The Cohomology of Monoids (RSME Springer Series, 12) 3031502574, 9783031502576

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RSME Springer Series  12

Antonio M. Cegarra Jonathan Leech

The Cohomology of Monoids

RSME Springer Series Volume 12

Editor-in-Chief Maria A. Hernández Cifre, Departamento de Matemáticas, Universidad de Murcia, Murcia, Spain Series Editors Nicolas Andruskiewitsch, FaMAF - CIEM (CONICET), Universidad Nacional de Córdoba, Córdoba, Argentina Francisco Marcellán, Departamento de Matemáticas, Universidad Carlos III de Madrid, Leganés, Madrid, Spain Pablo Mira, Departamento de Matemática Aplicada y Estadística, Universidad Politécnica de Cartagena, Cartagena, Spain Timothy G. Myers, Centre de Recerca Matemàtica, Barcelona, Spain Joaquín Pérez, Departamento de Geometría y Topología, Universidad de Granada, Granada, Spain Marta Sanz-Solé, Department of Mathematics and Computer Science, Barcelona Graduate School of Mathematics (BGSMath), Universitat de Barcelona, Barcelona, Spain Karl Schwede, Department of Mathematics, University of Utah, Salt Lake City, UT, USA

As of 2015, RSME - Real Sociedad Matemática Española - and Springer cooperate in order to publish works by authors and volume editors under the auspices of a co-branded series of publications including advanced textbooks, Lecture Notes, collections of surveys resulting from international workshops and Summer Schools, SpringerBriefs, monographs as well as contributed volumes and conference proceedings. The works in the series are written in English only, aiming to offer high level research results in the fields of pure and applied mathematics to a global readership of students, researchers, professionals, and policymakers.

Antonio M. Cegarra • Jonathan Leech

The Cohomology of Monoids

Antonio M. Cegarra Department of Álgebra Facultad de Ciencias, Universidad de Granada Granada, Spain

Jonathan Leech Department of Mathematics and Computer Science Westmont College Santa Barbara, CA, USA

ISSN 2509-8888 ISSN 2509-8896 (electronic) RSME Springer Series ISBN 978-3-031-50257-6 ISBN 978-3-031-50258-3 (eBook) https://doi.org/10.1007/978-3-031-50258-3 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Paper in this product is recyclable.

Preface

The cohomology of monoids, that is, of semigroups with an identity element, has been of interest for some time. Actually there are multiple cohomology theories. Given a monoid S, there is a theory for left S-modules, or equivalently, left modules of the semigroup ring .ZS. There is also a dual theory for right S-modules and a theory for S-bimodules, where S operates on both sides of the module so that .(xa)y = x(ay) for all .x, y in S and a in the relevant module. The outcome xay is thus unambiguous. Studies in this area have also focused on the cohomology of certain structures derived from monoids, that encodes fundamental information about the monoid, but are mathematical object in its own right. These structures are typically small categories related to some aspect of division in the monoid. (This “small category” aspect also holds in the case of S-modules since after all, monoids are simply categories with exactly one object.) One of the first such cases occurred in the 1969 UCLA dissertation of the American coauthor of this monograph. The context was that of a subgroup N of the group of units in a monoid S, that was normal in the entire monoid. That is .xN = N x held for all .x ∈ S. The various cosets formed under pointwise multiplication a homomorphic image .S/N of S. Thus, one had an extension situation, .N ͨ→ S ↠ S/N, that in some ways was similar to the typical normal subgroup extension case for groups, but with one major difference that leads to other differences: in general, here the cosets need not have the same size so that .N × S/N could not in all cases form the underlying set of a reconstruction of S. Long story short, for any monoid S, the relevant small category in this case is the division category .D(S) whose set of objects is the set of S, whose morphisms are triples .〈u, x, v〉 : x → uxv, and whose composition is given by .〈u' , uxv, v ' 〉 〈u, x, v〉 = 〈u' u, x, vv ' 〉. Clearly, the corresponding relevant cohomology does not involve left S-modules, but rather the category .D(S)−Mod of .D(S)-modules whose objects are functors from .D(S) to .Ab,

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the category of abelian groups, and whose morphisms are natural transformations between them. Three early papers were: –P.A. Grillet, Left coset extensions, Semigroup Forum 7 (1974), 200–263. –J. Leech, .H-coextensions of monoids, Memoirs AMS 157 (1975), 1–66. –Ch. Wells, Extension theories for monoids, Semigroup Forum 16 (1978), 13–35. The first two dealt with more general situations than that encountered in Leech’s 1969 dissertation, but the cohomology introduced there was still relevant. The paper by Charles Wells studied this cohomology from the perspective of a general approach to cohomology theories due to Jonathan Mock Beck. The outcome: within the context considered by Leech, both Leech and the Beck cohomology agreed to with a simple dimension shift. In saying this we are not asserting that Shreier-type constructions of monoids are of no interest! Rather, they are special cases in a broader class of constructions. Likewise, Eilenberg-Mac Lane cohomology and indeed Hochschild-Mitchell cohomology applied to monoids can be viewed as special cases of .D-cohomology (known also as Leech cohomology). All we are saying is that there is a bigger picture. In the final section of Chap. 4, where we use groups as illustrative examples, this broader perspective will lead us back to Otto Schreier and also back to Eilenberg and Mac Lane. Around this time, input began coming from Australia and India. But in this case the cohomologies were based on small categories that were derived from semigroups. –H. Lausch, Cohomology of inverse semigroups, J. Algebra 35 (1975), 273–303. –M. Loganathan, Cohomology of inverse semigroups, J. Algebra 70 (1981), 375– 393. –M. Loganathan, Idempotent-separating extensions of regular semigroups with abelian kernel, J. Aust. Math. Soc. Ser. A 32 (1982), 104–113. –M. Loganathan, Cohomology and extensions of regular semigroups with abelian kernel, J. Aust. Math. Soc. Ser. A 35 (1982), 178–193. Moving to into the late 1980s and towards the new millennium, there were two new developments. First, Pierre Antoine Grillet, who for some time had studied commutative semigroups, began a serious study of their cohomology that eventually drew in other researchers. Unlike the cases above, cohomology for just commutative semigroups or monoids does not always blend in with the general case. Thus, the cohomology of monoids and semigroups split into two separate areas: the strictly commutative area and the not-necessarily-commutative area. Much of Grillet’s research, and that of others who joined in, is collected in his monograph –P.A. Grillet, The Cohomology of Commutative Semigroups. Lecture Notes in Mathematics Vol. 2037 (2022), Springer. The second one was the emergence of a research group at the University of Granada with interest in both cohomologies. It has been led the Spanish coauthor of this monograph. Indeed, in Grillet’s monograph, two of the seven nonintroductory

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chapters are based on published work by Antonio M. Cegarra and his colleagues. Their contributions include: –E. Aznar and A. Sevilla, Beck, .H and Leech Coextensions, Semigroup Forum 60 (2000), 385–404. –M. Calvo, A.M. Cegarra and B.A. Heredia, Structure and classification of monoidal groupoids. Semigroup Forum 87 (2013), 35–79. –M. Calvo, Cohomologies of monoids and the classification of monoidal groupoids, Thesis (Ph.D.) Universidad de Granada, 2016. –M. Calvo and A.M. Cegarra, (Co)homology of cyclic monoids. International J. Algebra and Computation 26 (2016), 887–910. –M. Calvo and A.M. Cegarra, Higher cohomologies of commutative monoids. J. Pure and Applied Algebra 223 (2019), 131–174. Clearly interest in the cohomology of monoids, commutative or otherwise, has not been lacking. But before surveying the content of this monograph a few remarks about the term coextension may be in order. Heuristically viewed, given objects A and B of the same type, B is an extension of A if A may be embedded in B in a manner compatible with their structure. That is, there is a one-to-one morphism .A ͨ→ B. Dually, B is a coextension of A (or a dual extension of A) if there is a surjection .B ↠ A that is compatible with their structure. It is possible in some cases for objects A and B to have both occur. (Caveat: some authors use “extension” to refer to a coextension. Be careful!) A relevant example: Let S be a monoid and let .A : D(S) → Ab a .D(S)-module. Set .S ⋊ A = {(x, a) | x ∈ S, a ∈ A(x)}. We endow .S ⋊ A with a monoid product .(x, a)(y, b) = (xy, A〈1, x, y〉a + A〈x, y, 1〉b). Of course, .(1, 01 ) is the identity. Let .S ͨ→ S ⋊ A be defined by .x I→ (x.0x ) and define .S ⋊ A ↠ S by sends .(x, a) I→ x. Clearly .S ⋊ A is both an extension of S and a coextension of S. In a different context, in a normal extension of a group by a monoid .N ͨ→ S ↠ S/N, S is both an extension of N and a coextension of .S/N. An .H-coextension of a monoid S is a coextension .p : E ↠ S in which the p-induced congruence on E is contained in the fundamental Green’s relation .H on E. Then, .H is connected to both the explicit and implicit global subgroup structure of a monoid and E can be reconstructed from S, a group-valued functor on .D(S), and a factor set. Cohomology arises when the groups in this construction are all abelian. .H-coextensions, with the resulting cohomology, was the topic of Leech’s 1975 publication, the above works by Lausch and Loganathan, and also appear in Grillet’s 1974 paper. Returning to the case of a now abelian normal subgroup A extended by a monoid, .A ͨ→ S ↠ S/A, we have an .H-coextension S of .S/A. With the A-cosets in S possibly shrinking in size in S as one moves away from A, the usual monoid actions of the form .S × A → A, or .S/A × A → A, will not be part of a general description of things. One needed to work with some scheme where varying abelian groups are acted upon. What works here are functors .A from .D(S), or even from .D(S/A), to the category of abelian groups with .A(1) = A. And so it goes in general. We now turn to the content of this monograph.

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Chapter 1 surveys basic information on the cohomology of small categories. The attempt is to present the pertinent facts in a coherent, logical fashion. Some can skip this and go directly to Chap. 2. Chapter 2 gives a formal introduction to the category .D(S) of a monoid S, and the category .D(S)−Mod of .D(S)-modules consisting of functors from .D(S) to .Ab, the category of abelian groups, (the objects) and natural transformations between them (the morphisms), and finally the cohomology theory for S, with coefficients in .D(S)-modules, that is, the .D-cohomology of S. Special attention is given to the role of .H 0 (S, A), .H 1 (S, A) and .H 2 (S, A) in certain situations, and especially to those involving twisted semidirect product coextensions .S ⋊α A of S, where .A is a .D(S)-module used to form the monoid .S ⋊ A given above that is the platform for a coextension of S and .α is a 2-cocycle (factor set) used to twist the natural multiplication of .S ⋊A. Following a look at cohomological dimension, the chapter concludes by calculating the cohomology groups for a cyclic monoid, i.e., monoids that can be generated from a single generator. Chapter 3 studies the relationship of other cohomologies to the .D-cohomology. Both the Eilenberg-Mac Lane cohomology of left S-modules and the HochschildMitchell cohomology of S-bimodules are quickly seen to be essentially subcohomologies of .D-cohomology. Thus .dimEM (S) ≤ dimHM (S) ≤ Dim(S). But if S is cancellative, then .Dim(S) = dimHM (S). And if S is a free commutative monoid on n generators, then the dimension of both is n. In the special case when S is a group, the .D-category is a groupoid and the .D-cohomology quickly reduces to the Eilenberg-Mac Lane cohomology. We then introduce the category .D(S). It is a canonical homomorphic image of .D(S) that is important in the study of .H-coextensions. Indeed, the .D(S)-functors of interest here there are .D(S)-functors that factor through .D(S). In general .D(S) and .D(S) do not induce equivalent cohomology theories. But .D(S)-cohomology does become a sub-cohomology of .D(S)-cohomology for the elegant class of inverse monoids. The precise details are given in this chapter. As a pleasant consequence, for inverse monoids, the .D-cohomology (via .D(S) functors) is equivalent to that given by M. Loganathan and H. Lausch. This only adds to the “Beck Certification” provided by the late Charles Wells. The chapter continues with three sections devoted to cohomologies that properly include the main cohomology of this monograph. The first such section shows how the .D-cohomology is a particular instance of the cohomology of simplicial sets given by Gabriel and Zisman. The next section contains a natural generalization of the .D-cohomology of monoids to a cohomology theory for small categories that was first given by Charles Wells in an unpublished 1980 article. It is essentially the same cohomology published later by Baues and Wirsching in 1985. A retyped version of Wells’ paper became available on the internet in 2001. The final section returns to the .D-cohomology proper and proves its equivalence, modulo a simple dimension shift, with relevant instances of Grothendieck sheaf cohomology and also its equivalence, also up to a dimension shift, with relevant instances of Beck cohomology, the latter being a proof of Wells’ Theorem mentioned above.

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Chapter 4 is a continuation of Chap. 2, but with an emphasis on .H-coextensions. The first section looks at things from the inside. The main tool is the group-valued kernel functor .Σ of any congruence contained in the canonical equivalence .H. It plays a role similar to that of the relevant normal subgroup associated with a congruence on a group. The case when .Σ is abelian group-valued is the subject in the second section, with the third returning to the general case. Cohomological aspects are studied in the fourth section. In particular .H 3 comes into play in studying obstructions to an extension. This obstruction theory is more recent and was not included in Leech’s 1975 Memoir. In a final fifth section we use groups as an illustrative example. Chapter 5 considers the cohomology of monoids on which a monoid of operators .𝚪 acts. For groups, the .𝚪-cohomology theory we develop here goes back to that first introduced by J.H.C. Whitehead in his seminal 1950 paper. When the monoid of operators is trivial, this .𝚪-cohomology theory agrees with the ordinary Leech’s .Dcohomology theory for monoids. Computation by cocycles, connections with other known cohomology theories and applications to the classification of equivariant coextensions of monoids with operators are topics discussed in this chapter. In the concluding Chap. 6 our attention shifts from coextensions to monoidal groupoids. We analyze the structure of monoidal abelian groupoids by developing a 3-dimensional Schreier-Grothendieck theory of factor sets for their classification using third cohomology classes. We also consider monoidal abelian groupoids with a coherent monoid .𝚪-action. Granada, Spain Boynton Beach, FL, USA February 2024

Antonio M. Cegarra Jonathan Leech

Acknowledgements

The authors are much indebted to the anonymous referee, whose useful observations greatly improved our exposition.

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Contents

1

Functor Categories and Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 The Category of Modules Over a Small Category. . . . . . . . . . . . . . . . . . . . . 1.2 Cohomology of Small Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 7 10

2

The .D-Cohomology of Monoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 The D-Category of a Monoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 The D-Cohomology of a Monoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 H 0 and H 1 . Derivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Cohomology and Coextensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 The Cohomological D-Dimension of Monoids . . . . . . . . . . . . . . . . . . . . . . . . 2.6 The Cohomology of Cyclic Monoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11 12 16 26 32 42 44 53

3

Other Cohomologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.1 Eilenberg-Mac Lane Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.2 Hochschild-Mitchell Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 3.3 The D-Category of a Monoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 3.4 The D-Cohomology of Inverse Monoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 3.5 Gabriel-Zisman Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 3.6 Wells Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 3.7 Grothendieck and Beck Cohomologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

4

Cohomology and .H-Coextensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 The Kernel of an H-Coextension. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 The Category of H-Coextensions with Abelian Kernels . . . . . . . . . . . . . . 4.3 The Category of H-Coextensions with Arbitrary Kernel . . . . . . . . . . . . . 4.4 Obstructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Nonabelian Group Coextensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

111 112 120 126 136 144 151

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Contents

5

Cohomology of Monoids with Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 The Cohomology of 𝚪-Monoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 The Linking Long Exact Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 The Whitehead Cochain Complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Cohomology and 𝚪-Coextensions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

153 154 157 161 166 170

6

Cohomology and Monoidal Groupoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Monoidal Groupoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Monoidal Groupoids with Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Realizing 3-Cohomology Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 The Classification Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

171 171 175 177 190 203

Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 Symbols & Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

Chapter 1

Functor Categories and Cohomology

In this chapter we briefly review the cohomology of small categories. The cohomology theories for monoids that we will encounter are either specific instances of this cohomology, or are readily derived from it. The topics covered divide roughly into two parts. In the first part we review some basic properties of the category of .C-modules, that is, of abelian group valued functors from .C, where .C is any given small category: objects and morphisms; kernels and cokernels; exactness; products and coproducts; and the existence of enough injectives and enough projectives. In the second section, we assume that the reader is familiar with the basics of homological algebra in abelian categories, and review the cohomology theory of a small category .C with coefficients in .Cmodules. None of this is new. Indeed, all facts here are implicit in Grothendieck’s seminal Tohoku paper [2]. This chapter just provides the necessary background for what follows. For more background we refer to the books by Cartan and Eilenberg [1], Mac Lane [6], Hilton and Stammbach [3] or Weibel [9].

1.1 The Category of Modules Over a Small Category Let .Ab denote the category of abelian groups. If .C is a small category, the category of (left) .C-modules, denoted by .C-Mod, is the category whose objects are the functors from .C to .Ab and whose morphisms the natural transformations between them; that is, C-Mod = AbC .

.

If .A : C → Ab is a .C-module, for each morphism .u : x → y in .C and each a ∈ A(x), we shall often denote by ua the value of .A(u) at a. Thus, a .C-module .A

.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 A. M. Cegarra, J. Leech, The Cohomology of Monoids, RSME Springer Series 12, https://doi.org/10.1007/978-3-031-50258-3_1

1

2

1 Functor Categories and Cohomology

provides of abelian groups .A(x), one for each .x ∈ ObC, and homomorphisms A(u) : A(x) → A(y),

.

a I→ ua,

one for each morphism .u : x → y in .C, satisfying .v(ua) = (vu)a and .1a = a. A morphism of .C-modules .F : A → A' consists of homomorphisms Fx : A(x) → A' (x),

.

one for each .x ∈ ObC, such that, for every morphism .u : x → y in .C and .a ∈ A(x), Fy (ua) = u Fx (a).

.

The Category C-Mod Is Abelian Let .C be a small category. The category of .C-modules is an abelian category. The set of morphisms between two .C-modules .A and .A' , denoted by HomC (A, A' ),

.

is an abelian group under pointwise addition of natural transformations: given morphisms .F, F' : A → A' , the morphism .F + F' : A → A' is given, at each ' ' ' .x ∈ ObC, by the homomorphism .Fx + Fx : A(x) → A (x), .a I→ Fx (a) + Fx (a). With this addition on hom-sets, .C-Mod becomes an additive category, that is, each hom-set is an abelian group and composition of morphisms distributes over addition. The zero .C-module is the constant functor .0 : C → Ab in which every .0(x) is the trivial abelian group 0. A .C-module .A' is a submodule of a .C-module .A, denoted .A' ≤ A, whenever for all .x ∈ ObC, .A' (x) is a subgroup of .A(x) and for all .u : x → y in .C, .A' (u) = A(u)|A' (x) . The quotient .C-module .A/A' has (A/A' )(x) = A(x)/A' (x)

.

for all objects x of .C; for every morphism .u : x → y, .(A/A' )(u) is induced from .A(u). If .F : A → A' is a .C-module morphism, its kernel .KerF is the submodule of .A determined pointwise by setting KerF(x) = Ker(F : A(x) → A' (x))

.

1.1 The Category of Modules Over a Small Category

3

for all objects x of .C. The image of .F is the submodule .ImF ≤ A' defined pointwise by ImF(x) = Im(Fx : A(x) → A' (x)).

.

As usual, .A/KerF ∼ = ImF under the natural isomorphism induced by .F. The cokernel of .F is the quotient .CokerF = A' /ImF. A sequence of .C-modules .A' → A → A'' is exact if and only if at all objects x of .C the sequence .A' (x) → A(x) → A'' (x) is an exact sequence of abelian groups. .C-Mod is a complete and cocomplete category with pointwise limits and colimits. In particular, if .{Ai , i ∈ I } is a family of .C-modules, the .C-modules   . A and . A are respectively defined by i i i∈I i∈I  .

  Ai (x) = Ai (x),

i∈I



i∈I

  Ai (x) = Ai (x),

i∈I

i∈I

at each .x ∈ ObC.

The Category C-Mod has Enough Projectives and Injectives Let Z[−] : Set → Ab,

.

X I→ Z[X],

be the free abelian group functor. For each object x of .C, let Px = Z[C(x, −)] : C → Ab, .

Ix = HomZ (Z[C(−, x)], Q/Z) : C → Ab,

(1.1)

be the .C-modules obtained by composing the hom-fuctors C(x, −) : C → Set and C(−, x) : Cop → Set

.

with the free abelian group functor. For every .C-module .A, the Yoneda Lemma [5, Chap. III, §2] yields a natural isomorphism HomC (Px , A) ∼ = A(x),

.

(1.2)

which assigns to each morphism of .C-modules .F : Px → A the element .Fx (1x ), and a natural isomorphism HomC (A, Ix ) ∼ = HomZ (A(x), Q/Z)

.

(1.3)

4

1 Functor Categories and Cohomology

˜ : that sends each morphism of .C-modules .F : A → Ix to the homomorphism .F ˜ A(x) → Q/Z given by .F(a) = Fx (a)(1x ). From the isomorphisms (1.2), we see that .Px is a projective .C-module: For any epimorphism of .C-modules .F : A → A' , the induced map F∗ : HomC (Px , A) → HomC (Px , A' )

.

is surjective since the square

.

commutes and .Fx : A(x) → A' (x) is surjective. The functors .Px (.x ∈ ObC) constitute a set of projective generators of the category of .C-modules, so that this category has enough projective objects. In fact, given .A any .C-module, for each .x ∈ ObC and .a ∈ A(x), let .Fx,a : Px → A be the morphism that correspond by the isomorphism (1.2) to a, that is, such that .Fx,a (1x ) = a. These morphisms together define a morphism of .C-modules F:

  

.

x∈ObC

Px −→ A

(1.4)

a∈A(x)

which is clearly an epimorphism. Dually, from the isomorphisms (1.3), it follows that, for every object x of .C, the .C-module .Ix is injective. In effect, let .F : A' → A be a monomorphism of ' .C-modules. Then .Fx : A (x) → A(x) is injective, and therefore F∗x : HomZ (A(x), Q/Z) → HomZ (A' (x), Q/Z)

.

is surjective since the abelian group .Q/Z is injective. Hence, the induced F∗ : HomC (A, Ix ) → HomC (A' , Ix )

.

is also surjective since the diagram

.

commutes.

1.1 The Category of Modules Over a Small Category

5

Moreover, the .C-modules .Ix (.x ∈ ObC) constitute a set of injective cogenerators of the category .C-Mod, whence this category has enough injectives. In fact, given .A any .C-module, for each .x ∈ ObC and .0 /= a ∈ A(x), we can choose a homomorphism of abelian groups .f x,a : A(x) → Q/Z with .f x,a (a) /= 0. (See [3, p. 33].) Therefore, by the isomorphism (1.3), we can create a homomorphism of .C-modules .Fx,a : A → Ix such that .Fx,a x (a) /= 0. Clearly, these morphisms together define a monomorphism of .C-modules F : A −→

 

Ix .



.

x∈ObC

0/=a∈A(x)

Free C-Modules Let .ObC also denote the discreet subcategory of .C consisting of just the objects of C and their identities. Thus, a functor .X : ObC → Set is simply a family of sets indexed by the set .ObC. By restricting to .ObC and composing with the forgetful functor .Ab → Set, we obtain a forgetful functor

.

U : C-Mod → SetObC ,

.

This carries each .C-module .A to the functor .UA : ObC → Set which attaches to each object x of .C the underlaying set of the abelian group .A(x). This functor .U has a left adjoint, the free .C-module functor, F : SetObC → C-Mod,

.

which is as follows. If .X : ObC → Set is any functor, then the free .C-module on X, denoted by .FX, is the projective .C-module

.

FX =





.

x∈ObC α∈X(x)

(1.1)

Px =





Z[C(x, −)].

x∈ObC α∈X(x)

Thus, for each object y of .C, FX(y) = Z[{(u, α) | u ∈ C(x, y), α ∈ X(x)}]

.

is the free abelian group on the set of all pairs .(u, α), where .u : x → y is a morphism in .C and .α ∈ X(x). For any morphism .v : y → z in .C, the homomorphism FX(v) : FX(y) → FX(z)

.

is defined on generators by .v(u, α) = (vu, α).

6

1 Functor Categories and Cohomology

Theorem 1.1 The functor .F is left adjoint to the functor .U. Thus, for any functor X : ObC → Set and any .C-module .A, there is a natural isomorphism of abelian groups

.



HomC (FX, A) ∼ =



A(x),

.

  F I→ Fx (α) .

x∈ObC α∈X(x)

Proof This follows from the isomorphisms .(1.2): For each functor .X : ObC → Set and each .C-module .A, there are natural isomorphisms HomC (FX, A) = HomC

 



.

 Px , A ∼ =

x∈ObC α∈X(x) (1.2)

∼ =





A(x) =

x∈ObC α∈X(x)



HomC (Px , A



x∈ObC α∈X(x)



  HomSet X(x), A(x)

x∈ObC

= HomSetObC (X, UA). Note that the counit of the adjunction, at each .C-module .A, is precisely the epimorphism .F : FUA → A in (1.4). ⨆ ⨅ For a functor .X : ObC → Set, when .x ∈ ObC and .α ∈ X(x), we often denote the generator .(1x , α) of .FX(x) simply by .α, so that every element .α ∈ X(x) is regarded as a generator of .FX(x), and for any .y ∈ ObC, a generator .(u : x → y, α) of .FX(y) can be written uniquely in the form uα

.



 = (u, α) = FX(u)(1x , α) .

If .ϕ : X → X' is a morphism in .SetObC , then the induced morphism of .Cmodules .Fϕ : FX → FX' , at every .y ∈ ObC, is the homomorphism .Fϕ : FX(y) → FX' (y) such that Fϕ(uα) = uϕ(α),

.

for every morphism .u : x → y in .C and .α ∈ X(x). Note that the unit of the adjunction .(F, U) in Theorem 1.1 is, at each functor .X : ObC → Set, just the inclusion .X ͨ→ UFX (regarding each element .α ∈ X(x) as a generator of .FX(x)).

1.2 Cohomology of Small Categories

7

1.2 Cohomology of Small Categories Let .C be a small category. The inverse limit functor (or global section functor) lim : C-Mod → Ab

.

←

carries each .C-module .A to the abelian group

  limA = (ax ) ∈ A(x) | uax = ay for every u : x → y in C .

.

←

x∈ObC

This is a left exact functor, whose right derived functors are the cohomology groups of .C with coefficients in .C-modules, H n (C, A) = (R n lim)(A),

.

←

n = 0, 1, 2, . . . ,

studied by Roos [7] and Watts [8] among other authors. These cohomology groups define a cohomology theory on the category of .C-modules which is characterized, up to natural isomorphisms, by the following basic properties (see, e.g., Weibel [9, II, Theorem 2.4.7]): (i) .H 0 (C, A) = limA.

.

←

(ii) .H n (C, I) = 0 when .n > 0 and .I is injective. .(iii) For each short exact sequence of .C-modules .

G F S : 0 → A' → A → A'' → 0

.

there is a long exact exact sequence

.

which is natural on .S. The cohomology groups .H n (C, A) can also be defined as follows. Let Z : C → Ab,

.

id

(x → y) I→ (Z → Z),

8

1 Functor Categories and Cohomology

be the constant .C-module defined by the abelian group of integers .Z. Then, lim = HomC (Z, −) : C-Mod → Ab

.

←

and therefore H n (C, A) = (R n HomC (Z, −))(A) = ExtnC (Z, A) = (R n HomC (−, A))(Z). (1.5)

.

Thus, the cohomology groups .H n (C, A) can be computed as the n-th cohomology groups of both cochain complexes .HomC (Z, I• ) and .HomC (P• , A), for 0 → A → I0 → I1 → · · ·

.

an (any) injective resolution of the .C-module .A and .

· · · → P1 → P0 → Z → 0

a (any) projective resolution of .Z in the category of .C-modules. Every functor between small categories .κ : C' → C induces, by composition, an exact functor κ ∗ : C-Mod → C' -Mod,

.

A I→ κ ∗ A | (κ ∗ A)(x) = A(κ(x)).

Theorem 1.2 Let .κ : C' → C be a functor between small categories. For any C-module .A there are induced natural homomorphisms

.

κ ∗ : H n (C, A) → H n (C' , κ ∗A),

.

n = 0, 1, 2, . . . ,

which are compatible with the connecting homomorphisms; that is, for any short exact sequence .0 → A' → A → A'' → 0 of .C-modules, the squares below commute.

