 We want to determine what the localization of A at a from Example 3.1.49 in the subcategory C∞ Ringlc of germ-determined C∞ -rings looks like. Using Proposition 3.1.50 we find A[a−1 ]∧ = C ∞ (R2 )/(I, xy − 1)∧ = 0, i.e. the localization of A at a in C∞ Ringlc corresponds to the zero ring.

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The following lemma relates R-points of C∞ -rings to the R-points of their image under the reflection functor. As a consequence, we will see in the next section that A ∈ C∞ Ringft and A∧ ∈ C∞ Ringlc give the same topological space. In some way this is similar to Algebraic Geometry where the prime spectrum of a ring coincides with the prime spectrum after modding out all nilpotent elements. Lemma 3.1.52 The R-points of A ∈ C∞ Ringft and its image A∧ ∈ C∞ Ringlc under the reflection functor coincide. Proof: This is just an application of the fundamental theorem on homomorphisms of C∞ -rings (cf. Proposition 3.1.7). q.e.d. Because of Lemma 3.1.46 all C∞ -rings of finite presentation belong to C∞ Ringlc . One can say even more: The full subcategory C∞ Ringfp of finitely presented C∞ -rings is stable under all pushouts. Thus, C∞ Ringfp is closed under localization. So far, we have a chain C∞ Ringfp ⊂ C∞ Ringlc ⊂ C∞ Ringft ⊂ C∞ Ring of full subcategories. We end the section with a special property of the embedding C ∞ : Man → C∞ Ring. Theorem 3.1.53 The functor C ∞ : Man → C∞ Ring turns fibered products in pushouts. Proof:

This can be found for example in [Koc 06, Ch. III, § 8, Thm. 8.1]. q.e.d.

3.2

C∞-rings and Topology

In this section we switch to the geometric point of view, i.e. we are going to introduce affine spectra of C∞ -rings which will be organized in the category Aff ∞ of affine schemes. We construct the topological realization functor from affine schemes Aff ∞ to Top and investigate some properties of the resulting topological spaces. Moreover, we will equip Aff ∞ with a Grothendieck topology and we will point out the connection of the Grothendieck topology and the topological realization of C∞ -rings. All this serves as a preparation for the functorial approach to topology via smooth functors we will pursue in the next section.

Definition 3.2.1 The category Aff ∞ of affine schemes is given by
◦ Aff ∞ := C∞ Ringfp ,

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the dual of the category of C∞ -rings of finite presentation. The spectrum Spec : C∞ Ringfp → Aff ∞ is the duality functor. An important advantage of passing from the category of C∞ -rings to the category Aff ∞ of affine schemes is that we can discuss the theory from the geometric angle, i.e. manifolds become correctly, that is covariantly, embedded. Example 3.2.2 By Theorem 3.1.53 the embedding Man → Aff ∞ preserves fibered products. Using R-points we can give the follwing Definition 3.2.3 The covariant functor | · | : Aff ∞ → Top , Spec A 7→ |Spec A| := HomAff ∞ (Spec C ∞ (R0 ), Spec A) from the category of affine schemes to the category of topological spaces defined by the initial topology on |Spec A| is called the topological realization functor. First of all let us explain that the topological realization gives indeed a functor: By duality we see that |Spec A| is the set HomC∞ Ring (A, C ∞ (R0 )) of all R-points of A. Every element a ∈ A of the underlying R-algebra of A determines a map a∗ : |Spec A| → R by sending a R-point φ ∈ |Spec A| to a∗ φ = φR (a). We equip |Spec A| with the initial topology, i.e. the coarsest topology such that all a∗ are continuous. To be precise, we take the topology on |Spec A| which is generated by the subbasis [  (a∗ )−1 (U ) | U ∈ Op(R) . τA := a∈j ∗ (A)

Now let ϕ : A → R be a morphism in C∞ Ringfp . Define a map via fϕ : |Spec R| → |Spec A| ,

p 7→ ϕ∗ (p) = p ◦ ϕ.

Let U ⊂◦ |Spec A| be an open subset, i.e. it is of the form [\ (a∗ji )−1 (Vji ) i∈I j∈Ji

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for a∗ji : A → C ∞ (R0 ) and Vji ⊂ R open. Taking the preimage under fϕ we find fϕ−1 (U ) =

[\

fϕ−1 (a∗ji )−1 (Vji ) =

i∈I j∈Ji

[\

(a∗ji ◦ fϕ )−1 (Vji ),

i∈I j∈Ji

where for every R-point ψ of R we have (a∗ji ◦ fϕ )(ψ) = a∗ji (fϕ (ψ)) = ψR ◦ ϕR (a) = (ϕR (a))∗ (ψ), consequently a∗ji ◦ fϕ is given by an element of R and thus the preimage of U under fϕ is open. Remark. It is important to note that the functor | · | : Aff ∞ → Top can easily be extended to a functor | · | : C∞ Ring◦ → Top by the very same construction sending ϕ : A → R to fϕ : |R◦ | → |A◦ |. This allows to construct C∞ -schemes for arbitrary C∞ -rings as it was done by D. Joyce in [Joy 09]. We did ∞◦ not pursue this since we are interested in deriving C∞ -schemes from functors in SetAff and therefore restrict ourselves to C∞ Ringfp to ensure that the functor category we are working with is locally small. In view of our main goal to obtain derived manifolds from functors this will be sufficient since derived manifolds are covered by affines that are ”homotopically” finitely presented. Note that because of Lemma 3.1.37 in both cases the set of all D(a) with a ∈ A forms a basis for the topology. Lemma 3.2.4 The functor | · | : Aff ∞ → Top factors through the full subcategory of Hausdorff spaces, i.e. the initial topology from above on |Spec A| is Hausdorff. Proof: Let x 6= y in |Spec A|. Hence there is an a ∈ A such that a∗ (x) 6= a∗ (y) in the real line. As the real line is Hausdorff we find an open subset V containing a∗ (x) and U containing a∗ (y) such that V ∩ U = ∅ in R. Using the continuity of a∗ we see that (a∗ )−1 (V ) is an open neighbourhood of x and (a∗ )−1 (U ) is an open neighbourhood of y and (a∗ )−1 (V ) ∩ (a∗ )−1 (U ) = ∅. Hence we are done. q.e.d. This holds also true for the extension | · | : C∞ Ring◦ → Top and is in contrast to conventional Algebraic Geometry where the topology in general is not separated. Definition 3.2.5 Let M be a manifold and C ∞ (M ) its associated C∞ -ring. If I ⊂ C ∞ (M ) is an ideal we let Z(I) := {x ∈ M | ∀f ∈I f (x) = 0} be the zero-set of I in M .

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Consider the following situation in Algebraic Geometry: Let K be a algebraically closed field and A = K[X1 , . . . , Xn ]/I be a finitely generated K-algebra. Then there is a bijection between closed points of the prime spectrum of A and  Z(I) = (x1 , . . . , xn ) ∈ K n | ∀f (X1 ,...,Xn )∈A f (x1 , . . . , xn ) = 0 . Since every point of a topological space in the image of the topological realization functor is closed by Lemma 3.2.4, the following proposition can be seen as the analogue in the C∞ -setting. Proposition 3.2.6 Let A ∈ C∞ Ring be a C∞ -ring of finite type with epimorphism α : C ∞ (Rn ) → A and kernel I ⊂ C ∞ (Rn ). Then the map α∗ : |Spec A| → Rn ,

ϕ 7→ α∗ ϕ := (π1∗ α∗ ϕ, . . . , πn∗ α∗ ϕ)

with πk being the k-th generator of C ∞ (Rn ) gives a homeomorphism on Z(I). Proof: We have to check that the image of α∗ equals Z(I), that α∗ is injective, continuous, and open. For every f ∈ I and every R-point p ∈ |Spec A| we have α∗ p∗ f = 0 since, using the representation of A, every R-point of A can be turned into a R-point of C ∞ (Rn ) via α. Thus, using Theorem 3.1.30 we see that the image of |Spec A| under α∗ is given by Z(I), the zero set of all functions of I. Because of its construction α∗ is continuous. Again, by Theorem 3.1.30 one deduces injectivity. For the last part let V ⊂ |Spec A| be open. Thus, V is of the form [\ (a∗ji )−1 (Vji ). i∈I j∈Ji

Applying α∗ yields α∗ (V ) =

[\

α∗ (a∗ji )−1 (Vji )

i∈I j∈Ji

and we are to show that each α∗ (a∗ji )−1 (Vji ) is of the form Vj0i ∩ Z(I) for some Vj0i ⊂ Rn open. But this is true since A is of finite type. Hence, for all p ∈ (a∗ji )−1 (Vji ) there is a f : Rn → R such that a∗ji (p) = pR (aji ) = pR (A(f )(a1 , . . . , an )) = C ∞ (Rn )(f )(pR (a1 ), . . . , pR (an )) = f (pR (a1 ), . . . , pR (an )), and we see that α∗ (a∗ji )−1 (Vji ) consists of all x ∈ Rn with f (x) ∈ Vji for f from above. Thus, [ α∗ (a∗ji )−1 (Vji ) = f −1 (Vji ) ∩ Z(I)

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where the union runs over all such f : Rn → R is open. q.e.d. Corollary 3.2.7 The functor | · | : Aff ∞ → Top factors through the full subcategory of paracompact locally compact Hausdorff spaces. Proof: By Lemma 3.1.52 and Proposition 3.2.6 the topological space |Spec A| is homeomorphic to closed subsets of some Rn for A ∈ C∞ Ring of finite type. q.e.d. Note that this is not true for the general case | · | : C∞ Ring◦ → Top. Corollary 3.2.8 Let A ∈ C∞ Ringft a C∞ -ring of finite type and A∧ ∈ C∞ Ringlc its image under the reflection functor R : C∞ Ringft → C∞ Ringlc . Then |A| = |A∧ |. In particular, we have Z(I) = Z(I ∧ ), where I and I ∧ are the representing ideals, i.e. A = C ∞ (Rd )/I and A∧ = C ∞ (Rd )/I ∧ . Proof:

This is just Lemma 3.1.52 together with Proposition 3.2.6. q.e.d.

Proposition 3.2.9 Let R ∈ C∞ Ring be of finite type. Then every open subset U ⊂ |Spec R| admits a characteristic function. Proof: Because of the homeomorphism |Spec R| = Z(I) from above we can choose for an open subset U ⊂ |Spec A| a homeomorphic subset V ⊂ Z(I). Because of the subspace topology we can write V = V 0 ∩ Z(I) with V 0 ⊂ Rn open. Due to Lemma 3.1.23 we can choose a characteristic function f 0 ∈ CR∞n (Rn ) for V 0 . The restriction f 0 |Z(I) of f 0 to Z(I) gives a characteristic function for V . Composing with the homeomorphism |Spec R| = Z(I) gives a characteristic function for U . q.e.d. Corollary 3.2.10 For every manifold M ∈ Man there is an isomorphism |Spec C ∞ (M )| → M of topological spaces.

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Proof: Using the isomorphism |Spec C ∞ (M )| ∼ = M of Lemma 3.1.31 and the existence of characteristic functions for both M and |Spec C ∞ (M )| (cf. Lemma 3.1.23 and Proposition 3.2.9) we translate characteristic functions of M into characteristic functions of |Spec C ∞ (M )| and vice versa. Thus, we can choose the isomorphism |Spec C ∞ (M )| ∼ =M to be our desired homeomorphism. q.e.d. Definition 3.2.11 Let A be a C∞ -ring. For U ⊂ |Spec A| define the vanishing ideal of U to be  IU := a ∈ A | ∀p∈|Spec A| a∗ p = pR (a) = 0 . The other way round for I ⊂ A the assignment V(I) := {p ∈ |Spec A| | ∀f ∈I pR (f ) = 0} gives a subset of the set |Spec A| of all R-points, the vanishing set of I. Lemma 3.2.12 A subset W ⊂ |Spec A| is closed if and only if W = V(IW ). Proof: Note that W ⊂ V(IW ) is always true. Assume W is a closed subset of |Spec A|, hence U := |Spec A|\W is open. Choose a characteristic function g for U . By construction we see that for all q ∈ W the equation g ∗ q = 0 holds true. Thus, g ∈ IW and hence g ∗ p = 0 for all p ∈ V(IW ). So we end up with V(IW ) ⊂ W . The other way round is easy. Assume V(IW ) = W , then W is closed because V(IW ) is the intersection of the zero sets of all functions of IW . q.e.d. Corollary 3.2.13 Let M be a manifold and W ⊂ M . Then W = Z(IW ) if and only if W ⊂ M is closed. Proof: Using |Spec C ∞ (M )| = M and Theorem 3.1.30 we find V(I) = Z(I) which gives the result. q.e.d. Lemma 3.2.14 Let I ⊂ C ∞ (M ) and A := C ∞ (M )/I. Then we have I = IZ(I) if and only if A is point-determined. Proof: First assume I = IZ(I) and suppose g is in C ∞ (M ) such that p∗ g = 0 for all R-points p ∈ |Spec A|. But this is equivalent to g(x) = 0 for all x ∈ Z(I) and hence since IZ(I) = I we see that A is point-determined. For the other direction we merely prove the non trivial part. Let g ∈ IZ(I) . Thus, we have g(x) = 0 for all x ∈ Z(I) and because of the isomorphism Z(I) = |Spec A| we see for all

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p ∈ |Spec A| that pR (g) = 0 and hence g = 0 in A because of the point-determinedness, where g denotes the image of g ∈ C ∞ (M ) in A. Thus we have IZ(I) ⊂ I. q.e.d. Corollary 3.2.15 Let A ∈ C∞ Ringft be of finite type represented by an ideal I. Then A is point-determined if and only if I = IZ(I) . Theorem 3.2.16 Let M ∈ Man be a manifold. Then the assignments Z( · ) and I( · ) form a Galois correspondence, i.e. there is a one-to-one correspondence between closed subsets W of M and ideals I of C ∞ (M ) such that the quotient C ∞ (M )/I is point-determined. Proof:

This is just a combination of the preceeding lemmas. q.e.d.

Remark. From this angle the germ-radicals introduced in Definition 3.1.42 correspond √ to the formation of radical ideals I 7→ I in conventional Algebraic Geometry. We now describe a pretopology on Aff ∞ by describing co-coverings on the category C∞ Ringlc of C∞ -rings. Definition 3.2.17 An open co-cover of A ∈ C∞ Ringlc is a family (A → A[s−1 ] | s ∈ S) with S ⊂ A such that a Hom(A[s−1 ], C ∞ (R0 )) → Hom(A, C ∞ (R0 )) s∈S

is a surjection. We will call elements of an open co-cover open maps. The family is jointly surjective as far as R-points are involved. Thus, any C∞ -ring without R-points is covered by the empty family (cf. [Dub(b) 81, Def. 3 and Prop. 11]). Lemma 3.2.18 The definition of open co-covers gives a Grothendieck pretopology on the category (C∞ Ringlc )◦ . The idea for the proof comes originally from [Dub(b) 81, Prop. 1, Prop. 2, Def. 3 and the following]. We proceed in a very same way as it was done in [Koc 06, Ch. III, §7, Prop.7.3]. Since we cannot assume that every localization of germ-determined C∞ rings is germ-determined again (cf. Example 3.1.49), we have to make it so by applying the reflection functor from Proposition 3.1.50 which we will do in the following proof without mentioning it. Note that due to Lemma 3.1.52 this will not cause any effect to the (gerneralized) topological realization functor | · | : C∞ Ring◦ → Top. Proof: We have to check three points. The first thing to verify is that every isomorphism is a co-covering which is obvious. The second point is to check whether the cobase change

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of an open co-covering (ξi : A → A[a−1 i ] | i ∈ I) ∈ Cov(A) along an arrow f : A → R gives an open co-covering of R. Lets have a look at the cocartesian diagram A   ξi y

f

−−−→ R  f (ξ ) y∗ i

A[a−1 i ] −−−→ Ci . −1 The maps (ξi )∗ (f ) : A[a−1 i ] → Ci and f∗ (ξi ) : R → Ci factor through R[f (ai ) ] because of the universal property of localization (since f∗ (ξi )R (f (ai )) ∈ Ci× ). From this and the universal property of pushouts we get Ci = R[f (ai )−1 ]. Hence we have checked that f∗ (ξi ) is an open map. It remains to verify the point surjectivity. For this take a morphism g : R → C ∞ (R0 ). Precomposition with f yields an arrow A → C ∞ (R0 ) which gives at least ∞ 0 one morphism A[a−1 i ] → C (R ). Thus, the universal property of pushouts gives a map Ci → C ∞ (R0 ) and hence it follows point surjectivity and (f∗ (ξi ) : R → Ci | i ∈ I) ∈ Cov(R) is an open co-cover. The hard part is to check that open co-covers are stable under composition. We already know that open maps are stable under cobase change. Of course, the composition of jointly surjective families is jointly surjective since surjectivity is preserved under composition. We merely have to bother with the problem that composition of open inclusions forms an open inclusion again, i.e. if there is a s ∈ A such that (A[a−1 ])[α−1 ] = A[s−1 ] for a ∈ A and α ∈ A[a−1 ]. Now this is the point where our restriction to C∞ Ringlc comes in. Since A ∈ C∞ Ringlc ⊂ C∞ Ringft , we can choose a presentation i

π

I −−−→ C ∞ (Rn ) −−−→ A. There is a c ∈ C ∞ (Rn ) mapping to a ∈ A under π. We look at the cocartesian square C ∞ (Rn ) −−−→ C ∞ (Rn )[c−1 ] = C ∞ (U )     πy yπ A

−−−→

A[a−1 ],

where U ⊂ Rn is the open subset equal to {x ∈ Rn | c(x) ∈ R× }. Using the general fact that epimorphisms are stable under cobase change we find an element h ∈ C ∞ (Rn )[c−1 ] mapping to α ∈ A[a−1 ] via π. In C∞ Ringlc we now have a sequence of two pushout diagrams C ∞ (Rn ) −−−→ C ∞ (U ) −−−→ C ∞ (Rn )[c−1 ][h−1 ]       πy πy y A

−−−→ A[a−1 ] −−−→

A[a−1 ][α−1 ].

Again, defining V := {x ∈ U | h(x) ∈ R× } enables us to write C ∞ (Rn )[c−1 ][h−1 ] = C ∞ (U )[h−1 ] = C ∞ (V ).

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Now we choose a characteristic function χ ∈ CR∞n (Rn ) for V ⊂◦ Rn . This leads to a pushout C ∞ (Rn ) −−−→ C ∞ (Rn )[χ−1 ]     πy yπ A

−−−→ A[(π(χ))−1 ].