.

Proof Let .P• be a projective resolution of .Z in .C-Mod and let .P'• be a projective resolution of .Z in .C' -Mod. Since .κ ∗ is exact, .κ ∗ P• is an acyclic resolution of .κ ∗ Z = Z in .C' -Mod. Hence, there exists a complex morphism .F• : P'• → κ ∗ P• , unique

1.2 Cohomology of Small Categories

9

up to homotopy, that induces the identity on .Z. Given a .C-module .A, this yields the morphism F∗• : HomC' (κ ∗ P• , κ ∗A) → HomC' (P'• , κ ∗ A)

.

which, by composition with κ ∗ : HomC (P• , A) → HomC' (κ ∗ P• , κ ∗A),

.

induces the homomorphisms in the statement.

⨆ ⨅

Categories of Cohomological Dimension 0 Let .C be a small category. The cohomological dimension of .C, .cd(C) = m ≤ ∞, is defined by the condition that .cd(C) ≤ m whenever .H n (C, A) = 0 for .n > m and all .C-modules .A. Lemma 1.1 The following are equivalent: .(i) (ii) .(iii) .

cd(C) ≤ m. H m+1 (C, −) = 0 m .H (C, −) is right exact. . .

Proof Clearly .(i) ⇒ (ii) ⇒ (iii). If .0 → A → I → A' → 0 is a short exact sequence of .C-modules with .I injective, the corresponding long exact sequence of cohomology groups yields

.

⎧ 0 ⎨ H m (C, A' ) → H m+1 (C, A) → H m+1 (C, I) = 0, ⎩ H k (C, A' ) ∼ = H k+1 (C, A), for all k ≥ m + 1.

Thus .H k (C, A) = 0 for .k ≥ m + 1 by the above and induction.

⨆ ⨅

Proposition 1.1 If .C is a small category with an initial object, then .cd(C) = 0. Proof Let .ι be the initial object of .C. For any .C-module .A, we have a natural isomorphism .limA ∼ = A(ι). It follows that the functor .H 0 (C, −) = lim is exact and ← ← the result follows from the above lemma. ⨆ ⨅ The above condition is not a necessary one for the cohomological dimension of C to be 0. For connected categories we have the following useful criterium.

.

Theorem 1.3 If .C is a connected small category, then .cd(C) = 0 if and only if there exists a pair .(ι, u) with .ι ∈ ObC and .u ∈ End(ι) = C(ι, ι) such that: .(i) For all .x ∈ ObC, .C(ι, x) /= ∅. (ii) Every diagram

.

10

1 Functor Categories and Cohomology

.

in .C can be completed to a commutative square

.

(iii) u is a right zero in the monoid .End(ι), that is, .vu = u for all .v ∈ End(ι).

.

Proof See Laudal [4].

⨆ ⨅

We shall call a pair .(ι, u) satisfying the above conditions of Theorem 1.3 an initial pair. Thus, we have that if .C is connected, then .cd(C) = 0 if and only if .C possesses an initial pair. Proposition 1.2 Let .C be a small connected category, let .ι an object of .C and let u ∈ End(ι). Then .(ι, u) is an initial pair if and only if

.

.(i) For all .x ∈ ObC, .C(ι, x) = / ∅. (ii) For all .x ∈ ObC and all .v, v ' ∈ C(ι, x), .vu = v ' u.

.

Proof This is clear.

⨆ ⨅

References 1. Cartan, E., nd Eilenberg, S.: Homological Algebra. Princeton Landmarks in Mathematics. Princeton University Press, Princeton (1999) 2. Grothendieck, A.: Sur quelques points d’algère homologique. Tohoku Math. J. 9, 119–221 (1957) 3. Hilton, P.J., Stammbach, U.: A Course in Homological Algebra. Graduate Texts in Mathematics, vol. 4, 2nd edn. Springer, New York (1997) 4. Laudal, O.A.: Note on the projective limit on small categories. Proc. Am. Math. Soc. 33, 307– 309 (1972) 5. Mac Lane, S.: Categories for the Working Mathematician, vol. 5. Springer, New York (1971) 6. Mac Lane, S.: Homology. Classics in Mathematics. Springer, Berlin (1995) 7. Roos, J.E.: Sur les foncteurs dérivés de lim. Applications. C. R. Acad. Sci. Paris 252, 3702–3704 ← − (1961) 8. Watts, C.E.: A homology theory for small categories. In: 1966 Proceedings Conference Categorical Algebra, La Jolla, pp. 331–335. Springer, New York (1965) 9. Weibel, C.A.: An Introduction to Homological Algebra. Cambridge Studies in Advanced Mathematics, vol. 38. Cambridge University Press, Cambridge (1994)

Chapter 2

The D-Cohomology of Monoids .

In this chapter we present the main cohomology theory of this monograph, the .Dcohomology of a monoid, also referred to as Leech cohomology in some of the literature. Its importance is at least 2-fold: its connection to certain structural issues in the study of monoids (the subject of Chap. 4); and the fact that other cohomology theories of interest can be seen as special cases of the .D-cohomology (the subject of Chap. 3). Section 2.1 constructs a functor .D that assigns to each monoid S a small category .D(S), which can be viewed as a representing the division structure of S (in case S is a multiplicative monoid). The .D-cohomology of S uses .D(S)-modules as coefficients and is the cohomology of .D(S): H n (S, −) = H n (D(S), −),

.

n = 0, 1, 2, . . . .

The cohomology itself is introduced in Sect. 2.2 where we construct a free resolution of .Z in the category of .D(S)-modules. For any .D(S)-module .A, the cohomology groups .H n (S, A) are the cohomology groups of this resolution with coefficients in .A. Various aspects of this cohomology are discussed, with the section ending with the long exact cohomology sequence induced by a surjective homomorphism of monoids. In the third section the focus is on .H 0 and .H 1 . For any .D(S)-module .A, H 0 (S, A) = {a ∈ A(1) | ax = xa for all x ∈ S}.

.

As was in the case for group cohomology, .H 1 (S, A) is the result of factoring out a group of inner derivations from the full group of derivations: n .Der(S, A)/IDer(S, A). We also show that for any free monoid S, .H (S, A) = 0 for all .n > 1. In the fourth section, we show that the second cohomology groups of S classify certain coextensions of S. Of particular interest are, of course, .H-coextensions.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 A. M. Cegarra, J. Leech, The Cohomology of Monoids, RSME Springer Series 12, https://doi.org/10.1007/978-3-031-50258-3_2

11

2 The .D-Cohomology of Monoids

12

D(S)-modules and their 2-cocycles (i.e., factor systems) are used to create such structures. These connections are further studied in Chap. 4. Section 2.5 is dedicated to the cohomological .D-dimension of monoids, and the chapter ends in Sect. 2.6 with the calculation of the cohomology groups of cyclic monoids.

.

2.1 The D-Category of a Monoid If S is a monoid, then .D(S) is the small category whose set of objects is S, such that for every .x, y ∈ S, D(S)(x, y) = {〈u, x, v〉 | uxv = y},

.

and morphisms compose as follows: 〈u' , uxv, v ' 〉〈u, x, v〉 = 〈u' u, x, vv ' 〉.

.

Composition in .D(S) is associative, since multiplication in S. Since S is a monoid, for all .x ∈ S, idx = 〈1, x, 1〉.

.

Every homomorphism .f : S → T of monoids induces a functor D(f ) : D(S) → D(T ),

.

defined on objects by .D(f )(x) = f (x) and on morphisms by D(f )〈u, x, v〉 = 〈f (u), f (x), f (v)〉.

.

This makes .D a functor from the category .Mon of monoids to the category .Cat of small categories: D : Mon → Cat.

.

Lemma 2.1 A morphism .〈u, x, v〉 : x → y is an isomorphism in .D(S) if and only if .u, v ∈ G(S), the group of units of S. Proof If .u, v ∈ G(S), then .〈u−1 , y, v −1 〉 = 〈x, u, v〉−1 is clear. Conversely, let −1 = 〈u' , y, v ' 〉. Then, .〈u, x, v〉 be an isomorphism and let .〈u, x, v〉 .

〈1, x, 1〉 = 〈u' , y, v ' 〉〈u, x, v〉 = 〈u' u, x, v ' v〉, 〈1, y, 1〉 = 〈u, x, v〉〈u' , y, v ' 〉 = 〈uu' , y, vv ' 〉,

so that .u' u = 1 = uu' and .v ' v = 1 = vv ' follow.

⨆ ⨅

2.1 The D-Category of a Monoid

13

.D(S) contains two important subcategories, .L(S) and .R(S). Both .L(S) and R(S) have S as their set of objects. .L(S) is characterized by the fact that all its morphisms are of the form .〈u, x, 1〉, while .R(S) is characterized by the fact that all its morphisms are of the form .〈1, x, v〉. .L(S) and .R(S) together generate .D(S). If .〈u, x, v〉 is a morphism in .D(S), then .

〈u, x, v〉 = 〈u, xv, 1〉 〈1, x, v〉 = 〈1, ux, v〉 〈u, x, 1〉.

.

〈u, xv, 1〉〈1, x, v〉 is the .L R decomposition of .〈u, x, v〉, while its .R L decomposition is .〈1, ux, v〉〈u, x, 1〉.

.

Lemma 2.2 The .L R and .R L decompositions of .〈u, x, v〉 are unique. Proof Given .〈u, x, v〉 = α β with .α a morphism in .L(S) and .β a morphism in .R(S), then .β must have the form .〈1, x, v ' 〉 and .α must have the form .〈u' , xv ' , 1〉. Hence 〈u, x, v〉 = 〈u' , xv ' , 1〉〈1, x, v ' 〉 = 〈u' , x, v ' 〉,

.

so that .u' = u and .v ' = v. The .R L part follows by the dual argument.

⨆ ⨅

Let .K be a category. If .F : D(S) → K a functor, then so are its restrictions FL = F |L(S) : L(S) → K,

.

FR = F |R(S) : R(S) → K. Whenever a pair of functors, say .FL : L(S) → K and .FR : R(S) → K, are restrictions of a functor .F : D(S) → K, then we say that F extends the pair .(FL , FR ) to .D(S). Since .L(S) and .R(S) together generate .D(S), such an extension is unique. Theorem 2.1 Let .FL : L(S) → K and .FR : L(S) → K be functors. Then there exists a functor .F : D(S) → K that extends .FL and .FR to .D(S) if and only if for all .u, x, v ∈ S: .(i) .FL (x) = FR (x), (ii) .FR 〈1, ux, v〉 FL 〈u, x, 1〉 = FL 〈u, xv, 1〉 FR 〈1, x, v〉.

.

If the extension exists, then it is uniquely determined by .F (x) = FR (x) and F 〈u, x, v〉 = FR 〈1, ux, v〉 FL 〈u, x, 1〉 = FL 〈u, xv, 1〉 FR 〈1, x, v〉.

.

If .F, F ' : D(S) → K are functors, then a family σ = {σx : F (x) → F ' (x); x ∈ S}

.

of morphisms in .K is a natural transformation .σ : F → F ' if and only if both ' ' .σL = σ : FL → F and .σR = σ : FR → F are natural transformations. L R

2 The .D-Cohomology of Monoids

14

Proof If F extends .FL and .FR , then .(i) and .(ii) hold. Conversely, assume that .(i) and .(ii) hold and let .F : D(S) → K be defined by .F (x) = FR (x) and .F 〈u, x, v〉 being the above composition of morphisms. If .〈u, x, v〉 and .〈u' , uxv, v ' 〉 are any two composable morphisms in .D(S), then F 〈u' , uxv, v ' 〉 F 〈u, x, v〉 = FR 〈1, u' uxv, v ' 〉 FL 〈u' , uxv, 1〉 FR 〈1, ux, v〉 FL 〈u, x, 1〉 = FR 〈1, u' uxv, v ' 〉 FR 〈1, u' ux, v〉 FL 〈u' , ux, 1〉 FL 〈u, x, 1〉

.

= FR 〈1, u' ux, vv ' 〉 FL 〈u' u, x, 1〉 = F 〈u' u, x, vv ' 〉. Hence, F respects morphism composition. .F (idx ) = idF (x) follows from .FR (x) = FL (x) and .idx = idx idx ∈ L(S) ∩ R(S). If .σ : F → F ' is a natural transformation, then so are .σL : FL → FL' and ' .σR : FR → F . Conversely, in the following diagram, if the inner squares commute, R then so does the outer rectangle.

.

⨆ ⨅ In what follows we shall be study (left) .D(S)-modules, that is, functors in D(S)-Mod = AbD(S) .

.

When .A is a .D(S)-module, it is convenient to have S act on the abelian groups A(x) by

.

 .

ua = A〈u, x, 1〉(a) ∈ A(ux), av = A〈1, x, v〉(a) ∈ A(xv),

for all .u, x, v ∈ S and .a ∈ A(x), so that u' (ua) = (u' u)a, u(av) = (ua)v, (av)v ' = a(vv ' ), 1a = a = a1.

.

Then .A〈u, x, v〉(a) = (ua)v = u(av) ∈ A(uxv), so that uav is unambiguous; thus A〈u, x, v〉(a) = uav ∈ A(uxv).

.

(2.1)

2.1 The D-Category of a Monoid

15

Of course, in situations where we are interested in the behavior of several .D(S)modules simultaneously, it will be sometimes more convenient to work with the longer notation. In this notation, a morphism of .D(S)-modules (that is, a natural transformation) ' .F : A → A consists of homomorphisms F = Fx : A(x) → A' (x),

.

one for each .x ∈ S, such that F(ua) = uF(a), F(av) = F(a)v,

.

for any .u, x, v in S and .a ∈ A(x). For each functor .X : ObD(S) = S → Set, we denote by FX

.

the free .D(S)-module on .X: for each .x ∈ S, .FX)(x) is the free abelian group FX(x) = Z[{(u, α, v) ∈ S × X(z) × S | uzv = x}].

.

If .〈u' , x, v ' 〉 : x → y is a morphism in .D(S), then the induced homomorphism FX〈u' , x, v ' 〉 : FX(x) → FX(y),

.

a I→ u' av ' ,

is defined on generators by u' (u, α, v) v ' = (u' u, α, vv ' ).

.

Remark 2.1 As in Sect. 1.1, for every .z ∈ S and .α ∈ X(z), we usually write the generator .(1, α, 1) of .FX(z) simply as .α, so that every element .α ∈ X(z) is regarded as an element .α ∈ FX(z). Thus, every generator of .FX(x) can be written uniquely in the form uαv

.

(= (u, α, v) = FX〈u, z, v〉(1, α, 1)).

with .u, v ∈ S and .α ∈ X(z), for a unique .z ∈ S such that .uzv = x. By Theorem 1.1, there is a natural isomorphism of abelian groups HomD(S) (FX, A) ∼ =

 

.

x∈S α∈X(x)

for every .D(S)-module .A.

A(x),

  F I→ Fx (α) .

(2.2)

2 The .D-Cohomology of Monoids

16

2.2 The D-Cohomology of a Monoid The .D-cohomology of a monoid S with coefficients in .D(S)-modules is defined by the functors H n (S, −) := H n (D(S), −) = ExtnD(S) (Z, −) : D(S)-Mod → Ab,

.

where .Z is the constant .D(S)-module given by the group of integers. This cohomology, together with certain variations of it, is the main topic of this monograph. We begin by constructing the standard free resolution of .Z in the category of .D(S)modules.

The Standard Resolution of Z. Standard Cochains Underlying the following construction is the classifying simplicial set BS of the monoid S. (See Sect. 3.5.) For each .n ≥ 0, let .Xn : ObD(S) → Set be defined, if .n ≥ 1, by Xn (x) = {[u1 | . . . | un ] ∈ S n | u1 · · · un = x}

.

(2.3)

and, for .n = 0, by .X0 (1) = {[1]} and .X0 (x) = ∅ if .x /= 1. Then, an augmented chain complex of free .D(S)-modules F• S → Z

.

(2.4)

is defined as follows: For each .n ≥ 0, .Fn S = FXn is the free .D(S)-module on .Xn . Thus, for each .x ∈ S, Fn S(x) = Z[{(u0 , [u1 | . . . | un ], un+1 ) ∈ S × Xn (z) × S | u0 z un+1 = x}]

.

and, for each .u, x, v ∈ S, the homomorphism .Fn S〈u, x, v〉 : Fn S(x) → Fn S(y) is given on generators by Fn S〈u, x, v〉((u0 , [u1 | . . . | un ], un+1 ) = (uu0 , [u1 | . . . | un ], un+1 v).

.

The boundary .∂n : Fn S → Fn−1 S is the unique morphism of .D(S)-modules such that, for any .x ∈ S and .[u1 | . . . | un ] ∈ Xn (x), n−1  .∂n [u1 | . . . | un ] = u1 [u2 | . . . | un ] + (−1)i [u1 | . . . | ui ui+1 | . . . | un ] i=1

+ (−1)n [u1 | . . . | un−1 ]un , and the augmentation .∂0 : F0 S → Z is given by .∂0 [1] = 1.

2.2 The D-Cohomology of a Monoid

17

We shall use the simpler description for the abelian groups .Fn S(x): Fn S(x) = Z[{(u0 , u1 , . . . , un , un+1 ) ∈ S n+2 | u0 u1 · · · un un+1 = x}],

.

(2.5)

the homomorphism .Fn S〈u, x, v〉 : Fn S(x) → Fn S(y): Fn S〈u, x, v〉(u0 , u1 , . . . , un , un+1 ) = (uu0 , u1 , . . . , un+1 v)

.

the boundary .∂n : Fn S(x) → Fn−1 S(x): ∂n (u0 , . . . , un+1 ) =

n 

.

(−1)i (u0 , . . . , ui ui+1 , . . . , un+1 ),

(2.6)

i=0

and the augmentation .∂0 : F0 S(x) → Z: ∂0 (u0 , u1 ) = 1.

(2.7)

.

Theorem 2.2 .F• S → Z is a free resolution (hence projective) of .Z in .D(S)-Mod. Proof It suffices to prove that, for every .x ∈ S, ∂

F• S(x) → Z :

∂

∂

· · · → F2 S(x) −→ F1 S(x) −→ F0 S(x) −→ Z → 0

.

is an augmented chain complex with a contracting homotopy (whence it is exact). For, let .Фx−1 : Z → F0 S(x) and .Фxn : Fn S(x) → Fn+1 S(x) be the homomorphisms of abelian groups defined on generators by  .

Фx−1 (1) = (1, x), Фxn (u0 , . . . , un+1 ) = (1, u0 , . . . , un+1 ),

(2.8)

where we are using the descriptions in (2.5), (2.6), and (2.7). That, for all .n ≥ 1 and .x ∈ S, .∂0 Фx−1 = idZ and .Фxn−1 ∂n + ∂n+1 Фxn = idFnS(x) hold, only requires a straightforward check on generators which we leave the reader. Assuming .∂n ∂n+1 = 0, we see that the identity map in this complex is homotopic to the zero chain map on this complex. Thus, its homology groups vanish. To show .∂n ∂n+1 = 0, first verify that .∂0 ∂1 = 0 by straightforward computation. For .n ≥ 1, we proceed by induction. If .∂n−1 ∂n = 0, then ∂n ∂n+1 Фxn = ∂n (id − Фxn−1 ∂n ) = ∂n − ∂n Фxn−1 ∂n

.

= ∂n − (id − Фxn−2 ∂n−1 )∂n = Фxn−2 ∂n−1 ∂n = 0

2 The .D-Cohomology of Monoids

18

given that .∂n−1 ∂n = 0. Thus .∂n ∂n+1 Фxn = 0 : Fn S(x) → Fn−1 S(x), for all .n ≥ 1 and .x ∈ S. Now, let .(u0 , u1 , . . . , un+2 ) ∈ Fn+1 S(x) be any generator. If .u1 · · · un+2 = y in S, then ∂n ∂n+1 (u0 , u1 , . . . , un+2 ) = ∂n ∂n+1 (u0 (1, u1 , . . . , un+2 ))

.

y

= u0 ∂n ∂n+1 (1, u1 , . . . , un+2 ) = u0 ∂n ∂n+1 Фn (u1 , . . . , un+2 ) = u0 0 = 0. Thus .∂n ∂n+1 = 0.

⨆ ⨅

If .A is any .D(S)-module, then the cohomology groups of S with coefficients in .A can be computed as the cohomology groups of the cochain complex .HomD(S) (F• S, A). By (2.2), we have an isomorphism of cochain complexes HomD(S) (F• S, A) ∼ = C • (S, A),

(2.9)

.

where δ

C • (S, A) :

δ

δ

0 → C 0 (S, A) → C 1 (S, A) → C 2 (S, A) → · · ·

.

(2.10)

is defined as follows. If .n ≥ 1 C n (S, A) =



.

A(u1 · · · un )

(u1 ,...,un )∈S n

and .δ : C n (S, A) → C n+1 (S, A) is given by (δϕ)(u1 , . . . , un+1 ) = u1 ϕ(u2 , . . . , un+1 )+

n 

.

(−1)i ϕ(u1 , . . . , ui ui+1 , . . . , un+1 )

i=1

+ (−1)n+1 ϕ(u1 , . . . , un )un+1 , and .C 0 (S, A) = A(1) with .δa(x) = xa − ax. Therefore, there are natural isomorphisms H n (S, A) ∼ = H n C • (S, A),

.

n = 0, 1, . . . .

(2.11)

The free resolution of .Z given in (2.4) is the standard resolution of .Z, and, for any given .D(S)-module .A, .C • (S, A) is the standard cochain complex of S with

2.2 The D-Cohomology of a Monoid

19

coefficients in .A. As usual, elements of .C n (S, A) are n-cochains, those in Z n (S, A) = Ker(δ n )

.

are n-cocycles, and those in B n (S, A) = Img(δ n−1 )

.

are n-coboundaries. If .ϕ ∈ Z n (S, A) is a n-cocycle, then [ϕ] ∈ H n (S, A)

.

denotes its cohomology class in .H n (S, A) = Z n (S, A)/B n (S, A). The assignment .A I→ C • (S, A) is functorial and the isomorphisms (2.9) and (2.11) are natural in .A. If .F : A → A' is a morphism of .D(S)-modules, the induced cochain map F∗ : C • (S, A) → C • (S, A' )

.

is defined by composition with .F, that is, every .a ∈ A(1) = C 0 (S, A) goes to ' ' 0 n .F(a) ∈ A (1) = C (S, A ), and if .ϕ ∈ C (S, A) with .n ≥ 1, then .F∗ ϕ ∈ ' n C (S, A ) is given by (F∗ ϕ)(u1 , . . . , un ) = F(ϕ(u1 , . . . , un )).

.

Thus we see that if .[ϕ] ∈ H n (S, A), then .F∗ [ϕ] = [F∗ ϕ].

The Normalized Resolution of Z. Normalized Cochains We present a useful variation of the standard resolution. Let .F• S → Z be our standard resolution (2.4) of .Z in the category of .D(S)modules. We define a subcomplex .F'• S ⊆ F• S as follows: Let .F'0 S = 0. For each .n ≥ 1, define a subfunctor .X'n of .Xn : ObD(S) → Set, (see (2.3)) by X'n (x) = {[u1 | . . . | un ] ∈ Xn (x) | ui = 1 for some i}

.

and let .F'n S = FX'n ≤ Fn S be the free .D(S)-module on .X'n . So defined, it is an easy check to see that .∂F'n S ⊆ F'n−1 S, so that .F'• S becomes a subcomplex of .F• S. Indeed, .F'• S is a free exact resolution of the .D(S)-module 0, since, for all .x ∈ S, the contracting homotopy .Фx on .F• S(x) in (2.8) restricts to a contracting homotopy on ' .F• S(x). The standard normalized resolution of .Z is the quotient complex .

F˜ • S = F• S/F'• S.

2 The .D-Cohomology of Monoids

20

˜ n of .Xn with Equivalently, .F˜ n S is the free .D(S)-module on the subfunctor .X .

˜ n (x) = {[u1 | . . . | un ] ∈ Xn (x) | ui /= 1 for all i}. X

Hence .F˜ • S is a free resolution of .Z. Therefore, if .A is a .D(S)-module, there are natural isomorphisms H n (S, A) ∼ = H n HomD(S) (F˜ • S, A)

.

n = 0, 1, . . .

For each .D(S)-module .A, let .

C˜ • (S, A) ≤ C • (S, A)

(2.12)

be the cochain subcomplex of normalized cochains of S with coefficients in .A, that is, the subcomplex of the standard cochain complex whose n-cochains, called normalized, are those .ϕ ∈ C n (S, A) with the property that .ϕ(u1 , . . . , un ) = 0 whenever some .ui = 1. Elements of .C˜ n (S, A) are normalized n-cochains, those in .

Z˜ n (S, A) = Ker(δ n )

are normalized n-cocycles, and those in .

B˜ n (S, A) = Img(δ n−1 )

are normalized n-coboundaries. The projection .F• S ↠ F˜ • S induces a cohomology isomorphism inclusion ˜ • S, A) ͨ→ HomD(S) (F• S, A), through which the natural isomorphism .HomD(S) (F .HomD(S) (F• S, A) ∼ = C • (S, A) restricts by giving a natural isomorphism HomD(S) (F˜ • S, A) ∼ = C˜ • (S, A).

.

Hence, there are natural isomorphisms H n (S, A) ∼ = H n C˜ • (S, A),

.

n = 0, 1, . . . .

(2.13)

Let us stress that every cocycle in .C • (S, A) differs from a normalized cocycle by a coboundary, since the inclusion .C˜ • (S, A) ͨ→ C • (S, A) induces isomorphisms on cohomology groups.

2.2 The D-Cohomology of a Monoid

21

Change of Monoid Let .f : S → T be a homomorphism of monoids. We denote by f ∗ : D(T )-Mod → D(S)-Mod

.

the exact functor induced by the functor .D(f ) : D(S) → D(T ): for any .D(T )module .A, the .D(S)-module .f ∗A is given, at any .x ∈ S, by .(f ∗A)(x) = A(f (x)), and uav = f (u) a f (v),

.

for any .u, v ∈ S and .a ∈ (f ∗A)(x). By Theorem 1.2, the natural homomorphism f ∗ : HomD(T ) (Z, A) → HomD(S) (Z, f ∗A),

.

F I→ f ∗F,

uniquely determine natural homomorphisms f ∗ : H n (T , A) → H n (S, f ∗A),

.

n = 0, 1, 2, . . . ,

(2.14)

which are compatible with the connecting homomorphisms: for any short exact sequence .0 → A' → A → A'' → 0 of .D(T )-modules, the squares below commute.

.

We now look at how one computes the homomorphisms .f ∗ in (2.14) from the standard cochain complexes. Since .f ∗ : D(T )-Mod → D(S)-Mod turns the standard free resolution of .Z in .D(T )-Mod into a (usually not projective) resolution of .Z in .D(S)-Mod, there exits a morphism of augmented complexes of .D(S)-modules

.

2 The .D-Cohomology of Monoids

22

and any two such morphisms are homotopic. The standard morphism .F• is the one uniquely determined by  .

F0 [1] = [1], Fn [u1 | . . . | un ] = [f (u1 ) | . . . | f (un )] for n ≥ 1,

and we have the following natural morphisms of cochain complexes

.

If, for any .D(T )-module .A, we identify, by means of the isomorphisms (2.9), HomD(T ) (F• T , A) with .C • (T , A) and .HomD(S) (F• S, f ∗ A) with .C • (S, f ∗ A), we obtain a standard cochain map induced by f :

.

f ∗ : C • (T , A) → C • (S, f ∗ A).

.

Thus, .f ∗ = idA(1) : C 0 (T , A) → C 0 (S, f ∗A), and if .ϕ ∈ C n (T , A) with .n ≥ 1, then (f ∗ ϕ)(u1 , . . . , un ) = ϕ(f (u1 ), · · · , f (un )).

.

(2.15)

Theorem 2.3 Let .f : S → T be a homomorphism of monoids. The induced functor f ∗ : D(T )-Mod → D(S)-Mod has a left adjoint

.

f∗ : D(S)-Mod → D(T )-Mod.

.

Proof For any .D(S)-module .A, .f∗ A is the .D(T )-module defined as follows: For each .t ∈ T , .f∗ A(t) is the abelian group generated by all pairs .(a, x) where x is an element of S such that .f (x) = t and .a ∈ A(x), subject to the defining relations (a + b, x) = (a, x) + (b, x)

.

f (u) (a, x) f (v) = (u a v, uxv)

.

(x ∈ S, a, b ∈ A(x)),

(2.16)

(u, x, v ∈ S, a ∈ A(x)).

(2.17)

Every morphism .F : A → A' of .D(S)-modules induces a morphism of .D(T )modules f∗ F : f∗ A → f∗ A'

.

defined on generators by f∗ F(a, x) = (F(a), x)

.

(x ∈ S, a ∈ A(x)).

2.2 The D-Cohomology of a Monoid

23

For any .D(S)-module .A and .D(T )-module .B, mutually inverse bijections .

θ : HomD(S) (A, f ∗ B) → HomD(T ) (f∗ A, B),

F I→ θF ,

λ : HomD(T ) (f∗ A, B) → HomD(S) (A, f ∗ B),

G I→ λG ,

are defined as follows: For each morphism of .D(S)-modules .F : A → f ∗ B, θF : f∗ A → B

.

is the morphism of .D(T )-modules which acts on generators by θF (a, x) = F(a).

.

and is well-defined since θF (a + b, x) = F(a + b) = F(a) + F(b) = θF (a, x) + θF (b, x)

.

= θF ((a, x) + (b, x)), θF (uav, uxv) = F(uav) = f (u) F(a) f (v) = f (u) θF (a, x) f (v)   = θF f (u)(a, x)f (v) . Conversely, every morphism of .D(T )-modules .G : f∗ A → B induces for each x ∈ S the homomorphism

.

λG : A(x) → f ∗ B(x),

.

a I→ G(a, x).

Then, .λG : A → f ∗ B is a morphism of .D(S)-modules since (2.16)

λG (a + b) = G(a + b, x) = G((a, x) + (b, x)) = G(a, x) + G(b, x)

.

= λG (a) + λG (b),

 (2.17)  λG (uav) = G(uav, uxv) = G f (u)(a, x)f (v) = f (u) G(a, x)f (v) = f (u) λG (a)f (v) = u λG (a)v. ⨆ ⨅

2 The .D-Cohomology of Monoids

24

The Long Exact Sequence Let .p : E ↠ S be a monoid homomorphism of E onto S. In what follows we denote by .Kp = E ×p E the congruence relation on E induced by p. Not only is .Kp a congruence on E, but it is also a monoid, indeed a submonoid of .E × E. Let p1 , p2 : Kp → E,

.

p1 (x, y) = x, p2 (x, y) = y,

be the coordinate projections. Clearly, .pp1 = pp2 . Then, if .A is any .D(S)-module, p1∗ p∗ A and .p2∗ p∗ A are the same .D(Kp )-module, which we denote simply by .p∗A. There is an induced diagram of cochain complexes

.

.

where .p1∗ p∗ = p2∗ p∗ and .p∗ is injective. In fact .p∗ it is the equalizer of .p1∗ and .p2∗ . Indeed, let .α ∈ C n (E, p∗A). If .p1∗ α = p2∗ α, then α(x1 , . . . , xn ) = (p1∗ α)((x1 , y1 ), . . . , (xn , yn ))

.

= (p2∗ α)((x1 , y1 ), . . . , (xn , yn )) = α(y1 , . . . , yn ). Hence .α is in the image of .p∗ . The cochain complex .C • (Kp , p∗A) has an important subcomplex, the complex of transitive cochains, denoted by T C • (Kp , p∗A),

.

which consists of those cochains .α ∈ C n (Kp , p∗A) such that, for all .xi Kp yi Kp zi with .1 ≤ i ≤ n,   α (x1 , z1 ), . . ., (xn , zn ) (2.18)     = α (x1 , y1 ), . . . , (xn , yn ) + α (y1 , z1 ), . . . , (yn , zn ) ,

.

and .T C 0 (Kp , p∗A) = 0. Lemma 2.3 Let .α ∈ T C • (Kp , p∗A). For any .xi Kp yi with .1 ≤ i ≤ n we have: α((x1 , x1 ), . . . , (xn , xn )) = 0, .

(2.19)

α((x1 , y1 ), . . . , (xn , yn )) = −α((y1 , x1 ), . . . , (yn , xn )).

(2.20)

.

2.2 The D-Cohomology of a Monoid

25

Proof Equation (2.19) follows from (2.18) by replacing .yi and .zi by .xi . Then, that (2.20) also follows from (2.18) can be seen after replacing .zi by .xi . ⨆ ⨅ The cohomology groups of .T C • (Kp , p∗A) are the congruence cohomology groups. They will be denoted by T H n (Kp , p∗A).

.

Let d ∗ = p1∗ − p2∗ : C • (E, p∗A) → C • (Kp , p∗A)

.

be the cochain difference morphism. An n-cochain in .C • (Kp , p∗A) looks like ∗ .β((x1 , y1 ), . . . , (xn , yn )) with .xi Kp yi in E. Thus, .d = 0 if .n = 0 and, for .n ≥ 1 • ∗ and .α ∈ C (E, p A), (d ∗ α)((x1 , y1 ), . . . , (xn , yn )) = α(y1 , . . . , yn ) − α(x1 , . . . , xn ).

.

Form this, we easily see that the image of .d ∗ is contained in .T C • (Kp , p∗A). Theorem 2.4 The sequence of cochain complexes

.

is a short exact sequence, and induces a long exact sequence of cohomology groups

.

Proof That .p∗ = Ker(d ∗ ) is a direct consequence of .p∗ being the equalizer of .p1∗ and .p2∗ . To prove that .d ∗ is surjective, let .α ∈ T C n (Kp , p∗A) be any transitive n-cochain. For each .Kp -class in E let us choose a particular congruence class representative, say .x. ˜ Define .β ∈ C n (E, p∗A) by β(x1 , . . . , xn ) = α((x˜1 , x1 ), . . . , (x˜n , xn )).

.

Replacing .xi by .x˜i in (2.18) and solving for .α((y1 , z1 ), . . . , (yn , zn )) we obtain α = d ∗ β. ⨆ ⨅

.

The cohomology group .T H 1 (Kp , p∗A) is of interest because it is the middle term of the so called five term exact sequence, which is just the beginning of the long exact sequence above. (See [16, 4.1] for a slightly different 5-term exact sequence.)

2 The .D-Cohomology of Monoids

26

Remark 2.2 Suppose E and S are groups and .p : E ↠ S is a group epimorphism with kernel N . Then, the congruence kernel .Kp of p is a group isomorphic to the semidirect product group .E ⋊ N , .(x, n) I→ (x, nx). If A is an S-module regarded as a constant on objects .D(S)-module with trivial right action, then there is a natural isomorphism   T H 1 (Kp , p∗A) ∼ = HomS N/[N, N ], A .

.

Explicitly, this isomorphism carries each .α ∈ T H 1 (Kp , p∗A) = T Z 1 (E ⋊ N, p∗A) to the homomorphism of S-modules .fα : N/[N, N ] → A given by fα (n[N, N]) = α(1, n).

.

Cf. [14, Chapter VI, (8.2)].

2.3 H 0 and H 1 . Derivations The groups .Z 0 (S, A), .Z 1 (S, A) and .B 1 (S, A) are connected to the construction of monoid coextensions. Let S be a monoid. If .A is a .D(S)-module, then H 0 (S, A) = Z 0 (S, A) = {a ∈ A(1) | xa = ax for all x ∈ S}

.

= {a ∈ A(1) | uav = u' av ' whenever uv = u' v ' }. Cocycles in .Z 1 (S, A) are maps .d : S → and, for all .x, y ∈ S,

 x∈S

A(x) such that .d(x) ∈ A(x)

d(xy) = x d(y) + d(x) y

.

and are also called derivations of S in .A. Thus, .Z 1 (S, A) is also denoted by Der(S, A).

.

Its subgroup .B 1 (S, A) is also denoted by IDer(S, A).

.

Its elements are the inner derivations of S in .A. Thus, a derivation d is an inner derivation if there exists an element .a ∈ A(1) such that for all .x ∈ S, d(x) = xa − ax.

.

2.3 H 0 and H 1 . Derivations

27

Then, H 1 (S, A) ∼ = Der(S, A)/IDer(S, A).

.

Let .Mon ↓S denote the slice category of monoids over S, that is, the category whose objects are the monoid homomorphisms .f : T → S and whose morphisms are the commutative triangles of monoid homomorphisms

.

For any given .D(S)-module .A, we have a contravariant functor Der(−, A) : Mon ↓S → Ab,

.

f

T → S I→ Der(T , f ∗A).

In Proposition 2.1 below, we show that this functor .Der(−, A) is correpresentable. The semidirect product of S by a .D(S)-module .A, denoted by S ⋊A,

(2.21)

.

is the monoid whose underlying set is .{(x, a) | x ∈ S, a ∈ A(x)} with multiplication (x, a)(y, b) = (xy, ay + xb).

.

This product is easily seen to be associative and to have an identity element .(1, 0). The projection π : S ⋊ A ↠ S,

.

(x, a) I→ x.

makes .(S ⋊ A, π ) a coextension of S (actually, a coextension of S by .A, as defined in Sect. 2.4). The map .s : S → S ⋊A defined by .s(x) = (x, 0) is a monoid homomorphism. Actually, s is a cross section of the congruence induced by .π; it also splits .π (makes .π a split surjection). Clearly the finding of all such cross sections is equivalent to finding the morphisms in .Mon ↓S from .idS : S → S to .π : S ⋊ A → S. These morphisms over S are classified in the following theorem. Proposition 2.1 For any monoid homomorphism .f : T → S and any .D(S)-module A, there is a natural bijection

.

f

Der(T , f ∗A) ∼ = HomMon↓S (T → S, S ⋊ A → S).

.

π

2 The .D-Cohomology of Monoids

28

Proof The assignment .(x, a) I→ a is a derivation .η : S ⋊ A → π ∗A:   η (x, a)(y, b) = η(xy, ay + xb) = ay + xb = η(x, a)π(y, b) + π(x, a)η(y, b).

.

Then, every homomorphism of monoids .g : T → S ⋊ A over S determines a derivation .ηg : T → f ∗A. Conversely, for each derivation .d : T → f ∗A, there exists a unique homomorphism .g : T → S ⋊ A over S such that .ηg = d. In fact, g is defined by .g(t) = (f (t), d(t)), for every .t ∈ T , and it is straightforward to check that g is actually a homomorphism: g(tt ' ) = (f (tt ' ), d(tt ' )) = (f (t)f (t ' ), f (t)d(t ' ) + d(t)f (t ' ))

.

= (f (t), d(t))(f (t ' ), d(t ' )) = g(t)g(t ' ). g(1) = (f (1), d(1)) = (1, 0). for every .t, t ' ∈ T .

⨆ ⨅

Remark 2.3 If .A is a .D(S)-module, then the semidirect product coextenπ sion (2.21), .S ⋊A → S, is an internal abelian group object in the comma category .Mon ↓S of monoids over S. The internal group operation + : (S ⋊A) ×S (S ⋊A) → S ⋊A

.

is given by .(x, a) + (x, b) = (x, a + b), for .x ∈ S and .a, b ∈ A(x). In fact, π Wells [21] showed that the assignment .A I→ (S ⋊ A → S, +) is the function on objects of an equivalence of categories between the category of .D(S)-modules and the category of abelian group objects in .Mon ↓S . For every .D(S)-module .A, the natural bijection in Proposition 2.1, f

Der(T , f ∗A) ∼ = HomMon↓S (T → S, S ⋊ A → S), π

.

f

π

is an isomorphism of abelian groups when .HomMon↓S (T → S, S ⋊A → S) becomes an abelian group with the natural induced addition: .g + g ' : t I→ g(t) + g(t ' ). Corollary 2.1 If S is free on a set X, then, for any .D(S)-module .A, there is a natural isomorphism Der(S, A) ∼ =



.

x∈X

A(x),

  d I→ d(x) .

2.3 H 0 and H 1 . Derivations

29

Proof The claimed isomorphism is the composite of the bijections id π Der(S, A) ∼ = HomMon↓S (S → S, S ⋊ A → S) ∼ =



.

A(x),

x∈X

  where, if .g(x) = (x, d(x)) for each .x ∈ X, the second one is given by .g I→ d(x) . ⨆ ⨅ From the above proposition, we see easily how to compute the first cohomology groups of a free monoid. Theorem 2.5 Let S be a free monoid on X. For any .D(S)-module .A, there is a natural exact sequence of abelian groups δ

0 → H 0 (S, A) → A(1) −→



.

A(x) → H 1 (S, A) → 0,

x∈X

where .δ(a) = (xa − ax)x∈X , for every .a ∈ A(1).

□

.

Proposition 2.2 below states that the functor Der(S, −) : D(S)-Mod → Ab

.

is representable. Let the augmentation ideal .IS of S be the kernel of the augmentation morphism .∂0 : F0 S → Z of the standard resolution (2.4) of S, so that there is a short exact sequence of .D(S)-modules 0 → IS → F0 S → Z → 0.

.

For every .x ∈ S, the abelian group .IS(x) is freely generated by the elements (u0 , u1 ) − (x, 1),

.

where .u0 , u1 ∈ S are elements such that .u0 u1 = x and .(u0 , u1 ) /= (x, 1). Every morphism .〈u, x, v〉 : x → y in .D(S) induces the homomorphism IS〈u, x, v〉 : IS(x) → IS(y),

.

a I→ uav,

such that   u (u0 , u1 ) − (x, 1) v = (uu0 , u1 v) − (ux, v)     = (uu0 , u1 v) − (y, 1) − (ux, v) − (y, 1) ,

.

for every generator .(u0 , u1 ) − (x, 1) of .IS(x).

(2.22)

2 The .D-Cohomology of Monoids

30

Let .ξ : S → IS be the derivation defined by ξ(x) = (1, x) − (x, 1),

.

which induces for every .D(S)-module .A a homomorphism ξ ∗ : HomD(S) (IS, A) → Der(S, A),

.

F I→ F ξ.

Proposition 2.2 For any .D(S)-module .A, .ξ ∗ is a natural isomorphism HomD(S) (IS, A) ∼ = Der(S, A).

.

Proof Each derivation .d : S → A gives rise to a morphism .Fd : IS → A of .D(S)modules which, at every .x ∈ S, consists of the homomorphism .Fd : IS(x) → A(x) such that Fd ((u0 , u1 ) − (x, 1)) = u0 d(u1 ),

.

for every generator of .IS(x). In fact, .Fd is a morphism of .D(S)-modules and the maps .d I→ Fd and .F I→ F ξ are mutually inverse. Indeed, if .u0 u1 = x and .uxv = y,    Fd u (u0 , u1 ) − (x, 1) v     = Fd (uu0 , u1 v) − (y, 1) − (ux, v) − (y, 1)

.

= uu0 d(u1 v) − uxd(v)   = uu0 u1 d(v) + d(u1 )v − uxd(v) = uu0 u1 d(v) + uu0 d(u1 )v − uxd(v) = uu0 d(u1 )v   = uFd (u0 , u1 ) − (x, 1) v,     (F ξ )d (u0 , u1 ) − (x, 1) = u0 (F ξ )(u1 ) = u0 F (1, u1 ) − (u1 , 1)    = F u0 (1, u1 ) − (u1 , 1)   = F (u0 , u1 ) − (x, 1) ,   (Fd ξ )(x) = Fd (1, x) − (x, 1) = d(x). ⨆ ⨅

2.3 H 0 and H 1 . Derivations

31

Corollary 2.2 For any .D(S)-module .A and .n ≥ 2, there are natural isomorphisms ∼ n−1 Der(S, −)(A). H n (S, A) ∼ = Extn−1 D(S) (IS, A) = R

.

Proof The first isomorphism .H n (S, A) ∼ = Extn−1 D(S) (IS, A) comes from the long exact cohomology sequence induced by the projective presentation .0 → IS → F0 S → Z → 0, in the category of .D(S)-modules: .

n · · · → 0 → Extn−1 D(S) (IS, A) → ExtD(S) (Z, A) → 0 → · · ·

∼ n−1 Der(S, −)(A) follows from the The second isomorphism .Extn−1 D(S) (IS, A) = R natural isomorphisms .HomD(S) (IS, −) ∼ Der(S, −) in Proposition 2.2. ⨅ ⨆ = Proposition 2.3 If S is a free monoid, then .IS is a free .D(S)-module. Proof Suppose S be free on the set X and let .A be any .D(S)-module. Then, by Corollary 2.2 and Proposition 2.1, we have isomorphisms HomD(S) (IS, A) ∼ = Der(S, A) ∼ =



.

A(x),

x∈X

whose composition gives a natural isomorphism HomD(S) (IS, A) ∼ =



.

A(x),

F I→ (Fξ(x)).

x∈X

Hence, .IS is the free .D(S)-module on the functor .X : ObD(S) → Set, where X(x) = {ξ(x)} if .x ∈ X and .X(x) = ∅ if .x ∈ / X. ⨆ ⨅

.

Proposition 2.3 and Corollary 2.2 imply that the cohomology groups of free monoids vanish in dimensions higher than 1: Theorem 2.6 If S is a free monoid, then .H n (S, A) = 0 for every .D(S)-module .A when .n ≥ 2. .□ Let S be a monoid. The functor of differentials over S DiffS : Mon ↓S → D(S)-Mod

.

is defined by f

DiffS (T → S) = f∗ (I T ),

.

where .I T is the augmentation ideal of T and .f∗ : D(T )-Mod → D(S)-Mod is the left adjoint functor to the functor .f ∗ : D(S)-Mod → D(T )-Mod. (See Theorem 2.3.)

2 The .D-Cohomology of Monoids

32

For every homomorphism of monoids .f : T → S and any .D(S)-module .A, Theorem 2.3 and by Proposition 2.1 yield isomorphisms f HomD(S) (DiffS (T → S), A) ∼ = HomD(T ) (I T , f ∗ A) ∼ = Der(T , f ∗ A).

.

Thus, the functor .Der(T , f ∗ −) is representable by the .D(S)-module of differentials f

DiffS (T → S). Furthermore, since, by Proposition 2.1, we have a natural bijection

.

f π Der(T , f ∗A) ∼ = HomMon↓S (T → S, S ⋊ A → S),

.

there is a natural bijection f f π HomD(S) (DiffS (T → S), A) ∼ = HomMon↓S (T → S, S ⋊ A → S).

.

Hence, Theorem 2.7 The functor .DiffS : Mon ↓S → D(S)-Mod is left adjoint to the functor D(S)-Mod → Mon ↓S ,

.

π

A I→ S ⋊ A → S. □

.

2.4 Cohomology and Coextensions Let S be a monoid. A coextension of S is a surjective homomorphism of monoids p : E ↠ S. If .A is a .D(S)-module, a coextension

.

(A, E, p, +)

.

of S by .A is a coextension .p : E ↠ S of S endowed, for each .x ∈ S, with a simply-transitive right action + : p−1 (x) × A(x) → p−1 (x),

.

(w, a) I→ w + a,

of the abelian group .A(x) on the fibre subset .p−1 (x) ⊆ E of .p : E → S at x (i.e., for each .w, w ' ∈ p−1 (x) there is a unique .a ∈ A(x) such that .w ' = w + a), such that, for all .w ∈ p −1 (x), .w ' ∈ p −1 (x ' ), .a ∈ A(x), and .a ' ∈ A(x ' ), (w + a)(w' + a ' ) = ww ' + (a x ' + x a ' ).

.

(2.23)

2.4 Cohomology and Coextensions

33

The category of abelian group coextensions of the monoid S, denoted by ExtD (S),

.

(2.24)

is the category whose objects are the coextensions of S by .D(S)-modules. A morphism between two such coextensions (F, F ) : (A, E, p, +) → (A' , E ' , p' , +)

.

consists of a morphism of .D(S)-modules .F : A → A' together with a homomorphism of monoids .F : E → E ' , such that ' .(i) .p F = p, so that the square below commutes.

.

(ii) .F (w + a) = F (w) + F(a), for all .x ∈ S, .w ∈ p −1 (x) and .a ∈ A(x). Composition in .ExtD (S) is defined by .(F ' , F ' )(F, F ) = (F ' F, F ' F ). The identity of a coextension .(A, E, p, +) is the morphism .(idA , idE ). Also, for any fixed .D(S)-module .A, let .

Ext(S, A)

.

denote the subcategory of .ExtD (S) whose objects are the coextensions of S by .A and whose morphisms F = (idA , F ) : (A, E, p, +) → (A, E ' , p' , +)

.

have .idA as its first component. Thus, we have an inclusion functor Ext(S, A) ⊆ ExtD (S).

.

The category .Ext(S, A) is actually a groupoid, by Corollary 2.4 below. Isomorphisms in the category .Ext(S, A) are equivalences of coextensions, and we denote by Ext(S, A) = Ext(S, A)/∼

.

the class (actually a set, see Theorem 2.9) of equivalence classes of coextensions of S by .A.

2 The .D-Cohomology of Monoids

34

The Category of 2-Cocycles of a Monoid With Schreier and Eilenberg-Mac Lane theory for the cohomology classification of coextensions of groups with abelian kernel in mind, we shall link the above categories of coextensions with corresponding ones defined in terms of 2-cocycles. Let S be a monoid. According to the definitions in Sect. 2.2, if .A is a .D(S)module, a 2-cochain .α ∈ C 2 (S, A) assigns an element .α(x, y) ∈ A(xy) to every 2 .x, y ∈ S; a 2-cocycle .α ∈ Z (S, A) is a 2-cochain such that xα(y, z) − α(xy, z) + α(x, yz) − α(x, y)z = 0,

.

for all .x, y, z ∈ S; and two such 2-cocycles .α, α ' ∈ Z 2 (S, A) are cohomologous when they differ by a 2-coboundary, equivalently, whenever there exists a 1-cochain of .φ ∈ C 1 (S, A) (that assigns an element .φ(x) ∈ A(x) to each element .x ∈ S) such that α ' (x, y) = α(x, y) + xφ(y) − φ(xy) + φ(x)y.

.

for all .x, y ∈ S. The category of (abelian) 2-cocycles of the monoid S is the category AbZ2 (S),

.

defined as follows. The objects of .AbZ2 (S) are pairs .(A, α) of a .D(S)-module .A and a 2-cocycle .α ∈ Z 2 (S, A). In .AbZ2 (S), a morphism is a pair (F, φ) : (A, α) → (A' , α ' )

.

of a morphism of .D(S)-modules .F : A → A' and a 1-cochain .φ ∈ C 1 (S, A' ) such that F∗ α = α ' + δφ.

.

Composition of .(F, φ) with .(F ' , φ ' ) : (A' , α ' ) → (A'' , α '' ) is given by (F ' , φ ' )(F, φ) = (F ' F, φ ' + F '∗ φ).

.

The result is a morphism from .(A, α) to .(A'' , α '' ) since (F ' F )∗ (α) = F '∗ F ∗ α = F '∗ (α ' + δφ) = F '∗ α ' + F '∗ δφ = α '' + δφ ' + δF '∗ φ

.

= α '' + δ(φ ' + F '∗ φ). One verifies easily that this composition of morphisms is associative and that identity morphisms exist, namely .id(A,α) = (idA , 0).

2.4 Cohomology and Coextensions

35

Recall that, if .A is a .D(S)-module, then .Der(S, A) = Z 1 (S, A). Let EndD(S) (A) ⋊ Der(S, A),

.

AutD(S) (A) ⋊ Der(S, A)

be the corresponding semidirect products with respect to the natural action (F, d) I→ F∗ d.

.

Then, a direct observation gives EndAbZ2 (S) (A, 0) = EndD(S) (A) ⋊ Der(S, A), .

(2.25)

AutAbZ2 (S) (A, 0) = AutD(S) (A) ⋊ Der(S, A).

(2.26)

.

Proposition 2.4 A morphism .(F, φ) : (A, α) → (A' , α ' ) in .AbZ2 (S) is an isomorphism if and only if .F is an isomorphism of .D(S)-modules. ' ' Proof If .F is an isomorphism, then .(F−1 , −F−1 ∗ φ) : (A , α ) → (A, α) is an inverse of .(F, φ). The converse is clear. ⨅ ⨆

For any fixed .D(S)-module .A, let Z2 (S, A)

.

denote the subcategory of .AbZ2 (S) whose objects’ first components is .A and whose morphisms’ first component is .idA : φ = (idA , φ) : (A, α) → (A, α ' ).

.

Thus, we have an inclusion functor Z2 (S, A) ⊆ AbZ2 (S).

.

Let us stress that, by Proposition 2.4, every morphism in the category .Z2 (S, A) is an isomorphism, so that .Z2 (S, A) is a groupoid. For any .α ∈ Z 2 (S, A), AutZ2 (S,A) (A, α) = Der(S, A),

.

(2.27)

and that its set of connected components is H 2 (S, A) = Z2 (S, A)/∼ =.

.

(2.28)

2 The .D-Cohomology of Monoids

36

The Twisted Semidirect Product Construction Let S be a monoid. For any given .D(S)-module .A, every 2-cocycle .α ∈ Z 2 (S, A) gives rise to a twisted semidirect product of S by .A, which is the monoid denoted by S ⋊α A,

.

(2.29)

whose underlying set is .{(x, a) | x ∈ S, a ∈ A(x)}, with multiplication (x, a)(y, b) = (xy, ay + α(x, y) + xb).

.

This multiplication is associative since .α is a 2-cocycle. Furthermore, from the equations .δα(x, 1, 1) = 0 and .δα(1, 1, x) = 0, it follows that .xα(1, 1) = α(x, 1) and .α(1, 1)x = α(1, x), whence the multiplication in .S⋊αA is unitary, with identity .(1, −α(1, 1)). Note that .S ⋊0 A = S ⋊A, the semidirect product monoid introduced in (2.21). We now have a coextension of S by .A Δ(A, α) = (A, S ⋊α A, π, +),

.

where .π : (x, a) I→ x is the projection on S and the simply transitive action + : π −1 (x) × A(x) → π −1 (x)

.

is given by .(x, a) + b = (x, a + b), for each .x ∈ S. The assignment .(A, α) I→ Δ(A, α) is the function on objects of a functor Δ : AbZ2 (S) → ExtD (S),

.

which acts on morphisms as follows. Let .(F, φ) : (A, α) → (A' , α ' ) be a morphism in .AbZ2 (S). Define (F, φ)∗ : S ⋊α A → S ⋊α ' A'

.

by .(F, φ)∗ (x, a) = (x, F(a)+φ(x)). Then, .(F, φ)∗ is a homomorphism of monoids: for every .(x, a), (y, b) ∈ S ⋊α A,     (F, φ)∗ (x, a)(y, b) = (F, φ)∗ xy, ay + xb + α(x, y)   = xy, F(a)y + xF(b) + F(α(x, y)) + φ(xy)   = xy, F(a)y + xF(b) + α ' (x, y) + xφ(y) + φ(x)y

.

2.4 Cohomology and Coextensions

37

= (x, F(a) + φ(x))(y, F(b) + φ(y)) = (F, φ)∗(x, a) (F, φ)∗(y, b), and .(F, φ)∗ (1, −α(1, 1)) = (1, −α ' (1, 1))), since Fα(1, 1) = α ' (1, 1) + ∂φ(1, 1) = α ' (1, 1) + φ(1).

.

Moreover, .π ' (F, φ)∗ = π and, for every .x ∈ S, .(x, a) ∈ S ⋊α A and .b ∈ A(x), (F, φ)∗ ((x, a) + b) = (F, φ)∗ (x, a + b) = (x, F(a) + F(b) + φ(x))

.

= (x, F(a) + φ(x)) + F(b) = (F, φ)∗ (x, a) + F(b). Thus, .(F, (F, φ)∗ ) : Δ(A, α) → Δ(A' , α ' ) is a morphism of coextensions of S. Let Δ(F, φ) = (F, (F, φ)∗ ).

.

Then, .Δ preserves identity morphisms, since .(idA , 0)∗ = idS⋊α A , and preserves composition: If .(F, φ) : (A, α) → (A' , α ' ) and .(F ' , φ ' ) : (A' , α ' ) → (A'' , α '' ) are morphisms in .AbZ2 (S), then, for every .(x, a) ∈ S ⋊α A,  .

 (F ' , φ ' )(F, φ) ∗ (x, a) = (F ' F, φ ' + F∗ φ)∗ (x, a) = (x, F ' F(a) + φ ' (x) + F ' (φ(x)) = (F ' , φ ' )∗ (x, F(a) + φ(x)) = (F ' , φ ' )∗ (F, φ)∗ (x, a),

  so that . (F ' , φ ' )(F, φ) ∗ = (F ' , φ ' )∗ (F, φ)∗ . Thus, .Δ is now a functor. Restricting .Δ to .Z2 (S, A) yields a functor Δ : Z2 (S, A) → Ext(S, A)

.

which carries each isomorphism .φ = (idA , φ) : (A, α) → (A, α ' ) in the category 2 .Z (S, A) to the equivalence φ∗ : (A, S ⋊α A, π, +) → (A, S ⋊α ' A, π ' , +)

.

defined by the monoid isomorphism .φ∗ : S ⋊α A ∼ = S ⋊α A such that φ∗ (x, a) = (x, a + φ(x)).