Using C ∞ (Rn )[χ−1 ] = C ∞ (V ) and taking into account that the total square of the two pushouts above is cocartesian as well we end up with A[a−1 ][α−1 ] = A[π(χ)−1 ] as a consequence of the uniqueness of colimits. q.e.d. Remark. The proof shows that we could also have topologized the full subcategory of C∞ -rings of finite type. The additional assuption that the kernel of each presentation is of local character is not necessary. It will be important later on to show that all representable presheaves are in actual fact sheaves. Since the category C∞ Ringfp of finitely presented C∞ -rings is closed under pushouts, the very same procedure equips C∞ Ringfp with open maps and enables us to give the following Definition 3.2.19 The site SAff ∞ := (Aff ∞ , CovAff ∞ ) obtained by endowing the category Aff ∞ with the Grothendieck topology induced by open co-covers is called the C ∞ -site. Manifolds are the prototype for affine schemes. The following proposition compares open covers of manifolds to open co-covers of the associated affine C∞ -schemes. It is remarkable that the covers determine each other. Proposition 3.2.20 The natural embedding i : Man → Aff ∞ preserves and reflects open covers. Proof:

We define i := Spec ◦ C ∞ ( · ) and using the fact from Corollary 3.1.33 that C ∞ ( · ) : Man → C∞ Ringfp

is fully faithful, we see that i is an embedding. Let (Ui → M | i ∈ I) be an open cover of M ∈ Man, i.e. a jointly surjective family

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of open immersions. Applying the functor C ∞ ( · ) : Man → C∞ Ring gives a family of morphisms C ∞ (M ) → C ∞ (Ui ). Choosing characteristic functions fi for Ui gives a family ∞ (fi )i∈I ⊂ CM (M ) of global sections such that C ∞ (Ui ) = C ∞ (M )[fi−1 ] for every i ∈ I. Using partition of unity it can be shown that the family (C ∞ (M ) → C ∞ (M )[fi−1 ] | i ∈ I) is indeed an open co-cover, i.e. jointly surjective as far as R-points are involved. Hence (Spec C ∞ (M )[fi−1 ] → Spec C ∞ (M ) | i ∈ I) ∈ CovAff ∞ (Spec C ∞ (M )) is an open cover. The other way round let (C ∞ (M ) → C ∞ (M )[gj−1 ] | j ∈ J) ∈ CovAff ∞ (C ∞ (M )) be an open co-cover of C ∞ (M ). Define Vj := gj−1 (R× ) to be the preimage. We have to show that the family (Vj → M | j ∈ J) is jointly surjective. Since R-points of C ∞ (M ) are in bijective correspondence to points x ∈ M of the manifold M by Corollary 3.1.31, the assertion follows from the point-surjectivity of the open co-cover. q.e.d. Proposition 3.2.21 The topology of open co-covers is subcanonical, i.e. every representable presheaf Hom(A, · ) : C∞ Ringlc → Set is already a sheaf. The original proof is from [Dub(b) 81, Prop. 4, Prop. 8, Prop. 9] and was overtaken by [Koc 06, Ch. III, § 7, Thm. 7.4]. We follow the later. Proof: Let A ∈ C∞ Ringlc be a finitely generated germ-determined C∞ -ring and consider an arbitrary open covering (αi : A → A[a−1 i ] | i ∈ I) ∈ CovAff ∞ (Spec A). Because of the left exactness of the Hom-functor the sheaf condition can be proved simultaneously for all HomC∞ Ringlc (R, · ) by showing that Y Y −1 A→ A[a−1 ] ⇒ A[a−1 i i , aj ] i∈I

i,j∈I

is an equalizer in C∞ Ringlc . Because A is of finite type, we can proceed as in the proof of Lemma 3.2.18: We choose an epimorphism π : C ∞ (Rn ) → A and open subsets Ui ⊂◦ Rn for every i ∈ I such that C ∞ (Rn ) −−−→ C ∞ (Ui )     η πy y A

−−−→ A[a−1 i ]

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Chapter 3: Smooth Functors as C∞ -Schemes

is cocartesian. The point-surjectivity condition forces every R-point p : A → C ∞ (R0 ) to ∞ 0 factor through some q : A[a−1 i ] → C (R ). Using |Spec A| = Z(I) and the commutativity of the diagram above we see that the pullback of a R-point q ∈ |Spec A| to a R-point p ∈ |Spec C ∞ (Rn )| factors through some C ∞ (Ui ). Hence, we have

Z(I) ⊂ U :=

[

Ui

i∈I

for the zero set of Q I. Now ∈ A[a−1 i )i∈I Q i ] be a compatible family, i.e. (xi )i∈I is equalized by the arrows Q let (x −1 −1 −1 −1 −1 i∈I A[ai ] ⇒ i,j∈I A[ai , aj ]. This implies that for all R-points of A[ai , aj ], (xi )p = (xj )p . Thus, for all p ∈ |Spec A| there is a well-defined x(p) ∈ Ap such that x(p) = (xi )p for any i ∈ I such that p belongs to |Spec A[a−1 i ]|. We shall construct an element x ∈ A such that xp = x(p) for all R-points p ∈ |Spec A| of A. ∞ Since C ∞ (Ui ) → A[a−1 i ] is an epimorphism, we choose hi ∈ C (Ui ) such that η(hi ) = xi . Let (φi )i∈I be a locally finite partition of unity subordinate to the open cover (Ui )i∈I of U . The function φi · hi is definded and smooth on the whole of U because the support supp(φi ) of φi is closed and contained in Ui . Hence, we can define

h :=

X

φi · hi ,

i∈I

which is therefore an element of C ∞ (U ). In order to show that h ∈ CR∞n (U ) provides an element of A we are going to prove that the map π : C ∞ (Rn ) → A factors accross the restriction map r : C ∞ (Rn ) → C ∞ (U ). To achieve this let first of all g ∈ CR∞n (Rn ) be a characteristic function for U . We show that πR (g) is invertible in A. Then the universal property of the localization forces π to factor through C ∞ (Rn )[g −1 ] = C ∞ (U ). Define a := (πR (g)) to be the ideal generated by the image of g under π. Now we come to the point where we have to use the germ-determinedness: To prove a = A it suffices to show ap = Ap for all R-points p ∈ |Spec A|, hence for all points p ∈ Z(I). But pR (πR (g)) ∈ R× since g is nowhere zero on Z(I). Thus, we get a map π : C ∞ (U ) → A such that π = π ◦ r. We let x := π(h) ∈ A. Given any p ∈ |Spec A| we choose V ⊂ |Spec A| around p meeting only finitely many supp(φ) (those with index J ⊂ I). Let K := {i ∈ I | p ∼ = x ∈ Ui }. Then

3.3 C∞ -Schemes from Smooth Functors

93

we get ! xp = π

X

φi · hi

i∈J

=

X

p

π(φi )p · π(hi )p

i∈J

=

X

π(φi )p · (xi )p

i∈J

=

X

π(φi )p · x(p)

i∈J

! =π

X

· x(p)

φi

i∈I

p

= 1 · x(p). In (A[a−1 i ])p we then have (αi (x))p = (xi )p for all R-points p ∈ A[a−1 i ] and therefore (αi (x))p −(xi )p ∈ (0)p which implies αi (x)−xi ∈ (0) −1 since A[ai ] is germ-determined. But then αi (x) = xi in A[a−1 i ] and so the second sheaf axiom is satisfied. The uniqueness of x follows in a similar fashion since all R-points of A are in some A[a−1 i ]. Assume another y ∈ A such that (αi (y))p = (xi )p . Because yp = x(p) = xp we conclude x = y from the germ-determinedness of A. Hence we are done. q.e.d. Corollary 3.2.22 The C∞ -site SAff ∞ is subcanonical, i.e. the functor of points h( · ) : Aff ∞ → sSetAff

∞◦

factors through the category Sh(SAff ∞ ) of set-valued sheaves on the C∞ -site.

3.3

C∞-Schemes from Smooth Functors

In this section we introduce the category of C∞ -schemes as a full subcategory of the category of smooth functors. We first define the functor category of smooth functors and call its objects C∞ -schemes iff they are sheaves with respect to the C∞ -site satisfying certain covering conditions one suspects for schemes. It turns out that the category C∞ Sch of C∞ -schemes can be organized into a site using the notion of open subfunctors which we will explain following [To¨e 07]. The set of all open subfunctors associated to a C∞ -scheme forms a locale, i.e. it behaves somehow like the frame of opens of a topological space. In

94

Chapter 3: Smooth Functors as C∞ -Schemes

contrast to [To¨e 07] we will spell that out, provide proofs, and give examples for the case of C∞ -schemes. Note that the idea to define C∞ -open morphisms is taken from [To¨e 07] and woven together with E. Dubuc’s Grothendieck topology from [Dub(b) 81] we discussed in the previous paragraph. Definition 3.3.1 The category

∞ Ring

SetC

fp

of functors from the category of finitely presented C∞ -rings to the category of sets is the category of smooth functors. ∞ An object F ∈ SetC Ringfp is called a smooth functor. The category Sh(SAff ∞ ) is the full subcategory of smooth functors which are sheaves with respect to the topology of the C∞ -site. Corollary 3.3.2 The functor ι : Man → SetC

∞ Ring

fp

defined by ι := h( · ) ◦ Spec ◦ C ∞ ( · ) from the category of smooth manifolds to the category of smooth functors is fully faithful and factors through Sh(SAff ∞ ). Proof: The functor ι is the composition of i : Man → Aff ∞ from Proposition 3.2.20 followed by the functor of points. Thus, ι is fully faithful as the composition of fully faithful functors. Since the C∞ -site is subcanonical, the image of the embedding is actually a sheaf, i.e. an object of Sh(SAff ∞ ). q.e.d. ∞

Definition 3.3.3 A smooth functor F ∈ SetC Ringfp is called an affine C ∞ -scheme if it is in the essential image of h( · ) : Aff ∞ → Sh(SAff ∞ ) the functor of points. By the Yoneda lemma affine schemes correspond to affine C∞ -schemes and we will not tend to differentiate between them. Example 3.3.4 Every manifold is an affine C∞ -scheme.

3.3 C∞ -Schemes from Smooth Functors

95

Definition 3.3.5 (C∞ -open Subfunctors) ∞

(1) Let X ∈ SetC Ringfp be an affine C∞ -scheme and U → X be a subsheaf. The subfunctor U is called C ∞ -open in X if and only if there is a family (Xi → X | i ∈ I) of open covers in Aff ∞ such that U is the image of the morphism of sheaves a Xi → X. i∈I

(2) A morphism f : F → F in Sh(SAff ∞ ) is C ∞ -open if and only if for all affine C∞ ∞ schemes Y ∈ SetC Ringfp and all morphisms of sheaves ψ : Y → F the morphism ψ ∗ (f ) : ψ −1 F → Y obtained by base change along ψ is a monomorphism which is C∞ -open in Y . In this case we call F a C ∞ -open subfunctor. We write SubC∞ (F) for the set of all C∞ -open subfunctors of F. One can immediately check that C∞ -opens are monomorphisms in Sh(SAff ∞ ). Lemma 3.3.6 The C∞ -open morphisms are stable under change of base and under composition. Proof: We first prove the stability of C∞ -open morphisms under change of base. Therefore let f : F → F be a C∞ -open subfunctor and g : G → F any morphism in Sh(SAff ∞ ). We are to show that g ∗ (f ) : g −1 F → G is C∞ -open again. Now let Y be an affine C∞ scheme and ψ : Y → G a morphism of sheaves for the C∞ -site. Composing with g we get g ◦ ψ : Y → F. Since F is an open subfunctor of F, there is a family (Yj → Y | j ∈ J) of ` −1 open covers such that (g ◦ ψ) F = im( j∈J Yj → Y ). Using (g ◦ ψ)−1 F = ψ −1 g −1 F we see that ψ ∗ (g ∗ (f )) : ψ −1 (g −1 F ) → Y is C∞ -open and hence we are done with the first part. Let f : F → F be a C∞ -open subfunctor of F and g : G → F an open subfunctor of F . −1 For every affine C∞ `-scheme and every morphism φ : Y → F the pullback φ F is in the essential image of Xi → Y for some open maps (Xi → Y | i ∈ I). For every i ∈ I we have a morphism αi : Xi → F and we can look at the pullback αi−1 G along g : G → F . Since G is an open ` subfunctor of F , we find open maps (Zij → Xi | j ∈ Ji ) such that αi−1 G is in the image of j Zij → Xi . Thus, (Zij → Y | j ∈ Ji , i ∈ I) defines a family of open maps such that a  φ−1 G = im Zij → Y and we have proven the second assertion. q.e.d.

Chapter 3: Smooth Functors as C∞ -Schemes

96

Definition 3.3.7 Let F ∈ Sh(SAff ∞ ) be a sheaf. Define on the partially ordered set SubC∞ (F) two operations, the intersection U ∧ V := U ×F V of open subfunctors U and V in SubC∞ (F) and the open union ! _ a Fi := im Fi → F i∈I

i∈I

of a family (Fi )i∈I of open subfunctors in SubC∞ (F). Proposition 3.3.8 Let F ∈ Sh(SAff ∞ ) be a sheaf for the C∞ -site. Then the open union and the intersection of open subfunctors satisfy the infinite distributive law, i.e. _ _ U ∧ Vi = (U ∧ Vi ) i∈I

i∈I

for every open subfunctor U → F, and every family (Vi )i∈I ⊂ SubC∞ (F). W W of F and U ∧ Proof: Note that at first glance i∈I (U ∧ Vi ) is an open subfunctor i∈I Vi W is an open subfunctor of U . By Lemma 3.3.6 the functor U ∧ i∈I Vi is also an object of SubC∞ (F). Since now equality can be checked object-wise, it is a tedious but straight forward exercise to verify the asserted isomorphism. q.e.d. Thus, for every sheaf F we see that the partially ordered set SubC∞ (F) has finite meets and all joins which satisfies the infinite distributive law and is therefore a frame. Morphisms between frames which respect the partial order and respect both joins and meets and the largest and the smallest element of the frame are called morphisms of frames (consult [Moe 91, Ch. IX, § 1] for a detailed introduction to this subject). The class of all frames and their morphisms forms a category whose dual is said to be a locale. In particular, SubC∞ (F) is a frame which we will consider as a category in the very same way as we do with Op(X) for a topological space X ∈ Top. Example 3.3.9 Let ξa : A → A[a−1 ] and ξb : A → A[b−1 ] two open maps of a C∞ -ring A and Xa := hSpec A[a−1 ] and Xb := hSpec A[−1 ] ∈ SubC∞ (F) the associated C∞ -open subfunctors of the affine C∞ -scheme X := hSpec A . Using the universal properties of colimits and of localization one checks that A[(a · b)−1 ] fits into a co-cartesian square induced by the open maps ξa and ξb . Thus, one sees Xa ∧ Xb = (ξb )−1 hSpec

A[a−1 ]

= hSpec

A[(a·b)−1 ] .

On the level of topological spaces this corresponds to D(a) ∩ D(b) = D(a · b) as can be immediately verified, cf. Corollary 3.1.36 and Lemma 3.1.37. Thus, we can consider every affine C∞ -scheme as a site itself.

3.3 C∞ -Schemes from Smooth Functors

97

Definition 3.3.10 (C∞ -Schemes) ∞ A smooth functor F ∈ SetC Ringfp is a C ∞ -scheme if and only if F is a sheaf for the C∞ site and if there exists a family (Xj )j∈J of affine schemes together with an epimorphism of sheaves a Xj → F j∈J

such that for every j ∈ J the map Xj → F is C∞ -open. The family (Xj → F | j ∈ J) from above is called affine C ∞ -cover of F . ∞ The category C∞ Sch denotes the full subcategory of C∞ -schemes in the category SetC Ringfp of smooth functors. Example 3.3.11 Every affine scheme is a C∞ -scheme, i.e. there is a fully faithful functor Aff ∞ → C∞ Sch from the category of affine schemes to the category of C∞ -schemes. Every manifold is a C∞ -scheme since we can compose the two embeddings: Man → Aff ∞ → C∞ Sch. Proposition 3.3.12 The category C∞ Sch is closed under finite limits. Proof: Let X : I → C∞ Sch be a diagram with I a finite category. Since the sheafifica∧ ∼ tion functor a : SAff ∞ → SAff ∞ preserves finite limits (cf. [Moe 91, Ch. II, § 5, Thm. 1]), we know that ← lim −−X is a sheaf. Since every Xi has an affine cover, one can show by the universal property of limits that ← lim −−X can also be covered by affines. q.e.d. Theorem 3.3.13 The topological realization functor | · | : Aff ∞ → Top can be extended to a functor | · | : C∞ Sch → Top, where a subset V ⊂ |F | := F (C ∞ (R0 )) is defined to be open if and only if there is a C∞ -open subfunctor U → F such that |U | = V . Proof: Since every C∞ -scheme can be covered by affines it suffices to show that for an affine C∞ -scheme, say F ∼ = hSpec A , a subfunctor U → F is open if and only if |U | ⊂ |F | = |Spec A| is open for the initial topology on |Spec A|. Let V ⊂ |F | be open, i.e. it is of the form V =

[ j∈J

D(aj )

Chapter 3: Smooth Functors as C∞ -Schemes

98

since the D(a) form a basis for the topology of |Spec A|. Define Xj := hSpec j ∈ J and let _ U := Xj .

A[a−1 j ]

for all

j∈J

Thus we have |U | = V by construction. The other way round is given by the same procedure. Since the D(a) form a basis for the topology of |hSpec A | and because of the point surjectivity of the open co-cover topology it is clear that both assignments are inverse to each other. q.e.d. Corollary 3.3.14 Let Spec A be an affine scheme and X its associated affine C∞ -scheme. Then we have an isomorphism SubC∞ (X) ∼ = Op(|Spec A|) of categories. More generally, let Y ∈ C∞ Sch be a C∞ -scheme. Then SubC∞ (Y ) ∼ = Op(|Y |) is an isomorphism of categories. Example 3.3.15 Given a smooth manifold M ∈ Man and a family of open subesets (Ui )i∈I in M . Due to Proposition 3.2.20 the family (Ui → M | i ∈ I) is an open cover of M if and only if (Spec C ∞ (Ui ) → Spec C ∞ (M ) | i ∈ I) is an open cover of Spec C ∞ (M ) for the C∞ -site. Looking at the associated affine C∞ -schemes, i.e. X = hSpec C ∞ (M ) and Xi = hSpec C ∞ (Ui ) for all i ∈ I, and taking Corollary 3.3.14 into account we see that (Xi → X | i ∈ I) is an affine C∞ -cover of X ∈ C∞ Sch. The theory woven so far gives us nice formulae connecting our topologies. For instance, using Corollary 3.2.10 we have _ Xi = M i∈I

and |Xi | = Ui . Moreover, using Example 3.3.9 we have |Xi ∧ Xj | = Ui ∩ Uj . Remark. Using the intersection and the open union of Definition 3.3.7 as well as the ∞-distributive law we can define a Grothendieck pretopoloy on C∞ Sch coming from open subfunctors. For an object F ∈ C∞ Sch we define a family W of morphisms (Xi → F | i ∈ I) to be covering if and only if every Xi ∈ SubC∞ (F) and Xi = F. By Lemma 3.3.6 this equips C∞ Sch with a pretopoloy. The Grothendieck topology associated to this pretopoloy turns C∞ Sch into a site: the big site of C∞ -schemes.