.

2 The .D-Cohomology of Monoids

38

Theorem 2.8 The functor .Δ : AbZ2 (S) → ExtD (S) is an equivalence of categories. For any .D(S)-module .A, the restricted functor .Δ : Z2 (S, A) → Ext(S, A) is also an equivalence of categories. Proof First, we show that every coextension .(A, E, p, +) of S by .A is equivalent to some .(A, S⋊α A, π, +). For, let us choose a map .s : S → E such that .ps = idS and .s1 = 1. Then, every element of E can now be written in the form .s(x) + a for some .x ∈ S and .a ∈ A(x), so that the map .F : S ⋊α A → E, .(x, a) I→ s(x) + a is a bijection. For every .x, y ∈ S define .α(x, y) ∈ A(xy) by s(x) s(y) = s(xy) + α(x, y).

.

(2.30)

Then, (2.30)

(2.23)

(s(x) s(y)) s(z) = (s(xy) + α(x, y)) s(z) = s(xy) s(z) + α(x, y)z (2.30)

= s(xyz) + α(x, y)z + α(xy, z),

.

(2.30)

(2.23)

s(x) (s(y) s(z)) = s(x) (s(yz) + α(y, z)) = s(x) s(yz) + xα(y, z) (2.30)

= s(xyz)) + α(x, yz) + xα(y, z)

whence, by comparison, we see that .α is a 2-cocycle. The bijection .F : S⋊αA → E is in fact an isomorphism:   F (x, a)(y, b) = F (xy, ay + α(x, y) + xb) = s(xy) + ay + α(x, y) + xb

.

(2.23)

= (s(x) + a)(s(y) + b) = F (x, a)F (y, b),

F (1, 0) = s(1) = 1, which establishes an equivalence F : (A, S ⋊α A, π, +) → (A, E, p, +).

.

We now prove that .Δ : AbZ2 (S) → ExtD (S) is full and faithful. Let .(A, α) and ' ' 2 .(A , α ) be objects of .AbZ (S) and let (F, F ) : (A, S ⋊α A, π, +) → (A' , S ⋊α ' A' , π ' , +)

.

be a morphism in .ExtD (S). We show that there is a unique .φ ∈ C 1 (S, A' ) such that ' ' 2 .(F, φ) : (A, α) → (A , α ) is a morphism in .AbZ (S) and .(F, φ)∗ = F . ' 1 If .φ ∈ C (S, A ) is such a 1-cochain, then, for any .x ∈ S, F (x, 0) = (F, φ)∗ (x, 0) = (x, φ(x)),

.

2.4 Cohomology and Coextensions

39

so .φ is uniquely determined by F . Conversely, let .φ be defined by .F (x, 0) = (x, φ(x)). For any .x, y ∈ S,       F (x, 0)(y, 0) = F xy, α(x, y) = F (xy, 0) + α(x, y)

.

= F (xy, 0) + Fα(x, y) = (xy, φ(xy)) + Fα(x, y) = (xy, Fα(x, y) + φ(xy)), F (x, 0) F (y, 0) = (x, φ(x))(y, φ(y)) = (xy, x φ(y) + φ(x) y + α ' (x, y)), whence we obtain that .F∗ α = α ' + ∂φ. Therefore, .(F, φ) : (A, α) → (A' , α ' ) is a morphism in the category .AbZ2 (S). Finally, we see that .(F, φ)∗ = F since F (x, a) = F ((x, 0) + a) = F (x, 0) + F(a) = (x, φ(x)) + F(a)

.

= (x, F(a) + φ(x)) = (F, φ)∗ (x, a), for every .x ∈ S and .a ∈ A(x).

⨆ ⨅

Corollary 2.3 Let S be a monoid. For any .D(S)-module .A, the functor .Δ induces isomorphisms EndD(S) (A) ⋊ Der(S, A) ∼ = EndExtD (S) (A, S ⋊A, π, +),

.

AutD(S) (A) ⋊ Der(S, A) ∼ = AutExtD (S) (A, S ⋊A, π, +). ⨆ Proof This follows from Theorem 2.8 and the isomorphisms (2.25) and (2.26). ⨅ Corollary 2.4 Let S be a monoid. For any .D(S)-module .A, the category .Ext(S, A) is a groupoid; that is, every morphism in .Ext(S, A) is an equivalence. Proof .Ext(S, A) is equivalent to .Z2 (S, A), which is a groupoid.

⨆ ⨅

The following theorem, which follows from (2.27) and (2.28), mainly states that H 2 (S, A) classifies coextensions of S by .A.

.

Theorem 2.9 Let S be a monoid. For any .D(S)-module .A, the equivalence of groupoids .Δ : Z2 (S, A) ≃ Ext(S, A) induces a bijection between the sets of connected components H 2 (S, A) ∼ = Ext(S, A)

.

(2.31)

and, for any 2-cocycle .α ∈ Z 2 (S, A), an isomorphism of groups Der(S, A) ∼ = AutExt(S,A) (A, S ⋊α A, π, +).

.

□

.

2 The .D-Cohomology of Monoids

40

Remark 2.4 In [13, Chapter 3], a Baer sum of coextensions is defined such that the bijection (2.31) becomes an isomorphism of abelian groups. Cf., also, [8] and [19].

Dependence on S of ExtD (S) Given a homomorphism of monoids .f : S → T be a homomorphism of monoids, we construct a functor f ∗ : AbExt(T ) → ExtD (S)

.

as follows. First, every coextension .(A, E, p, +) of T by a .D(T )-module .A yields the coextension of S by the .D(S)-module .f ∗A f ∗ (A, E, p, +) = (f ∗A, f ∗ E, pS , +)

.

where .f ∗ E = E ×T S = {(w, x) ∈ E × S | pw = f x} is the pullback monoid on p and f , .pS : f ∗ E → S is the projection, .pS (w, x) = x, and, for each .x ∈ S, every −1 ∗ .a ∈ (f A)(x) = A(f (x)) acts on the fibre .p S (x) by .(w, x) + a = (w + a, x). If ' ' ' .(F, F ) : (A, E, p, +) → (A , E , p , +) is a morphism in .AbExt(T ), then f ∗ (F, F ) = (f ∗ F, f ∗ F ) : (f ∗A, f ∗ E, pS , +) → (f ∗A' , f ∗ E ' , pS' , +)

.

where .f ∗ F : f ∗ E → f ∗ E ' is the homomorphism defined by (f ∗ F )(w, x) = (F (w), x).

.

For any fixed .D(T )-module .A, the functor .f ∗ above restricts to a functor f ∗ : Ext(T , A) → Ext(S, f ∗A).

.

Similarly, the homomorphism .f : S → T induces a functor f ∗ : AbZ2 (T ) → AbZ2 (S)

.

defined on objects of .AbZ2 (T ) by .f ∗ (A, α) = (f ∗A, f ∗ α) (see (2.15)) and on morphisms .(F, φ) : (A, α) → (A' , α ' ) by f ∗ (F, φ) = (f ∗ F, f ∗ φ) : (f ∗ A, f ∗ α) → (f ∗ A' , f ∗ α ' ),

.

the latter is a morphism .AbZ2 (S) since (f ∗ F)∗ (f ∗ α) = f ∗ (F∗ α) = f ∗ (α ' + δφ) = f ∗ α ' + f ∗ δφ = f ∗ α ' + δf ∗ φ.

.

2.4 Cohomology and Coextensions

41

Thus constructed, .f ∗ is a functor: indeed .f ∗ (idA , 0) = (idf ∗ A , 0), so that .f ∗ preserves identities. If .(F, φ) : (A, α) → (A' , α ' ), .(F ' , φ ' ) : (A' , α ' ) → (A'' , α '' ) are two composable morphisms in .AbZ2 (T ), then f ∗ (F ' , φ ' ) f ∗ (F, φ) = (f ∗ F ' , f ∗ φ ' )(f ∗ F, f ∗ φ)   = f ∗ F ' f ∗ F, f ∗ φ ' + (f ∗ F)∗ (f ∗ φ)

.

= (f ∗ (F ' F), f ∗ (φ ' + F∗ φ)) = f ∗ (F ' F, φ ' + F∗ φ)   = f ∗ (F ' , φ ' )(F, φ) , so that .f ∗ preserves compositions. For any fixed .D(T )-module .A, the functor .f ∗ : AbZ2 (T ) → AbZ2 (S) restricts to a functor f ∗ : Z2 (T , A) → Z2 (S, f ∗A).

.

Theorem 2.10 For any monoid homomorphism .f : S → T and any .D(T )-module A, the squares below commute.

.

.

.

Proof For any 2-cocycle .α ∈ Z 2 (T , A), after the identification (T ⋊α A) ×T S = S ⋊f ∗α f ∗A,

.

  (f (x), a), x = (x, a),

the proof only requires a straightforward verification which we leave to the reader. ⨆ ⨅

2 The .D-Cohomology of Monoids

42

2.5 The Cohomological D-Dimension of Monoids The cohomological .D-dimension of a monoid S is the cohomological dimension Dim(S) = cd(D(S)) of the category .D(S), defined by the condition that .Dim(S) ≤ m whenever .H n (S, A) = 0 for all .n > m and all .D(S)-modules .A.

.

Theorem 2.11 Let S be a monoid. The following statements are equivalent .(i) .Dim(S) = 0. (ii) .D(S) has an initial object. .(iii) .S = {1}. .

Proof Since .(iii) ⇒ (ii) ⇒ (i), we prove that .(i) ⇒ (iii). For every .x ∈ S, we have .D(S)(1, x) /= ∅; hence .D(S) is a connected small category and Laudal’s Theorem 1.3 applies and .D(S) has an initial pair .(x, 〈u, x, v〉), where .〈u, x, v〉 : x → x, so that .uxv = x. By Proposition 1.2, .D(S)(x, y) /= ∅ for all .y ∈ S, and 〈u' , x, v ' 〉〈u, x, v〉 = 〈u'' , x, v '' 〉〈u, x, v〉

.

for all .〈u' , x, v ' 〉, 〈u, x, v〉 : x → y, so that u' u = u'' u, vv ' = vv ''

.

for all .u' , u'' , v ' , v '' ∈ S. In particular, .u2 = 1u = u and .v 2 = v1 = v. Hence .ux = u(uxv) = uxv = x; similarly .xv = x. Then yx = yux = 1ux = x

.

for all .y ∈ S; dually .xz = x for all .z ∈ S. Since .D(S)(x, 1) /= ∅ we also have .yxz = 1 for some .y, z ∈ S, so that .x = yxz = 1 and .s = sx = x = 1 for all .s ∈ S. Thus S is trivial. ⨅ ⨆ We now look at monoids of cohomological .D-dimension is 1. We begin with a consequence of Theorem 2.8. Lemma 2.4 Let S be a monoid and let .A be a .D(S)-module. For any 2-cocycle α ∈ Z 2 (S, A), the following are equivalent:

.

(i) .[α] = 0 in .H2 (S, A).  .(ii) .HomExt(S,A) (A, S ⋊A, π, +), (A, S ⋊α A, π, +) is nonempty.   .(iii) .HomExt (S) (0, S, idS , +), (A, S ⋊α A, π, +) is nonempty. D .

id

π

(iv) .HomMon↓S (S → S, S ⋊α A → S) is nonempty.

.

Proof By Theorem 2.8 there are bijections of the set .{φ ∈ C 1 (S, A) | α = −δφ} onto the sets of morphisms in .(ii) and .(iii). Hence .(i) ⇔ (ii) ⇔ (iii). The

2.5 The Cohomological D-Dimension of Monoids

43

implication .(iii) ⇒ (iv) is trivial. To prove that .(iv) implies .(i), assume that there is a homomorphism .f : S → S ⋊αA such that .πf = idS . Define .φ ∈ C 1 (S, A) by .f (x) = (x, φ(x)). From the equality .f (xy) = f (x) f (y), it follows that φ(xy) = φ(x) y + α(x, y) + x φ(y).

.

for all .x, y ∈ S. Thus .α = −δφ and .[α] = 0 in .H 2 (S, A).

⨆ ⨅

Theorem 2.12 Let S be a nontrivial monoid. Then the following are equivalent: Dim(S) = 1. The functor .H 1 (S, −) : D(S)-Mod → Ab is right exact. The functor .Der(S, −) : D(S)-Mod → Ab is exact. The ideal augmentation .IS is a projective .D(S)-module. 2 .H (S, A) = 0 for all .D(S)-module A. For any coextension .(A, E, p, +) of S by a .D(S)-module .A, the terminal morphism .(0, p) : (A, E, p, +) → (0, S, idS , +) is a retraction in .ExtD (S). .(vii) For any coextension .(A, E, p, +) of S by a .D(S)-module .A, the terminal morphism .p : E → S is a retraction in .Mon ↓S . .(i) (ii) .(iii) .(iv) .(v) .(vi)

.

.

Proof Since S is nontrivial, we have .Dim(S) ≥ 1. Thus .(i), .(ii) and .(v) are equivalent by Lemma 1.1. The functor .Der(S, −) is always left exact and .H 1 (S, −) is always middle exact. Since H 1 (S, −) = Der(S, −)/IDer(S, −),

.

we have that .(ii) is equivalent with .(iii). The equivalence .(iii) ⇔ (iv) follows from Proposition 2.2. Finally, the equivalence of .(v), .(vi) and .(vii) follows from Lemma 2.4 and Theorem 2.8. ⨆ ⨅ As above, let .G(S) denote the group of units of a monoid S. A monoid S is free on a subset X of S with inverses in a subset .X' of .X ∩ G(S) if every mapping f of X into a monoid T such that .f (X' ) ⊆ G(T ) extends uniquely to a morphism .f¯ : S → T . If S is free on some subset X with inverses in some subset .X' of X, then S is a semifree monoid. If a pair .(X, X' ) is  given with .X' ⊆ X, then the submonoid of the free group on X generated by .X (X' )−1 is free on X with inverses in .X' . Moreover, for a given pair .(X, X' ) any two monoids which are free on X with inverses in .X' are isomorphic by the usual argument. By Theorem 2.6, we know that if S is a free monoid then .Dim(S) = 1. We extend this result to semifree monoids. We first need a lemma, Lemma 2.5 Let S be a monoid and let .(A, E, p, +) be a coextension of S by a D(S)-module .A. If .x ∈ G(S), then .p−1 (x) ⊆ G(E).

.

2 The .D-Cohomology of Monoids

44

Proof Let .w ∈ p−1 (x) and let us choose any .w ' ∈ p−1 (x −1 ). Since both .ww ' and ' −1 (1), we have .ww ' = 1 + a and .w ' w = 1 + b for some .a, b ∈ A(1). .w w are in .p Then, by (2.23), w(w ' + x −1 (−a)a) = ww ' + (−a) = 1 + (a − a) = 1,

.

(w' + (−bx −1 )w = w' w + (−b) = 1 + (−b + b) = 1, whence .w −1 = w ' + x −1 (−a) = w' + (−bx −1 ) is an inverse of w in E.

⨆ ⨅

Theorem 2.13 Let S be a nontrivial semifree monoid. Then .Dim(S) = 1. Proof Suppose S free on X with inverses in .X' . If .(A, E, p, +) is any coextension of a monoid S by a .D(S)-module .A, we can choose an element .wx ∈ p −1 x for each .x ∈ X and define the map .f : X → E by .f (x) = wx . By Lemma 2.5 above, ' .f (X ) ⊆ G(E). Therefore f extends to a monoid homomorphism .f¯ : S → E. ⨆ ⨅ Then .p f¯ = idS . Hence, by Theorem 2.12 .(vii), .Dim(S) = 1.

2.6 The Cohomology of Cyclic Monoids The structure of cyclic monoids was first stated by Frobenious [9] and Moore [20]. (See [7].) Briefly, let .C = 〈t〉 = {t n | n ∈ N} be a cyclic monoid with a generator t. If .t m /= t n for all .m /= n, then C is the free monoid with generator t and we denote it by .C∞ : C∞ = {1, t, t 2 , . . .} ∼ = N.

.

If there exists m such that .t m = t n for some .n /= m, the last such m is the index of C, and the last .q ≥ 1 such that .t m = t m+q is its period. The index and the period uniquely determines the set C: t x = t y if and only if either x = y < m, or x, y ≥ m and x ≡ y mod q,

.

so that the underlying set of this monoid can be described as the set C = {1, t, . . . , t m , t m+1 , . . . , t m+q−1 }.

.

The index and the period also determine the multiplication of C: t x t y = t ℘ (x+y) ,

.

where .℘x = x if .x < m + q, otherwise .℘x is the integer such that m ≤ ℘x < m + q

.

and

℘x ≡ x mod q.

2.6 The Cohomology of Cyclic Monoids

45

Up to isomorphism there is therefore only one cyclic monoid with index m and period q; we denote it by .Cm,q . Note that .Cq = {t m , t m+1 , . . . , t m+q−1 } is a cyclic of order q subgroup of .Cm,q . In this section, we compute the cohomology groups of cyclic monoids. A computation of the homology groups for these monoids can be found in [3]. Theorems 2.5 and 2.6 yield the cohomology groups of .C∞ : Theorem 2.14 For any .D(C∞ )-module .A, there is a natural exact sequence of abelian groups δ

0 → H 0 (C∞ , A) → A(1) → A(t) → H 1 (C∞ , A) → 0,

.

where the homomorphism .δ is defined by .δa = ta − at. Furthermore, H n (C∞ , A) = 0,

.

for all n ≥ 2. □

.

In what follows we focus on the finite cyclic monoids .Cm,q and assume that m + q ≥ 2, so that .Cm,q is not the trivial monoid. For any .D(Cm,q )-module .A, homomorphisms .S, .Q and .T are defined as follows. For every .0 ≤ j, k < m + q, let

.

S : A(t k ) → A(t k+1 )

.

be given by a I→ S(a) = ta − at, .

Qj : A(t k ) → A(t k+j ) be given by a I→ Qj (a) =

j 

t i a t j −i ,

(2.32) (2.33)

i=0

and, for each .0 < k < m + q, let the “trace map” T : A(t k ) → A(t m+k−1 ) be given by a I→ Qm+q−1 (a) − Qm−1 (a).

.

(2.34)

Lemma 2.6 For any left .D(Cm,q )-module .A and .0 ≤ k, j < m + q, the triangular regions in diagram below commute (whence the outside square too)

.

where the diagonal .A(t k ) → A(t k+j +1 ) is the homomorphism given by a I→ t j +1 a − a t j +1 .

.

2 The .D-Cohomology of Monoids

46

Proof .SQj (a) =

j

t i+1 a t j −i −

i=0

we see that .Qj S : a I→ t j +1 a

j

t i a t j −i+1 = t j +1 a − a t j +1 . Analogously,

i=0 − a t j +1 .

⨆ ⨅

Lemma 2.7 For every .D(Cm,q )-module .A, the sequences S

T

A(t k ) −→ A(t k+1 ) −→ A(t m+k ),

.

T

S

A(t k ) −→ A(t m+k−1 ) −→ A(t m+k ),

are null, that is, .T S = 0 and .S T = 0. Proof By Lemma 2.6, .TS = ST. Furthermore,   ST(a) = S Qm+q−1 (a) − Qm−1 (a) = SQm+q−1 (a) − SQm−1 (a)

.

= t m+q a − at m+q − t m a + at m = t m a − at m − t m a + at m = 0 for every .a ∈ A(t k ).

⨆ ⨅

Lemma 2.8 For every .D(Cm,q )-module .A, the following squares commute:

.

.

Proof The first square commutes since .TS = 0. The other two squares commute since .Cm,q is commutative. ⨆ ⨅ Lemma 2.9 For every morphism .F : A → A' of .D(Cm,q )-modules, the following squares commute, so that .S, .Qj and .T are natural homomorphisms;

.

2.6 The Cohomology of Cyclic Monoids

47

Proof This is straightforward.

⨆ ⨅

A Resolution of Z by Free D(Cm,q )-modules It is possible to calculate efficiently the (co)homology of finite cyclic monoids by a clever choice of resolution. We construct below a specific free resolution of the trivial .D(Cm,q )-module .Z, .

ϵ F¯ • → Z :

∂ ∂ ϵ · · · → F¯ 2 → F¯ 1 → F¯ 0 → Z → 0,

(2.35)

as follows. For each integer .r ≥ 0, we choose symbols .vr and .wr , define functors ¯ 2r : ObD(Cm,q ) → Set and .X ¯ 2r+1 : ObD(Cm,q ) → Set by X

.

 .

¯ 2r (t ) = X k

{vr } if t k = t rm ∅ otherwise,

 ¯ 2r+1 (t ) = X k

{wr } if t k = t rm+1 , ∅ otherwise,

¯ 2r and .F¯ 2r+1 = FX ¯ 2r+1 to be the free .D(Cm,q )-modules on and define .F¯ 2r = FX ¯ ¯ .X2r and .X2r+1 , respectively. By Remark 2.1, .vr can be treated as a generator of .F¯ 2r (t rm ), and every generator of .F¯ 2r (t k ) can be written uniquely in the form .t u vr t v for some .u, v such that u+rm+v = t k ; similarly, .w can be treated as a generator of .F ¯ 2r+1 (t rm+1 ), and .t r k u ¯ every generator of .F2r (t ) can be written uniquely in the form .t wr t v for some .u, v such that .t u+rm+1+v = t k . The differential .∂ : F¯ 2r+2 → F¯ 2r+1 is the morphism of .D(Cm,q )-modules such that ∂(vr+1 ) = T(wr ),

.

where .T : F¯ 2r+1 (t rm+1 ) → F¯ 2r+1 (t (r+1)m ) is the trace map (2.34), and the differential .∂ : F¯ 2r+1 → F¯ 2r is the morphism of .D(Cm,q )-modules such that ∂(wr ) = S(vr ),

.

where .S : F¯ 2r (t rm ) → F¯ 2r (t rm+1 ) is the homomorphism (2.32). The augmentation ¯ 0 → Z is the morphism such that .ϵ : F ϵ(v0 ) = 1 ∈ Z(1) = Z.

.

We are now ready to establish the main result of this subsection. Theorem 2.15 .F¯ • → Z, defined as above, is a free resolution (hence projective) of the .D(Cm,q )-module .Z.

2 The .D-Cohomology of Monoids

48

Proof First, we prove that .F¯ • → Z is an augmented complex of .D(Cm,q )-modules. ∂ ϵ ¯ 0→ Z is null since, by Lemma 2.9, The sequence .F¯ 1 → FF 2.9

ϵ∂(w0 ) = ϵS(v0 ) = Sϵ(v0 ) = t1 − 1t = 1 − 1 = 0.

.

Similarly, .∂∂ = 0 since, for any .r ≥ 1, 2.9

2.7

∂∂(wr ) = ∂S(vr ) = S∂(vr ) = ST(wr−1 ) = 0,

.

2.9

2.7

∂∂(vr+1 ) = ∂T(wr ) = T∂(wr ) = TS(vr ) = 0. To prove exactness, for any fixed k with .0 ≤ k < m + q, we construct a contracting homotopy on the augmented complex of abelian groups .

· · · → F¯ 2 (t k ) → F¯ 1 (t k ) → F¯ 0 (t k ) → Z → 0.

ϵ ¯ k) → Z: F(t

∂

∂

ϵ

(2.36)

Let .φ : Z → F¯ 0 (t k ), .Ф : F¯ 2r+1 (t k ) → F¯ 2r+2 (t k ) and .Ф : F¯ 2r (t k ) → F¯ 2r+1 (t k ) be the homomorphisms such that φ(1) = v0 t k ,

.

Ф(t wr t ) = u



0

v

Ф(t u vr t v ) =

vr+1 u−1 

tv

if u < m + q − 1, if u = m + q − 1,

t i wr t u+v−i−1 = Qu−1 (wr ) t v .

i=0

These homomorphisms constitute a contracting homotopy of (2.36). Indeed: k k k .ϵφ = idZ , since .ϵφ(1) = ϵ(v0 t ) = ϵ(v0 )t = 1 t = 1. u+v = t k , then .∂Ф + φ ϵ = id ¯ F (t x ) , since, if .0 ≤ u, v < m + q and .t 0

2.9

∂Ф(t u v0 t v ) = ∂(Qu−1 (w0 )t v ) = ∂(Qu−1 (w0 )t v ) = (Qu−1 ∂(w0 ))t v

.

2.6

= (Qu−1 S(v0 ))t v = (t u v0 − v0 t u )t v = t u v0 t v − v0 t u+v = t u v0 t v − v0 t k = t u v0 t v − φ(1) = t u v0 t v − φ ϵ(t u v0 t v ) = (idF¯ ∂Ф + Ф∂ = idF¯

.

m + q − 1, then

2r+1 (t

0 (k)

k)

− φϵ)(t u v0 t v ).

, since if .t u wr t v is a generator of .F¯ 2r+1 (t k ) and .u
1, are given by ⎧ ⎪ (x , . . . , xn ) if i = 0, ⎪ ⎨ 2 .di (x1 , . . . , xn ) = (x1 , . . . , xi xi+1 , . . . , xn ) if 0 < i < n, ⎪ ⎪ ⎩ (x , . . . , x ) if i = n, 1 n−1 There is a canonical functor π : Δ(BS) → D(S),

.

which acts on objects by π(x1 , . . . , xn ) = x1 · · · xn

.

3.5 Gabriel-Zisman Cohomology

93

and carries a morphism .α : (y1 , . . . , ym ) → (x1 , . . . , xn ) in .Δ(BS) to the morphism π(α) : y1 · · · ym → x1 · · · xn of .D(S) defined by

.

⎧ 〈x1 · · · xα(0) , y1 · · · ym , xα(m)+1 · · · xn 〉 ⎪ ⎪ ⎪ ⎪ ⎨ 〈1, y1 · · · ym , xα(m)+1 · · · xn 〉 .π(α) = ⎪ 〈x1 · · · xα(0) , y1 · · · ym , 1〉 ⎪ ⎪ ⎪ ⎩ 〈1, y1 · · · ym , 1〉

if 0 < α(0) and α(m) < n, if 0 = α(0) and α(m) < n, if 0 < α(0) and α(m) = n, if 0 = α(0) and α(m) = n.

Composition with .π yields a functor π ∗ : D(S)-Mod → Δ(BS)-Mod,

.

and thus each .D(S)-module .A gives a coefficient system .π ∗A on BS and Gabrieln (BS, π ∗A) are defined. Zisman cohomology groups .HGZ Theorem 3.26 Let S be a monoid. For any .D(S)-module .A there are natural isomorphisms n H n (S, A) ∼ (BS, π ∗A), = HGZ

.

n = 0, 1, . . . .

• (BS, π ∗A). For every .n ≥ 0, Proof Let us analyze the cochain complex .CGZ n CGZ (BS, π ∗A) =

.



A(π(x1 , . . . , xn )) =

(x1 ,...,xn )∈S n



A(x1 · · · xn ).

(x1 ,...,xn )∈S n

n−1 n (BS, π ∗A), observe that To describe the coboundary .δ : CGZ (BS, π ∗A) → CGZ i the morphism .d : di (x1 , . . . xn ) → (x1 , . . . , xn ) of .Δ(BS), for every .0 ≤ i ≤ n and .(x1 , . . . , xn ) ∈ S n , induces the homomorphism

d∗i = Aπ(d i ) : Aπ(di (x1 , . . . , xn )) → Aπ(x1 , . . . , xn ),

.

namely ⎧ ⎪ A〈x1 , x2 · · · xn , 1〉 : A(x2 · · · xn ) → A(x1 · · · xn ) i = 0, ⎪ ⎨ i .d∗ = 0 < i < n, A〈1, x1 · · · xn , 1〉 : A(x1 · · · xn ) → A(x1 · · · xn ) ⎪ ⎪ ⎩ A〈1, x · · · x , x 〉 : A(x · · · x ) → A(x · · · x ) i = n. 1 n−1 n 1 n−1 1 n

94

3 Other Cohomologies

n−1 n (BS, π ∗A) is given by Hence, the coboundary .δ : CGZ (BS, π ∗A) → CGZ

(δϕ)(x1 , . . . , xn ) =

.

n  (−1)i d∗i ϕ(di (x1 , . . . , xn )) i=0

= x1 ϕ(x2 , . . . , xn ) +

n−1 

(−1)i ϕ(x1 , . . . , xi xi+1 , . . . , xn )

i=1

+ (−1)n ϕ(x1 , . . . , xn−1 ) xn . • (BS, π ∗A) is the standard cochain complex .C • (S, A), (2.10), that Thus, .CGZ computes the .D-cohomology of the monoid S with coefficients in .A. The result follows from the isomorphisms (2.11) and (3.10). ⨆ ⨅

3.6 Wells Cohomology This cohomology is a natural generalization to small categories of the .Dcohomology of monoids. Though commonly attributed to Baues and Wirsching [4], it can be found in an unpublished 1980 article by Wells [26] (the internet link is to a copy of a retyped version made available in 2001). Let .C be a small category. Its division category, .D(C), is the category whose objects are the morphisms of .C, with a morphism from .x : a → b to .y : c → d a triple of arrows of .C 〈u, x, v〉

.

such that .uxv = y; equivalently, such that the following square commutes:

.

Composition is given by the formula 〈u' , uxv, v ' 〉〈u, x, v〉 = 〈u' u, x, vv ' 〉,

.

and the identity arrow at any .x : a → b is the triple .〈1b , x, 1a 〉 : x → x.

(3.11)

3.6 Wells Cohomology

95

The cohomology groups .HWn (C, A) of the small category .C with coefficients in a .D(C)-module .A (a natural system in the terminology of [4]) are those of its division category, that is, .

If .C = S is a monoid, regarded as a category with one object, then .D(C) is the division category .D(S) of S and the Wells cohomology of .C reduces to the .D-cohomology of S.

Cocycles We construct an explicit cochain complex that computes the cohomology groups HWn (C, A). Recall that the classifying simplicial set or nerve of .C is the simplicial set

.

BC : Δop → Set

.

whose n-simplices are all composable sequences x1

xn

a0 ← a1 ← · · · ← an−1 ← an

.

of n arrows in .C if .n > 0, objects .a0 of .C if .n = 0, and in which .α ∗ : BCn → BCm is defined, for any map .α : m → n in .Δ, by  x1  y1 ym xn α ∗ a0 ← · · · ← an = aα(0) ← · · · ← aα(m)

.

where, for each .i = 0, . . . , m − 1,  yi+1 =

xα(i)+1 · · · xα(i+1) if α(i) < α(i + 1),

.

if α(i) = α(i + 1).

1aα(i)

Thus the face maps .d0 , d1 : BC1 → BC0 are x1

d0 (a0 ← a1 ) = a1 ,

.

x1

d1 (a0 ← a1 ) = a0 ,

96

3 Other Cohomologies

and, for .n > 1,

.

As in the case of a monoid, there is a canonical functor π : Δ(BC) → D(C)

.

defined on objects by .

.

If y1

ym

x1

xn

α : (aα(0) ← · · · ← aα(m) ) −→ (a0 ← · · · ← an )

.

is a morphism of .Δ(BC), then

.

is the morphism of .D(S) described by the commutative square below:

.

Composition with .π yields a functor π ∗ : D(C)-Mod → Δ(BC)-Mod

.

n (BC, π ∗A) are defined for every and Gabriel-Zisman cohomology groups .HGZ .D(C)-module .A. Let • • CW (C, A) = CGZ (BC, π ∗A)

.

be the standard cochain complex that computes these cohomology groups.

3.6 Wells Cohomology

97 v

x

u

For every morphisms .c → a → b → c of .C and .ξ ∈ A(x), let  .

uξ = A〈u, x, 1a 〉(ξ ) ∈ A(ux),

(3.12)

ξ v = A〈1b , x, v〉(ξ ) ∈ A(xv).

• (C, A) is as follows: For every .n ≥ 0, The cochain complex .CW

.

x1

xn

For every .0 ≤ i ≤ n and every .(a0 ← · · · ← an ) ∈ BCn , the morphisms x1

x1

xn

xn

d i : di (a0 ← · · · ← an ) → (a0 ← · · · ← an )

.

of .Δ(BC) induce homomorphisms x1

xn

x1

xn

d∗i = Aπ(d i ) : Aπ(di (a0 ← · · · ← an )) → Aπ(a0 ← · · · ← an )

.

given by .

.

and, for .0 < i < n, .

.

n−1 n (BC, π ∗A) is given by Hence the coboundary .δ : CW (C, A) → CW

.

n−1 for every .ϕ ∈ CW (C, A).

,

98

3 Other Cohomologies

Theorem 3.27 Let .C be a small category. For any .D(C)-module .A there are natural isomorphisms • HWn (C, A) ∼ (C, A), = H n CW

n = 0, 1, . . . ,

.

Proof Construct a free resolution .P• of the constant .D(C)-module .Z, as follows. For every .n ≥ 1, define .Xn : ObD(C) → Set by x1

x

xn

Xn (b ← a) = {(b = a0 ← · · · ← an = a) ∈ BCn | x1 · · · xn = x}

.

x

and, for .n = 0, define .X0 : ObD(C) → Set by .X0 (b ← a) = ∅ if .x /= 1a and .X0 (1a ) = {1a }. Let .Pn = FXn be the free .D(C)-module on .Xn . The boundary .∂ : Pn → Pn−1 is the unique morphism of .D(C)-modules such that

,

.

x1

x1 ···xn

xn

for every .(a0 ← · · · ← an ) ∈ Xn (a0 ←− an ). And the augmentation .∂ : P0 → Z is the morphism of .D(C)-modules such that .∂[1a ] = 1. For every morphism .x : a → b of .C, .Pn (x) is therefore the free abelian group x0

x

x1

xn+1

xn

Pn (b ← a) = Z[{(b ← a0 ← · · · ← an ←− a) | x0 x1 · · · xn xn+1 = x}];

.

for every morphism .〈u, x, v〉 : x → y in .D(C), as in (3.11), y

x

Pn 〈u, x, v〉 : Pn (b ← a) → Pn (d ← c), ξ I→ uξ v,

.

is the homomorphism such that x0

x1

xn+1

xn

ux0

x1

xn

xn+1 v

u(b ← a0 ← · · · ← an ←− a)v = (d ← a0 ← · · · ← an ←− a),

.

x0

x1

xn

xn+1

x

for every generator .b ← a0 ← · · · ← an ←− a of .Pn (b ← a). The boundary x

x

∂ : Pn (b ← a) → Pn−1 (b ← a)

.

3.6 Wells Cohomology

99

is the homomorphism defined on generators by

,

.

x0

x

x1

and the augmentation .∂ : P0 (b ← a) → Z by .∂(b ← a0 ← a) = 1. .P• is actually a projective resolution of .Z in the category of .D(C)-modules, since x every augmented chain complex .P• (b ← a) → Z has a contracting homotopy, which consists of the homomorphisms x

Ф−1 : Z → P0 (b ← a)

.

x

x

and Фn : Pn (b ← a) → Pn+1 (b ← a),

such that ⎧ 1b x ⎨ Ф−1 (1) = (b ← b ← a), . xn+1 xn+1 x0 1b x0 x1 x1 xn xn ⎩ Фn (b ← a0 ← · · · ← an ←− a) = (b ← b ← a0 ← · · · ← an ←− a). x

for every generator of .Pn (b ← a). Therefore, .P• → Z → 0 is exact, and is a projective resolution of .Z. By (2.2), we have an isomorphism of cochain complexes • HomD(C) (P• , A) ∼ (C, A), = CW

.

⨆ ⨅

whence the theorem follows.

Theorem 3.27 implies that Wells cohomology of small categories, like the Leech cohomology of monoids, is a particular case of the Gabriel-Zisman cohomology of simplicial sets: Corollary 3.10 Let .C be a small category. For any .D(C)-module .A there are natural isomorphisms n HWn (C, A) ∼ (BC, π ∗A), = HGZ

.

n = 0, 1, . . . .

Coextensions Let .C be a small category. A coextension of .C is a functor .P : E ↠ C which is the identity on objects and surjective on arrows. If .A is a .D(C)-module, a coextension (A, E, P , +)

.

100

3 Other Cohomologies

of .C by .A (a linear extension in Baues-Wirsching’s terminology) is a coextension P : E ↠ C of .C endowed, for each morphism .x : a → b of .C, with a simplytransitive right action .+ of .A(x) on .P −1 (x), denoted by

.

(f : a → b, η) I→ f + η : a → b,

.

y

x

such that, for any two composable arrows .a → b → c of .C, .f ∈ P −1 (x), .g ∈ P −1 (y), .η ∈ A(x), and .ζ ∈ A(y), (g + ζ )(f + η) = gf + (ζ x + yη),

.

where .ζ x and .yη are as in (3.12). For instance, every 2-cocycle .α ∈ Z 2 (C, A) yields a (twisted crossed product) coextension of .C by .A, (A, C⋊α A, P , +)

(3.13)

.

as follows: .C⋊α A is the category with the same objects as .C and whose morphisms (x, η) : a → b are pairs with .x : a → b a morphism in .C and .η ∈ A(x). The composition of two morphisms

.

.

is defined by the formula y

x

(y, ζ )(x, η) = (yx, ζ x + yη + α(c ← b ← a)).

.

A straightforward verification shows that this composition is associative thanks to the fact that .α is a 2-cocycle. Furthermore, for any morphism .x : a → b in .C, we x

1a

1b

1a

1b

x

x

1a

1b

x

have .δα(b ← a ← a ← a) = 0 and .δα(b ← b ← b ← a) = 0, so that 1a

1a

x α(a ← a ← a) = α(b ← a ← a)

.

and 1a

1a

α(a ← a ← a) x = α(b ← b ← a);

.

1a

1a

hence the composition in .C⋊α A is also unitary, with .(1a , −α(a ← a ← a)) the identity morphism of every object a.

3.6 Wells Cohomology

101

The coextension .(A, C⋊α A, P , +) of .C by .A, is then defined by the projection functor .P : C⋊α A → C, .P (x, η) = x, together the actions + : π −1 (x) × A(x) → π −1 (x)

.

given by .(x, η) + η' = (x, η + η' ). Let .(A, E, P , +) and .(A, E' , P ' , +) be coextensions of .C by .A. An equivalence between them is an isomorphism .F : E ∼ = E' such that .P ' F = P and, for any −1 .x : a → b in .C, .f ∈ P (x) and .η ∈ A(x), F (f + η) = F (f ) + η.

.

For every .D(C)-module .A, let Ext(C, A)

.

denote the collection (actually a set, by the classification theorem below) of equivalence classes .[A, E, P , +] of coextensions .(A, E, P , +) of category .C by a .A. Theorem 3.28 (Classification) Let .C be a small category. For any .D(C)-module A there is a natural bijection

.

Ext(C, A) ∼ = HW2 (C, A).

.

Proof Let .(A, E, P , +) be an extension of .C by a .D(C)-module .A. For each s(x)

x

morphism .a ← b of .C, let us choose a morphism .a ←− b in .P −1 (x). For every y x .(a ← b ← c) ∈ BC2 there is a unique x

y

αs (a ← b ← c) ∈ A(xy)

.

such that .

.

Then, .αs is a 2-cocycle αs = αs (A, E, p, +) ∈ Z 2 (C, A),

.

102

3 Other Cohomologies x

y

z

since, for every .(a ← b ← c ← d) ∈ BC3 , associativity in .E implies

.

.

The cohomology class .[αs ] ∈ H 2 (C, A) does not depend on the choice of the x morphisms .s(x) in .P −1 (x) for each .a ← b of .C. Indeed, if .s ' (x) is any other selection, then there is a unique 1-cochain .φ ∈ C 1 (C, A) such that s(x) = s ' (x) + φ(x)

.

x

x

y

for any .a ← b. For every .(a ← b ← c) ∈ BC2 ,

,

.

whence x

y

x

y

x

y

αs ' (a ← b ← c) = αs (a ← b ← c) + δφ(a ← b ← c),

.

So that .αs ' = αs + δφ. If .(A, E, P , +) and .(A, E' , P ' , +) are two coextensions of .C by .A and .F : E∼ = E' is an equivalence between them, then .αFs (A, E' , P ' , +) = αs (A, E, P , +). Hence, we have a well-defined map Ψ : Ext(C, A) → HW2 (C, A)

.

given by .Ψ[A, E, P , +] = [αs (A, E, P , +)]. This map .Ψ is surjective: For any 2-cocycle .α ∈ Z 2 (C, A), let .(A, C⋊αA, P , +) be the twisted crossed product coextension of .C by .A, as in (3.13). If, for every

3.7 Grothendieck and Beck Cohomologies

103

x

morphism .a ← b of .C, we choose the morphism .s(x) = (x, 0) in .P −1 (x), we just obtain that αs (A, C⋊α A, π, +) = α,

.

and thus .Ψ[A, C⋊α A, P , +] = [α]. Finally, the map .Ψ is injective: If .(A, E, P , +) is any coextension of .C by .A, and .αs = αs (A, E, P , +), then there is an equivalence of coextensions between .(A, C⋊α A, P , +) and .(A, E, P , +), which is established by the identity on objects isomorphism .C ⋊αs A ∼ = E such that .(x, η) I→ s(x) + η. Furthermore, if two 2cocycles .α, α ' ∈ Z 2 (C, A) are cohomologous, say .α ' = α + δφ for some .φ ∈ C 1 (C, A), then an equivalence of coextensions between .(A, C ⋊α A, P , +) and .(A, C⋊α ' A, P , +) is defined the identity on objects isomorphism .C⋊αA ∼ = C⋊α 'A such that (x, η) I→ (x, η + φ(x)).

.

Hence, the injectivity of .Ψ follows.

⨆ ⨅

Wells introduced this cohomology motivated by the study of coextensions of small categories through a generalization of semigroup and monoid theory techniques. By contrast, Baues and Wirsching’s interest in this cohomology was motivated by its relation to various topological and algebraic problems. In particular, secondary homotopy operations or first k-invariants of classifying spaces.

3.7 Grothendieck and Beck Cohomologies It was pointed out by Quillen in [22] that the Grothendieck cohomology of sheaves over a site (i.e., over a category with a Grothendieck topology, see [10]) can be used as a general method to define a cohomology theory for every class of universal algebras. Thus, a natural definition of the cohomology of a monoid S with coefficients in a .D(S)-module .A arises by specializing this method, as follows. The class of surjective epimorphisms (i.e., coextensions) in the comma category .Mon ↓S of monoids over S is stable under composition and pullbacks, so that we have a Grothendieck topology on .Mon ↓S in which the coverings are the families consisting of a single surjective epimorphism over S

.

104

3 Other Cohomologies

With this covering topology on .Mon ↓S , sheaves are simply left-exact (i.e., preserving coequalizers) contravariant functors. In particular, for every .D(S)-module .A, the functor Der(−, A) : Mon ↓S → Ab

(3.14)

.

defined by f

Der(P → S, A) = Der(P , f ∗A)

.

is a sheaf of abelian groups on .Mon ↓S . Recall from Proposition 2.1 and Remark 2.3 that .A defines an internal abelian group .S ⋊ A → S in .Mon ↓S and the functors .Der(−, A) and .HomMon↓S (−, S ⋊ A → S) are naturally equivalent. Hence Grothendieck cohomology groups of the site .Mon ↓S with coefficients in the sheaf .Der(−, A) are defined. These cohomology groups, which we denote by n HGT (S, A),

.

n = 0, 1, . . . ,

(3.15)

can be computed from flask resolutions 0 → Der(−, A) → F0 → F1 → · · ·

(3.16)

.

of the sheaf .Der(−, A) by  id id n HGT (S, A) = H n 0 → F0 (S → S) → F1 (S → S) → · · · ).

.

f

Exactness of (3.16) means that for any free monoid over S, say .F → S, the cochain complex f

f

0 → Der(F, f ∗ A) → F0 (F → S) → F1 (F → S) → · · ·

.

is exact [22, Chapter II, 5, Lemma 1.1]; and that each .Fn in (3.16) is a flask sheaf means that, for every surjective epimorphism .p : P ↠ Q over S, the Czech cochain complex δ1

δ2

Fn (P → S) → Fn (P ×Q P → S) → Fn (P ×Q P ×Q P → S) → · · ·

.

3.7 Grothendieck and Beck Cohomologies

105

is exact. (See [22, Chapter II, 5, Proposition 1].) The coboundary homomorphisms of the Czech complex are δk =

k+1 

.

(−1)i+1 Fn (π1 , · · · , πi−1 , πi+1 , · · · , πk+1 ),

k ≥ 1,

i=1

where .πi : P ×Q · · · ×Q P → P denotes the ith projection: .(u1 , . . . , uk+1 ) I→ ui . The Grothendieck cohomology groups (3.15) are, up to a dimension shift, the Leech cohomology groups of S with values in .A. More precisely, Theorem 3.29 Let S be a monoid. For every .D(S)-module .A, there are natural isomorphisms  HGT (S, A) ∼ =

.

n

Der(S, A)

if n = 0,

H n+1 (S, A) if n ≥ 1. f

Proof Recall that for every monoid over S, .P → S,   δ1 Ker C 1 (P , f ∗A) → C 2 (P , f ∗A) = Z 1 (P , f ∗A) = Der(P , f ∗A).

.

Then, by (2.11), it suffices to prove that 0 → Der(−, A) → C 1 (−, A) → C 2 (−, A) → · · ·

.

(3.17)

is a flask resolution of the sheaf .Der(−, A) on .Mon ↓S . Above, every C n (−, A) : Mon ↓S → Ab

.

f

is the sheaf that carries every monoid over S, .P → S, to the abelian group f

C n (P → S, A) = C n (P , f ∗A)

.

of n-cochains of P with coefficients in the .D(P )-module .f ∗A. f

The complex (3.17) is exact, since if .F → S is any free monoid over S, then, by Theorem 2.6, .H n C • (F, f ∗A)) = H n (F, f ∗ A) = 0 for all .n ≥ 2. To show that every .C n (−, A) is a flask sheaf on .Mon ↓S , let .p : P ↠ Q be any surjective epimorphism of monoids over S. The associated Czech cochain complex is δ1

δ2

C n (P → S, A) → C n (P ×Q P → S, A) → C n (P ×Q P ×Q P → S, A) → · · · ,

.

106

3 Other Cohomologies

in which the coboundaries     k k+1 δ k : C n P ×Q · · · ×Q P → S, A → C n P ×Q · · · ×Q P → S, A

.

are defined by

 δ k ϕ (u11 , . . . , u1k+1 ), . . . , (un1 , . . . , unk+1 )

.

=

k+1

 (−1)i+1 ϕ (u11 , . . . , u1i−1 , u1i+1 , . . . , u1k+1 ), . . . , i=1

 . . . , (un1 , . . . , uni−1 , uni+1 , . . . , unk+1 ) ,

  k for every n-cochain .ϕ ∈ C n P ×Q · · · ×Q P → S, A . This Czech complex is exact, since it has a contracting homotopy which consists of the homomorphisms     k+1 k Фk : C n P ×Q · · · ×Q P → S, A → C n P ×Q · · · ×Q P → S, A

.

defined by   Фk ϕ (u11 , . . . , u1k ), . . . ,(un1 , . . . , unk )   = (sp(u11 ), u11 , . . . , u1k+1 ), . . . , (sp(un1 ), un1 , . . . , unk+1 )

.

  k+1 for every .ϕ ∈ C n P ×Q · · · ×Q P → S, A , where .s : Q → P is a chosen map such that .ps = idQ . Indeed, the equality .d k−1 Фk−1 + Фk d k = id, for every .k ≥ 2, only requires a straightforward verification. ⨆ ⨅ On the other hand, the triple cohomology theory, referred to as Beck cohomology, was developed for any tripleable (=monadic) category by Beck [5] and Barr and Beck [3]. The category .Mon of monoids is tripleable over the category .Set of sets [17, Chapter VI, §4]. In fact, the underlaying set functor .U : Mon → Set together with its left adjoint, the free monoid functor .F : Set → Mon, induces a triple .T on .Set such that an Eilenberg-Moore’s .T-algebra is just a monoid. The monoid particularization of Beck cohomology was studied by Wells [25]. Briefly: Let .η : idSet → UF and .ε : FU → idMon denote the unit and the counit of the adjunction .(F, U), respectively. For every given monoid S, there is an induced cotriple .(G, ε, ν) in the comma category of monoids over S, which is as follows. The endofunctor G : Mon ↓S → Mon ↓S

.

3.7 Grothendieck and Beck Cohomologies

107

is defined by

.

f

The counit .ε : G → id sends each .P → S to the surjective epimorphism over S

.

f

and the comultiplication .ν : G → G2 carries each .P → S, to the injective homomorphism over S

.

This cotriple .(G, ε, ν) yields an augmented simplicial object in the category of endofunctors in .Mon ↓S , ε

G• → id,

.

called the (Beck, Godement) standard cotriple resolution, which is defined by .Gn = Gn+1 , with face and degeneracy natural transformations di = Gn−i εGi : Gn → Gn−1 ,

.

si = Gn−i νGi : Gn → Gn+1 , for .0 ≤ i ≤ n. Then, for any .D(S)-module .A, composition with the derivations functor (3.14), .Der(−, A) : Mon ↓S → Ab, yields an augmented cochain complex in the (abelian) category of abelian group valuated functors from .Mon ↓S to .Ab ϵ∗

δ0

δ1

0 → Der(−, A) → Der(G, A) → Der(G2 , A) → · · ·

.

(3.18)

108

3 Other Cohomologies

whose coboundaries are δ n−1 =

.

n  (−1)i di∗ : Der(Gn , A) → Der(Gn+1 , A). i=0

By definition, the Beck (or Barr-Beck) cohomology groups of S with coefficients in .A, which we denote by n HBB (S, A),

.

n = 0, 1, . . . ,

are the cohomology groups of the cochain complex .Der(G• (idS ), A):   id id n HBB (S, A) = H n 0 → Der(G(S → S), A) → Der(G2 (S → S), A) → · · · .

.

This brings us to Well’s theorem. In [25], Wells proved that with a dimension shift both Beck and Leech cohomology of monoids are the same: Theorem 3.30 Let S be a monoid. For every .D(S)-module .A, there are natural isomorphisms  HBB (S, A) ∼ =

.

n

Der(S, A)

if n = 0,

H n+1 (S, A) if n ≥ 1.

Proof This follows from Theorem 3.29, since (3.18) is a flask resolution of the sheaf .Der(−, A) in the category of sheaves of abelian groups on .Mon ↓S , so that n (S, A) ∼ H n (S, A) for all .n ≥ 0. there are natural isomorphism .HBB = GT In fact, (3.18) is exact since, for every free monoid over S, .F(X) → S, the homomorphisms over S Фn = Gn F(ηX ) : Gn (F(X) → S) → Gn+1 (F(X) → S),

.

n ≥ 0,

yield a contracting homotopy     Ф∗n : Der Gn+1 (F(X) → S), A → Der Gn (F(X) → S), A ,

.

on the cochain complex 0 → Der(F(X) → S, A) → Der(G(F(X) → S), A)

.

→ Der(G2 (F(X) → S), A) → · · · .

n ≥ 0,

References

109

Moreover, for every .n ≥ 1, the sheaf .Der(Gn , A) is flask: If .p : P ↠ Q is a surjective epimorphism of monoids over S and we chose .s : P → Q a section map of p, then the homomorphisms over S, for .k ≥ 1, k

Ψk = Gn−1 F(spπ1 , π1 , · · · , πk ) : Gn (P ×Q · · · ×Q P → S)

.

k+1

→ Gn (P ×Q · · · ×Q P → S) yield a contracting homotopy     k+1 k Ψk∗ : Der Gn (P ×Q · · · ×Q P → S), A → Der Gn (P ×Q · · · ×Q P → S), A

.

on the associated Czech cochain complex δ1

δ2

Der(Gn (P → S), A) → Der(Gn (P ×Q P → S), A) → · · ·

.

whence it is exact.

⨆ ⨅

References 1. Aznar, E.A., Sevilla, A.M.: Equivalences between D-categories of inverse monoids. In: Semigroup Forum, vol. 59, pp. 435–452 (1999) 2. Aznar, E.A., Sevilla, A.M.: Beck, H and Leech coextensions. In: Semigroup Forum, vol. 61, pp. 385–404 (2000) 3. Barr, M., Beck, J.: Homology and standard constructions. In: Seminar on Triples and Categorical Homology. Lecture Notes in Math., vol. 80, pp. 245–335. Springer, Berlin (1969) 4. Baues, H.-J., Wirsching, G.: Cohomology of small categories. J. Pure Appl. Algebra 38 (1985), 187–211. 5. Beck, J.: Triples, Algebras and Cohomology. Thesis (Ph.D.) Columbia University, 1967 (Repr. Theory Appl. Categ. 2 (2003), 1–59). 6. Clifford, A.H., Preston, G.B.: The Algebraic Theory of Semigroups. Vol. I. Mathematical Surveys, No. 7. Amer. Math. Soc., Providence (1961) 7. Fiedorowicz Z.: Classifying spaces of topological monoids and categories. Amer. J. Math. 106, 301–325 (1984) 8. Gabriel, P., Zisman, M.: Calculus of Fractions and Homotopy Theory. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 35. Springer New York, New York (1967) 9. Illusie, L.: Complex Cotangent et Déformations II. Lecture Notes in Math. 283. Springer, Berlin (1972) 10. Johnstone, P.T.: Topos Theory. Academic Press (1977) 11. Lausch, H.: Cohomology of inverse semigroups. J. Algebra 35, 273–303 (1975) 12. Leech, J.: The D-category of a semigroup. In: Semigroup Forum, vol. 11, pp. 283–296 (1975/1976) 13. Leech, J.: The D-category of a monoid. In: Semigroup Forum, vol. 34, pp. 89–116 (1986) 14. Leech, J.: Constructing inverse monoids from small categories. In: Semigroup Forum, vol. 36, pp. 89–116 (1987)

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15. Leech, J.: On the foundations of inverse monoids and inverse algebras. Proc. Edinb. Math. Soc. 41, 1–21 (1998) 16. Loganathan, M.: Cohomology of inverse semigroups. J. Algebra 70, 375–393 (1981) 17. Mac Lane, S.: Categories for the Working Mathematician, vol. 5. Springer, New York (1971) 18. Mac Lane, S.: Homology. Classics in Mathematics. Springer, Berlin (1995) 19. Martins-Ferreira, N., Montoli, A., Sobral, M.: Baer sums of special Schreier extensions of monoids. In: Semigroup Forum, vol. 93, pp. 403–415 (2016) 20. Mitchell, B.: On the dimension of objects and categories. I. Monoids. J. Algebra 9, 314–340 (1968) 21. Mitchell, B.: Rings with several objects. Adv. Math. 8, 1–161 (1972) 22. Quillen, D.: Homotopical Algebra, Lecture Notes in Math., vol. 43. Springer (1967) 23. Rédei, L.: Die Verallgemeinerung der Schreierschen Erweiterungstheorie. Acta Sci. Math. (Szeged) 14, 252–273 (1952) 24. Steinberg, B.: Twists, crossed products and inverse semigroup cohomology. J. Australian Math. Soc., Cambridge University Press, published online October (2021) 25. Wells, Ch.: Extension theories for monoids. In: Semigroup Forum, vol. 16, 13–3 (1978) 26. Wells, Ch.: Extension theories for categories. Preliminary report (1980). Revised version (2001). Available from https://www.academia.edu/4333242

Chapter 4

Cohomology and H-Coextensions .