99

3.4 The Big Structure Sheaf

3.4

The Big Structure Sheaf

The big structure sheaf O for the C∞ -site we are going to introduce now enables us to derive structure sheaves for the topological realization of a C∞ -scheme. This way we will obtain C∞ -ringed spaces. In order to prepare the introduction of the structure sheaf for the C∞ -site we need the following lemma which states the fact that every (finitely presented) C∞ -ring is recoverable from its spectrum. Lemma 3.4.1 Let A ∈ C∞ Ringfp be a C∞ -ring and n ∈ N. Then there is a bijection An = HomAff ∞ (Spec A, Spec C ∞ (Rn )) of sets. Proof: The above bijection is just the Yoneda lemma. To be more explicit we follow [Moe 91, Ch. I, § 1, Proposition 1.1] and write πk : Rn → R for the projection on the k-th component. A sequence of n elements a1 , . . . , an ∈ A determines a morphism ϕ : C ∞ (Rn ) → A of C∞ -rings with ϕR (πk ) = ak by ϕ(f ) := A(f )(a1 , . . . , an ) for every f : Rn → R. The functoriality of A and C ∞ (Rn ) and the naturality of the morphisms involved forces ϕ to be unique. q.e.d. The previous lemma reminds us why we call C ∞ (Rn ) the free C ∞ -ring in n generators, these generators being the projections. Corollary 3.4.2 The underlying R-algebra functor is representable. To be precise there exists a natural isomorphism j ∗ = HomC∞ Ring (C ∞ (R), · ) of functors. Proposition 3.4.3 Let F be a smooth functor. For A ∈ C∞ Ring we have a canonical isomorphism F (A) = Hom(hSpec A , F ). Proof:

This is the covariant Yoneda lemma using the duality HomAff ∞ ( · , Spec A) = HomC∞ Ringfp (A, · ). q.e.d.

Chapter 3: Smooth Functors as C∞ -Schemes

100

The C∞ -ring in one generator is of particular interest. Thus, we define ∞ Ring

A : Euk → SetC via

Rd 7→ Ad := hSpec C ∞ (Rd ) Q and we observe that Ad = A1 . Using Proposition 3.2.21 we can define a new sheaf. Definition 3.4.4 The structure sheaf for the C ∞ -site is the functor ∞ Ring

O : SetC

fp

→ C∞ Ring

given by O(X) := Hom(X, A). Note that O(X) is indeed a C∞ -ring in the obvious way, i.e. d

O(X)(R ) =

d Y

1

Hom(X, A ) =

k=1

d Y

O(X).

k=1

∞

Example 3.4.5 Let X ∈ sSetC Ringfp be an affine C∞ -scheme, say X = hSpec A . Using Proposition 3.4.3 one easily checks that O(X) = A. We will write O(Spec A) instead of O(X) when O is restricted to affine C∞ -schemes. In this case the global structure sheaf can be considered as a contravariant functor O : Aff ∞ → C∞ Ring. The following proposition is the reason why O is called the structure sheaf for the C∞ -site. Proposition 3.4.6 The presheaf O : Aff ∞ ◦ → C∞ Ring is a sheaf for the C∞ -site. Proof: Since the topology on C∞ Ringlc is subcanonical, it suffices to show that O is representable. But O is representable by definition since it is represented by the C∞ -ring in one generator (after obvious identifications). q.e.d. We can deduce a criterion for a smooth functor being an affine scheme: ∞ Ring

Lemma 3.4.7 A functor X ∈ SetC

fp

is affine if and only if there is an isomorphism

X∼ = hSpec O(X) .

101

3.4 The Big Structure Sheaf

Proof: Assume X ∼ = hSpec O(X) , then X is affine by definition. On the other hand let C∞ Ringfp X ∈ Set be affine, say X = hSpec A for A ∈ C∞ Ringfp a C ∞ -ring. Then we have O(X)(R) = Hom(hSpec A , A1 ) = Hom(Spec A, Spec C ∞ (R)) = Hom(C ∞ (R), A) = A. Hence we are done. q.e.d. Let X be an affine C∞ -scheme represented by A ∈ C∞ Ring, i.e. X is isomorphic to the functor of points given by hSpec A . The set of all open subschemes SubC∞ (X) of X is isomorphic to the set Op(|Spec A|) of all open morphisms by Corollary 3.3.14, so we can identify a principal open subset D(a) ⊂◦ |X| with the open subfunctor hSpec A[a−1 ] . The induced functor ι† : Op(|X|) → (Aff ∞ ↓ Spec A) mapping D(a) to (Spec A[a−1 ] → Spec A) gives a morphism of sites ι : (Aff ∞ ↓ Spec A) → Op(|X|).

On the other hand, by Example 2.7.11 there is a morphism of sites t : Aff ∞ → (Aff ∞ ↓ Spec A) obtained by base change along the terminal map Spec A → Spec C ∞ (R0 ). Taken all together we end up with a sequence of geometric morphisms SetEuk


(Aff ∞ )◦

(t∗ ,t∗ )

−−−→ SetEuk


(Aff ∞ ↓Spec

(ι∗ ,ι∗ )

A)◦

−−−→ SetEuk


Op(X)◦

.

By the help of Corollary 2.3.19 we define OX := t∗ O ∈ (SetEuk )(Aff

∞ )◦


↓X .

Example 3.4.8 Again, for an arbitrary C∞ -scheme the base change functor of the terminal map induces a morphism t : C∞ Sch → (C∞ Sch ↓ Y ) of sites by Example 2.7.11. Using the isomorphism Op(|Y |) = SubC∞ (Y ) of Corollary 3.3.14 we get another morphism j : (C∞ Sch ↓ Y ) → Op(|Y |) of sites. The induced morphisms of presheaves can be combined to the sequence SetEuk


(C∞ Sch)◦

(t∗ ,t∗ )

−−−→ SetEuk


(C∞ Sch↓Y )◦

(j ∗ ,j∗ )

−−−−→ SetEuk


Op(|Y |)◦

.

Chapter 3: Smooth Functors as C∞ -Schemes

102

(Be aware that we abused notation for typocraphical reasons since C∞ Sch is not a small category.) We define OYpetit := j∗ t∗ O ∈ Sh(|Y |) and call it the structure sheaf of |Y | associated to the C∞ -scheme Y . Remark.

If X is affine, say X = hSpec A , we have an isomorphism petit ∼ OX = ι∗ OX

of sheaves by Corollary 2.3.19. It is important to note that by construction we have petit OX (D(a)) = O(Spec A[a−1 ]) = A[a−1 ] petit on principal opens D(a) ⊂◦ |X| where a ∈ OX (|X|) = A. petit petit In this case we will write occasionally OA instead of OX . ∞ Ring

Proposition 3.4.9 For X ∈ SetC

fp

and R ∈ C∞ Ring there is a natural bijection

HomSetC∞ Ringfp (X, hSpec R ) ∼ = HomC∞ Ring (R, O(X)) of sets. Proof: We use the description of the set of natural transformations between two functors as an end to ease notation. Thus we get Z HomSetC∞ Ring (X, hSpec R ) = HomSet (X(A), HomC∞ Ring (R, A)) A





Z =

Z HomSet X(A),

A

HomSet (Rd , Ad )

Rd

Z Z =

HomSet (X(A), HomSet (Rd , Ad ))

A Rd

∼ =

Z Z

HomSet (Rd , HomSet (X(A), Ad ))

Rd A

 Z =

HomSet Rd ,

 Z

HomSet (X(A), Ad (A))

A

Rd



 Z

= HomC∞ Ring R,

HomSet (X(A), A(A)) A

= HomC∞ Ring (R, O(X)),

103

3.4 The Big Structure Sheaf

where A runs over all finitely presented C∞ -rings and Rd runs over all objects of Euk. We heavily use the fact that properties for the preservation of limits carry over to the preservation of ends because ends may be displayed as limits. Details to this subject can be found in [Mac 91, Ch. IX, § 5]. q.e.d.

Corollary 3.4.10 Let (Ai )i∈I be a directed system of C∞ -rings. Then we have Spec (− colim lim −−−→Ai ) = ← −− (Spec Ai ). Example 3.4.11 We already know that binary coproducts in C∞ Ringfp exist. Thus, we have ∞ Spec (A ⊗ B) = Spec A × Spec B. Corollary 3.4.12 Let A ∈ C∞ Ring be finitely presented, i.e. A = C ∞ (Rn )/I with I = (a1 , . . . , ad ) generated by finitely many elements of A. Due to Example 3.1.14 this is equivalent to the fact that A fits into a cocartesian diagram C ∞ (Rd ) −−−→ C ∞ (R0 )     ψy y C ∞ (Rn ) −−−→

A.

Thus, using the previous corollary we can conclude the following. A C∞ -scheme X ∈ C∞ Sch is a finitely presented affine C∞ -scheme if and only if it can be written as a fibred product in C∞ Sch X −−−→ Spec C ∞ (R0 )     y y0 φ

Spec C ∞ (Rn ) −−−→ Spec C ∞ (Rd ) where φ = Spec ψ is induced by some smooth map f : Rn → Rd and 0 by the zero map R0 → Rk . Corollary 3.4.13 The full subcategory of affine C∞ -schemes is closed under finite limits in the category C∞ Sch of C∞ -schemes. Proof:

The functor of points h(·) : Aff ∞ → Sh(SAff ∞ ) is left exact. q.e.d.

Chapter 3: Smooth Functors as C∞ -Schemes

104

3.5

C∞-ringed Spaces as C∞-Schemes

In this section we introduce the notion of C∞ -ringed spaces following the approach of conventional Algebraic Geometry. We give an explicit construction of the conventional spectrum functor from the category of C∞ -rings to the category of C∞ -ringed spaces and show that our construction agrees with the one given in literature. At the end of the section we prove that these conventional constructions can be obtained from our smooth functor approach. Recall that a ring object in a category C with finite limits is an object R ∈ C equipped with morphisms 0, 1 : 1 → R , +, · : R × R → R in C for which the usual identities for a commutative ring with unit hold. In a similar way one can define other algebraic objects in any topos. For example, in the Grothendieck topos Sh(X, SetEuk ) a C∞ -ring object EX is a sheaf which gives a product-preserving functor from the euclidian category to the category of sets on every open subset of the topological space X. More succinctly, one can consider it as a functor Euk → Op(X)∼ using the exponential law. We start with the definition of C∞ -ringed spaces following [Dub(a) 81, Def. 6]. Definition 3.5.1 A C ∞ -ringed space X = (X, EX ) is a C∞ -ring object EX in the category of sheaves over a topological space X ∈ Top. A morphism of C ∞ -ringed spaces (f, f # ) : (X, EX ) → (Y, EY ) is a continuous map f : X → Y together with a morphism f # : f ∗ EY → EX in Sh(X, C∞ Ring). A C∞ -ringed space is local if its stalks are local C∞ -rings. We will denote LC∞ RS the category of locally C ∞ -ringed spaces. We give the fundamental example which is of particular interest. Example 3.5.2 Let M ∈ Man be a smooth manifold. Define a sheaf HM of C ∞ -rings on M by HM (U ) := HomMan (U, · ) for every open subset U ⊂◦ M . Note that HM is a ∞ local sheaf of C ∞ -rings since its underlying R-algebra CM is the sheaf of smooth functions on M which is indeed a local sheaf of rings. Hence X := (M, HM ) is a locally C ∞ -ringed space. Let f : M → N be a smooth map between manifolds. Then for Y := (N, HN ) there is a morphism (f, f # ) : X → Y of local C ∞ -ringed spaces, where fV# : HN (V ) = HomMan (V, · ) → HomMan (f −1 (V ), · ) = HM (f −1 (V ))

3.5 C∞ -ringed Spaces as C∞ -Schemes

105

is given by precomposition, i.e. for all smooth maps α ∈ HN (V, Rd ) we have fV# (Rd )(α) = α ◦ f . This way we get a morphism of sheaves of C ∞ -rings f [ : E N → f∗ H M which leads to the morphism (f, f # ) : (M, HM ) → (N, HN ) of C ∞ -ringed spaces. This morphism is automatically a local morphism of locally C∞ ringed spaces. Remark.

We want to explain the classical construction spec : C∞ Ring◦ → LC∞ RS

which is given in [Joy 09, § 4.2] and follows [Har 06]. Let us recall this construction. D. Joyce defines the spectrum spec A of A ∈ C∞ Ring in [Joy 09, § 4.2] as follows: He topologizes the R-points |Spec A| of A in the same way as we do (cf. Proposition 3.2.3). For ` each open subset U ⊂ |Spec A| he defines E|Spec A| (U ) to be the set of functions s : U → p∈U Ap with s(p) ∈ Ap for all p ∈ U , and such that U may be covered by open sets V ⊂ |Spec A| for which there exist a, d ∈ A with p(d) 6= 0 for all p ∈ V , such that s(p) equals the quotient Prp (a)/ Prp (d), where Prp : A → Ap denotes the localization map. Thus the C∞ ring structure of Ap turns E|specA| (U ) into a C∞ -ring which is easily be seen to be a sheaf of C∞ -rings on |Spec A| whose restriction map ρU V : E|Spec A| (U ) rightarrowE|Spec A| (V ) for V ⊂ U open is given by s 7→ s|V and whose stalks E|Spec A|,p are isomorphic to Ap . The tuple (|Spec A|, E|Spec A| ) is the spectrum spec A. For a morphism ϕ : A → B the map fϕ : |Spec B| → |Spec A| from Proposition 3.2.3 is continuous. Let ϕp : Afϕ (p) → Bp be the induced map of local C∞ -rings. Then for # U ⊂ |Spec A| open one defines fϕ# (U ) : E|Spec A| (U ) → E|Spec B| (ϕ−1 p (U )) by fϕ (U )(s) : p 7→ ϕp (s(fϕ (p))). Hence fϕ# : E|Spec A| → (fϕ )∗ E|Spec B| is a morphism of sheaves of local C∞ rings on |Spec A|. Thus, (fϕ , fϕ# ) : (|Spec B|, E|Spec

B| )

→ (|Spec A|, E|Spec

A| )

is a morphism of locally C∞ -ringed spaces. Since the construction is functorial in A ∈ C∞ Ring, we see that spec : C∞ Ring◦ → LC∞ RS defines indeed a functor. It agrees with the construction performed in [Dub(a) 81, Thm. 8] and [Quˆe 87, § 3]. We will prove that this construction coincides with our’s later. In [Quˆe 87, § 1] there is another definition of a spectrum in which the points are not maximal ideals of the underlying C∞ -ring but C∞ -radical prime ideals. We do not pursue this approach since the latter is less common and appears to be less natural. The next definition is due to [Joy 09, Def. 4.23].

Chapter 3: Smooth Functors as C∞ -Schemes

106

Definition 3.5.3 A locally C∞ -ringed space (X, OX ) is a C ∞ -scheme if X can be covered by a family of open sets (Ui )i∈I ⊂ X such that (Ui , OX |Ui ) is an affine C∞ -scheme. In this context we refer to affine C∞ -schemes in the sense of D. Joyce, i.e. (Ui , OX |Ui ) is an affine C∞ -scheme if it is in the essential image of the functor spec : C∞ Ring◦ → LC∞ RS we discussed above. It is important to note that the theory of C∞ -schemes differs from conventional scheme theory in Algebraic Geometry in one specific point: the spectrum functor for general C∞ -rings loses information, and that not only up to isomorphism. In classical Algebraic Geometry one has Γ(Spec R) ∼ = R for an arbitrary ring R (cf. [Har 06, Ch. II, Prop. 2.2]). That is not so for C∞ -rings as we will see in the following Example 3.5.4 For C∞ -rings A ∈ C∞ Ringft of finite type with representation I → C ∞ (Rd ) → A we can check Ip = Ip∧ for all R-points p of A. In addition we have Z(I) = Z(I ∧ ) and hence it follows that spec A ∼ = spec A∧ . Applying this to A = C0∞ (R) and a = x0 from −1 ∧ Example 3.1.49 we have A[a ] = 0. Thus
A[a−1 ]∧ → Γ spec A[a−1 ]∧ ) is not an isomorphism. From this we conclude that spec : (C∞ Ringft )◦ → LC∞ RS is in general neither full nor faithful. As a consequence, it is not possible to introduce general affine C∞ -schemes as the dual of C∞ Ring. The following theorem is from E. Dubuc [Dub(a) 81, Thm. 13] and can be understood as follows: C∞ -rings of local character are the geometrically interesting ones because there is no loss of information when passing to locally C∞ -ringed spaces. Theorem 3.5.5 The functor spec : (C∞ Ringlc )◦ → LC∞ RS is full and faithful. We give another definition of the spectrum functor which follows the approach to schemes given in [Liu 06, Ch. 2, § 2.3] and [Joh 72, Ch. V, § 3]. For this purpose we sum up some technical facts in the following lemmas that are immediate consequences from the theory developed in the first paragraph of this chapter.

3.5 C∞ -ringed Spaces as C∞ -Schemes

107

Lemma 3.5.6 Let A ∈ C∞ Ringlc be a C∞ -ring. We write D(a) for the principal open subset of |Spec A| which is given by a ∈ A and A[a−1 ] the localization at a ∈ A. Then the following is true. (1) The inclusion D(f ) ⊂ D(g) induces a natural map A[g −1 ] → A[f −1 ]. (2) The equality D(f ) = D(g) gives a natural isomorphism A[f −1 ] ∼ = A[g −1 ]. Proof: (1) For D(f ) ⊂ D(g) the map A[g −1 ] → A[f −1 ] is given by the Yoneda lemma using the isomorphism D(f ) ∼ = HomC∞ Ring (A[f −1 ], C ∞ (R0 )). (2) The isomorphism is given by the universal property of the localization. q.e.d. Now we define ESpec A (D(a)) := A[a−1 ]

(3.5.1)

for every principal open subset D(a). This will be the candidate for our structure sheaf on |Spec A|, where |Spec A| denotes the topological realization (cf. Definition 3.2.3 and the following remark). Lemma 3.5.7 With the definiton from above the following is true. (1) The object ESpec A is a sheaf of C∞ -rings on |Spec A|. (2) The stalk at x ∈ |Spec A| is given by ESpec A,x ∼ = Ax . (3) Every morphism ϕ : A → R in C∞ Ring gives fϕ : |Spec R| → |Spec A|, x 7→ x ◦ ϕ in Top. Proof: (1) This is already proven in Proposition 3.2.21. (2) We use the universal properties of localization and of colimits to find −1 ESpec A,x = −colim colim −−−→ESpec A (D(f )) = − −−−→ A[f ] = Ax , D(f )3x

f ∈A, D(f )3x

which is local by Corollary 3.1.41. (3) This is just Definition 3.2.3. q.e.d. Summing up the results developed so far give the following

Chapter 3: Smooth Functors as C∞ -Schemes

108

Proposition 3.5.8 The construction which assigns to every C∞ -ring A the tuple (|Spec A|, ESpec A ), where ESpec A ist determined by equation 3.5.1 and |Spec A| is given by Definition 3.2.3, is functorial. Thus we get a functor spec : (C∞ Ringlc )◦ → LC∞ RS from the category of C∞ -rings to the category of locally C∞ -ringed spaces which we will call the classical spectrum functor.

3.6

From Smooth Functors to C∞-Ringed Spaces

In this section we will see that every C∞ -scheme in the sense of Definition 3.3.5 gives a locally C∞ -ringed space. More precisely, we construct a functor Φ that sends smooth functors that are C∞ -schemes to locally C∞ -ringed spaces. This functor possesses a right adjoint and it will be shown that Φ gives an equivalence of categories in the appropriate sense. The following example will be useful for the next theorem. Example 3.6.1 Let f : X → Y be a morphism of C∞ -schemes. Let U → X and V → Y petit be two open subfunctors such that |U | ⊂ |f |−1 (|V |). Recall that OX = j ∗ t∗ O as described in Example 3.4.8. Then the restriction of f on U induces a unique morphism petit f ∗ : OYpetit (|V |) → OX (|U |)

which factors through |f |−1 OYpetit (|U |) =

colim − − −− → ◦ −1

|U |⊂ |f |

OYpetit (|V |). Thus, we get a morphism

(|V |)

petit f # : |f |−1 OYpetit → OX ∈ Sh(|X|, SetEuk )

of sheaves on |X|. Note that f # is automatically a morphism of locally ringed spaces by Proposition 3.1.28. Theorem 3.6.2 There is a fully faithful functor Φ : C∞ Sch → LC∞ RS petit ). sending X to (|X|, OX

Proof: Let f : Y → Z be a morphism of C∞ -schemes. Because of Theorem 3.3.13 there is a morphism |f | : |Y | → |Z| of topological spaces. Example 3.4.8 and Example 3.6.1 teach us how to obtain the structure sheaves and the local morphism for the corresponding C∞ -schemes. Note that OYpetit is a local sheaf of C∞ -rings for, since Y can be covered by

3.6 From Smooth Functors to C∞ -Ringed Spaces

109

affines and stalks are a local thing it suffices to retreat to the affine case which is treated in Lemma 3.5.7. Thus, the construction of LC∞ RS involves Top in a covariant way and C∞ Ring in a contravariant way and hence Φ defines indeed a functor. It remains to check that the functor is fully faithul. The functor Φ is faithful: Let f, g : Y → Z be two morphisms of C∞ -schemes such that their images (|f |, |f |# ) = (|g|, |g|# ) under Φ coincide. We will show f = g. Without loss of generality we may assume Y to be affine since f −1 V = g −1 V for every open subscheme V → Y . Thus, we are to show f ∗ = g ∗ : O(Z) → O(Y ) which is the case since # petit |f |# (|Z|) → OYpetit (|Y |). |Y | = |g||Y | : OZ

    The functor Φ is full: We are to show that a morphism (f, f # ) : |Y |, OYpetit → |Z|, OZpetit of locally C∞ -ringed spaces is induced by a morphism g : Y → Z of schemes such that (|g|, |g|# ) = (f, f # ). Choose C∞ -open covers (Zj → Z | j ∈ J) and (Yi → Y | i ∈ I) of Y and Z such that f (|Yi |) ⊂ |Zj |. We are to find morphisms gij : Yi → Zj inducing f |Yi that we can glue together to get our desired morphism. Thus, we are able to retreat to the case where both Z and Y are affine, i.e. Y = hSpec A and Z = hSpec B . Considering # global sections of f # : f ∗ OZpetit → OYpetit we get a map φ : B → A from f|Y | : O(Z) → O(Y ). We are to show that g = Spec (φ) : Spec A → Spec B does the job. We check that for x ∈ |Spec A| we have |Spec (φ)|(x) = f (x) ∈ |Spec B|. But this is an easy consequence of Yoneda’s lemma. petit It remains to show |g|# = f # , i.e. we must check that both maps |g|# , f # : OSpec B (V ) → petit OSpec A (U ) coincide on principal open sets V ⊂ |Spec B| and U ⊂ |Spec A| with U ⊂ f −1 V . But this is the case since the map is of the form B[b−1 ] → A[(φ(b))−1 ] due to Lemma 3.1.21. q.e.d. Remark. Because of the previous theorem we may identify the category of C∞ -schemes with the full subcategory of locally C∞ -ringed spaces which are locally isomorphic to local petit C∞ -spaces of type (|hSpec A |, OA ). Corollary 3.6.3 Every manifold M ∈ Man gives a locally C∞ -ringed space. Proof:

This is Example 3.3.11 together with Theorem 3.6.2. q.e.d.

Proposition 3.6.4 The functor Ψ : LC∞ RS → SetC

∞ Ring

fp

given by   petit Y 7→ A 7→ HomLC∞ RS ((|Spec A|, OSpec ), Y) A

Chapter 3: Smooth Functors as C∞ -Schemes

110 is right adjoint

Ψ`Φ to Φ. Proof:

We are to prove that there exists a natural bijection HomLC∞ RS (Φ(X), Y) = HomSetC∞ Ringfp (X, Ψ(Y)).

Given a morphism (f, f # ) : Φ(X) → Y of locally C∞ -ringed spaces we define a morphism g : X → Ψ(Y) of smooth functors on objects A ∈ C∞ Ring to be the composition Φ

petit X(A) = Hom(hSpec A , X) −−−→ Hom((|Spec A|, OSpec A ), Φ(X)) (f,f # )∗

−−−−→

petit Hom((|Spec A|, OSpec A ), Y),

where we have used Proposition 3.4.3. The resulting arrow gA : X(A) → Ψ(Y)(A) is easily seen to be natural in A. The other way round assume there is a morphism g : X → Ψ(Y) of smooth functors. We want to construct a morphism Φ(X) → Y = (Y, OY ) of locally ringed spaces. First we construct |g| : |X| → Y : Let |g| be the composition |X| = Hom(hSpec C ∞ (R0 ) , X) → Hom(hSpec C ∞ (R0 ) , Ψ(Y)) = |Ψ(Y)| = Y. petit To obtain a map g # : |g|∗ OY → OX we define a morphism petit gV# : OY (V ) → OX (|U |) = O(U )

for every open subset V ⊂◦ Y and every C∞ -open subfunctor U → X such that |U | ⊂ |g|−1 (V ). Then g # is the morphism obtained by the universal property of the colimit defining |g|∗ OY on opens subsets of |X| as a left Kan extension. It suffices to fix g ∗ on R, i.e. to determine gR∗ : OY (V )(R) → O(U )(R) = Hom(U, A1 ), since both functors are product-preserving. Now let R ∈ C∞ Ring be such that the induced affine C∞ -scheme is an open subfunctor of U , i.e. |Spec R| ⊂ |g|−1 (V ). Let x ∈ U (R) and write εx for its image in Hom(hSpec R , U ) under the Yoneda embedding. Composing with the restriction gU of g on U yields gU ◦ εx ∈ Hom(hSpec R , Ψ(Y)). Evaluation at idSpec R gives    petit # gU ◦ εx (idSpec R ) = (χ, χ ) ∈ Hom |Spec R|, OSpec R , (Y, OY ) . Note that |g| |Spec R| = χ for, since hSpec R was supposed to be an open subfunctor of U , we have χR (p) = p∗ χ = p∗ g(εx (idSpec R )) = gR (p). Thus, we have ιV

OY (V ) −−−→

colim − −−−→ −1

|Spec R|⊂χ

χ# |Spec R|

petit OY (W ) = χ OY (|Spec R|) −−−−−→ OSpec R (|Spec R|) = R. ∗

(W )

3.6 From Smooth Functors to C∞ -Ringed Spaces

111

Now for f ∈ OY (V ) define gR∗ (f ) ∈ Hom(U, A1 ) to be gR∗ (f )(x) := χ# |Spec R|,R (ιV,R (f )). It follows immediately that the assignments are well-defined and computation shows that they provide the desired natural bijection. We show one direction. Assume   petit # (h, h ) : |X|, OX → (Y, OY ) is a morphism of locally C∞ -ringed spaces. Furthermore, let g : X → Ψ((Y, OY )) be the petit associated morphism of smooth functors and (|g|, g # ) : (|X|, OX ) → (Y, OY ) its corre∞ sponding morphism of locally C -ringed spaces. We are to show that |g| is equal to h and that g # coincides with h# . Let p ∈ |Spec A| be a R-point and Spec A → X be an open subfunctor. Then we have |g| |Spec A| (p) = p∗ (g ◦ εx (idSpec A )) = p∗ h(idSpec A ) = h(p). To petit petit show that h# : h∗ OY → OX and g # : g ∗ OY → OX coincide it suffices to verify that petit g ∗ = h∗ : OY (V ) → OX (|U |) for U → X an open subfunctor and V ⊂◦ Y an open subset petit such that |U | ⊂ h−1 (V ). Recall that in this situation g ∗ : OY (V ) → OX (|U |) is given by # ∗ ∗ f 7→ g (f ) = (y 7→ g (f )(y) = χ|Spec R| (ιV (f ))) where petit (χ, χ# ) = g ◦ εy (idSpec R ) = (h, h# ) ◦ Φ(εy (idSpec R )) ∈ Hom((|Spec R|, OSpec R ), (Y, OY )). # # # Thus, we have χ# |Spec R| (ιV (f )) = h ◦ Φ(εy (idSpec R )) (ιV (f )) = h (ιV (f )) and we are done. q.e.d.

∞ Ring

Corollary 3.6.5 For every diagram X : I → SetC natural isomorphism

fp

of smooth functors we have a

! colim colim − −−−→Φ(X) = Φ − −−−→X I

.

I

Remark. Following Corollary 2.3.23 one can prove that L a Ψ. Thus, we conclude by the uniqueness of adjoints that Φ ∼ = L. Corollary 3.6.6 Let X : I → SetC there is a natural isomorphism

∞ Ring

fp

be finite diagram of smooth functors. Then !

lim lim ← −−Φ(X) = Φ ← −−X I

I

of locally C∞ -ringed spaces. Proof: By the previous remark the functor Φ is isomorphic to a Kan extension which is defined by a filtered colimit. q.e.d.

Chapter 3: Smooth Functors as C∞ -Schemes

112 Theorem 3.6.7 The functor

Φ ◦ Spec : C∞ Ringfp


◦

→ LC∞ RS

from the category of finitely represented C∞ -rings to the category of locally C∞ -ringed spaces is isomorphic to the classical spectrum functor, i.e. there is a natural isomorphism Φ ◦ Spec = spec of functors. Moreover, the restriction of the classical spectrum functor to C∞ Ringfp is isomorphic to the spectrum functor spec defined in [Dub(a) 81, Thm. 8], [Joy 09, § 4.2] and [Quˆe 87, § 3]. As a consequence, all C∞ -schemes that possess a cover by finitely presented C∞ -rings can be derived from the smooth functor approach. Proof: The statement follows by construction since for every A ∈ C∞ Ringfp we have the same underlying topological space |Spec A| and the structure sheaf is given by petit OA (D(a)) = O(A[a−1 ]) = A[a−1 ] = ESpec A (D(a)).

In [Joy 09, § 4.2] D. Joyce also chooses the initial topology for all R-points mapping to the real line for his underlying topological space. He proves in [Joy 09, § 4.2, Prop 4.20] that for his spectrum functor with A ∈ C∞ Ringfp , one has EX (D(a)) = A[a−1 ]. Thus, his spectrum functor coincides with our classical spectrum functor and due to [Joy 09, § 4.2, Remark 4.6] with [Dub(a) 81, Thm. 8] and [Quˆe 87, § 3]. q.e.d. ∞

Corollary 3.6.8 The full subcategory C∞ Sch of smooth functors SetC Ringfp is equivalent to the full subcategory C∞ Schfp of schemes covered by finitely presented C∞ -rings in locally C∞ -ringed spaces LC∞ RS in the sense of D. Joyce. We finish the chapter with a comparison of the various ways how to associate C∞ -ringed spaces to manifolds which were the initial point for all of our considerations. Because of Theorem 3.6.7 and since for every smooth manifold the C∞ -ring C ∞ (M ) is finitely presented, we merely have to compare one of the spectrum constructions with Example 3.5.2. Proposition 3.6.9 Let M be a manifold. We write X for the locally C ∞ -ringed space (M, HM ) obtained by the construction of Example 3.5.2 and spec C ∞ (M ) for the one given in Theorem 3.6.7 applied to the C ∞ -ring C ∞ (M ) (since C ∞ (M ) ∈ C∞ Ringfp by Proposition 3.1.45). Then we have X∼ = spec C ∞ (M ) an isomorphism of locally C ∞ -ringed spaces.

3.6 From Smooth Functors to C∞ -Ringed Spaces Proof:

113

In order to specify an isomorphism (f, f # ) : (M, HM ) → (|Spec C ∞ (M )|, ESpec C ∞ (M ) )

we have to define a homeomorphism f : M → |Spec C ∞ (M )| and an isomorphism f # : HM → ESpec C ∞ (M ) of sheaves of local C∞ -rings. Corollary 3.2.10 gives the homeomorphism. Looking at principal open subsets U := D(x), we find HM (U ) = HomMan (U, · ) and ESpec C ∞ (M ) (U ) = C ∞ (M )[x−1 ] = C ∞ (U ) = HomMan (U, · ) and see that both sheaves of local C∞ -rings agree on a basis for the topology and hence it follows that they are isomorphic. q.e.d.

Chapter 4 Derived Manifolds as Functors In this chapter we will pursue a way to extend the notion of C∞ -rings to so called smooth rings and C∞ -schemes to derived manifolds. Using the injective model structure on sSetEuk one can define a lax version of product preserving functors that are the building blocks for derived manifolds and organize them into a category we denote by SR. Note that the model category sSetEuk is left proper and so is SR, which is necessary for further localization that we use to force the simplicial spectrum functor we encounter in paragraph four to become a homotopy sheaf. The lax version of product preserving functors and the notion of homotopy sheaves are the key points in which the theory of derived manifolds differs from the theory of C∞ -schemes. In the first paragraph we shortly recall the lax version of product-preserving functors given by D. Spivak and organize them into the category SR of smooth rings. The following section is devoted to the category dAff ∞ of derived spectra. We introduce affine derived manifolds as the dual of the full subcategory sC∞ of SR, the so called homotopically finitely presented smooth rings. We will turn dAff ∞ into a site and construct a topological realization functor that associates to every open cover of an affine derived manifold a honest topological space with a honest cover by open subsets. In paragraph three we define our version of derived manifolds as homotopy sheaves that possess an affine cover in the category of smooth-simplicial functors which is basically the ∞◦ category sSetdAff of simplicial presheaves on dAff ∞ with an appropriate model structure. It turns out that the set of open subfunctors Sub∞ (F) for any derived manifold F is in actual fact a locale and we define a functor from the category of derived manifolds to the category of topological spaces. In paragraph four and five we introduce the simplicial spectrum functor and the global structure sheaf and investigate their behaviour with respect to the model structure of the underlying model categories. We will see that both functors form a Quillen pair and that they form even a simplicial adjunction. Section six is a recollection of D. Spivak’s way to define derived manifolds as a full subcategory of locally smooth-ringed spaces. To do so we recall his definition of the simplicial model category LRS of locally smooth-ringed spaces. Moreover, we show that there is a functor that associates to every derived manifold in the sense of paragraph three a locally 115

116

Chapter 4: Derived Manifolds as Functors

smooth-ringed space. In the last section, using the achievements gained so far, we define the affine spectrum functor that associates a locally smooth-ringed space to a homotopically finitely presented smooth ring. By means of some technical preparations we are capable of proving our main result which states the fact that both approaches to derived manifolds with a finite affine cover are essentially equivalent. To improve the readability of the last chapter we give a short overview of all important categories and functors we are going to encounter. Starting point is the functor category sSetEuk from which we obtain the category of smooth functors SR by Bousfield localization. The category SR possesses a full subcategory sC∞ which will be of particular importance for our purposes and whose dual dAff ∞ gives the affine building blocks for derived manifolds. We will write dSpec for the dualization functor sC∞ → dAff ∞ . Thus, we have the diagram dSpec

sSetEuk −−−→ SR ←−−− sC∞ −−−−→ dAff ∞ . ∞

The functorial approach towards derived manifolds starts with the functor category sSetsC . Its Bousfield localization Shv(S∞ ) is the category of smooth-simplicial functors which contains the category of derived manifolds dMan as a full model subcategory: sSetsC

∞

−−−→ Shv(S∞ ) ←−−− dMan.

Another way to look at derived manifolds is given by D. Spivak who introduces the category RS of ringed spaces whose structure sheaves are smooth ring-valued homotopy sheaves. The category dManS of derived manifolds in this context is a full model subcategory of the category LRS of locally smooth-ringed spaces in RS: dManS −−−→ LRS −−−→ RS. These categories are linked by functors Shv(S∞ ) o

O sSpec

/

sC∞

Sp

/

LRS o

Φ

/

Ψ

Shv(S∞ )

as we will show in this chapter.

4.1

The Layout for Smooth Rings

In this section we define the simplicial model category SR of smooth rings that is the basic setup for derived manifolds. We follow [Spi 10]. We say that F : Euk → sSet is weakly product-preserving if and only if the natural map F (Rd × Rn ) → F (Rd ) × F (Rn )

4.1 The Layout for Smooth Rings

117

is a weak equivalence. In order to find an appropriate model structure we merely have to look at the simplicial Yoneda lemma which gives Map(C ∞ (Rd ), F ) = F (Rd ) for the C∞ -ring C ∞ (Rd ) in d generators. Thus we define p(d,n) : C ∞ (Rd ) q C ∞ (Rn ) → C ∞ (Rd+n ) to be the natural map out of the coproduct in sSetEuk induced by the coordinate projections Rd+n → Rd and Rd+n → Rn . Note that I : Euk → sSet defined by sending Rd to the initial simplicial set ∅ is the initial smooth ring. Now let S be the set S := {p(d,n) | d, n ∈ N0 } ∪ {I → C ∞ (R0 )} of morphisms in sSetEuk of all such maps. Definition 4.1.1 The category SR := LS (Pre(Euk), sSet) obtained by left Bousfield localization of the injective model structure on sSetEuk at the set S is called the category of smooth rings. The category SR is a left proper, cofibrantly-generated simplicial model category whose fibrant objects are precisely those smooth rings F ∈ SR which are local with respect to all p(d,n) ∈ S (cf. chapter two, paragraph six), i.e. MapSR (C ∞ (Rd+n ), F ) → MapSR (C ∞ (Rd ) q C ∞ (Rn ), F ) is a weak equivalence which is equivalent to F (Rd+n ) ' F (Rd ) × F (Rn ) by the simplicial Yoneda lemma. Looking at Proposition 2.8.8 we have the following Example 4.1.2 Every C∞ -ring A ∈ C∞ Ring is a fibrant smooth ring. As in the case of C∞ -rings we tend to write F d instead of F (Rd ). Again by Proposition 2.8.8 we have Lemma 4.1.3 There is an embedding c∗ : C∞ Ring → SR of C∞ -rings into the category of smooth rings such that π0 ◦ c∗ ∼ = 1, where π0 is the simplicial path component functor.