In this chapter we return to the primary motivation for the cohomology of monoids: the structure and classification of .H-coextensions. While studying monoid cohomology for its own sake is certainly of interest, .H-coextensions and their mild variations were almost always part of any initial research. This was true for Grillet and Leech. It was also for Lausch and Loganathan in the case of inverse semigroups, and more generally, regular semigroups. A natural generalization to .H-coextensions of small categories can be found in the paper by Wells [9]. Papers on regular and inverse semigroups often use a different terminology. An idempotent-separating congruence is a congruence with at most one idempotent in each congruence-class. Congruences contained in the .H-relation are clearly idempotent-separating. The converse need not be true. For regular monoids and in particular in the inverse case, we have Lallement’s Theorem: A congruence on a regular semigroup is contained in the relation .H if and only if it is idempotentseparating. .H-coextensions of regular monoids are thus often called “idempotent separating coextension”, or even “idempotent separating extension” in the literature. So, reader be aware!. Once again, an .H-coextension of a monoid S is a coextension .(E, p) of S such that the congruence on E induced by the surjective epimorphism .p : E ↠ S is contained in .H. Section 4.1 introduces its kernel, which is a functor .Σp : D(E) → Gr that plays the same role for .(E, p) that is played by the kernel subgroup of a group coextension. It will be our main tool for analyzing objects in the category .HExt(S) of .H-coextensions of S. Section 4.2 looks at the abelian case where .Σp : D(E) → Ab, the category of abelian groups. The class of such coextensions induces a full subcategory .AbHExt(S) of .HExt(S), whose analysis this section is mainly dedicated to. In Sect. 4.3 we return to a further analysis of the general case. Our knowledge of kernel functors is used to define semifunctors from .D(S) to the category of groups which along with possibly nonabelian variations of 2-cocycles can be used to construct coextensions .(E, p) of S. We also give necessary and sufficient conditions for such coextensions to be .H-coextensions. Indeed, to within © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 A. M. Cegarra, J. Leech, The Cohomology of Monoids, RSME Springer Series 12, https://doi.org/10.1007/978-3-031-50258-3_4

111

4 Cohomology and .H-Coextensions

112

isomorphism, all .H-coextensioins of S arise in this way. Cohomological aspects are studied in the fourth section. In particular the third cohomology group appears in the study of obstructions. In a final Sect. 4.5 we use groups as an illustrative example. We show how the classical theorems on the classification of nonabelian extensions of groups by Schreier and Eilenberg-Mac Lane arise as instances of the previous results on .H-coextensions with nonabelian kernel and obstructions.

4.1 The Kernel of an H-Coextension An .H-coextension .(E, p) of a monoid S is a surjective homomorphism of monoids p : E ↠ S that induces on E a congruence contained in the Green’s relation .H (.p(x) = p(y) implies .xHy). We denote by

.

HExt(S) ⊆ Mon ↓S

.

the full subcategory of the category of monoids over S whose objects are all the H-coextensions of S. To begin our study of .H-coextensions we recall the definition and main properties of Schützenberger groups. We refer the reader to [1] and [4] for more details.

.

The Schützenberger Group For every .H-class H of S, R(H ) = {v ∈ S | H v ⊆ H }

.

/ ∅, then .H v ⊆ H and .v ∈ R(H ). is a submonoid of S; moreover, if .H v ∩ H = Every .v ∈ R(H ) induces a mapping .[v] : H → H , given by .x I→ xv. Because H is an .H-class, Σ(H ) = {[v] | v ∈ R(H )}

.

is a simply transitive group of permutations of H , the right Schützenberger group of H . For each .x, y ∈ H there is a unique .[v] ∈ Σ(H ) such that .y = xv [4, Chapter II, Proposition 3.1]. We denote by .

H × Σ(H ) → H,

.

(x, [v]) I→ x . [v] = xv,

4.1 The Kernel of an H-Coextension

113

the action of .Σ(H ) on H . The dual left Schützenberger group of H is isomorphic to Σ(H ) and will not used hereafter; this lets us call .Σ(H ) the Schützenberger group of H . If .x ∈ S, it is convenient to regard .Σ(Hx ) as the Schützenberger group of x and to denote it by

.

Σ(x) = Σ(Hx ).

.

If e is an idempotent of S, then its .H-class .He is a (maximal) subgroup of S, and He ∼ = Σ(e). In general, Schützenberger groups need not always appear as actual subgroups of the relevant monoid or semigroup.

.

Left Invariance If .x L y, then .R(Hx ) = R(Hy ) and there is a canonical isomorphism .Σ(x) ∼ = Σ(y) that sends .[v] ∈ Σ(x) to .[v] ∈ Σ(y) induced by the same .v ∈ R(Hx ) = R(Hy ) [4, Chapter II, Proposition 3.4]. We may therefore identify .Σ(x) and .Σ(y) whenever .x L y; we say that the assignment .x I→ Σ(x) is left invariant. If .ux L x, then .(ux)v = u(xv) for all .v ∈ R(Hx ) = R(Hux ), so that (ux) . g = u(x . g)

.

(4.1)

for all .g ∈ Σ(x) = Σ(ux). The concept of left invariance can now be expressed as the following commuting diagram of group actions, where .λu denotes multiplication on the left by u.

(4.2)

.

Unfortunately, .H is not in general a left congruence; so .λu can send one .x ∈ Hx to one .y = ux ∈ Hy , but it probably does not send all of .Hx into .Hy . There is no .λu : Hx → Hux and no diagram (4.2). However, if .ux ∈ Hx , then .uHx ⊆ Hx and (4.2) holds.

Conjugation Given .x ∈ S, every element y of .Hx can be written in the form .y = x . g for some g ∈ Σ(x). How does g depend on x? The answer is provided by conjugation:

.

(x . g) . h = (x . h) . g h ,

.

where .g h = h−1 gh.

(4.3)

4 Cohomology and .H-Coextensions

114

Since .xv ∈ Hx implies .Hx v ⊆ Hx , with .h = [v], (4.3) yields (y . g)v = (yv) . g [v]

.

(4.4)

and we have the following commuting diagram of group actions

(4.5)

.

where .ρv denotes right multiplication by v.

Functorial Properties Let .f : S → T be a morphism of monoids. If .x ∈ S, then .f (Hx ) is contained in a single .H-class .Hf (x) of T and .f (R(Hx )) ⊆ R(Hf (x) ), since .Hx v ⊆ Hx implies .f (Hx )f (v) ⊆ f (Hx ), .Hf (x) f (v) ∩ Hf (x) /= ∅, and .Hf (x) f (v) ⊆ Hf (x) . Moreover, if .v, v ' ∈ R(Hx ) and .[v] = [v ' ] ∈ Σ(x), then .xv = xv ' , .f (x)f (v) = f (x)f (v ' ) and .[f (v)] = [f (v ' )] ∈ Σ(f (x)). Hence f induces a homomorphism fx : Σ(x) → Σ(f (x))

.

such that f (x) . fx (g) = f (x . g)

.

(4.6)

for all .g ∈ Σ(x). If .u, x ∈ S and .ux L x, then .f (ux) L f (x) in T . We saw that .Σ(x) and .Σ(ux) can be identified so that .(ux) . g = u(x . g) for all .g ∈ Σ(x) = Σ(ux); similarly, .Σ(f (x)) and .Σ(f (ux)) can be identified so that .f (ux) . g = f (u)(f (x) . g) for all .g ∈ Σ(f (x)) = Σ(f (ux)). Then f (ux) . fux (g) = f ((ux) . g) = f (u(x . g)) = f (u)f (x . g)

.

= f (u)(f (x) . fx (g)) = (f (u)f (x)) . fx (g) = f (ux) . fx (g) for all .g ∈ Σ(x), by (4.6). Therefore fux = fx .

.

We say that the assignment .x I→ fx is left invariant.

4.1 The Kernel of an H-Coextension

115

If .ux L x, then .fx = fux . Thus, trivially, both homomorphisms have the same kernel. This will turn out to be useful! It should be quite clear that .g ∈ Kerfx if and only if .f (x . g) = f (x). Thus we see that if .uxLx in S, then .f (x . g) = f (x) if and only if .f (ux . g) = f (ux). Finally, we mention that if .ux L x hence also .f (u)f (x) L f (x), then f induces a morphism from the diagram (4.5) for .(u, x) to the corresponding diagram for .(f (u), f (x)). Similarly, if .x H xv hence also .f (x) H f (x)f (v), then f induces a morphism from the diagram (4.5) for .(x, v) to the corresponding diagram for .(f (x), f (v)).

The Kernel of an H-Coextension The kernel of an .H-coextension .(E, p) of S is the functor from .D(E) to the category Gr of groups

.

Σp : D(E) → Gr

.

defined as follows. For every object .x ∈ E of .D(E),   Σp (x) = Ker px : Σ(x) → Σ(p(x))

.

where .px is induced as above by the surjective epimorphism .p : E ↠ S. If .x L y in E, then we saw that .px = py , so that p(x) = p(y) =⇒ Σp (x) = Σp (y).

.

(4.7)

We now define .Σp on morphisms. Lemma 4.1 For every .x ∈ E, the action of .Σ(x) on .Hx induces a simply transitive action of .Σp (x) on .p−1 (p(x)). Proof If .p(y) = p(x), then .y = x . k for some .k ∈ Σ(x) and then p(x) = p(y) = p(x) . px (k)

.

by (4.6), so that .px (k) = 1 and .k ∈ Σp (x). If also .g ∈ Σp (x), then .kg ∈ Σp (x) and (4.6) yields p(y . g) = p(x . kg) = p(x) . px (kg) = p(x).

.

Thus the action of .Σ(x) induces an action of .Σp (x) on .p−1 (p(x)). Now .Σp (x) acts transitively on .p−1 (p(x)), by the first part of the proof, and acts simply on −1 (p(x)), since .Σ(x) does. .p ⨆ ⨅

4 Cohomology and .H-Coextensions

116

Lemma 4.2 For every morphism .〈u, x, v〉 of .D(E), there is a homomorphism Σp 〈u, x, v〉 : Σp (x) → Σp (uxv)

.

such that (uxv) . Σp 〈u, x, v〉g = u(x . g)v

.

(4.8)

for all .g ∈ Σp (x). Proof Let .u, x, v ∈ E. Since p(u(x . g)v) = p(u)p(x . g)p(v) = p(u)p(x)p(v) = p(uxv),

.

there is for every .g ∈ Σp (x) a unique .Σp 〈u, x, v〉g ∈ Σp (uxv) such that the above Eq. (4.8) is satisfied, by Lemma 4.1. We show that .Σp 〈u, x, v〉 : g I→ Σp 〈u, x, v〉g is a homomorphism. Let .g, h ∈ Σp (x). Then .gh ∈ Σp (x). Since .p−1 (p(x)) ⊆ Hx we also have .x . g = ax for some .a ∈ E. Similarly, .x . h = xb and .u(x . g) = cux for some .b, c ∈ E. Then (4.1) yields  .

 (uxv) . Σp 〈u, x, v〉g .Σp 〈u, x, v〉h = (u(x . g)v) . Σp 〈u, x, v〉h = (cuxv) . Σp 〈u, x, v〉 = c((uxv) . Σp 〈u, x, v〉h) = c(u(x . h)v) = cuxbv = u(x . g)bv = uaxbv = ua(x . h)v = u((ax) . h)v = u((x . g) . h)v = (uxv) . Σp 〈u, x, v〉(gh);

therefore .(Σp 〈u, x, v〉g)(Σp 〈u, x, v〉h) = Σp 〈u, x, v〉(gh). Theorem 4.1 If .(E, p) is an .H-coextension of a monoid S, then a functor Σp : D(E) → Gr

.

is well defined by Σp (x) = Ker(px ) ⊆ Σ(x)

.

and (uxv) . Σp 〈u, x, v〉g = u(x . g)v,

.

for all .u, x, v ∈ E and .g ∈ Ker(px ).

⨆ ⨅

4.1 The Kernel of an H-Coextension

117

Moreover, for every .x ∈ E, the action of .Σ(x) on .Hx induces a simply transitive action of .Σp (x) on .p−1 (p(x)). Proof For all .x ∈ E and .g ∈ Σp (x), we have .x . Σp 〈1, x, 1〉g = x . g, so that Σp 〈1, x, 1〉 is the identity on .Σp (x). Moreover, for all .u, u' , x, v, v ' ∈ E and .g ∈ Σp (x), we have, by (4.8),

.

((u' (uxv)v ' ) . Σp 〈u' , uxv, v ' 〉).Σp 〈u, x, v〉g

.

= u' ((uxv) . Σp 〈u, x, v〉g)v ' = u' u(x . g)vv ' = (u' uxvv ' ) . Σp 〈u' u, x, vv ' 〉g, so that .(Σp 〈u' , uxv, v ' 〉Σp 〈u, x, v〉)g = Σp 〈u' u, x, vv ' 〉g. Thus .Σp is a functor.

⨆ ⨅

The functor .Σp : D(E) → Gr is the the kernel functor of .(E, p). Proposition 4.1 Given .x . g and .y . h in E, where g is in .Σp (x) and h is in .Σp (y), then:   (x . g)(y . h) = xy . (Σp 〈1, x, y〉g)(Σp 〈x, y, 1〉h) .

.

Proof Since .p−1 (p(x)) ⊆ Hx , we have .ax = x . g for some .a ∈ E. By (4.1) and (4.8), (x . g)(y . h) = (ax)(y . h)

.

= a(xy . Σp 〈x, y, 1〉h) = axy . Σp 〈x, y, 1〉h = ((x . g)y) . Σp 〈x, y, 1〉h = (xy . Σp 〈1, x, y)g) . Σp 〈x, y, 1〉h   = xy . (Σp 〈1, x, y〉g)(Σp 〈x, y, 1〉h) . ⨆ ⨅ Proposition 4.2 For every .H-coextension .(E, p) of S, the kernel functor .Σp is smooth in that it factors through the canonical functor .ϱ : D(E) ↠ D(E). Proof We show that .[u, x, v] = [u' , x, v ' ] implies .Σp 〈u, x, v〉 = Σp 〈u' , x, v ' 〉. By Lemma 3.4, it suffices to show that .Σp 〈u, x, v〉 = Σp 〈u' , x, v ' 〉 if .ux = u' x and ' ' ' .xv = xv . Indeed this implies .uxv = u xv . Next, for every .g ∈ Σp (x), we have

4 Cohomology and .H-Coextensions

118

x . g ∈ p −1 (p(x)) ⊆ Hx , so that .x . g = xa = bx for some .a, b ∈ E. If .ux = u' x and .xv = xv ' , then .uxv = u' xv ' and

.

u(x . g)v = uxav = u' xav = u' bxv = u' bxv ' = u' (x . g)v ' ;

.

hence .Σp 〈u, x, v〉 = Σp 〈u' , x, v ' 〉, by (4.8).

⨆ ⨅

We also denote by .Σp the functor Σp : D(E) → Gr

.

such that .Σp = Σp ϱ. Thus .Σp [u, x, v] = Σp 〈u, x, v〉 for all .u, x, v ∈ E. The properties of left invariance and conjugation are expressed as follows: Lemma 4.3 (i) If .ux L x, then .Σp (ux) = Σp (x) and .Σp 〈u, x, 1〉 = idΣp (x) . (ii) If .xv H x with .xv = x . h for some h in .Σ(x), then .Σp 〈1, x, v〉 = ( )h . Proof If .ux L x, then .Σp (ux) = Σp (x) by (4.7) and (4.8), (4.1) yield ux . Σp 〈u, x, 1〉g = u(x . g) = (ux) . g

.

for all .g ∈ Σp (x). If .xv H x and .xv = x . h for some .h ∈ Σ(x), then .[v] = h and (4.8), (4.4) yield xv . Σp 〈1, x, v〉g = (x . g)v = xv . g [v] = xv . g h

.

for all .g ∈ Σp (x). Lemma 4.4 If .p(x) =

⨆ ⨅ p(x ' ),

then, for every .u ∈ E,

(i) .Σp 〈1, x, u〉 = Σp 〈1, x ' , u〉. (ii) .Σp 〈u, x, 1〉 = Σp 〈u, x ' , 1〉. Proof Let .a ∈ E such that .x = ax ' . By left invariance, .Σp (x) = Σp (x ' ), Σp 〈a, x ' , 1〉 = idΣp (x) = Σp 〈1, x, 1〉 = Σp 〈1, x ' , 1〉

.

and, for every .u ∈ E, .p(xu) = p(x ' u) and .p(ux) = p(ux ' ), so that also .Σp (xu) = Σp (x ' u) and .Σp (ux) = Σp (ux ' ), and Σp 〈1, x, v〉 = Σp 〈1, x, v〉Σp 〈a, x ' , 1〉 = Σp 〈a, x ' , v〉 = Σp 〈1, x ' , v〉Σp 〈a, x ' , 1〉

.

= Σp 〈1, x ' , v〉, Σp 〈u, x, 1〉 = Σp 〈u, x, 1〉Σp 〈1, x, 1〉 = Σp 〈u, x, 1〉Σp 〈1, x ' , 1〉 = Σp 〈u, x ' , 1〉.

.

⨆ ⨅

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119

We call the kernel functor .Σp abelian if the group .Σp (x) is abelian for all .x ∈ E. Then .Σp is a .D(E)-module (or a .D(E)-module). Theorem 4.2 If .(E, p) is an .H-coextension of S, then .Σp factors through the functor .D(p) : D(E) → D(S) if and only if .Σp is abelian. Proof Let .x ∈ E and let .g ∈ Σp (x), so that .x . g = xv for some .v ∈ E and ( )g = Σp [1, x, v] by Lemma 4.3. Since .x . g ∈ p −1 (p(x)) we have .p(xv) = p(x) and

.

D(p)[1, x, v] = [1, p(x), p(v)] = [1, p(x), 1] = D(p)[1, x, 1].

.

If .Σp factors through .D(p), then .( )g : h I→ ghg −1 equals .Σp [1, x, v] = Σp [1, x, 1] and is the identity on .Σp (x). Since this holds for every .x ∈ E and every .g ∈ Σp (x), it follows that every group .Σp (x) is abelian. Conversely, assume that .Σp is abelian. By Lemma 3.4 it suffices to prove the following: if .u, u' , x, x ' , v, x ' ∈ E and .p(x) = p(x ' ), .p(ux) = p(u' x ' ), and .p(xv) = p(x ' v ' ), then .Σp (x) = Σp (x ' ), .Σp (uxv) = Σp (u' x ' v ' ), and ' ' ' .Σp [u, x, v] = Σp [u , x , v ]. The first two equalities follow from Lemma 4.3, ' since .p(x) = p(x ) and .p(uxv) = p(u' x ' v ' ) imply .x L x ' and .uxv L u' x ' v ' . Since −1 (p(uxv)) ⊆ H ' ' .p uxv we have .u xv = auxv for some .a ∈ E; similarly, .x = xb ' ' and .xv = x v c for some .b, c ∈ E. If .Σp is abelian, then .Σp [a, uxv, 1], .Σp [1, x, b], and .Σp [1, u' x ' v ' , c] are identity homomorphisms, by Lemma 4.3, and Σp [u' , x ' , v ' ] = Σp [1, u' x ' v ' , c] Σp [u' , x ' , v ' ] = Σp [u' , x ' , v ' c]

.

= Σp [u' , x ' , v ' c] Σp [1, x, b] = Σp [u' , x, bv ' c] = Σp [u' , x, v] = Σp [a, uxv, 1] Σp [u, x, v] = Σp [u, x, v]. Thus, .Σp factors through .D(S).

⨆ ⨅

If .Σp is abelian, we also denote by Σp : D(S) → Ab

.

the unique functor (i.e., .D(S)-module) such that

.

commutes, and call it the kernel of the .H-coextension .(E, p).

4 Cohomology and .H-Coextensions

120

In general, every monoid E has a greatest congruence .K ⊆ H (that contains every congruence .C ⊆ H). The kernel functor .ΣC of a congruence .C ⊆ H (actually, the kernel functor .Σp of the projection .p : E ↠ E/C) is then a subfunctor of the kernel functor .ΣK of K: for all .x, u, v ∈ E, .ΣC (x) is a subgroup of .ΣK (x) and .ΣC 〈u, x, v〉 is the restriction of .ΣK 〈u, x, v〉 to .ΣC (x), for all morphisms .〈u, x, v〉 in .D(E). Conversely, every such subfunctor of .ΣK is the kernel functor of a unique congruence .C ⊆ H. The one-to-one correspondence between congruences .C ⊆ H and subfunctors of .ΣK is, in fact, an isomorphism of complete lattices.

4.2 The Category of H-Coextensions with Abelian Kernels The .H-coextensions of a monoid S and their morphisms constitute a full subcategory of the category .Mon ↓ S of monoids over S; we denote this subcategory by HExt(S).

.

Similarly, the .H-coextensions of S with abelian kernels (and their morphisms) constitute a full subcategory of .HExt(S), which we denote by AbHExt(S).

.

Thus, AbHExt(S) ⊆ HExt(S) ⊆ Mon ↓ S.

.

The coextensions .(A, E, p, +) of a monoid S by .D(S)-modules .A and their morphisms constitute a category ExtD (S)

.

that was defined in Sect. 2.4. (See (2.24).) In Sect. 3.3 we showed that the canonical functor .ϱ : D(S) ↠ D(S) induces an exact embedding .ϱ∗ of .D(S)-Mod into ∗ .D(S)-Mod. (See (3.9).) We shall regard .ϱ as an inclusion, so that .D(S)-Mod becomes a subcategory of .D(S)-Mod. A coextension of a monoid S by a .D(S)module .A is a coextension .(A, E, p, +) of S by .A, regarded as a .D(S)-module. These coextensions and their morphisms constitute a full subcategory ExtD (S)

.

of .ExtD (S). We denote by HExtD (S)

.

4.2 The Category of H-Coextensions with Abelian Kernels

121

the full subcategory of .ExtD (S) whose objects are the .H-coextensions of S by D(S)-modules. Thus

.

HExtD (S) ⊆ ExtD (S) ⊆ ExtD (S).

.

If .(E, p) is an .H-coextension of S with abelian kernel .Σp , then Theorem 4.2 states that .Σp is actually a .D(S)-module; then Lemma 4.1 and Proposition 4.1 show that .(E, p) is a coextension of S by the .D(S)-module .Σp , so that we can identify .(E, p) and .(Σp , E, p, .). This lets us identify .AbHExt(S) with a full subcategory of .HExtD (S). Thus, AbHExt(S) ⊆ HExtD (S) ⊆ ExtD (S).

.

In this section we take a closer look at the inclusions above. Is .HExtD (S) = ExtD (S)? In other words, is every coextension of S by a .D(S)module an .H-coextension? In general the answer is negative, unless (as we see below) S is regular. We give necessary and sufficient criterion for a coextension of S by a .D(S)-module to be an .H-coextension. Lemma 4.5 Let .A be a .D(S)-module and let .α ∈ Z 2 (S, A) be a 2-cocycle. Then .(S ⋊α A, π ) is an .H-coextension if and only if       A(u)x + α(u, x) = A(x) = xA(v) + α(x, v) . (4.9) u∈l(x)

v∈r(x)

for all x in S, where l(x) = {u ∈ S | ux = x} and r(x) = {v ∈ S | xv = x}.

.

Proof The coextension .(S⋊αA, π ) is an .H-coextension if and only if, for all .x ∈ S, π −1 (x) ⊆ H(x,0) : if and only if, for every .x ∈ S and .a, b ∈ A(x), there exist .(u, c), (v, d) ∈ S ⋊α A such that .

(u, c)(x, a) = (x, b) = (x, a)(v, d),

.

equivalently (ux, cx + α(u, x) + ua) = (x, b) = (xv, av + α(x, v) + xd).

.

(4.10)

If (4.10) holds, then .ux = x = xv, so that .u ∈ l(x) and .v ∈ r(x). Conversely, if .u ∈ l(x) and .v ∈ r(x), then .[u, x, 1] = [1, x, 1] = [1, x, v], .ua = a = av, and (4.10) holds if and only if cx + α(u, x) + a = b = a + α(x, v) + xd.

.

Equation (4.11) can be solved for c and d if and only if (4.9) holds for x.

(4.11) ⨆ ⨅

4 Cohomology and .H-Coextensions

122

Theorem 4.3 If S is regular, then .HExtD (S) = ExtD (S). Proof By Theorem 2.8, every coextension of S by a .D(S)-module .A is equivalent to .(A, S ⋊α A, π, +) for some 2-cocycle .α ∈ Z 2 (S, A). Hence it suffices to show that every .(S ⋊α A, π ) is an .H-coextension. If S is regular, then for every .x ∈ S there exist idempotents .e ∈ Rx and .f ∈ Lx . Then .e ∈ l(x), .f ∈ r(x), and .[1, e, x], .[x, f, 1] are isomorphisms, by Theorem 3.17, so that .A(e)x = A(x) = xA(f ). Hence (4.9) holds. ⨆ ⨅ In case .(A, E, p, +) is an .H-coextension, we have: Theorem 4.4 If .(A, E, p, +) is an .H-coextension of S by a .D(S)-module .A, then Σp ∼ = (Σp , E, p, .). = A is abelian and .(A, E, p, +) ∼

.

Proof By Theorem 2.8, we can assume that .(A, E, p, +) = (A, S ⋊α A, π, +) for some 2-cocycle .α ∈ Z 2 (S, A), so that p−1 (x) = π −1 (x) = {(x, a) | a ∈ A(x)},

.

for every .x ∈ S. Let .s = (x, a), t = (y, b) ∈ E. Since the congruence induced by p on E is contained in .H, we have .s H t in E if and only if .p(s) H p(t) in S, if and only if .x H y. Hence .(v, c) ∈ R(Hs ) if and only if .xv = x, and (x, a) . [(v, c)] = (x, a)(v, c) = (xv, av + α(x, v) + xc).

.

Then (4.6) yields xv = p((x, a) . [(v, c)]) = p(x, a) . ps ([(v, c)]) = x . ps ([(v, c)]);

.

hence .[(v, c)] ∈ Σp (s) = Ker(ps ) if and only if .xv = x, and then .[1, x, v] = [1, x, 1], .av = a, and (x, a) . [(v, c)] = (x, a + α(x, v) + xc) = (x, a) + (α(x, v) + xc).

.

In particular, .Σp (x, a) = Σp (x, a ' ) for all .a, a ' ∈ A(x), and .[(v, c)] = [(v ' , c' )] implies .α(x, v) + xc = α(x, v ' ) + xc' . A mapping .θx : Σp (s) → A(x) is now well-defined by θx [(v, c)] = α(x, v) + xc,

.

so that .(x, a) . g = (x, a) + θx (g) for all .a ∈ A(x) and .g ∈ Σp (s) (and the same θx serves for all .a ∈ A(x)). Then .θx is bijective, since both .A(x) and .Σp (s) act simply and transitively on .p−1 (x). Moreover, for all .g, h ∈ Σp (s),

.

(x, a) + θx (gh) = (x, a) . (gh) = ((x, a) . g) . h = (x, a + θx (g)) . h

.

= (x, a + θx (g) + θx (h)) = (x, a) + (θx (g) + θx (h)). Thus .θx is an isomorphism. In particular, .Σp (s) ∼ = A(x) is abelian.

4.2 The Category of H-Coextensions with Abelian Kernels

123

By Theorem 4.2, .Σp factors through .D(p) : D(E) → D(S), which yields a D(S)-module, also denoted by .Σp : D(S) → Ab, such that .Σp (x) = Σp (x, a) and .Σp [u, x, v] = Σp [(u, c), (x, a), (v, d)] for all .u, x, v ∈ S and all .c ∈ A(u), .a ∈ A(x), and .d ∈ A(v). Thus, applying p to (4.2): .

(u, c)(x, a)(v, d) . Σp [(u, c), (x, a), (v, d)]g = (u, c)((x, a) . g)(v, d),

.

where .(u, c), (x, a), (v, d) ∈ E and .g ∈ Σp (x), yields uxv . Σp [u, x, v]g = u(x . g)v

.

for all .u, x, v ∈ S and .g ∈ Σp (x). Since .A is already a .D(S)-module we already have .A[u, x, v]a = (ua)v = u(av) for all .a ∈ A(x) and all .u, x, v ∈ S. It follows that .θ : x I→ θx : Σp (x) → A(x) is an isomorphism of .D(S)-modules: indeed let .u, x, v ∈ S and let .u ¯ = (u, 0), .x¯ = (x, 0), and .v¯ = (v, 0) in E, so that u¯ x¯ v¯ = (u, 0)(x, 0)(v, 0) = (u, 0)(xv, α(x, v)) = (uxv, α(u, xv) + uα(x, v)).

.

For every .g ∈ Σp (x) we have .x¯ . g = (x, θx (g)), and (4.2) yields u¯ x¯ vu ¯ + θuxv (Σp [u, ¯ x, ¯ v]g) ¯ = u¯ x¯ v¯ . Σp [u, ¯ x, ¯ v]g ¯

.

= u( ¯ x¯ . g)v¯ = (u, 0)(x, θx (g))(v, 0) = (u, 0)(xv, α(x, v) + (θx (g))v) = (uxv, α(u, xv) + uα(x, v) + u((θx (g))v)) = u¯ x¯ v¯ + A[u, x, v]θx (g). Thus the square

.

commutes for every .u, x, v ∈ S, and .θ is a morphism of .D(S)-modules. Then (θ, idE ) : (Σp , E, p, .) ∼ = (A, E, p, +) is an isomorphism of .H-coextensions of S. ⨆ ⨅

.