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The lemma reads as follows: A discrete smooth ring is fibrant if and only if it is a C -ring in the classical sense. ∞

Lemma 4.1.4 The discrete smooth ring C ∞ (R0 ) is the homotopy initial object in SR. Proof:

For F ∈ SR fibrant one has MapSR (C ∞ (R0 ), F ) → MapSR (I, F ) = ∆[0],

where the arrow is a weak equivalence of simplicial sets. q.e.d.

4.2

Topology for Smooth Rings

For our purposes it suffices to focus on a full subcategory of SR, namely the category sC∞ of homotopically finitely presented smooth rings. The dual category dAff ∞ of sC∞ will be turned into a site and the dualization functor dSpec will enable us to define affine derived manifolds in the next section. Since SR is a model category, it is closed under homotopy pushouts by Corollary 2.5.11. Hence the following definition is sensible. Definition 4.2.1 Let A, B, C ∈ SR smooth rings. Assume there are morphisms φ : A → B and ψ : A → C of smooth rings. Then we write φ

ψ

B ⊗hA C := −hocol −−−→(B ← A → C) for the homotopy pushout of ψ along φ. Note, that the homotopy colimit of the diagram (C ∞ (R0 ) ← C ∞ (R) → C ∞ (R0 )), induced by the diagram (R0 → R ← R0 ) (sending the point to the origin) is an example of a non-discrete smooth ring. Thus, SR possesses not only discrete smooth rings and therefore contains nontrivial homotopical information. Definition 4.2.2 Define sC∞ to be the subcategory of SR of all C ∞ (Rd ) for d ∈ N0 that is closed under all colimits ∞ C ∞ (Rn ) ⊗C ∞ (Rd ) C ∞ (R0 ) and homotopy colimits of type C ∞ (Rn ) ⊗hC ∞ (Rd ) C ∞ (R0 ) for all d, n ∈ N0 . The category SC∞ is defined to be the left Bousfield localization SC∞ := LD SR

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4.2 Topology for Smooth Rings

n o Q ∞ at the set of morphisms D = A → ← holim A | A ∈ sC that are induced by hyperin −−−− covers on the smooth site (cf. Definition 4.3.1). Considering sC∞ with respect to the model structure of SC∞ , we call it the category of homotopically finitely presented smooth rings. Remark. At first glance there is no reason to perform the left Bousfield localization at D. However, we wish to force the simplicial enriched adjunction O a sSpec we are going to introduce in section 4.3 to be a Quillen adjunction. By [Hir 03, Ch. 3, § 3.3, Thm. 3.3.20 (1) ] this can be accomplished by localizing at D. We will discuss this issue later in more detail.

By construction all finitely presented C∞ -rings belong to sC∞ since sC∞ is closed under colimits. Proposition 4.2.3 The functor π0 : sSetEuk → SetEuk defined by π0 (F ) = (Rd 7→ π0 (F (Rd ))) sends fibrant objects in SR as well as fibrant objects in sC∞ to C∞ -rings. Moreover, π0 turns homotopy colimits to colimits in the case of their existence. In this context we sometimes refer to π0 as the smooth-ring functor. Proof:

For all fibrant objects F in sC∞ we have (π0 F ) (Rd ) = π0 MapsC∞ (C ∞ (Rd ), F ) ∼ = π0 ∼ = =

d Y

MapsC∞ (C ∞ (R), F )

k=1 d Y

π0 MapsC∞ (C ∞ (R), F )

k=1 d Y

π0 F (R).

k=1

Thus, π0 sends fibrant smooth rings to C∞ -rings. Now let I be an indexing category and Ψ ∈ (sC∞ )I be a diagram and Ψ0 ∈ (sC∞ )I its cofibrant replacement for the I-projective model structure. Then we find ! ! π0 − hocol −−−→Ψ I

= π0

0 colim − −−−→Ψ I

0 = −colim colim −−−→π0 (Ψ ) = − −−−→π0 (Ψ) , I

I

where we have used that π0 possesses a right adjoint and sends weak equivalences to isomorphisms. q.e.d.

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Corollary 4.2.4 The smooth ring functor π0 takes homotopy pushouts of smooth rings to pushouts of C∞ -rings. In particular, for every F ∈ sC∞ we have that π0 (F ) ∈ C∞ Ringfp is a finitely presented C∞ -ring. Corollary 4.2.5 There exists a functor U : SR → R-Alg which sends smooth rings to their underlying R-Algebra via F 7→ j ∗ (π0 F ), the underlying R-Algebra functor for smooth rings. Proof: The underlying R-algebra functor for smooth rings is the composition of π0 and the underlying R-algebra functor j ∗ : C∞ Ring → R-Alg for C∞ -rings. q.e.d. Definition 4.2.6 A smooth ring A ∈ SR is called local if and only if its image π0 (A) ∈ C∞ Ring under the smooth-ring functor is a local C∞ -ring. A morphism ϕ : A → F in SR between local smooth rings is local iff π0 (ϕ) ∈ HomC∞ Ring (π0 A, π0 F ) is a local morphism of local C∞ -rings. The following lemma gives the link to Spivak’s definition (cf. [Spi 10, def. 5.10]) where a smooth ring is defined to be local iff its underlying R-algebra is local in the usual sense and where a morphism of local smooth rings is called local iff its morphism of R-algebras is local in the usual sense. Lemma 4.2.7 A smooth ring A ∈ SR is local if and only if its image under the underlying R-algebra functor for smooth rings is a local R-algebra. A morphism ϕ : A → F between local smooth rings is local if and only iff its image U (ϕ) under the underlying R-algebra functor is a local morphism of R-algebras. Proof: The assertion is an immediate consequence of the definition, Lemma 3.1.26 and Proposition 3.1.28. q.e.d. For the reader’s convenience, we recall the description of finitely presented C∞ -rings in terms of colimits from chapter three. By Example 3.1.14 a C∞ -ring A ∈ C∞ Ring is finititely presented if and only if ∞

A∼ = C ∞ (Rn ) ⊗C ∞ (Rd ) C ∞ (R0 ),

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4.2 Topology for Smooth Rings ∞

d where C ∞ (Rn ) ⊗C ∞ (Rd ) C ∞ (R0 ) suppresses the maps and stands for a colimit − colim −−−→Γψ,n with ψ

0

Γdψ,n := (C ∞ (Rn ) ←−−− C ∞ (Rd ) −−−→ C ∞ (R0 )). ∞

Thus, the localization A[a−1 ] is given by C ∞ (Rn+1 ) ⊗C ∞ (Rd+1 ) C ∞ (R0 ), i.e. by a colimit d+1 d+1 d colim − −−−→Γψ,n+1 where the diagram Γψ,n+1 arises from Γψ,n as explained in Example 3.1.20. With this in mind we can give the following Definition 4.2.8 Let F ∈ sC∞ be a smooth ring and x ∈ U (F ). By Corollary 4.2.4, π0 F ∞

is isomorphic to C ∞ (Rn ) ⊗C ∞ (Rd ) C ∞ (R0 ) as explained above. The h-localization of F at x is defined to be F [x−1 ]h := C ∞ (Rn+1 ) ⊗hC ∞ (Rd+1 ) C ∞ (R0 ) as in the case of C∞ -rings simply by replacing the colimit by the homotopy colimit. Note that by Proposition 2.5.10 and the following remark one has a morphism F → F [x−1 ]h which need not be unique unless the cofibrant replacement is functorial. But the map inherits the universal property of the homotopy colimit and by that it is unique up to weak equivalence. Lemma 4.2.9 The category sC∞ of homotopically finitely presented smooth rings is closed under h-localisation. Proof:

This is an immediate consequence of the definition. q.e.d.

The following proposition is the reason why sC∞ is called the category of homotopically finitely presented smooth rings. Proposition 4.2.10 All objects in sC∞ are compact, i.e. if I is a cofiltered category and X : I → sC∞ is a diagram of homotopically finitely presented smooth rings, then for all F ∈ sC∞ the natural map colim colim − −−−→MapsC∞ (F, X) → MapsC∞ (F, − −−−→X) I

I

is a weak equivalence. Proof: Every F ∈ sC∞ is weakly equivalent to a homotopy colimit F ' − hocol −−−→Fi ∞ because sC is formed by freely adding homotopy colimits as described above. Thus,

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general nonsense gives holim colim colim −−−−MapsC∞ (Fi , X) −−−→← − −−−→MapsC∞ (F, X) ' − I

I

lim '− colim −−RMapsC∞ (Fi , X) −−−→← I

'← lim colim −−− −−−→RMapsC∞ (Fi , X) I

'← holim colim −−−−− −−−→MapsC∞ (Fi , X) I

'← holim colim −−−−MapsC∞ (Fi , − −−−→X) I

where we have used Proposition 2.8.5, the fact that the directed colimit of fibrant objects is again fibrant and that C ∞ (Rd ) is small for every d ∈ N0 which is easy to prove using the simplicial Yoneda lemma. q.e.d.

Proposition 4.2.11 All homotopically finitely presented smooth rings are local smooth rings. Proof: This is immediate taking into account that all objects of sC∞ are homotopy colimits of some diagram of C ∞ (Rd ) and the fact that the smooth ring functor takes homotopy colimits of homotopically finitely presented smooth rings to finitely presented C∞ rings. Thus, Corollary 3.1.41 applies. q.e.d. We want to turn sC∞ into a site. Definition 4.2.12 The open co-cover pretopology on sC∞ is given in the following way. Define h (αi : F → F [x−1 i ] | i ∈ I) ∈ cov∞ (F ) to be an open co-cover in sC∞ if and only if its image h (π0 (αi ) : π0 (F ) → π0 (F [x−1 i )] ) | i ∈ I) ∈ cov∞ (π0 (F ))

under the smooth ring functor π0 is an open co-cover in the sense of Definition 3.2.17. Definition 4.2.13 We denote by dAff ∞ the dual of sC∞ and call it the category of (affine) derived spectra. We will write S∞ := (dAff ∞ , cov∞ )

4.2 Topology for Smooth Rings

123

for the site obtained by endowing dAff ∞ with the Grothendieck topology induced by open co-covers of sC∞ and call it the smooth site. We write dSpec : sC∞ → dAff ∞ for the dualization functor and call it the derived spectrum. A morphism dSpec Ai → dSpec A is said to be a standard open map if it belongs to some open co-cover. Be aware that the model structure of dAff ∞ is the dual model structure of sC∞ . To be precise, a morphism A → B is a cofibration, fibration, weak equivalence in sC∞ if it is a cofibration, fibration, weak equivalence for the model structure of SC∞ which is precisely the case if dSpec B → dSpec A is a fibration, cofibration, weak euquivalence in dAff ∞ , i.e. in the model structure of (sC∞ )◦ and vice versa. The question is what is the simplicial structure of dAff ∞ ? The answer is given by the following Example 4.2.14 The dualization dSpec : sC∞ → dAff ∞ determines the simplicial structure of dAff ∞ as taught by Proposition 2.3.4. From this it readily follows that the simplicial action on dSpec A is given by
(dSpec A) ⊗ S = dSpec AS . Proposition 4.2.15 Consider the functor | · | : S∞ → Top, the (derived) topological realization, extending the topological realization functor for C∞ rings to the smooth site defined by dSpec F 7→ |dSpec F | := |Spec π0 (F )|. This functor turns homotopy pushouts of smooth rings into fibered products of the topological spaces associated to the underlying diagram. Proof: We use the fact that π0 (F ) is a finitely presented C∞ -ring for every F
∈ sC∞ and we abbreviate its presentation with Γdn := C ∞ (Rd ) ← C ∞ (Rd ) → C ∞ (R0 ) . Thus, we get |dSpec (F )| = |Spec π0 (F )|
d = Spec − colim −−−→π0 Γ n
d = Spec − colim −−−→ Γn 
∞ 0  d = HomC∞ Ring − colim Γ −−−→ n , C (R )
∞ d ∞ n ∞ 0 =← lim |Spec C (R )| → |Spec C (R )| ← |Spec C (R )| . −− q.e.d.

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Corollary 4.2.16 The derived topological realization functor factors through the full subcategory of locally compact, paracompact Hausdorff spaces. Proof: This is an immediate consequence of the observation that for every F ∈ sC∞ the C∞ -ring π0 F is of finite presentation and hence of finite type. Thus, Lemma 3.2.4 and Lemma 3.2.7 apply and we are done. q.e.d. ∞ 0 ∞ 1 ∞ 0 0 0 Example 4.2.17 Define A := −hocol −−−→ (C (R ) ← C (R ) → C (R )). Since R ×R R = R0 we have |dSpec A| = R0 .

Example 4.2.18 Let f = (f1 , . . . , fd ) : Rn → Rd be a smooth map and X ⊂ Rn be the vanishing set of f , i.e. X fits into the cartesian diagram X −−−→   y

R0   y0

f

Rn −−−→ Rd . Define I to be the ideal in C ∞ (Rn ) generated by f , i.e. I = (f1 , . . . , fd ). Then we have Z(I) = X due to Definition 3.2.5. We define the C∞ -ring A ∈ C∞ Ringfp to be the quotient A := C ∞ (Rn )/I (cf. Proposition 3.1.7), which is equivalent to the colimit ∞ n ∞ d ∞ 0 colim − −−−→(C (R ) ← C (R ) → C (R )) given by Example 3.1.14. Using Proposition 3.2.6 we have |Spec A| = X and using Proposition 3.2.9 there is a characteristic function gi ∈ C ∞ (Rn ) for every open subset Vi ⊂ X. Write ai for the image of gi in A, we −1 then have |Spec A[a−1 i ]| = Vi . Following Example 3.1.20 we see that A[ai ] can also be k+1 obtained by the colimit − colim −−−→Γψ,n+1 . Thus, replacing all colimits by homotopy colimits k+1 h F := C ∞ (Rn ) ⊗hC ∞ (Rd ) C ∞ (R0 ) and F [a−1 hocol i ] := − −−−→Γψ,n+1 we end up with an open cover h (dSpec F [a−1 i ] → dSpec F | i ∈ I) ∈ cov∞ (dSpec F )

of dSpec F for the smooth site S∞ whose topological realization |dSpec F | is homeomorphic to X.

4.3

Derived Manifolds from Smooth-Simplicial Functors

In this section we introduce the category Shv(S∞ ) of smooth-simplicial functors. To do so ∞◦ we equip the presheaf category sSetdAff with the projective model structure and then we perform the left Bousfield localization at the set of hypercovers induced from open covers ∞◦ in dAff ∞ . We have chosen to start with the projective model structure on sSetdAff

4.3 Derived Manifolds from Smooth-Simplicial Functors

125

in order to force the simplicial enriched adjunction O a sSpec we will encounter in the next section to be a morphism of simplicial model categories. Moreover, we define open morphisms in Shv(S∞ ) and give our definition of derived manifolds as certain smoothsimplicial functors. It turns out that the set of open subfunctors associated to a derived manifold forms a locale and that there is a topological realization on the category of derived manifolds which is an extension of the one introduced in section two. Definition 4.3.1 The category of smooth-simplicial functors Shv(S∞ ) := LS (Pro(dAff ∞ ), sSet) ∞◦

is the category sSetdAff of simplicial presheaves after the left Bousfield localization of the projective model structure at the set n o [ S := hocol U → r(dSpec F ) | U → r(dSpec F ) ∈ Hyp (dSpec F ) ∞  dAff −−−−→  dSpec F ∈dAff ∞

of all hypercovers that are induced by open covers (dSpec Ai → dSpec A | i ∈ I) ∈ cov∞ (dSpec A). An object F ∈ Shv(S∞ ) is called a smooth-simplicial functor. Fibrant objects of Shv(S∞ ) are homotopy sheaves with respect to the smooth site. Lemma 4.3.2 Manifolds form a fully faithful subcategory i : Man → Shv(S∞ ) of the category of smooth-simplicial functors. Proof:

Let i be the composition of C ∞ ( · ) : Man → C∞ Ring followed by Map( · ) : dAff ∞ → Shv(S∞ ) ,

dSpec F 7→ Map( · , dSpec F ).

Then i is fully faithful as the composition of fully faithful functors. q.e.d. The previous lemma enables us to to give the following Definition 4.3.3 A smooth-simplicial functor F ∈ sSetdAff manifold if it is in the essential image of Map( · ) : dAff ∞ → Shv(S∞ ) of the simplicially enriched Yoneda embedding.

∞◦

is called an affine derived

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Chapter 4: Derived Manifolds as Functors

Example 4.3.4 Every smooth manifold is an affine derived manifold by Lemma 4.3.2. Recall from category theory that a morphism F → G in a category C (which is assumed to have fibered products) is a monomorphism if and only if the canonical map F → F ×G F is an isomorphism in C. Thus, the following definition generalizes the notion of monomorphisms and enables us to define subobjects in the homotopical setting: A morphism F → G in a model category is a h-monomorphism if and only if F → F ×hG F is a weak equivalence (F ×hG F denotes the homotopy fibered product). For every smooth-simplicial functor X ∈ Shv(S∞ ) the sheaf associated to the presheaf π0 (X) of sets on S∞ is denoted π0S∞ (X). The h-monomorphisms induce monomorphisms on the level of the sheaves π0S∞ . Definition 4.3.5 (Open Subfunctors) (1) Let X ∈ Shv(S∞ ) be an affine derived manifold and U → X be a fibrant subobject. We call U open in X if and only if there is a family (Xi → X | i ∈ I) of open maps in dAff ∞ such that π0S∞ (U ) is in the image of the morphism a π0S∞ (Xi ) → π0S∞ (X) i∈I

of sheaves. (2) A morphism f : F → G in Shv(S∞ ) is open if and only if for all affine derived manifolds Y and all morphisms ψ : Y → G the morphisms ψh∗ (f ) : ψh−1 F → Y, where ψh−1 F := F ×hG Y denotes the homotopy fiber product, is a h-monomorphism which is open in Y . If F → G is open we call F an open subfunctor of G and we denote the set of all open subfunctors of G by Sub∞ (G). Lemma 4.3.6 Open morphisms in Shv(S∞ ) are stable under homotopy pullback and composition. Proof: We merely show the first claim since for the second the proof carries over from its C∞ -ring analogue in Lemma 3.3.6 because of the universal property of the homotopy limit. Assume U to be open subfunctor of F and V any subfunctor. We are to show that

V ×hF U → V ×hF U ×hV V ×hF U

is a h-monomorphism. But V ×hF U ×hV V ×hF U is weakly equivalent to V ×hF U ×hF U . Since U is an open subfunctor of F, the homotopy invariance makes sure that V ×hF U → V ×hF U ×hF U

4.3 Derived Manifolds from Smooth-Simplicial Functors

127

is a weak equivalence. Hence the claim follows from the 2-out-of-3 axiom for weak equivalences. q.e.d. Definition 4.3.7 Let U, V ∈ Sub∞ (F) two open subfunctors of F ∈ Shv(S∞ ) and (Ui )i∈I ⊂ Sub∞ (F) a family of open subfunctors. Then we define the intersection U ∧ V := U ×hF V and the open union _

Ui := − hocol −−−→Ui .

i∈I

i∈I

Proposition 4.3.8 The open union and the intersection of open subfunctors enjoy the ∞-distributive law, i.e. there is a weak equivalence _ _ (U ∧ Vi ) U ∧ Vi ' i∈I

i∈I

for every open subfunctor U → F and every family (Vi )i∈I ⊂ Sub∞ (F). Proof: The assertion comes down to the fact that directed colimits and finite limits commute. q.e.d. Corollary 4.3.9 Let F ∈ dMan be a derived manifold. Let Y be an affine derived manifold and ϕ : Y → F be an open map and (Ui )i∈I ⊂ Sub∞ (F). Then we have ! _ _ ϕ−1 Ui ' ϕ−1 (Ui ) . i∈I

i∈I

Definition 4.3.10 (Derived Manifolds) A smooth-simplicial functor F ∈ Shv(S∞ ) is a derived manifold if and only if it is a homotopy sheaf for the ∞-site S∞ and if there exists a family (Xi )i∈I of affine derived manifolds together with an epimorphism a π0S∞ (Xi ) → π0S∞ (F) i∈I

such that Xi → F is open for every i ∈ I. The family (Xi → F | i ∈ I) is called affine cover of F. We denote by dMan the category of derived manifolds in Shv(S∞ ). It is a non-trivial fact that every affine derived manifold is a derived manifold. This will be proven in the next section.

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Chapter 4: Derived Manifolds as Functors

Theorem 4.3.11 The functor | · | : dAff ∞ → Top can be extended to a functor | · | : dMan → Top where for F ∈ dMan a subset V ⊂ |F| := π0 F(C ∞ (R0 )) is defined to be open if and only if there is an open subfunctor U ∈ Sub∞ (F) such that |U | = V . Proof: It suffices to retreat to the affine case since every derived manifold can be covered by affines. Thus, assume F = MapdSpec F for F ∈ sC∞ and U → F to be an open subfunctor. We have |F| = |Spec π0 F |, since |F| = π0 MapSC∞ (F, C ∞ (R0 )) = HomC∞ Ring (π0 F, C ∞ (R0 )) = |Spec π0 F | = |dSpec F |, where for the second isomorphism we have used that the image of a fibrant simplicial set under π0 equals the set of 0-vertices, that π0 is left adjoint to the constant simplicial set functor und that π0 maps smooth rings to finitely presented C∞ -rings. Hence |U | ⊂◦ |Spec π0 F | where the right hand side is the classical topological realization functor of Definition 3.2.3. Hence the proof of Theorem 3.3.13 applies to this situation and we are done. q.e.d. Corollary 4.3.12 Let F be a derived manifold. Then we have an isomorphism Sub∞ (F) ∼ = Op(|F|) of categories. Example 4.3.13 The principal opens D(a) ⊂ |F| in this setting are given by D(a) = |dSpec A[a−1 ]h | = HomC∞ Ring (π0 A[a−1 ]h , C ∞ (R0 )), i.e. D(a) corresponds to dSpec A[a−1 ]h for the open subfunctor MapdSpec

A

→ F.

The next example shows that affine derived manifolds can be glued together by their standard opens. Example 4.3.14 Let X = Map( · , dSpec A) be an affine derived manifold and h (dSpec A[a−1 i ] → A | i ∈ I)

129

4.4 The Global Structure Sheaf

h be an open cover of dSpec A. Then the collection (Ui )i∈I with Ui = Map( · , dSpec A[a−1 i ] ) covers X by definition and its open union is weakly equivalent to X by construction. Moreover, for the topological realization we find _ −1 h ∞ 0 Ui = −colim −−−→π0 Map(dSpec C (R ), dSpec A[ai ] ) i∈I

i∈I

−1 h ∞ 0 = −colim −−−→Hom(π0 A[ai ] , C (R )) i∈I


Spec π0 A[a−1 ]h = −colim i −−−→ i∈I

= |Spec (π0 A)| = |X| . As we have already done in the case of C∞ -schemes, we can turn the category dMan of derived manifolds into W a site: For F ∈ dMan let (Xi → F | i ∈ I) be a covering iff Xi ∈ Sub∞ (F) and Xi = F. Using the ∞-distributive law and Lemma 4.3.6 one readily checks that these coverings define a Grothendieck pretopology on dMan. This way we consider dMan as a site when equipped with the Grothendieck topology induced by the pretopology above: The big site dMan of derived manifolds.

4.4

The Global Structure Sheaf

We define the global structure sheaf O and show that it is the left adjoint of a simplicially enriched adjunction. The global structure sheaf will do its job in the next section since it enables us to associate the structure sheaf to the topological realization of a derived manifold. At the end of this section we will show that O is in fact part of a Quillen adjunction. As a consequence, every affine derived manifold is in fact a derived manifold, i.e. a homotopy sheaf in Shv(S∞ ). Definition 4.4.1 The simplicial spectrum functor sSpec : (SC∞ )◦ → sSetsC

∞

is defined to be the composition of the simplicial mapping space functor A 7→ MapSC∞ (A, · ) ∞

followed by the restriction u∗ : sSetSC u : sC∞ → SC∞ .

→ sSetsC

∞

of functors induced by the inclusion ∞

Note that for the restriction u∗ (sSpec) : sC∞ ◦ → sSetsC of sSpec to sC∞ we simply write sSpec A = Map( · , dSpec A).

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Chapter 4: Derived Manifolds as Functors

In this case sSpec A is an affine derived manifold by Definition 4.3.3. To put the topological realization functor from the previous paragraph into this context we give the following Example 4.4.2 For any A ∈ sC∞ we have |sSpec A| = |dSpec A|. Definition 4.4.3 The global structure sheaf ∞

O : sSetsC → sSetEuk is given by
X 7→ Rd 7→ MapsSetsC∞ (X, sSpec C ∞ (Rd ) . ∞

Example 4.4.4 Let X ∈ sSetsC be in the essential image of the simplicial spectrum functor restricted to homotopically finitely presented smooth rings, say X ∼ = sSpec A for ∞ A ∈ sC . Then we have O(X) ∼ = A. We will write O(dSpec A) for O(X) in this case and we may consider the global structure sheaf as a functor O : dAff ∞◦ → sSetEuk when O is restricted to affine derived manifolds. ∞

Lemma 4.4.5 A smooth-simplicial functor X ∈ sSetsC and only if there is an isomorphism

is an affine derived manifold if

u∗ sSpec O(X) = X. Proof: If X = u∗ sSpec O(X) it readily follows from the definition that X is an affine derived manifold. The other way round is given by the simplicially enriched Yoneda lemma and the simplicial Yoneda lemma. To be precise, for X = sSpec A with A ∈ sC∞ we have O(sSpec A)(Rd ) = O(A)(Rd ) = Ad for all euclidian spaces Rd due to Example 4.4.4. q.e.d. Proposition 4.4.6 The pair (O, sSpec ) gives a simplicial enriched adjunction O a sSpec with the simplicial spectrum functor as the right adjoint.

131

4.4 The Global Structure Sheaf Proof:

We are to verify that there is an isomorphism MapsSetEuk (A, O(X)) = MapsSetsC∞ (X, sSpec A)

of simplicial sets natural in both X and A. To do so we will heavily use the simplicial enrichment of both categories and the properties of ends like in the proof of Proposition 3.4.9. Thus, we compute MapsSetEuk (A, O(X))n = HomsSetEuk (A ⊗ ∆[n], O(X)) Z
= HomsSet Ad × ∆[n], MapsSetsC∞ (X, MapSC∞ (C ∞ (Rd ), · )) Rd

Z =


HomsSetSC∞ X ⊗ (Ad × ∆[n]), MapSC∞ (C ∞ (Rd ), · )

Rd

Z Z =

HomsSet X(R) × (Ad × ∆[n]), Rd




Rd R

Z Z =


HomsSet X(R) × ∆[n], MapsSet (Ad , Rd )

Rd R

Z HomsSet (X(R) × ∆[n], MapSC∞ (A, R))

= R

= HomsSetsC∞ (X ⊗ ∆[n], sSpec A) = MapsSetsC∞ (X, sSpec A)n for all n ∈ N0 . q.e.d. Corollary 4.4.7 For X = dSpec A there is an isomorphism O (X ⊗ ∆[n]) = O∆[n] (X). in sSetEuk for every n ∈ N0 . Proof:

Let B ∈ SC∞ . Then
Map (B, O(dSpec A ⊗ ∆[n])) = Map B, O(dSpec (A∆[n] )) = Map sSpec (A∆[n] ), sSpec B





= Map dSpec (A∆[n] ), dSpec B = Map (B ⊗ ∆[n], A) = Map (B ⊗ ∆[n], O(dSpec A))
= Map B, O(dSpec A)∆[n] .

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Hence we are done. q.e.d. Now we are going to show that the adjoint pair (O, sSpec) forms a morphism of model categories, i.e. is in actual fact a Quillen adjunction. For every fibrant F ∈ SC∞ the smooth-simplicial spectrum sSpec F ∈ Pro(dAff ∞ , sSet) is fibrant. We prove that as a consequence we have the following: (O, sSpec) is a Quillen pair with respect to the projec∞ tive model structure on sSetsC . The next step is to prove that this Quillen adjunction can be extended to a Quillen adjunction with respect to the model structure obtained by the left Bousfield localization of the underlying model structure at the set of all hypercovers which we have used to define Shv(S∞ ). To achieve this we use a theorem of J. Lurie which says that a simpicially enriched adjunction (F, G) between simplicial model categories is Quillen if and only if F preserves cofibrations and G preserves fibrant objects. Lemma 4.4.8 For every A ∈ SC∞ the object sSpec A ∈ Pro(sC∞ , sSet) is fibrant. Proof: The simplicial set (sSpec A) (R) is fibrant for every R ∈ sC∞ since all objects of SC∞ are cofibrant and all objects of sC∞ are fibrant. Hence sSpec A is fibrant for the ∞ projective model structure on sSetsC . q.e.d. Proposition 4.4.9 The pair (O, sSpec) is a morphism (O, sSpec) : SC∞ → Pro(dAff ∞ , sSet) of simplicial model categories. Proof: We already know by Proposition 4.4.6 that the adjunction O a sSpec is simplicially enriched. It remains to show that the adjunction is Quillen. To do so recall that all objects of SC∞ are cofibrant and therefore by [Hir 03, Ch. 13, § 13.1, Corallary 13.1.3] SC∞ is left proper. Since the adjunction O a sSpec is simplicially enriched, we must show that sSpec preserves fibrant objects and O sends cofibrations to cofibrations due to [Lur 09, Corollary A.3.7.2]. Because of Lemma 4.4.8 it remains to show that O preserves cofibrations. By Lemma 2.2.3 it suffices to prove that sSpec preserves acyclic fibrations. This is what we will do. Let A → B be an acyclic fibration in SC∞ . We have to verify that sSpec B → sSpec A is an acyclic fibration. Since all objects F of sC∞ are fibrant, the map MapSC∞ (A, F ) → MapSC∞ (B, F ) is an acyclic fibration of simplicial sets due to equation (1.3.1) and hence sSpec B → sSpec A is an acyclic fibration in Pro(dAff ∞ , sSet). q.e.d. Note that the global structure sheaf O can be considered as a functor O : sC∞ → sSetEuk

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4.4 The Global Structure Sheaf

when restricted to affines in Aff ∞ as already done in Example 4.4.4. Using this we may ∞ identify the restriction of O as an object of (SR)sC . To be more precise, we have Lemma 4.4.10 The functor O : sC∞ → SR is the identity. Every object F ∈ sC∞ is fibrant and hence we have

Proof:


O(F ) = Rd 7→ Map(sSpec F, sSpec C ∞ (Rd ))
= Rd 7→ Map(C ∞ (Rd ), F ) = (Rd 7→ F d ) = F, which is an immediate consequence of Example 4.4.4 and the simplicial Yoneda lemma. q.e.d. Corollary 4.4.11 The global structure sheaf O ∈ Pro (dAff ∞ , SC∞ ) is fibrant. Lemma 4.4.12 For every fibrant F ∈ SC∞ the object sSpec F ∈ Shv(S∞ ) is fibrant. ∞

Proof: We know that sSpec F ∈ sSetsC is fibrant for the projective model structure. Hence it remains to show the S-locality of sSpec F by [Hir 03, Ch. 3, § 3.4, Proposition 3.4.1]. Because of Proposition 4.4.6 the map Map(φ, sSpec A) : Map(r(dSpec A), sSpec F ) → Map(−hocol −−−→U , sSpec F ) is equivalent to 

 Map(A, O(φ)) : Map(F, O(A)) → Map F, ← holim −−−−

Y

O(Ain )

in ∈I n+1

where φ ∈ S is induced by some open cover (dSpec Ai → dSpec QA | i ∈ I). Using the fact that O(A) = A we see that O(φ) is equivalent to A → ← holim −−−− Ain which is a morphism ∞ in SC that belongs to the localizing class D (cf. Definition 4.2.2). Thus, Map(F, O(φ)) is a weak equivalence and we are done. q.e.d.

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Theorem 4.4.13 The simplicial enriched adjunction O a sSpec is a morphism (O, sSpec ) : SC∞ → Shv (S∞ ) of simplicial model categories. Proof: The left Bousfield localization preserves cofibrations. Consequently, O preserves cofibrations by Proposition 4.4.9. Moreover, the left Bousfield localizations of left proper model categories are left proper by [Hir 03, Ch. 4, §4.1, Thm. 4.1.1]. Because of the previous lemma sSpec sends fibrant objects to fibrant objects, so again by [Lur 09, A.3.7.2] the simplicially enriched adjunction O a sSpec is Quillen. q.e.d. Corollary 4.4.14 The global structure sheaf O ∈ Shv (S∞ , sC∞ ) is a homotopy sheaf. Consequently, the name global structure sheaf is justified. The following corollary proves the non-trivial fact that affine derived manifolds are indeed derived manifolds, i.e. the embedding i : Man → Shv(S∞ ) from Lemma 4.3.2 factors through the category Shv (S∞ )fib of homotopy sheaves. The initial purpose to obtain SC∞ as a left Bousfield localization at D (cf. Definition 4.2.2) was to force affine derived manifolds to be homotopy sheaves. Corollary 4.4.15 For every A ∈ sC∞ its image under the simplicial spectrum functor sSpec A = MapdAff ∞ ( · , dSpec A) is a homotopy sheaf for the smooth site. As in the case of C∞ -schemes we have the following Example 4.4.16 Let X be a derived manifold and t : X → sSpec C ∞ (R0 ) the homotopy terminal map and t−1 : dMan → (dMan ↓ X ) be the associated change of base functor. Let ι : (dMan ↓ X ) → Op(|X |) be the natural morphism of sites induced by the functor ι† : Op(|X |) ∼ = Sub∞ (X ) → (dMan ↓ X ) sending an open subset V ⊂◦ |X | to the arrow

135

4.5 Derived Manifolds as Ringed Spaces

U → X where the topological realization |U | is V . Using Corollary 4.3.12 and that dMan can be seen as a site we get a sequence of functors (t−1 )∗

ι

∗ → Shv (|X |, SR) . Shv (dMan, SR) −−−→ Shv ((dMan ↓ X ) , SR) −−−

The global structure sheaf O offers the opportunity to associate a homotopy sheaf OXpetit on the topological realization |X | in a canonical way by defining OXpetit := ι∗ (t−1 )∗ O. We call OXpetit the structure sheaf of |X | associated to the derived manifold X . Indeed, by Proposition 2.8.10 and its following discussion, OXpetit ∈ Shv(|X |, SR) is fibrant, i.e. a sheaf of smooth rings. petit In the case of an affine derived manifold, i.e. X = sSpec A for A ∈ sC∞ , we write OA petit instead of OX .

4.5

Derived Manifolds as Ringed Spaces

This section serves as a review of D. Spivak’s approach to derived manifolds. We follow D. Spivak and define the category LRS of locally smooth-ringed spaces, quote some properties, and introduce the category of derived manifolds as a full subcategory having certain covering properties. The following definition is taken form [Spi 07, Ch. 2, §2.3, Def. 2.3.13]. Definition 4.5.1 A smooth-ringed space is a pair X = (X, OX ), where X is a topological T1 space (i.e. points are closed) and OX is a homotopy sheaf of smooth rings (i.e. a fibrant object in Shv(X, SR)). We call OX the structure sheaf of X. For a pair X = (X, OX ) and Y = (Y, OY ) of smooth-ringed spaces we define the simplicial mapping space via a Map(X , Y) := Map(g ∗ OY , OX ). g∈HomTop (X,Y )

A pair (f, f # ) : (X, OX ) → (Y, OY ) is a morphism of smooth-ringed spaces if it is a vertex in the simplicial set Map(X , Y). The category RS of smooth-ringed spaces together with its mapping spaces is a simplicial category. via

Using Lemma 2.3.2 one can readily verify that the simplicial structure of RS is given a ∆[n] Map(X ⊗ ∆[n], Y) = Map(g ∗ OY , OX ). g∈HomTop (X,Y )

Thus, the simplical structure of RS is induced by the simplical structure of the category Shv(X, SR).

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Chapter 4: Derived Manifolds as Functors

We need a criterion to compare our different approaches to derived manifolds. Therefore we recall the following definition from [Spi 10, Definition 6.5]: Definition 4.5.2 A morphism (f, f # ) : (X, OX ) → (Y, OY ) in RS is called an equivalence of smooth-ringed spaces if f : X → Y is an isomorphism of topological spaces and f # : f ∗ OY → OX ∈ Shv(X, SR) is a weak equivalence. In this way the category RS becomes a simplicial model category whose weak equivalences are precisely the equivalences of smooth-ringed spaces and whose morphisms (f, f # ) are cofibrations (respective fibrations) iff f # is a cofibration (respective fibration). Proposition 4.5.3 Let X = (X, OX ) and Y = (Y, OY ) be smooth-ringed spaces and suppose U = (U, OU ) and V = (V, OV ) are open subobjects of X and Y respectively which are equivalent. Then one can glue X and Y together along U, i.e. one gets a smooth-ringed space X ∪U Y which is the homotopy colimit of the underlying diagram in RS. Proof: This is taken from [Spi 10, Lemma 6.8] and proven there. The topological space of X ∪U Y is just given by gluing of X and Y along the homeomorphism U ∼ = V , i.e. by the colimit X qU Y in the category of topological spaces. The structure sheaf O of X qU Y is defined to be the homotopy limit O := ← holim −−−− (i∗ OX → k∗ OW ← j∗ OY ) in Shv(X qU Y, SR) where OW denotes the cylinder object of the weak equivalence OU → OV and j : Y → X qU Y , i : X → X qU Y and k : U → X qU Y denote the canonical open injections. The description as homotopy colimit follows from [Spi 10, Prop. 6.9] using that X ∪U Y is a homotopy colimit iff the natural map MapRS (X ∪U Y, Z) → MapRS (X , Z) ×hMapRS (U ,Z) MapRS (Y, Z) is a weak equivalence of simplical sets.

q.e.d.