4 Cohomology and .H-Coextensions

124

Proposition 4.3 Let .(A, E, p, +) and .(A' , E ' , p' , +) be coextensions of S by .D(S)-modules. If .(A, E, p, +) is an .H-coextension, then for any morphism F : (E, p) → (E ' , p' )

.

in .Mon ↓S , there exists a unique morphism of .D(S)-modules .F : A → A' such that (F, F ) : (A, E, p, +) → (A' , E ' , p' , +)

.

is a morphism of abelian group coextensions of S. Hence,     HomMon↓S (E, p), (E ' , p' ) ∼ = HomExtD (S) (A, E, p, +), (A' , E ' , p' , +) .

.

Proof Without loss of generality we may assume that, for some 2-cocycles α ∈ Z 2 (S, A) and .α ' ∈ Z 2 (S, A' ), .(A, E, p, +) = (A, S ⋊α A, π, +) and ' ' ' ' ' ' .(A , E , p , +) = (A , S ⋊α ' A , π , +), where .(S ⋊α A, π ) is an .H-coextension. Let .

F : (S ⋊α A, π ) → (S ⋊α ' A' , π ' )

.

be a morphism in the category .Mon ↓S , so that .π ' F = π. If .F : A → A' is a morphism of .D(S)-modules such that (F, F ) : (A, S ⋊α A, π, +) → (A' , S ⋊α ' A' , π ' , +)

.

is a morphism of abelian group coextensions of S, then necessarily F (x, a) = F ((x, 0) + a) = F (x, 0) + F(a)

.

for every .(x, a) ∈ E. Hence .F is unique. Conversely, for each .x ∈ S, there is a unique mapping .F : A(x) → A' (x) such that .F (x, a) = F (x, 0) + F(a) for all .a ∈ A(x). There is also a unique 1-cochain ˜ 1 (S, A) such that .F (x, 0) = (x, φ(x)). Then .φ ∈ C F (x, a) = (x, φ(x) + F(a)).

.

  for all .a ∈ A(x). Then .F(0) = 0, and .F (x, a)(y, b) = F (x, a)F (x, b) yields φ(xy) + F(ay+xb + α(x, y))

(4.12)

.

'

= φ(x)y + xφ(y) + F(a)y + xF(b) + α (x, y) for all .x, y ∈ S , .a ∈ A(x), and .b ∈ A(y). In particular, with .a = 0 and .b = 0, (4.12) yields φ(xy) + Fα(x, y) = φ(x)y + xφ(y) + α ' (x, y).

.

(4.13)

4.2 The Category of H-Coextensions with Abelian Kernels

125

Let .a, b ∈ A(x). By Lemma 4.5, we have .a = cx + α(u, x) for some .u ∈ l(x) and .c ∈ A(u). Since .ux = x we also have .[u, x, 1] = [1, x, 1], so that .ua = a for all a. Applying (4.12) to .(u, c) and .(x, b) then yields φ(x) + F(a + b) = φ(ux) + F(cx + ub + α(u, x))

.

= φ(u)x + uφ(x) + F(c)x + uF(b) + α ' (u, x), so that for all .a, b ∈ A(x) F(a + b) = φ(u)x + F(c)x + F(b) + α ' (u, x).

.

In particular (with .b = 0) .F(a) = φ(u)x + F(c)x + α ' (u, x). Hence .F(a + b) = F(a) + F(b). Thus .F : A(x) → A' (x) is a homomorphism for all x. Now (4.13) yields φ(xy) + F(ay + xb) + Fα(u, x) = φ(x)y + xφ(y) + F(a)y + xF(b) + α ' (x, y),

.

so that F(ay + xb) = F(a)y + xF(b),

.

for all .a ∈ A(x) and .b ∈ A(y), by (4.13). In particular, .F(ay) = F(a)y and F(xb) = xF(b). Thus .F is a morphism of .D(S)-modules. Then

.

(F, F ) : (A, S ⋊α A, π, +) → (A' , S ⋊α ' A' , π ' , +)

.

is a morphism of abelian group coextensions since F ((x, a) + b) = F (x, a + b) = (x, φ(x) + F(a + b))

.

= (x, φ(x) + F(a) + F(b)) = (x, φ(x) + F(a)) + F(b) = F (x, a) + F(b). for every .x ∈ S and .a, b ∈ A(x).

⨆ ⨅

Corollary 4.1 The category .HExtD (S) is a full subcategory of .Mon ↓S . Theorem 4.5 For any monoid S, the embedding AbHExt(S) → HExtD (S),

.

(E, p) I→ (Σp , E, p, .),

and the forgetful functor HExtD (S) → AbHExt(S),

.

(A, E, p, +) I→ (E, p),

constitute an equivalence of categories .AbHExt(S) ≃ HExtD (S).

4 Cohomology and .H-Coextensions

126

Proof This follows from Theorem 4.4 and Proposition 4.3.

⨆ ⨅

4.3 The Category of H-Coextensions with Arbitrary Kernel We begin by presenting a category .Z2 (S), which is the nonabelian analogue of the category .AbZ2 (S) of abelian 2-cocycles of a monoid S. (See Sect. 2.4.) As above, let .Gr denote the category of groups and homomorphisms between them. We write groups additively, even though the groups need not be abelian. Let S be a monoid. A (left) semifunctor G : D(S) → Gr

.

assigns a group .G(x) to each .x ∈ S and assigns a homomorphism G〈u, x, v〉 : G(x) → G(uxv)

.

to each morphism .〈u, x, v〉 of .D(S), such that: (i) .GL = G|L(S) : L(S) → Gr is a functor. (ii) For every morphism .〈u, x, v〉 of .D(S), G〈u, x, v〉 = G〈u, xv, 1〉 G〈1, x, v〉 = G〈1, ux, v〉 G〈u, x, 1〉.

.

It is convenient to let

.

ug = G〈u, x, 1〉g, gv = G〈1, x, v〉g, ugv = G〈u, x, v〉g,

for all .u, x, v ∈ S and .g ∈ G(x). By (i) and (ii) u' (ug) = (u' u)g, u(gv) = (ug)v, 1g = g = g1.

.

for all .u, u' , x, v, v ' and g. By Theorem 2.1, .G is a functor if and only if .GR = G|R(S) is a functor, that is, if and only if .(gv)v ' = g(vv ' ) for all .x, v, v ' ∈ S and .g ∈ G(x). If .G : D(S) → Gr is a semifunctor, then the group .

C˜ 1 (S, G),

of normalized 1-cochains of S with coefficients in .G, consists of all mappings φ:S→



.

x∈S

G(x)

4.3 The Category of H-Coextensions with Arbitrary Kernel

127

such that .φ(x) ∈ G(x) and .φ(1) = 0, and the group .

C˜ 2 (S, G),

of normalized 2-cochains of S with coefficients in .G, consists of all mappings α :S×S →



.

G(x)

x∈S

such that .α(x, y) ∈ G(xy) and .α(x, y) = 0 whenever .x = 1 or .y = 1. A 2-cochain .α ∈ C˜ 2 (S, G) is a 2-cocycle if α(ux, v) + α(u, x)v + (gx)v = α(u, xv) + g(xv) + uα(x, v)

.

for all .u, x, v ∈ S and .g ∈ G(u). Lemma 4.6 Let .G : D(S) → Gr be a semifunctor. A 2-cochain .α ∈ C˜ 2 (S, G) is a 2-cocycle if and only if the following two conditions hold: (1) (ii)

α(ux, v) + α(u, x)v = α(u, xv) + uα(x, v), .(gx)v = −uα(x, v) + g(xv) + uα(x, v), .

for all .u, x, v ∈ S and .g ∈ G(u). ⨆ ⨅

Proof Straightforward. Let .

Z˜ 2 (S, G) ⊆ C˜ 2 (S, G)

denote the set of 2-cocycles of S with coefficients in a semifunctor .G : D(S) → Gr. The category of (nonabelian, normalized) 2-cocycles of the monoid S is the category Z2 (S)

.

defined as follows. Its objects are pairs .(G, α) of a semifunctor .G : D(S) → Gr and a 2-cocycle .α ∈ Z˜ 2 (S, G). A morphism in .Z2 (S), is a pair (σ, φ) : (G, α) → (G' , α ' )

.

where .φ ∈ C˜ 1 (S, G' ) is a 1-cochain and .σ is a mapping that assigns to each .x ∈ S a homomorphism .σ = σx : G(x) → G' (x), such that   σ (ugv) = u xφ(v) + σ (g)v − xφ(v) , .

.

σ α(x, y) = φ(xy) + α ' (x, y) − φ(x)y − xφ(y), for all .u, x, y, v ∈ S and .g ∈ G(x).

(4.14) (4.15)

4 Cohomology and .H-Coextensions

128

Composition of .(σ, φ) with .(σ ' , φ ' ) : (G' , α ' ) → (G'' , α '' ) is given by (σ ' , φ ' )(σ, φ) = (σ ' σ, σ∗' φ + φ ' ),

.

(4.16)

where (σ ' σ )x = σx' σx : G(x) → G'' (x),

.

and (σ∗' φ + φ ' )(x) = σ ' φ(x) + φ ' (x) ∈ G'' (x),

.

for every .x ∈ S. The result is a morphism from .(G, α) to .(G'' , α '' ): for every .u, x, y, v ∈ S and .g ∈ G(x),   (4.14) (σ ' σ )(ugv) = σ ' uxφ(v) + uσ (g)v − uxφ(v)

.

(4.14)

= uxσ ' φ(v) + σ ' (uσ (g)v) − uxσ ' φ(v)

(4.14)

= uxσ ' φ(v) + uxφ ' (v) + σ ' σ (g) − uxφ ' (v) − uxσ ' φ(v)     = ux (σ∗' φ + φ ' )(v) + σ ' σ (g) − ux (σ∗' φ + φ ' )(v) ,   (4.15) σ ' σ α(x, y) = σ ' φ(xy) + α ' (x, y) − φ(x)y − xφ(y)

.

(4.14)

= σ ' φ(xy) + σ ' α ' (x, y) − σ ' (φ(x)y) − xσ ' φ(y)

(4.15),(4.14)

=

σ ' φ(xy) + φ ' (xy) + α '' (x, y) − φ ' (x)y − xφ ' (y)   − xφ ' (y) + σ ' φ(x)y − xφ ' (y)) − xσ ' φ(y)

= σ ' φ(xy) + φ ' (xy) + α '' (x, y) − φ ' (x)y − σ ' φ(x)y − xφ ' (y) − xσ ' φ(y) = (σ∗' φ + φ ' )(xy) + α '' (x, y) − (σ∗' φ + φ ' )(x)y − x(σ∗' φ + φ ' )(y). One verifies easily that this composition of morphisms is associative and that identity morphisms exist, namely .id(G,α) = (idG , 0) where, for every .x ∈ S, (idG )x = idG(x) : G(x) → G(x),

.

0(x) = 0 ∈ G(x). Hence, .Z2 (S) is a category. Proposition 4.4 A morphism .(α, φ) : (G, α) → (G' , α ' ) in .Z2 (S) is an isomorphism if and only if, for every .x ∈ S, .σx : G(x) → G' (x) is an isomorphism.

4.3 The Category of H-Coextensions with Arbitrary Kernel

129

Proof If .σx : G(x) → G' (x) is an isomorphism for every .x ∈ S, then an inverse of 2 −1 , −σ −1 φ) : (G' , α ' ) → (G, α), where .(σ, φ) in .Z (S) is .(σ ∗ (σ −1 )x = σx−1 : G' (x) → G(x),

.

(−σ∗−1 φ)(x) = −σx−1 (φ(x)) ∈ G(x). for every .x ∈ S. The converse is clear.

⨆ ⨅

For any fixed semifunctor .G : D(S) → Gr, let Z2 (S, G)

.

denote the subcategory of .Z2 (S) whose  objects’ firstcomponents is .G and whose morphisms’ first component is .idG = idG(x) , x ∈ S : φ = (idG , φ) : (G, α) → (G, α ' ).

.

In other words, .Z2 (S, G) has .Z˜ 2 (S, G) as its set of objects and a morphism φ : α → α'

.

in .Z2 (S, G) is a 1-cochain .φ ∈ C˜ 1 (S, G) such that gy = xφ(y) + gy − xφ(y),

.

α(x, y) = φ(xy) + α ' (x, y) − φ(x)y − xφ(y). for all .x, y ∈ S and .g ∈ G(x). Thus, we have an inclusion functor Z2 (S, G) ⊆ Z2 (S)

.

and, by Proposition 4.4, every morphism in the category .Z2 (S, G) is an isomorphism. Hence .Z2 (S, G) is a groupoid. By definition, the set of its connected components is the second nonabelian cohomology set of S with coefficients in .G: H 2 (S, G) = Z˜ 2 (S, G)/∼ =.

.

The set .H 2 (S, G) may be empty for some semifunctors .G. (See Sect. 4.4.)

(4.17)

4 Cohomology and .H-Coextensions

130

The Twisted Semidirect Product Construction Let S be a monoid. For a semifunctor .G : D(S) → Gr, every 2-cocycle .α ∈ Z˜ 2 (S, G) gives rise to a twisted semidirect product of S by .G, which is the monoid denoted by S ⋊α G,

.

whose underlying set is .{(x, g) | x ∈ S, g ∈ G(x)}, with multiplication (x, g)(y, h) = (xy, α(x, y) + gy + xh).

.

This multiplication is associative: if .(u, g), (x, h), (v, k) ∈ S ⋊α G, then  .

 (u, g)(x, h) (v, k) = (uxv, α(ux, v) + α(u, x)v + (gx)v + uhv + uxk),   (u, g) (x, h)(v, k) = (uxv, α(u, xv) + g(xv) + uα(x, v) + uhv + uxk).

And the multiplication is also unitary, with identity .(1, 0), since .α is normalized. Then (S ⋊α G, π ),

.

is a coextension of S, in which π : S ⋊α G → S,

.

(x, g) I→ x,

denotes the first coordinate projection. The assignment .(G, α) I→ (S ⋊α G, π ) is the function on objects of a functor Δ : Z2 (S) → Mon ↓ S ,

.

(4.18)

which acts on morphisms as follows. Let .(σ, φ) : (G, α) → (G' , α ' ) be a morphism in .Z2 (S). Define Δ(σ, φ) : S ⋊α G → S ⋊α ' G'

.

by Δ(σ, φ)(x, g) = (x, −φ(x) + σ (g)).

.

Then, .Δ(α, φ) is a homomorphism of monoids: for every .(x, g), (y, h) ∈ S ⋊α G  Δ(σ, φ) (x, g)(y, h)) = (σ, φ)∗ (xy, α(x, y) + gy + xh)

.

= (xy, −φ(xy) + σ α(x, y) + σ (gy) + σ (xh))

4.3 The Category of H-Coextensions with Arbitrary Kernel (4.15),(4.14)

=

131

(xy, α ' (x, y) − φ(x)y + σ (g)y − xφ(y) + xσ (h))

= (x, −φ(x) + σ (g))(y, φ(y) + σ (h))    = Δ(σ, φ)(x, g) Δ(σ, φ)(y, h) , and .Δ(σ, φ)(1, 0) = (1, 0), since .φ(1) = 0. .Δ preserves identity morphisms, since .Δ(id , 0) = id G S⋊α G , and preserves ' ' ' ' composition: If .(σ, φ) : (G, α) → (G , α ) and .(σ , φ ) : (G' , α ' ) → (G'' , α '' ) are morphisms in .Z2 (S), then, for every .(x, g) ∈ S ⋊α G,   (4.16) Δ (σ ' , φ ' )(σ, φ) (x, g) = Δ(σ ' σ, σ∗' φ + φ ' )(x, g)

.

= (x, −φ ' (x) − σ ' φ(x) + σ ' σ (g)) = Δ(σ ' , φ ' )(−φ(x) + σ (g)) = Δ(σ ' , φ ' )Δ(σ, φ)(x, g). Thus, .Δ is functor. Lemma 4.7 The functor .Δ : Z2 (S) → Mon ↓ S is faithful. Proof Suppose .(σ, φ), (σ ' , φ ' ) : (G, α) → (G' , α ' ) are morphisms in .Z2 (S) such that .Δ(σ, φ) = Δ(σ ' , φ ' ). For every .x ∈ S and .g ∈ G(x), (x, −φ ' (x) + σ ' (g)) = Δ(σ ' , φ ' )(x, g) = Δ(σ, φ)(x, g) = (x, −φ ' (x) + σ ' (g)).

.

⨆ Hence, setting .g = 0, we first obtain .φ ' (x) = φ(x) and then also .σ ' (g) = σ (g). ⨅

The Category HZ2 (S) Let .Gr denote the quotient category of .Gr obtained by identifying homomorphisms that differ only by inner automorphisms. Thus if .f, f ' : G → G' is a pair of group ' homomorphisms, then .[f ] = [f ' ] in .Gr if and only if .f ' = ( )g f for some .g ' ∈ G' . If .G : D(S) → Gr is a semifunctor, then we denote by .G the induced semifunctor G : D(S) → Gr.

.

Note that if .G is a semifunctor such that .Z˜ 2 (S, G) /= ∅ then .G is a functor, by Lemma 4.6. Lemma 4.8 Let .G : D(S) → Gr be a semifunctor such that (i) .GL = G|L(S) factors through .L(S), (ii) .G factors through .D(S).

4 Cohomology and .H-Coextensions

132

Then, for every 2-cocycle .α ∈ Z˜ 2 (S, G), the coextension .(S ⋊α G, π ) is an .Hcoextension if and only if (iii)

The equalities   .

    α(u, x) + G(u)x = G(x) = α(x, v) + gv + xG(v)

u∈l(x)

(4.19)

v∈r(x)

hold for all x in S and .g ∈ G(x), where l(x) = {u ∈ S | ux = x} and r(x) = {v ∈ S | xv = x}.

.

Proof This is proved like Lemma 4.5 and follows from (u, c)(x, g) = (ux, α(u, x) + cx + ug),

.

(x, g)(v, d) = (xv, α(x, v) + gv + xd)

in which .ux = x implies .ug = g, but .xv = v does not imply .gv = v.

⨆ ⨅

Let HZ2 (S) ⊆ Z2 (S)

.

denote the full subcategory of the category of nonabelian 2-cocycles of S whose objects are those .(G, α) of .Z2 (S) such that the conditions (i), (ii) and (iii) in Lemma 4.8 hold. Theorem 4.6 Let S be regular. If .G : D(S) → Gr is a semifunctor satisfying the conditions (i) and (ii) in Lemma 4.8, then for any .α ∈ Z˜ 2 (S, G) the equalities (4.19) of condition (iii) hold. Proof If S is regular then for every .x ∈ S there exist idempotents .e ∈ Rx and f ∈ Lx . Then .e ∈ l(x), .f ∈ r(x), and .[1, e, x], .[x, f, 1] are isomorphisms, by Theorem 3.17. Since .G factors through .D(S) and isomorphisms in .Gr come only from isomorphisms in .Gr, both .G〈1, e, x〉 and .G〈x, f, 1〉 are isomorphisms, so that .G(e)x = G(x) = xG(f ). Hence (4.19) holds. ⨆ ⨅ .

The proof of Theorem 4.6 shows that the following condition on a .(G, α) ∈ Z 2 (S), where the semifunctor .G satisfies the conditions (i) and (ii) of Lemma 4.8: For all .x ∈ S, .u, v exist in S such that .ux = x = xv and both homomorphisms G〈1, u, x〉 : G(u) → G(x) and .G〈x, v, 1〉 : G(v) → G(x) are surjective.

.

implies that .(G, α) ∈ HZ2 (S). This condition is automatically satisfied if S is regular. It is also precisely the condition needed if S is finite and .H is trivial, or more generally, if S is a compact Hausdorff monoid with .H being trivial.

4.3 The Category of H-Coextensions with Arbitrary Kernel

133

By Lemma 4.8, if .(G, α) ∈ HZ2 (S) then .Δ(G, α) = (S ⋊α G, π ) ∈ HExt(S). Hence, the embedding (4.18) restricts to an embedding Δ : HZ2 (S) ͨ→ HExt(S).

.

(4.20)

It turns out that this is an equivalence of categories. First we show: Theorem 4.7 Every .H-coextension of S is isomorphic in .HExt(S) to a coextension (S ⋊α G, π ) for some .(G, α) ∈ HZ2 (S).

.

Proof Let .(E, p) be an .H-coextension of S. For each .x ∈ S, choose .s(x) ∈ p −1 (x), with .s(1) = 1. Define a semifunctor, .G : D(S) → Gr, by  .

G(x) = Σp (s(x)), G〈u, x, v〉 = Σp 〈s(u), s(x), s(v)〉,

where .Σp is the kernel functor of .(E, p). Let .u, u' , x ∈ S. Since .p(s(ux)) = ux = p(s(u)s(x)), by Lemma 4.4, Σp 〈s(u' ), s(ux), 1〉 = Σp 〈s(u' ), s(u)s(x), 1〉

.

so that G〈u' , ux, 1〉 G〈u, x, 1〉 = G〈u' u, x, 1〉.

.

By Proposition 4.2, .Σp is smooth; hence, if .ux = u' x, then G〈u, x, 1〉 = G〈u' , x, 1〉.

.

Therefore .G|L factors through .L(S). Next, G〈u, xv, 1〉 G〈1, x, v〉 = Σp 〈s(u), s(xv), 1〉 Σp 〈1, s(x), s(v)〉,

.

G〈1, ux, v〉 G〈u, x, 1〉 = Σ〈1, s(ux), s(v)〉 Σ〈s(u), s(x), 1〉. By Lemma 4.4, Σp 〈s(u), s(xv), 1〉 = Σp 〈s(u), s(x)s(v), 1〉,

.

Σp 〈1, s(ux), s(v)〉 = Σp 〈1, s(u)s(x), s(v)〉. Hence .G is a semifunctor. For all .x, v, v ' ∈ S, by a similar argument, we have G〈1, xv, v ' 〉 G〈1, x, v〉 = Σp 〈1, s(x), s(v)s(v ' )〉,

.

G〈1, x, vv ' 〉 = Σp 〈1, s(x), s(vv ' )〉.

4 Cohomology and .H-Coextensions

134

Let .g ∈ Σp (s(x)s(v)s(v ' )) be such that .s(x)s(v)s(v ' ) . g = s(x)s(vv ' ). Then, Σp 〈1, s(x), s(vv ' )〉 = ( )g Σp 〈1, s(x), s(v)s(v ' )〉.

.

Hence .G|R(S) is a functor. Thus .G is a functor. Now .G|L(S) factors through .L(S), since .G does. If .x, v, v ' ∈ S and .xv = ' xv , then, as above, .Σp 〈1, s(x), s(v))〉 and .Σp 〈1, s(x), s(v ' )〉 differ by an inner automorphism. Thus, .G factors through .R(S) and .G factors through .D(S). By Lemma 4.1, .Σp (x) acts simply and transitively on .p−1 (x), for each .x ∈ S. Hence every element of E can be written in the form s(x) . g

.

for some unique .x ∈ S and .g ∈ G(x). In particular s(x)s(y) = s(xy) . α(x, y)

.

for some unique .α(x, y) ∈ G(xy). This defines a 2-cochain .α ∈ C˜ 2 (S, G), which is normalized since .s(1) = 1. Proposition 4.1 yields, for all .g ∈ G(x) and .h ∈ G(y), (s(x) . g)(s(y) . h) = (s(x)s(y)) . (G〈1, x, y〉g + G〈x, y, 1〉h)

.

= s(xy) . (α(x, y) + G〈1, x, y〉g + G〈x, y, 1〉h). Hence, for every .x, y, z ∈ S and .g ∈ G(x), 

   (s(x) . g)s(y) s(z) = (s(x)s(y)) . (α(x, y) + gy) s(z)   = s(xyz) . α(xy, z) + α(x, y)z + (gy)z , .   (s(x) . g)(s(y)s(z)) = (s(x) . g) s(yz) . α(y, z)  = s(xyz) . α(x, yz) + g(yz) + xα(y, z) . whence, α(xy, z) + α(x, y)z + (gy)z = α(x, yz) + g(yz) + xα(y, z),

.

and therefore .α ∈ Z˜ 2 (S, G) is a 2-cocycle. Finally, the map .F : S ⋊α G → E, .(x, g) I→ s(x) . g is an isomorphism of coextensions .(S ⋊α G, π ) ∼ = (E, p) in .Mon ↓S . Since .(E, p) is an .H-coextension, also is .(S ⋊α G, π ) whence, by Lemma 4.8, .(G, α) ∈ HZ2 (S). ⨆ ⨅ Theorem 4.8 The embedding .Δ : HZ2 (S) ͨ→ HExt(S) is full. Proof Let .(G, α), (G' , α ' ) ∈ HZ2 (S) and let .F : (S ⋊α G, π ) → (S ⋊α ' G' , π ) be a morphism in .Mon ↓S .

4.3 The Category of H-Coextensions with Arbitrary Kernel

135

If .x ∈ S and .g ∈ G(x), then .π F (x, g) = π(x, g) = x and there are unique φ(x), σ (g) ∈ G' (x) such that .F (x, 0) = (x, φ(x)) and .F (x, g) = (x, −φ(x) + σ (g)). This defines a 1-cochain .φ ∈ C˜ 1 (S, G' ) (.φ(1) = 0 since .F (1, 0) = (1, 0)) and a mapping .σ = σx : G(x) → G(x ' ) for each .x ∈ S.  Then .σ (0) = 0, for all .x ∈ S. Moreover, .F (x, g)(y, h) = F (x, g)F (x, h) yields

.

  σ α(x, y) + gy + xh = φ(xy) + α ' (x, y) − φ(x)y + σ (g)y − xφ(y) + xσ (h), (4.21)

.

for all .x, y ∈ S, .g ∈ G(x) and .h ∈ G(y), which in turn yields σ α(x, y) = φ(xy) + α ' (x, y) − φ(x)y − xφ(y)

.

(4.22)

if .g = 0 and .h = 0. Hence (4.15) holds. Let .g, h ∈ G(x). By (4.19), .g = α(u, x) + g ' x for some .u ∈ l(x) and .g ' ∈ G(u). Then .uh = h and .uσ (h) = h, since both .GL and .G'L factor through .L(S) and ' .ux = x. Hence (4.21), for .u, x and .g , h, yields σ (g + h) = φ(ux) + α ' (u, x) − φ(u)x + σ (g ' )x − uφ(x) + σ (h).

.

In particular, if .h = 0, σ (g) = φ(ux) + α ' (u, x) − φ(u)x + σ (g ' )x − uφ(x),

.

whence σ (g + h) = σ (g) + σ (h),

.

so that .σ : G(x) → G' (x) is a homomorphism for every .x ∈ S. Now (4.21) yields σ α(x, y) + σ (gy + xh) = φ(xy) + α ' (x, y) − φ(x)y + σ (g)y − xφ(y) + xσ (h)

.

= φ(xy) + α ' (x, y) − φ(x)y − xφ(y) + (xφ(y) + σ (g)y − xφ(y)) + xσ (h) (4.22)

= σ α(x, y) + (xφ(y) + σ (g)y − xφ(y)) + xσ (h),

so that σ (gy + xh) = (xφ(y) + σ (g)y − xφ(y)) + xσ (h).

.

With .g = 0, or .h = 0, this last equality yields σ (xh) = xσ (h),

.

σ (gx) = xφ(y) + σ (g)y − xφ(y),

4 Cohomology and .H-Coextensions

136

for every .x, ∈ S, .g ∈ G(x) and .h ∈ G(y). Then, for every .u, x, v ∈ S and .g ∈ G(x),   σ (ugv) = uσ (gv) = u xφ(v) + σ (g)v − xφ(v) .

.

Hence (4.14) holds. Therefore, .(σ, φ) : (G, α, ) → (G' , α ' ) is a morphism in 2 .HZ (S) and, clearly, .Δ(σ, α) = F . ⨅ ⨆ Theorems 4.7 and 4.8 prove Theorem 4.9 The embedding (4.20) establishes an equivalence of categories Δ : HZ2 (S) ≃ HExt(S).

.