The next lemma is a helpful tool to detect weak equivalences in the category SR of smooth rings. Lemma 4.5.4 Let F and G be fibrant objects of the category SR of smooth rings. Then a morphism ϕ ∈ HomSR (F, G) is a weak equivalence if and only if the induced map ϕR : F (R) → G(R) is a weak equivalence of simplicial sets. Proof: Let ϕ : F → G be a weak equivalence of smooth rings. Using the simplicial enrichment of SR and the fact that C ∞ (R) ∈ SR is cofibrant one sees that MapSR (C ∞ (R), F ) → MapSR (C ∞ (R), G)

137

4.5 Derived Manifolds as Ringed Spaces

is a weak equivalence of simplicial sets. The claim follows from the simplicial Yoneda lemma. Assume ϕR : F (R) → G(R) to be a weak equivalence of simplicial sets. The fibrancy of both F and G makes sure that ϕRd : F (Rd ) → G(Rd ) is a weak equivalence for all d ∈ N0 and therefore ϕ : F → G is a weak equivalence in SR. q.e.d. Proposition 4.5.5 The simplicial model category of smooth-ringed spaces possesses finite homotopy limits. Proof:

This is [Spi 07, Ch. 2, § 2.3, Proposition 2.3.21]. q.e.d.

The following definition is taken from [Spi 10, Def. 6.3]. Definition 4.5.6 Let X be a topological space. A sheaf F ∈ Shv(X, SR) is a sheaf of local smooth rings iff F is fibrant and for every point x ∈ X given by p : R0 → X, ∗ 7→ x the smooth ring p∗ F ∈ Shv(R0 , SR) = SR is a local smooth ring in the sense of Definition 4.2.6. A morphism f : F → G of homotopy sheaves of local rings on X is called a local morphism if and only if for all points p : R0 → X the morphism on stalks p∗ (f ) is a local morphism in the sense of Definiton 4.2.6. We denote by Maploc (F, G) the simplicial subset of Map(F, G) spanned the vertices f ∈ Map(F, G)0 which represent a local morphism f : F → G. Definition 4.5.7 The category LRS of locally smooth-ringed spaces is the simplicial model category with smooth-ringed spaces X = (X, OX ) ∈ RS whose structure sheaf is a local sheaf as objects and local morphisms as arrows. The simplicial mapping space of local smooth-ringed spaces X = (X, OX ) and Y = (Y, OY ) is given by MapLRS (X , Y) :=

a

Maploc (g ∗ OY , OX ).

g∈HomTop (X,Y )

Example 4.5.8 Let M be a smooth manifold. Define a C∞ -ring-valued sheaf HM on M via HM : Op(M )◦ → C∞ Ring , U 7→ HM (U ) := C ∞ (U ). Note that HM is simplicially constant. Thus, using Corollary 2.8.9 the presheaf HM ∈ ∞ Shv(Op(M ), SR) is indeed a homotopy sheaf. Because of U (p∗ HM ) = U (Cp∞ (M )) = CM,p the homotopy sheaf HM is local. In the very same way one verifies that every smooth map g : M → N induces a vertex in Maploc (g ∗ Hn , HM ). A detailed proof of that can be found in [Spi 07, Prop. 3.2.3].

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Chapter 4: Derived Manifolds as Functors

The glueing of locally smooth-ringed spaces along equivalent subobjects as in Proposition 4.5.3 can also performed in the category LRS of locally smooth-ringed spaces (cf. [Spi 10, Lemma 6.8] and [Spi 10, Prop. 6.9]). Derived manifolds in the sense of [Spi 07] form a full subcategory of the category LRS of locally smooth-ringed spaces which is spanned by the locally smooth-ringed spaces that can be covered by certain vanishing sets which are affine derived manifolds. We quote the definition from [Spi 07]. Definition 4.5.9 An affine derived manifold X is a locally smooth-ringed space X = (X, OX ) ∈ LRS, where OX ∈ Shv(X, SR) is fibrant, and which can be obtained as a homotopy limit of a smooth function f : Rn → Rd in a diagram of the form (X, OX ) −−−→ (R0 , HR0 )     y y0 f

(Rn , HRn ) −−−→ (Rd , HRd ). A derived manifold X is a locally smooth-ringed space X = (X, OX ) ∈ LRS, where OX ∈ Shv(X, SR) is fibrant, and for which there exists an open covering (Ui → X | i ∈ I) such that (Ui , OUi ) is an affine derived manifold and OX |Ui = OUi for every i ∈ I. We write dManS for the category of derived manifolds in the sense of D. Spivak. If (V, OV ) and (X, OX ) are derived manifolds, we refer to (V, OV ) as a derived submanifold of (X, OX ) if there exists an open subset U ⊂◦ X and an equivalence (V, OV ) ' (U, OX |U ). Affine derived manifolds correspond in the homotopical setting precisely to finitely presented affine C∞ -schemes in the C∞ -context (cf. Corollary 3.4.12) and are therefore automatically Hausdorff, paracompact and local compact. Example 4.5.10 For every euclidian space Rd the C∞ -scheme (Rd , C ∞ (Rn )) is an affine derived manifold. Since every smooth manifold M ∈ Man is local euclidian, its associated C∞ -ring (M, HM ) is a derived manifold by Example 4.5.8. Following D. Spivak the inclusion i : Man → LRS is a fully faithful functor of simplicial categories (considering Man to be a trivial simplicial category, i.e. a simplical category whose mapping spaces MapMan (M, N ) are the constant simplicial sets of smooth maps M → N , cf. [Spi 07, Ch. 3, § 3.2, Prop. 3.2.3]). Note that the embedding factors through the category C∞ Sch of C∞ -schemes via the constant simplicial set functor c∗ .

4.6 The Functor approach to LRS

4.6

139

The Functor approach to LRS

In this section we define the affine spectrum functor Sp that associates to every smooth ring a locally smooth-ringed space. More generally, we construct a functor Φ : dMan ⊂ Shv(S∞ ) → LSR sending derived manifolds to locally smooth-ringed spaces and we prove that Φ is a faithful functor of ∞-categories. Using Sp we are capable of showing that the restriction of Φ to the category dManF of derived manifolds in Shv(S∞ ) that possess a finite cover by affine derived manifolds factors through the subcategory dManSF of derived manifolds in the sense of D. Spivak that possess a finite cover by affines and exhibits an equivalence Φ : dManF → dManSF of ∞-categories. We begin with an example that shows how the topological realization and the structure sheaf from Example 4.4.16 fit into the picture of smooth-ringed spaces. Example 4.6.1 Let f : X → Y be a morphism of derived manifolds. Following Example 4.4.16 we get two smooth-ringed spaces (|X |, OXpetit ) and (|Y|, OYpetit ). Moreover, let V → X ∈ Sub∞ (X ) and U → Y ∈ Sub∞ (Y) be two open subfunctors such that V → f −1 U is open. The restriction f |U of f on U gives a morphism O(U ) → O(V ). Using O(V ) = OYpetit (|V |), the universal property of colimits, and |V | ⊂ |f |−1 (|U |) we get exactly one map

(|V |→|f |

colim − −−−→ −1

OYpetit (|W |) → OXpetit (|V |).

(|W |))∈(|V |↓|f |−1 )

Hence we end up with a morphism   |f |# ∈ HomShv(|X |,SR) |f |∗ OYpetit , OXpetit . Taken all together we have a morphism (|f |, f # ) : (|X |, OXpetit ) → (|Y|, OYpetit ) of smooth-ringed spaces. Lemma 4.6.2 Let X be an affine derived manifold, say X = sSpec A for A ∈ sC∞ . The structure sheaf OXpetit is given on principal opens for a ∈ π0 OXpetit (|X |)(R) = U (A) via OXpetit (D(a)) = A[a−1 ]h . Proof: Using Corollary 4.3.13, the definition of the global structure sheaf O as well as the simplicially enriched Yoneda lemma one finds OXpetit (D(a)) = O(dSpec A[a−1 ]h ) = A[a−1 ]h .

140

Chapter 4: Derived Manifolds as Functors q.e.d.

Since derived manifolds X in the sense of D. Spivak possess an affine cover (Ui )i∈I such that OX |Ui = OUi , we can conclude that OX on principal opens is always of the form as described in Lemma 4.6.2. Proposition 4.6.3 Let X be a derived manifold. The structure sheaf OXpetit ∈ Shv(|X |, SR) is a local homotopy sheaf of smooth rings. Proof: Since locality is a local property we can choose an affine neighbourhood to check locality. Let x ∈ X ⊂◦ |X | be a point given by p : {∗} → X where X ⊂◦ |X | is given by an affine derived manifold. We are to show that p∗ OXpetit ∈ Pre(R0 , SR) ∼ = SR is local which is the case if and only if U (p∗ OXpetit ) is local. But we have   petit ∗ petit = U( − colim U p OX −−−→ OX (W )) p∈W ⊂◦ |X |

  petit U O (D(a)) = −colim X −−−→ p∈D(a) D(a)⊂X

−1 h =− colim −−−→U F [a ]




p∈D(a) D(a)⊂X

= (π0 F )p , which is local by Corollary 3.1.41. q.e.d.   Corollary 4.6.4 The morphism |f |# ∈ HomShv(|X |,SR) |f |∗ OYpetit , OXpetit in Example 4.6.1 is a local morphism. Proof:

Let p : R0 → |X | be a point of |X |. Then p∗ f [ : p∗ f ∗ OXpetit → p∗ OXpetit

is the morphism petit petit OY,f (x) → OX ,x .

Because of the previous Proposition both the left and right hand side are local smooth rings. Due to Corollary 3.1.41 and Proposition 3.1.28 the map petit petit π0 OY,f (x) → π0 OX ,x

4.6 The Functor approach to LRS

141

is a local morphism of C∞ -rings. Hence we are done. q.e.d. Now we can introduce the affine spectrum which associates to every smooth ring a locally smooth-ringed space. Definition 4.6.5 The functor petit Sp : sC∞ → LRS , A 7→ (|sSpec A|, OA )

is called the affine spectrum functor. Lemma 4.6.6 The locally smooth-ringed space (Rd , HRd ) from Example 4.5.8 is in the essential image of the affine spectrum functor. To be precise one has Sp(C ∞ (Rd )) ∼ = (Rd , HRd ). Proof: We have a homeomorphism |sSpec Rd | = Rd by construction. It remains to show that OCpetit ∞ (Rd ) is weakly equivalent to HRd . To do so we will prove that both presheaves coincide on all open subsets V ⊂◦ Rd . Recall that ∞ OCpetit (Rd )[fV−1 ]h ∞ (Rd ) (V ) = C

for some characteristic function fV for V ⊂ Rd by Lemma 4.6.2. Because of the left properness of the category of smooth rings and the fact that the obvious map C ∞ (R) → C ∞ (Rd+1 ) given by Example 3.1.20 is a cofibration, we have C ∞ (Rd )[fV−1 ]h = C ∞ (Rd )[fV−1 ] = C ∞ (V ). Since HRd (V ) = C ∞ (V ) by definition, we are done. q.e.d. Theorem 4.6.7 (D. Spivak’s Structure Theorem) Let X = (X, OX ) be a locally smooth-ringed space and Rd ∈ Euk. Then there is a homotopy equivalence MapLRS (X , Sp C ∞ (Rd )) ' MapSR (C ∞ (Rd ), OX (X)) of simplicial sets. Proof: First of all recall that Map((X, OX ), Sp C ∞ (Rd )) = Map((X, OX ), (Rd , HRd )). Since MapsC∞ (C ∞ (Rd ), OX (X)) ∼ = OX (X)(Rd ), the original proof in [Spi 10, Thm. 9.11] gives the result. q.e.d. Define two functors tn : dMan → (dMan ↓ X ) and χn : dMan → (dMan ↓ X ⊗ ∆[n]) by tn (Y) := (pr2 : Y × (X ⊗ ∆[n]) → X ⊗ ∆[n]) and χn (Y) := (Y ⊗ ∆[n]) × X → X ). The following lemma will be helpful to prove the next theorem.

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Chapter 4: Derived Manifolds as Functors

Lemma 4.6.8 The assignment (Y × (X ⊗ ∆[n]) → X ⊗ ∆[n]) 7→ ((Y ⊗ ∆[n]) × X → X ) gives a morphism (tn ↓ (Ω → X ⊗ ∆[n])) −→ (χn ↓ (Ω → X )) of categories functorial in Ω ∈ dMan. Proof: Since the simplicial structure is defined objectwise, one can check that there is an isomorphism (Y ⊗ ∆[n]) × X = Y × (X ⊗ ∆[n]) giving a well-defined assignment tn (Y) 7→ χn (Y) that is clearly functorial. q.e.d. Theorem 4.6.9 There is a fully faithful functor Φ : dMan → LRS of simplicial categories sending a derived manifold X to the locally smooth-ringed space (|X |, OXpetit ). Proof: Let f : X → Y be a morphism of derived manifolds. We are to show that Φ defines indeed a simplicial functor which is fully faithful in the sense of [Lur 09, Ch. I, § 1, Def 1.2.10.1], i.e. we are going to show that MapdMan (X , Y) → MapLRS (Φ(X ), Φ(Y)) is a weak equivalence of simplicial sets. Because of Theorem 4.3.11 there is a continuous map |f | : |X | → |Y| between the corresponding topological spaces. Recall from Example 4.6.1   petit petit that we get a morphism |f |# ∈ HomShv(|X |,SR) |f |∗ OY , OX . We are to show that |f |# is local. But this has already been done in Corollary 4.6.4. Hence we have   |f |# ∈ Maploc |f |∗ OYpetit , OXpetit , i.e. the image Φ(f ) of f ∈ HomdMan (X , Y) determines a vertex in   petit petit MapLRS (|X |, OX ), (|Y|, OY ) . Thus, Φ defines indeed a functor. We now show that Φ respects the simplicial structure, i.e. defines a simplicial functor. Due to [Hir 03, Part II, Ch. 9, § 9.8, Thm. 9.8.5] the existence of a morphism Φ(X ) ⊗ S → Φ(X ⊗ S ) for every finite simplicial set S respecting the simplicial structure will do the job. Since every simplicial set can be displayed as a colimit of representables, it suffices to prove the existence of a natural morphism Φ(X ) ⊗ ∆[n] → Φ(X ⊗ ∆[n]) for every [n] ∈ ∆. We show that this is even an equivalence of locally smooth-ringed spaces. By definition we

4.6 The Functor approach to LRS

143

petit have Φ(X ⊗ ∆[n]) = (|X ⊗ ∆[n]|, OXpetit ) ⊗ ∆[n] = ⊗∆[n] ) and Φ(X ) ⊗ ∆[n] = (|X |, OX petit ∆[n] (|X |, (OX ) ) and hence we have to specify an appropriate morphism     petit ∆[n] petit # (q, q ) : |X |, (OX ) → |X ⊗ ∆[n]|, OX ⊗∆[n] .

We claim that q : |X | → |X ⊗ ∆[n]| is a homeomorphism induced by the canonical map petit ∆[n] X ⊗∆[n] → X and q # : q ∗ OXpetit ) is induced by the application of the global ⊗∆[n] → (OX structure sheaf to the canonical morphism dSpecA⊗∆[n] → dSpecA in the category dAff ∞ of affine derived manifolds. We will cope with q first. We compute
|X ⊗ ∆[n]| = π0 X (C ∞ (R0 )) × ∆[n]   ∞ 0 ∼ = π0 X (C (R )) × ∆[n] gr   ∼ = π0 X (C ∞ (R0 )) gr × |∆[n]|gr   ∞ 0 ∼ π X (C (R )) × π0 (∆n ) = 0 gr   ∞ 0 ∼ = π0 X (C (R )) gr = |X |, where | · |gr denotes the geometric realization functor from Example 2.2.4 using the wellknown isomorphism between the simplicial path component functor for simplicial sets and the topological path component functor applied to the geometric realization of the simplicial set (cf. [Hov 99, Ch. 3, § 3.4, Lem. 3.4.3]) as well as the contractibility of |∆[n]|gr ⊂ Rn+1 . Thus, the image |p| of the canonical map p : X ⊗ ∆[n] → X under the topological realization functor induces a homeomorphism and we define q := |p|−1 : |X | → |X ⊗ ∆[n]|. Turning our attention towards q # we let U ⊂ |X | be an open subset such that U = |dSpec B| for a smooth ring B ∈ sC∞ , hence we also have q(U ) ∼ = |dSpec B|. Let n t : dMan → (dMan ↓ X ⊗ ∆[n]) the morphism of sites from Corollary 2.7.10. By the very definiton of the structure sheaf on |X ⊗ ∆[n]| we find q ∗ OXpetit ⊗∆[n] (U ) =

O(V ) = O(dSpec B) = B,

lim ← −−

(tn ↓(dSpec

B→X ⊗∆[n]))

since dSpec B 0 → X ⊗ ∆[n] is terminal in (tn ↓ (dSpec B 0 → X ⊗ ∆[n])). Using Lemma4.6.8 one gets a morphism q ∗ OXpetit ⊗∆[n] (U ) =

lim ← −−

O(V ) →

(tn ↓(dSpec B→X ⊗∆[n]))

lim ← −−

O(W ) = B ∆[n] ,

(χn ↓(dSpec B→X ))

where we have used the fact that (dSpec B⊗∆[n])×X → X is terminal in (χn ↓ dSpec B → X ) and Corollary 4.4.7. Taking into account that   (OXpetit )∆[n] (U ) = O∆[n] (V ) = O∆[n] (dSpec B) = B ∆[n] , lim ← −− (t0 ↓(dSpec B→X ))

144

Chapter 4: Derived Manifolds as Functors

we have found a natural morphism q ∗ OXpetit ⊗∆[n] (U ) →



OXpetit

∆[n]

(U ) which is a weak

equivalence, for B → B ∆[n] is a weak equivalence which can be seen as follows: B → B ∆[n] is a weak equivalence in SR if and only if B(R) → B(R)∆[n] is a weak equivalence of simplicial sets by Lemma 4.5.4. But ∆[n] → ∆[0] is a weak equivalence of simplicial sets and since B ∈ sC∞ is fibrant the functor MapsSet ( · , B(R)) : sSet → sSet preserves weak equivalences between cofibrant objects. Thus, the claim follows from the Ken Brown–Lemma and  ∆[n] petit # ∗ petit q : q OX ⊗∆[n] → OX is a weak equivalence. Hence (q, q # ) is an equivalence of locally ringed-spaces. We will now prove that Φ is faithful. It suffices to show that for any morphism f : X → Y of derived manifolds, the simplicial function space Maploc (|f |∗ OYpetit , OXpetit ) is contractible. To see that, we can retreat to the affine case since derived manifolds are covered by affines and homotopy sheaves are weakly equivalent if they are weakly equivalent on all principal opens. Therefore we may assume X and Y to be affine, i.e. X ' sSpec A and Y ' sSpec B. The morphism f : X → Y gives a vertex φ ∈ MapSC∞ (B, A) by the simplicially enriched Yoneda lemma. Because of the universal property of the homotopy colimit we get precisely one map B[b−1 ]h → A[φ(b)−1 ]h in Ho(SC∞ ). This means that MapSC∞ (OYpetit (D(b)), OXpetit (D(ϕ(b)))) is contractible. This is true for all principal opens V ⊂ |Y| and W ⊂ |X | with W ⊂ |f |−1 (V ) and thus MapSC∞ (|f |∗ OYpetit (W ), OXpetit (W )) is contractible. Now let U ⊂ |X | be an arbitrary open subset. Choose an open cover ˇ (Ui → U | i ∈ I) where the Ui are principal opens. Let U → rU be the associated Cech hypercover. By Lemma 2.8.16 there is a weak equivalence     petit ∗ petit |f | O (U ), O (U ) . holim Map MapSR |f |∗ OYpetit (U ), OXpetit (U ) ' ←   SR Y X −−−− Since the homotopy limit of contractible spaces is contractible again, the left hand side is contractible, i.e. we end up with   petit = {∗} π0 Maploc |f |∗ OYpetit , OX by [Spi 07, Ch.3, § 3.3, Prop. 3.3.1] and [Lur 09, A.3.1.8]. It remains to show that Φ is full. Assume (g, g # ) : (|X |, OXpetit ) → (|Y|, OYpetit ) is a morphism given in dMan. Because of the contractibility of Maploc (g ∗ OYpetit , OXpetit ), it suffices to show that there is a morphism f : X → Y inducing g : |X | → |Y|. Since Op(|X |) ∼ = Sub∞ (X ), we can choose a cover (Ui → X | i ∈ I) by affine open subfunctors whose topological realization is an epimorphic family on |X |. The same reasoning holds for Y and thus we may assume without loss of generality X and Y to be affine, i.e. petit petit X ∼ = sSpec B. The morphism g # : g ∗ OY → OX gives g ∗ : B → A = sSpec A and Y ∼

4.6 The Functor approach to LRS

145

on global sections which in turn gives f : sSpec A → sSpec B. Consequently, we have two maps g, |f | = |sSpec (g ∗ )| : |sSpec A| → |sSpec B| and we are to show that they are homotopic. But

|sSpec A| = |dSpec A| = HomC∞ Ring π0 A, C ∞ (R0 ) = HomsC∞ A, c∗ C ∞ (R0 ) holds and we can conclude from the Yoneda lemma that g is induced by f and so Φ is full. q.e.d. Remark. Because of the previous theorem we may identify the category dMan of derived manifolds with the full subcategory of the category LRS locally smooth-ringed spaces LRS which are locally weakly equivalent to locally smooth-ringed spaces of type petit (|sSpec A|, OA ). Corollary 4.6.10 There is an isomorphism Φ ◦ sSpec ∼ = Sp of functors. In particular, the functor Sp is fully faithful. Proof:

This follows by definition. q.e.d.

Theorem 4.6.11 The functor Ψ : LRS → Shv(S∞ ) given by (Y, OY ) 7→ (A 7→ MapLRS (Sp(A), (Y, OY ))) possesses a left adjoint L : Shv(S∞ ) → LRS. The adjunction L a Ψ yields a morphism (L, Ψ) : LRS → Shv(dAff ∞ ) of simplicial model categories. Proof:

Define

∞

L : sSetsC → LRS as in Theorem 2.3.21 via X 7→ − colim −−−→ [(Φ ⊗ ∆) ◦ πX ((x, [n], A))] . R

sC∞

X

146

Chapter 4: Derived Manifolds as Functors

By Theorem 2.3.21 L is well-defined and the adjunction L a Ψ is simplicially enriched. It remains to show that the adjunction is Quillen. Let (X, OX ) → (Y, OY ) be an (acyclic) fibration of locally smooth-ringed spaces. Since for every smooth ring A ∈ sC∞ the induced morphism petit petit iddSpec A : (|dSpec A|, OA ) → (|dSpec A|, OA ) is a cofibration, the map MapLRS (Sp(A), (X, OX )) → MapLRS (Sp(A), (Y, OY )) is a(n) (acyclic) fibration in sSet. Hence it follows that Ψ((X, OX )) → Ψ((Y, OY )) is a(n) (acyclic) fibration in Shv(S∞ ). q.e.d. Corollary 4.6.12 Assume X is an affine derived manifold, i.e. X ∼ = sSpec A for A ∈ sC∞ . Then there is a canonical isomorphism '

Sp(A) −−−→ L(X ) of locally smooth-ringed spaces. Proof:

An object Z (x, [n], R) ∈

X sC∞

is just a morphism x : A ⊗ ∆[n] → R of smooth rings. Therefore the n-simplex idA,[n] of the identity, i.e. the morphism id ⊗t∗

A ⊗ ∆[n] −−A−−→ A ⊗ ∆[0], with t : [n] → [0] the terminal map in ∆, is the initial object at the n-th simplicial level and hence the structure morphism ιA (idA,[n] ) : Sp(A)n → L(X )n is an isomorphism. Now the claim follows from the fact that colimits of locally smoothringed spaces are computed object-wise. q.e.d. The following technical lemma and its proof is quoted from [Spi 10, Lem. 9.12]. It allows us to take homotopy limits component-wise in the model category of simplicial sets. Lemma 4.6.13 Let sSet be the category of simplicial sets. Let I, J and K denote sets and let a a a A= Ai , B = Bj , C = Ck i∈I

j∈J

k∈K

4.6 The Functor approach to LRS

147

be coproducts in sSet indexed by I, J and K. Suppose that f : I → J and g : K → J are functions and that F : A → B and G : C → B are morphisms of simplicial sets that respect f and g in the sense that F (Ai ) ⊂ Bf (i) and G(Ck ) ⊂ Bg(k) for all i ∈ I and k ∈ K. Let I ×J K = {(i, j, k) ∈ I × J × K | f (i) = j = g(k)} denote the fibered product of sets. For typographical reasons, we use ×h to denote the homotopy limit in sSet, and × to denote the 1-categorial limit. Then the natural map   a  Ai ×hBj Ck  → A ×hB C (i,j,k)∈I×J K

is a weak equivalence in sSet. Proof: If we replace F by a fibration, then each component Fi := F |Ai : Ai → Bf (i) is a fibration. We reduce to showing that the map   a  Ai ×Bj Ck  → A ×B C (i,j,k)∈I×J K

is an isomorphism of simplicial sets. Restricting to the n-simplices of both sides, we may assume that A, B, and C are (discrete simplicial) sets. Now one can readily verify that the map above is injective and surjective. q.e.d. We will also need the following technical Lemma 4.6.14 Let U and X be topological spaces. Assume g : U → X is an open injection. Then for every sheaf F ∈ Shv(X, SR) of smooth rings the morphism g! g ∗ F → F induced by the counit of the adjunction g! a g ∗ is a cofibration in Shv(X, SR). Proof:

Let V ⊂◦ X be an open subset. Then one has (g! g ∗ F) (V ) =

colim − −−−→

g ∗ F.

(V →g(W ))∈(V ↓g)

That is, (g! g ∗ F) (V ) is either the initial object in SR which is precisely the case if (V ↓ g) is empty, i.e. if V * U , and thus a cofibration since cofibrations are determinded object-wise, or it is equal to g ∗ F(V ) = F(V ) which is the case if V ⊂ U , because then V → g(V ) is initial in (V ↓ g). That is, in both cases we end up with an object-wise cofibration and we are done. q.e.d.

148

Chapter 4: Derived Manifolds as Functors

Proposition 4.6.15 The affine spectrum functor Sp preserves finite homotopy colimits in the case of their existence. To be precise, one has a weak equivalence Sp(−hocol holim −−−→F ) ' ← −−−−Sp F of locally ringed spaces. Proof: Let I be a finite category and F ∈ sC∞ I be a diagram such that the homotopy colimit is defined at F . We use the fact that L is defined via a filtered colimit and that filtered colimits of fibrant objects are fibrant again. Moreover, we take Proposition 2.8.5 into account and get the following chain of weak equivalences Sp(−hocol hocol −−−→F ) = L ◦ sSpec (− −−−→F ) = L(Map( · , dSpec (− hocol −−−→F ))) = L(← holim −−−−(sSpec F )) = L(← lim −−R(sSpec F )) =← lim −−L(R(sSpec F )) =← holim −−−−(L ◦ sSpec (F )) =← holim −−−−SpF where R stands for any fibrant replacement. q.e.d. Remark. As an immediate consequence of Proposition 4.6.15 one has the following slight generalization of the famous structure theorem Theorem 4.6.7: MapLRS (X , Sp A) ' MapSR (A, OX (X)) for any locally smooth-ringed space X = (X, OX ) and A ∈ sC∞ . Proposition 4.6.16 The affine spectrum functor Sp : sC∞ → LRS factors through the category of affine derived manifolds in the sense of D. Spivak. More precisely, let X = (X, OX ) be an affine derived manifold in the sense of Spivak, i.e. X fits into a homotopy cartesian diagram X −−−→ (R0 , HR0 )     y y0 f

(Rn , HRn ) −−−→ (Rd , HRd ). Then there is a smooth ring F ∈ sC∞ such that we have an equivalence Sp(F ) ' (X, OX ), i.e. there is a homeomorphism |dSpec F | = X and a weak equivalence OX ' OFpetit of the corresponding structure sheaves.

4.6 The Functor approach to LRS Proof:

149

From Proposition 4.6.15 one readily deduces  
∞ 0 ∞ n h Sp C (R ) ⊗C ∞ (Rd ) C (R ) ' Sp (C ∞ (Rn )) ×hSp(C ∞ (Rd )) Sp C ∞ (R0 ) . q.e.d.

Proposition 4.6.17 Let X = (X, OX ) ∈ LRS be a derived manifold in the sense of D. Spivak. Assume X admits a finite cover (Ui )i∈I of affine derived manifolds. Then X is equivalent to the homotopy colimit of its affine subobjects. Proof: S Let (Ui , OUi )i∈I be a finite set of affine derived manifolds that cover X , i.e. we have Ui = X and OUi = OX |Ui . We are to show that the natural map MapLRS (X , Y) → ← holim −−−− MapLRS ((Ui , OUi ), Y) is a weak equivalence of simplicial sets for every locally ringed space Y = (Y, OY ). Since the (Ui )i∈I cover X, we have an isomorphism X = −colim −−−→Ui of ∗topological spaces. Let gi : Ui → X be the natural open injection. Note that OX |Ui = gi OX . Consequently, we are to show that a a Maploc (f ∗ OY , OX ) → ← holim Maploc (fi∗ OY , OUi ) −−−− f ∈Hom(X,Y )

fi ∈Hom(Ui ,Y )

is a weak equivalence. We apply Lemma 4.6.14 to reduce to the case of a single index. Since (gi )! gi∗ f ∗ OY → f ∗ OY is a cofibration and all structure sheaves are assumed to be fibrant, the natural morphism Maploc (f ∗ OY , OX ) → Maploc ((gi )! gi∗ f ∗ OY , OX ) is a weak equivalence for every map f : X → Y . Using the homotopy invariance of the homotopy limit, the fact that the local mapping space Maploc (f ∗ OY , OX ) is a fibrant simpli∗ ∗ cial set and hence the universal map ← lim holim −−Maploc (f OY , OX ) → ← −−−−Maploc (f OY , OX ) is a weak equivalence, we end up with the following: For any locally smooth-ringed space Y = (Y, OY ) we have a holim Map ((U , O ), Y) = holim Maploc (fi∗ OY , OUi ) i U LRS i ←−−−− ←−−−− fi ∈Hom(Ui ,Y ) a ∗ ∗ ∗ ' holim ← −−−−Maploc (gi f OY , gi OX ) f ∈ lim Hom(Ui ,Y ) ←−−

=

a

∗ ∗ holim ← −−−−Maploc ((gi )! gi f OY , OX )

f ∈Hom(X,Y )

'

a

Maploc (f ∗ OY , OX )

f ∈Hom(X,Y )

= MapLRS (X , Y).

150

Chapter 4: Derived Manifolds as Functors

Hence the proof is complete. q.e.d. We need one last technical lemma whose proof is heavily inspired by [Spi 10, Prop. 6.9]. Lemma 4.6.18 The functor Φ preserves finite homotopy colimits of affine derived submanifolds. Proof: Let (X, OX ) ∈ dManS be a derived manifold and U1 and U2 be open subsets of some affine cover (Ui )i∈I of X such that U1 ∩ U2 = U12 is not empty. Define (Y, OY ) := − hocol −−−→ ((U1 , OU1 ) ← (U12 , OU12 ) → (U2 , OU2 )) , i.e. Y = U1 qU12 U2 is the colimit of topological spaces together with the canonical open injections i : U2 → Y , k : U12 → Y as well as j : U1 → Y and OY = ← holim −−−−(i∗ OU2 → k∗ OU12 ← j∗ OU1 ) according to Proposition 4.5.3. Then one can check that there are weak equivalences j ∗ OY ' OU1 , i∗ OY ' OU2 and k ∗ OY ' OU12 . By Example 4.2.18 and Proposition 4.6.16 we can associate to (U1 , OU1 ) a homotopically finitely presented smooth ring A1 ∈ sC∞ such that we have an equivalence SpA1 ' (U1 , OU1 ). Similarly we gain A2 and A12 in sC∞ . We define an object in Shv(S∞ ) by X := −hocol −−−→ (sSpec A1 ← sSpec A12 → sSpec A2 ) . We are to show that this yields Φ(X ) ' − hocol −−−→Φ(sSpec A1 ← sSpec A12 → sSpec A2 ) an equivalence of locally ringed spaces. For the topological realization one immediately finds |X | = − colim −−−→(|sSpec A1 | ← |sSpec A12 | → |sSpec A2 |) = U1 qU12 U2 the desired homeomorphism. The hard part is to prove the weak equivalence OXpetit ' OY of homotopy sheaves. Since j : U1 → U1 qU12 U2 is an open injection, we find for all principal opens j ∗ OXpetit (D(a)) = OXpetit (j(D(a))) = O(sSpec A1 [a−1 ]h ) = A1 [a−1 ]h , petit coincide on principal opens up to weak equiwhere a ∈ U (A1 ). Thus, j ∗ OXpetit and OA 1 valence. Hence it follows petit j ∗ OX ' OA ' OU1 . 1

4.6 The Functor approach to LRS

151

Symmetric reasoning implies that i∗ OX ' OU2 and k ∗ OXpetit ' OU12 are also weak equivalences. Let (Z, OZ ) be any locally smooth-ringed space and f : U1 qU12 U2 → Z be a continuous map. Then we have Maploc (f ∗ OZ , OY ) ' Maploc (f ∗ OZ , i∗ OU1 ×hk∗ OU j∗ OU2 ) 12

∗

' Maploc (f OZ , i∗ OU1 )

×hMaploc (f ∗ OZ ,k∗ OU ) 12

' Maploc (i∗ f ∗ OZ , OU1 ) ×hMaploc (k∗ f ∗ OZ ,OU ' Maploc (i∗ f ∗ OZ , i∗ OXpetit ) ×hMap

loc (k

' Maploc (i! i∗ f ∗ OZ , OXpetit ) ×hMap

∗f ∗O

loc (k! k

' Maploc (f ∗ OZ , OXpetit ) ×hMap

loc (f

∗O

12

)

Z ,k

∗f ∗O

Maploc (f ∗ OZ , j∗ OU2 ) Maploc (j ∗ f ∗ OZ , OU2 )

∗ O petit ) X

petit ) Z ,OX

petit ) Z ,OX

Maploc (j ∗ f ∗ OZ , j ∗ OXpetit )

Maploc (j! j ∗ f ∗ OZ , OXpetit )

Maploc (f ∗ OZ , OXpetit )

' Maploc (f ∗ OZ , OXpetit ). We will explain the computation briefly. By definition of OY , the first weak equivalence follows. The second follows from the fact that the property of a map being local is itself a local property. The third weak equivalence follows from the fact that the simplicial enriched adjunction carries over to local morphisms. The weak equivalence before the last one follows from Lemma 4.6.14 and the homotopy invariance of the homotopy limit. From all this follows MapLRS ((Y, OY ), (Z, OZ )) ' MapLRS ((|X |, OXpetit ), (Z, OZ )) by Lemma 4.6.13, where Φ(X ) = (|X |, OXpetit ). Thus, the claim is proved at last. q.e.d. Proposition 4.6.19 The restriction of the functor Φ : dMan → LRS to derived manifolds that possess finite covers by affine derived manifolds maps essentially surjectively to dManSF . Proof: Let X ∈ dManS be a derived manifold that admits a finite cover. Then by Proposition 4.6.17 we have X ' − hocol −−−→(Ui , Oi ). Using∞ Example 4.2.18 we can construct a homotopically finitely presented smooth ring Ai ∈ sC such that |sSpec Ai | = Ui for every i ∈ I. By Proposition 4.6.16 we then have an equivalence Sp(Ai ) ' (Ui , OUi ) of locally smooth-ringed spaces. Taking into account that L is left adjoint to Ψ by Proposition 4.6.11 and that Φ preserves finite homotopy colimits we can conclude MapLRS (X , Y) = ← holim i , OUi ), Y) −−−− MapLRS ((U   petit '← holim Map (|sSpec A |, O ), Y i LRS Ai −−−− '← holim Shv(S∞ ) (sSpec Ai , Ψ(Y)) −−−− Map   ' MapLRS Φ(−hocol sSpec A ), Y i −−−→

152

Chapter 4: Derived Manifolds as Functors

for all locally smooth-ringed spaces Y = (Y, OY ) ∈ LRS. Here we have used the isomorphisms Φ ◦ sSpec = Sp and L ◦ sSpec = Sp of functors in the last step. q.e.d. The results achieved until now culminate in the following Theorem 4.6.20 The functor Φ : dManF → dManSF is a fully faithful and essentially surjective functor, i.e. an equivalence of ∞-categories in the sense of [Lur 09, Ch. 1, § 1.2, 1.2.10]. Recall that dManF denotes the category of derived manifolds coming from smoothsimplicial functors and dManSF the category of derived manifolds coming from locally smooth-ringed spaces, both having a finite cover by affines.

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Curriculum Vitae

Name

Franz Vogler

Geburtsdaten

18. August 1982 in Weimar

Schule

Max Planck–Gymnasium Ludwigshafen 1993–2002

Studium

Johannes Gutenberg–Universit¨at Mainz 2002–2008

Promotion

Universit¨at Augsburg 2009–2013

Kontakt

Lehrstuhl f¨ ur Algebra und Zahlentheorie Universit¨atsstraße 14 D–86159 Augsburg

Augsburg, den 2. Januar 2013

In this thesis a functorial approach to the category of derived manifolds is developed. We use a similar approach as Demazure and Gabriel did when they described the category of schemes as a full subcategory of the category of sheaves on the big Zariski site. Their work is further developed leading to the definition of C#-schemes and derived manifolds as certain sheaves on appropriate big sites. The new description of C#-schemes and derived manifolds via functors is compared to the previous approaches via locally ringed spaces given by D. Joyce and D. Spivak. Furthermore, it is proven that both approaches lead to equivalent categories.

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ISBN 978-3-8325-3405-9 ISSN 1611-4256