4.4 Obstructions Let .G : D(S) → Gr be a semifunctor. We are interested in whether or not .Z˜ 2 (S, G) is nonempty, and if not, finding its elements. If .Z˜ 2 (S, G) is not empty, then necessarily .G : D(S) → Gr is a functor. By (ii) in Lemma 4.6, it is also necessary that there exist a 2-cochain .α ∈ C˜ 2 (S, G) that repairs the lack of full functoriality of .G|R(S) , in the sense that (gx)v = −uα(x, y) + g(xv) + uα(x, v)

.

for all .u, x, v ∈ S and .g ∈ G(u). Such a cochain is a normalized 2-precocycle; PZ˜ 2 (S, G)

.

denotes the set of all 2-precocycles S in .G. Thus .Z˜ 2 (S, G) ⊆ PZ˜ 2 (S, G). In what follows we generally assume that .PZ˜ 2 (S, G) /= ∅. Then .G is a functor, but not conversely in general. The center ZG : D(S) → Ab,

.

of .G is the .D(S)-module such that ZG(x) = {g ∈ G(x) | ugv ∈ Z(G(uxv)) for all u, v ∈ S}

.

for every .x ∈ S. Since .G is a functor, .ZG is well-defined. Then .ZG(x) ⊆ Z(G(x)), but the equality does not hold in general. Moreover, if .G factors through .D(S), then so does .ZG.

4.4 Obstructions

137

Let .G : D(S) → Gr be a semifunctor such that .PZ˜ 2 (S, G) /= ∅. There is a right action +

PZ˜ 2 (S, G) × C˜ 2 (S, ZG) → PZ˜ 2 (S, G),

.

(α, ξ ) I→ α + ξ,

of the abelian group of normalized 2-cochains on S in the center of .G (see (2.12)) on the set of 2-precocycles on S in .G: If .α ∈ PZ˜ 2 (S, G) and .ξ ∈ C˜ 2 (S, ZG), then (α + ξ )(x, y) = α(x, y) + ξ(x, y).

.

For every .u, x, v ∈ S and .g ∈ G(u), since .uξ(x, v) ∈ Z(G(uxv)), (gx)v = −uα(x, v) + g(xv) + uα(x, v)

.

= −uξ(x, v) − uα(x, v) + g(xv) + uα(x, v) + uξ(x, v) and therefore .α + ξ ∈ PZ˜ 2 (S, G). The orbits under this action of .C˜ 2 (S, ZG) on ˜ 2 (S, G) are the orbits of .PZ˜ 2 (S, G) and constitute a set .PZ OrPZ˜ 2 (S, G).

.

A normalized 3-cochain h on S with coefficients in .G assigns to each .x, y, z ∈ S an element .h(x, y, z) ∈ G(xyz) so that .h(x, y, z) = 0 if x, y or z equals 1. Under pointwise addition, these 3-cochains constitute a group C˜ 3 (S, G).

.

For every .α ∈ PZ˜ 2 (S, G), define .hα ∈ C˜ 3 (S, G) by hα (u, x, v) = α(ux, v) + α(u, x)v − uα(x, v) − α(u, xv).

.

Then hα+ξ = hα − δ(ξ ).

.

for every .ξ ∈ C˜ 2 (S, ZG). Therefore there is a well-defined mapping Obs : OrPZ˜ 2 (S, G) → C˜ 3 (S, G)/B˜ 3 (S, ZG)

.

that assigns to each orbit .Θ its obstruction Obs(Θ) = {hα | α ∈ Θ} = hα + B˜ 3 (S, ZG) ∈ C˜ 3 (S, G)/B˜ 3 (S, ZG),

.

for any .α ∈ Θ.

4 Cohomology and .H-Coextensions

138

If .PZ˜ 2 (S, G) consists of a single orbit .Θ, then .Obs(Θ) is the obstruction of the semifunctor .G, denoted by Obs(G).

.

Theorem 4.10 Let .G : D(S) → Gr be a semifunctor such that .PZ˜ 2 (S, G) /= ∅. Let ˜ 2 (S, G). .Θ be an orbit of .PZ If .Obs(Θ) = 0, .α ∈ Θ and .hα = δ(ξ ), then .α + ξ ∈ Θ ∩ Z˜ 2 (S, G). In fact, .

Z˜ 2 (S, G) ∩ Θ /= ∅ if and only if Obs(Θ) = 0.

The action of .C˜ 2 (S, ZG) on .PZ˜ 2 (S, G) induces a simply transitive action  + Θ ∩ Z˜ 2 (S, G) × Z˜ 2 (S, ZG) → Θ ∩ Z˜ 2 (S, G),

 .

(α, ξ ) I→ α + ξ.

Finally, if the obstruction .Obs(Θ) ∈ C˜ 3 (S, ZG)/B˜ 3 (S, ZG), then Obs(Θ) ∈ Z˜ 3 (S, ZG)/B˜ 3 (S, ZG) = H 3 (S, ZG).

.

Proof If .Z˜ 2 (S, G) ∩ Θ /= ∅ and .α ∈ Z˜ 2 (S, G) ∩ Θ /= ∅, then .hα = 0 since .α is a 2-cocycle, and Obs(Θ) = hα + B˜ 3 (S, ZG) = B˜ 3 (S, ZG) = 0 ∈ Z˜ 3 (S, G)/B˜ 3 (S, ZG).

.

Conversely, if .Obs(Θ) = B˜ 3 (S, ZG), then, for any .α ∈ Θ, we have .hα = ∂(ξ ) for some .ξ ∈ C˜ 2 (S, ZG), α(ux, v) + α(u, x)v − uα(x, v) − α(u, xv)

.

= uξ(x, v) − ξ(ux, v) + ξ(u, xv) − ξ(u, x)v. Thus .hα+ξ = 0 and .α + ξ ∈ Z˜ 2 (S, G) ∩ Θ, so that .Z˜ 2 (S, G) ∩ Θ /= ∅. If .α ∈ Θ ∩ Z˜ 2 (S, G), then as above .α + ξ ∈ Θ ∩ Z˜ 2 (S, G) for all .ξ ∈ Z˜ 2 (S, ZG). Thus the action of .C˜ 2 (S, ZG)C 2 on .PZ˜ 2 (S, G) induces an action of .Z˜ 2 (S, ZG) on ˜ 2 (S, G). If .α, β ∈ Θ ∩ Z˜ 2 (S, G), then .β lies in the orbit of .α and .β = α + ξ .Θ ∩ Z for some unique .ξ ∈ C˜ 2 (S, ZG), which must be a 2-cocycle since both .α and .β are 2-cocycles. Thus the action of .Z˜ 2 (S, ZG) on .Θ ∩ Z˜ 2 (S, G) is simply transitive. Finally, assume that .Obs(Θ) ∈ C˜ 3 (S, ZG)/B˜ 3 (S, ZG) and let .α ∈ Θ, so that ˜ 3 (S, ZG). For all .t, u, v, w ∈ S, .hα ∈ C δhα (t, u, v, w) = thα (u, v, w) − hα (tu, v, w) + hα (t, uv, w)

.

− hα (t, u, vw) + hα (t, u, v)w = [tα(uv, w) + tα(u, v)w − tuα(v, w) − tα(u, vw)] − [α(tuv, w) + α(tu, v)w − tuα(v, w) − α(tu, vw)]

4.4 Obstructions

139

+ [α(tuv, w) + α(t, uv)w − tα(uv, w) − α(t, uvw)] − [α(tu, vw) + α(t, u)vw − tα(u, vw) − α(t, uvw)] + [α(tu, v)w + (α(t, u)v)w − tα(u, v)w − α(t, uv)w] where each expression in square brackets lies in the center of .G(tuvw). To simplify this expression we use the fact that if .g1 + · · · + gn lies in the center of a group and .σ is circular permutation of .1, . . . , n, then .gσ (1) + · · · + gσ (n) also lies in the center. Thus upon performing the two outer inverse operations and making the obvious cancelations we are left with: ∂hα (t, u, v, w)

.

= [tα(uv, w) + tα(u, v)w − tuα(v, w) − tα(u, vw)] + [tuα(v, w) − α(tu, v)w + α(t, uv)w − tα(uv, w) + tα(u, vw) − α(t, u)vw] + [α(tu, v)w + (α(t, u)v)w − tα(u, v)w − tα(u, vw)] = [tα(uv, w) + tα(u, v)w − tuα(v, w) − tα(u, vw)] + [α(t, uv)w − tα(uv, w) + tα(u, vw) − α(t, u)vw + tuα(v, w) + (α(t, u)v)w − tα(u, v)w − α(t, uv)w] = [tα(u, v)w − tuα(v, w) − tα(u, vw) + tα(uv, w)] + [−tα(uv, w) + tα(u, vw) + tuα(v, w) − tα(u, v)w] = 0, since .α is a 2-precocycle and (α(t, u)v)w = −tuα(v, w) + α(t, u)vw + tuα(v, w).

.

Thus .∂hα = 0 and .Obs(Θ) ∈ H 3 (S, ZG).

⨆ ⨅

Here is an example of a monoid S with a semifunctor .G for which .PZ˜ 2 (S, G) is nonempty and has distinct orbits that intersect nontrivially with .Z˜ 2 (S, G): Let .FX be the free monoid on a nonempty set X. Let .{0, y} be a 2-element set that is disjoint from .FX . Let .S = FX ∪ {0, y}. Extend the multiplication of .FX to S so that .0w = w0 = 0, .y 2 = 0 and .wy = yw = y for all .w ∈ FX . Then .{0, y} is a nilpotent 0-minimal ideal of the monoid S. Let G be a nontrivial group with trivial center. Let .G : D(S) → Gr be the functor defined by .G(x) = G if .x = y, and .{0} otherwise, with all nonidentity morphisms going to the 0-morphisms. For each homomorphism .f : FX → G we construct a 2-cocycle .f∗ determined by .f∗ (y, w) = f∗ (w, y) = f (w) for all .w ∈ FX (any 2-cocycle vanishes off of

4 Cohomology and .H-Coextensions

140

{y} × FX ∪ FX × {y}). Then distinct morphisms give rise to distinct 2-cocycles in distinct orbits. Notice that .(S ⋊f∗ G, π ) is in .HExt(S) if and only if f is surjective.

.

Theorem 4.11 Let S be a monoid and let .G : D(S) → Gr be a semifunctor with PZ˜ 2 (S, G) /= ∅. Suppose also that .G|L(S) factors through .L(S) and that for all .x ∈ S there exists .u ∈ S such that .ux = x and .G〈1, u, x〉 : G(u) → G(u) is surjective. Then: .

(i) The action of .C˜ 2 (S, ZG) upon .PZ˜ 2 (S, G) is simply transitive, so that ˜ 2 (S, G) consists of a single orbit. .PZ (ii) .Obs(G) ∈ H 3 (S, ZG). (iii) If .Obs(G) = 0, then the action .

Z˜ 2 (S, G) × Z˜ 2 (S, ZG) → Z˜ 2 (S, G),

(α, ξ ) I→ α + ξ,

is simply transitive. (iv) Under the restricted action .

Z˜ 2 (S, G) × B˜ 2 (S, ZG) → Z˜ 2 (S, G)

of .B˜ 2 (S, ZG) on .Z˜ 2 (S, G), the orbits are the elements of .H 2 (S, G). Hence there is a one-to-one correspondence between the elements of the abelian group .H 2 (S, ZG) and the elements of the set .H 2 (S, G). Proof (i) Let .α, α ' ∈ PZ˜ 2 (S, G) and let .u, x, y, v ∈ S. By the hypothesis there exists .s ∈ S such that .suxyv = uxyv and   ((gx)y)v = g(xy)suα(x,y) v = (g(xy)v)suα(x,y)v = (g(xy)v)uα(x,y)v

.

for all .g ∈ G(su). Similarly, '

((gx)y)v = (g(xy)v)uα (x,y)v .

.

Since G〈1, s, uxyv〉 = G〈1, suxy, v〉G〈1, su, xy〉G〈1, s, u〉

.

is surjective, so is G〈1, suxy, v〉 G〈1, su, xy〉,

. '

so that .huα(x,y)v = huα (x,y)v for all .h ∈ G(suxyv) = G(uxyv), and u(α(x, y) − α ' (x, y))v ∈ ZG(uxyv),

.

for all .u, x, y, v ∈ S. Thus .ξ = α − α ' ∈ C˜ 2 (S, ZG). This proves (i).

4.4 Obstructions

141

Similarly, let .α ∈ PZ˜ 2 (S, G) and let .s, u, x, y, v ∈ S. By the hypothesis there exists .w ∈ S such that .wsuxyv = suxyv and .G〈1, w, suxyv〉 is surjective. For all .g ∈ G(ws),      .(((gu)x)y)v = (g(ux))wsα(u,x) y v = (g(ux))y)wsα(u,x)y v   = (g(uxy))wsα(ux,y) )wsα(u,x)y v   = (g(uxy))ws(α(ux,y)+α(u,x)y) v  ws(α(ux,y)+α(u,x)y)v = (g(uxy))v  s(α(ux,y)+α(u,x)y)v = (g(uxy))v , whereas  wsuα(x,y)v  (((gu)x)y)v = ((gu)(xy))wsuα(x,y) v = (((gu)(xy))v  wsuα(x,y)v  = (g(uxy))wsα(u,xy) v  ws(α(u,xy)+uα(x,y))v = (g(uxy))v  s(α(u,xy)+uα(x,y))v = (g(uxy))v .

.

Since G〈1, w, suxyv〉 = G〈1, wsuxy, v〉G〈1, ws, uxy〉G〈1, w, s〉

.

is surjective, so is G〈1, wsuxy, v〉G〈1, ws, uxy〉,

.

so that, as above, 

hs



α(ux,y)+α(u,x)y v

.

= hs





α(u,xy)+uα(x,y) v

for all .h ∈ G(wsuxyv) = G(suxyv). Hence .shα (u, x, y)v ∈ ZG(suxyv), for all s, u, x, y, v ∈ S, and .hα ∈ C˜ 3 (S, ZG). Then Theorem 4.10 yields .hα ∈ Z˜ 3 (S, ZG). This proves (ii). (iii) follows from Theorem 4.10, since .PZ˜ 2 (S, G) has only one orbit. Finally, recall from (4.17) that two .α, α ' ∈ Z˜ 2 (S, G) represent the same element in .H 2 (S, G) if there exists a 1-cochain .φ ∈ C˜ 1 (S, G) such that

.

gy = xφ(y) + gy − xφ(y), .

.

'

α(x, y) = φ(xy) + α (x, y) − φ(x)y − xφ(y).

(4.23) (4.24)

for all .x, y ∈ S and .g ∈ G(x). In particular, if .α = α ' + ξ for some .ξ ∈ B˜ 2 (S, ZG), then .α and .α ' represent the same element in .H 2 (S, G). Conversely, let .α, α ' ∈

4 Cohomology and .H-Coextensions

142

Z˜ 2 (S, G) be such that (4.23) and (4.24) hold for some .φ ∈ C˜ 1 (S, G), and let .u, x, v ∈ S. By the hypothesis there exists .s ∈ S such that .suxv = uxv and .G〈1, s, uxv〉 is surjective. By (4.23) (gx)v = ((gx)v)−suφ(x)v = ((gx)v)−uφ(x)v

.

for all .g ∈ G(su) and all .u, x, v ∈ S. As above, G〈1, s, uxv〉 = G〈1, sux, v〉G〈1, su, x〉G〈1, s, u〉

.

is surjective, so that G〈1, sux, v〉G〈1, su, x〉

.

surjective and .h = h−uφ(x)v for all .h ∈ G(suxv) = G(uxv). Therefore .uφ(x)v ∈ ZG(uxv) for all .u, x, v ∈ S, .φ ∈ C˜ 1 (S, ZG), and .α ' = α − δ(φ), where .δ(φ) ∈ B˜ 2 (S, ZG). This proves (iv). ⨆ ⨅ If S is regular and .G is a functor and factors through .D(S), then, for each .x ∈ S, there exists .u ∈ S such that .ux = x and .G〈1, u, x〉 is surjective: indeed .Rx contains an idempotent e, and then .ex = x and .G〈1, e, x〉 is an isomorphism in .Gr; since all isomorphisms of .Gr originate in .Gr, then .G〈1, e, x〉 is an isomorphism. Hence Theorem 4.11 yields: Theorem 4.12 Let S be a regular monoid and let .G : D(S) → Gr be a semifunctor such that .G|L(S) factors through .L(S) and .G is a functor and factors through .D(S). Let .α be in .PZ˜ 2 (S, G). Then hα (u, x, v) = α(ux, v) + α(u, x)v − uα(x, v) − α(u, xv)

.

defines a normalized 3-cocycle .hα ∈ Z 3 (S, ZG) whose cohomology class Obs(G) = [hα ] ∈ H 3 (S, ZG)

.

does not depend on the choice of .α ∈ PZ˜ 2 (S, G). Moreover, ˜ .Z

2 (S, G)

/= ∅ if and only if .Obs(G) = 0,

in which case there is a simply transitive action .

Z˜ 2 (S, G) × Z˜ 2 (S, ZG) → Z˜ 2 (S, G),

(α, ξ ) I→ α + ξ,

such that the orbits under the restricted action .

Z˜ 2 (S, G) × B˜ 2 (S, ZG) → Z˜ 2 (S, G)

are the elements of .H 2 (S, G). Hence there is a bijective map .H 2 (S, ZG) ∼ = H 2 (S, G).

4.4 Obstructions

143

Corollary 4.2 Let S be a regular monoid and let .G : D(S) → Gr be a semifunctor satisfying the conditions of Theorem 4.12 above. The following are equivalent: G : D(S) → Gr is a functor. Z˜ 2 (S, G) ∩ Z˜ 2 (S, ZG) /= ∅. ˜ 2 (S, G) = Z˜ 2 (S, ZG). .Z There is .α ∈ Z˜ 2 (S, G) such that .α(x, y) ∈ Z(G(xy)), for all .x, y ∈ S, .

(i) (ii) (iii) (iv)

. .

Proof We have (i) .⇒ (ii) ⇒ (iv); (ii) .⇔ (iii) since .Z˜ 2 (S, ZG) acts simply and transitively on .Z˜ 2 (S, G); and Lemma 4.6 yields (iii) .⇒ (i). We prove (iv) .⇒ (ii). Let .α ∈ Z˜ 2 (S, G) satisfy .α(x, y) ∈ ZG(xy) for all .x, y ∈ S. Let .u, x, y, v ∈ S. Since S is regular, .R(uxyv) contains an idempotent e. Then .e R eu R eux R euxy R euxyv. Since .α is a 2-cocycle α(eux, y) + α(eu, x)y = α(eu, xy) + euα(x, y).

.

Now .α(eux, y) ∈ ZG(euxy), .α(eu, xy) ∈ ZG(euxy), and .α(eu, x)y ∈ ZG(euxy) since .G〈1, eux, y〉 is an isomorphism. Hence .euα(x, y) ∈ ZG(euxy) and euα(x, y)v = uα(x, y)v ∈ ZG(euxyv) = ZG(uxyv),

.

since .G〈1, euxy, v〉 is also an isomorphism. Thus .α ∈ C˜ 2 (S, ZG); in fact .α ∈ Z˜ 2 (S, ZG) since .α is a 2-cocycle.

⨆ ⨅

Thus far our examination of the role of cohomology in the study of 2-cocycles has depended on assuming an existing 2-precocycles. This raises the question of the existence of a 2-precocycle, that is, whether or not for a given semifunctor ˜ 2 (S, G) is nonempty. Establishing sufficiently general .G : D(S) → Gr, the set .PZ criteria for a semifunctor .G to have a 2-precocycle system seems to be a hard problem. It is clearly necessary that .G be a functor, but by no means sufficient. No such problem arises in the cohomological study of group extensions. This is due to a special case of the following result. Proposition 4.5 Let .G : D(S) → Gr be a semifunctor such that .G : D(S) → Gr is a functor. If .G〈u, x, 1〉 is surjective for all .u, x ∈ S, then .PZ˜ 2 (S, G) is nonempty. Proof For each .x, v ∈ S there exists .α(x, v) ∈ G(xv) such that (gx)v = (g(xv))α(x,v) ,

.

for all .g ∈ G(1). If .h ∈ G(u), then .h = ug for some .g ∈ G(1) and (hx)v = ((ug)x)v = u((gx)v) = u(g(xv))α(x,v)

.

= (ug(xv))uα(x,v) = (h(xv))uα(x,v) . Thus .α ∈ PZ˜ 2 (S, G).

⨆ ⨅

4 Cohomology and .H-Coextensions

144

We conclude with an, until now, unpublished result that was communicated to the authors by Grillet: Corollary 4.3 Let .G : D(S) → Gr be a semifunctor such that .G is a functor and for all .u, x, v ∈ S, .G〈u, x, v〉 is surjective. Then .PZ˜ 2 (S, G) /= ∅. Moreover, if ˜ 2 (S, G), then .α ∈ PZ hα (u, x, v) = α(ux, v) + α(u, x)v − uα(x, v) − α(u, xv) ∈ G(uxv)

.

defines a normalized 3-cocycle .hα ∈ Z˜ 3 (S, ZG) whose cohomology class, Obs(G) = [hα ] ∈ H 3 (S, ZG),

.

the obstruction of .G, is does not depend on the choice of .α ∈ PZ˜ 2 (S, G). Finally, ˜ 2 (S, G) /= ∅ if and only if .Obs(G) = 0. .Z The surjectivity of every .G〈u, x, v〉 implies that every successful .G-induced coextension .(E, π ) must be a normal extension of the group .G(1) by the monoid S. More precisely, (i) .G(1) is a subgroup of its group of units, (ii) .G(1) is normal in the monoid E (i.e., .G(1)x = xG(1) for all .x ∈ E), (iii) the resulting monoid of cosets .E/G(1) is isomorphic to S. This brings us, of course, right back to the beginning, [6]. Normal extensions of groups by monoids form a natural bridge from classical group extension theory to .H-coextensions of monoids. Indeed, given such a normal extension .G ͨ→ E ↠ S, the monoid E is both a normal extension of the group G and an .H-coextension of the monoid S. Leech’s 1982 paper [6] on the subject of extending groups by monoids, was upgraded from his dissertation due to input from the subsequent research of both Grillet [3] and Leech [5].

4.5 Nonabelian Group Coextensions For illustrative purposes, we show in this section how the classical theorems on the classification of nonabelian extensions of groups by Schreier [8] and EilenbergMac Lane [2] arise as instances of the previous results on .H-coextensions with nonabelian kernel and obstructions. Let G be a group. We first look at the equivalence of categories (4.20), Δ : HZ2 (G) ≃ HExt(G),

.

  (G, α) I→ G ⋊α G ↠ G ,

Proposition 4.6 .HZ2 (G) = Z2 (G). Proof Since G is a group, every semifunctor .G : D(G) → Gr satisfies the conditions (i), (ii) and (iii) in Lemma 4.8. Thus, every nonabelian 2-cocycle of G is an .H-2-cocycle. ⨆ ⨅

4.5 Nonabelian Group Coextensions

145

Next, regarding the category .HExt(G) of .H-coextensions of G, let GrExt(G)

.

be the full subcategory of the category .Gr ↓G of groups over G whose objects are the coextensions of groups .(E, p) of G. Proposition 4.7 A (monoid) coextension .(E, p) of G is an .H-coextension if and only if E is a group. Hence HExt(G) = GrExt(G).

.

Proof Suppose .(E, p) is an .H-coextension. Let .w ∈ p −1 (x) and choose .w ' ∈ p −1 (x −1 ). Since both .ww ' and .w ' w are in .p−1 (1), .ww ' H 1 and .w ' w H 1. Then, for some .u, v ∈ E, .ww ' u = 1 and .vw ' w = 1. Therefore .w ' u = vw' = w−1 is an inverse of w in E. The converse is clear, since in a group E the relation .H is .E × E. ⨆ ⨅ The category .Z2 (G) of nonabelian 2-cocycles of G has a classic description: Let SchZ2 (G)

.

denote the category of Schreier 2-cocycles of G: Its objects (N, f, α)

.

consist of a group N and maps .f : G → Aut(N ) and .α : G × G → N such that f1 = idN ,

.

α(x, 1) = 0 = α(1, x), .

(4.25)

α(xy, z) + fz α(x, y) = α(x, yz) + α(y, z), .

(4.26)

fy fx (n) = −α(x, y) + fxy (n) + α(x, y),

(4.27)

for all .x, y, z ∈ G and .n ∈ N. Above, for every .x ∈ G, we denote by .fx the automorphism .f (x) ∈ Aut(N ). A morphism in .Sch(G), is a pair (σ, φ) : (N, f, α) → (N ' , f ' , α ' )

.

where .σ : N → N ' is a group homomorphism and .φ : G → N ' is a map such that φ(1) = 1,

.

σ α(x, y) = φ(xy) + α ' (x, y) − fy' φ(x) − φ(y), σfy (n) = φ(y) + fy' σ (n) − φ(y), for all .x, y ∈ G and .n ∈ N.

4 Cohomology and .H-Coextensions

146

Composition of .(σ, φ) with .(σ ' , φ ' ) : (G' , α ' ) → (G'' , α '' ) is given by (σ ' , φ ' )(σ, φ) = (σ ' σ, σ ' φ + φ ' ),

.

where .(σ ' φ + φ ' )(x) = σ ' φ(x) + φ ' (x), for every .x ∈ S. The identity of every object .(N, f, α) of .Schr(G) is .id(N,f,α) = (idN , 0). Every Schreier 2-cocycle .(N, f, α) of G yields an object .(Gf , α) of .Z2 (G), in which the semifunctor .Gf : D(G) → Gr is defined by  .

Gf (x) = N, Gf 〈u, x, v〉 = fv : N → N,

for every .x, u, v ∈ G. The assignment .(N, f, α) I→ (Gf , α) is the function on objects of a functor J : SchZ2 (G) → Z2 (G)

.

(4.28)

which carries every morphism .(σ, φ) : (N, f, α) → (N ' , f ' , α ' ) in .SchZ2 (G) to the (equally denoted) morphism in .Z2 (G) J (σ, φ) = (σ, φ) : (Gf , α) → (Gf ' , α ' )

.

in which .σx = σ : N → N ' for all .x ∈ G. Proposition 4.8 The functor .J : SchZ2 (G) → Z2 (G) is an equivalence of categories. Proof J is clearly a fully faithful functor. We show that any given object .(G' , α ' ) of 2 2 .Z (G) is isomorphic to .J (N, f, α) for some .(N, f, α) of .SchZ (G): Define ⎧ ⎪ N = G' (1) ⎪ ⎨ . fx = G' 〈x −1 , 1, x〉 : N → N, n I→ x −1 nx, ⎪ ⎪ ⎩ α(x, y) = G' 〈y −1 x −1 , xy, 1〉α ' (x, y) = y −1 x −1 α ' (x, y) ∈ N, for every .x, y ∈ G. Then .(N, f, α) is a Schreier 2-cocycle of G: Clearly .f1 = idN and, for every .x ∈ G, .α(x, 1) = 0 = α(1, x) since .α ' (x, 1) = 0 = α ' (1, x). For every .x, y, z ∈ G,   xyz α(xy, z) + fz α(x, y) − α(y, z) − α(x, yz)

.

= α ' (xy, z) + α ' (x, y)z − xα ' (y, z) − α ' (x, yz) = 0,

4.5 Nonabelian Group Coextensions

147

whence α(xy, z) + fz α(x, y) − α(y, z) − α(x, yz) = 0

.

since .G〈xyz, 1, 1〉; G(1) → G(xyz) is an isomorphism. Similarly, for every .x, y ∈ G and .n ∈ N,   xy fy fx (n) − α(x, y) − fxy (n) + α(x, y)

.

= (nx)y − α ' (x, y) − n(xy) + α ' (x, y) = 0 whence .fy fx (n) − α(x, y) − fxy (n) + α(x, y) = 0. Finally, an isomorphism .(σ, 0) : (Gf , α) → (G' , α ' ) in .Z2 (G) is defined by the group isomorphisms σx = G' 〈x, 1, 1, 〉 : Gf (x) → G' (x),

.

n I→ xn,

(x ∈ G, n ∈ N).

Condition (4.15) holds since, for every .x, y ∈ G, σxy α(x, y) = xyy −1 x −1 α ' (x, y) = α ' (x, y),

.

and condition (4.14) also holds since σuxv fv (n) = uxvv −1 nv = uxnv = uσx (n)v

.

for every .u, x, v ∈ G and .n ∈ N = GN,f (x).

⨆ ⨅

Hence, by Propositions 4.6, 4.7 and 4.8, Theorem 4.9 yields Theorem 4.13 (Schreier) For every group G, the functor ΔJ : SchZ2 (G) → GrExt(G)

.

is an equivalence of categories.

⨆ ⨅

Remark The functor .ΔJ carries every Schreier 2-cocycle .(N, f, α) to the group coextension of G by N N ͨ→ G ⋊α N ↠ G

.

in which the twisted semidirect product .G ⋊α N is .G × N with multiplication (x, n)(y, m) = (xy, α(x, y) + fy (n) + m)

.

for every .x, y ∈ G and .n, m ∈ N .

4 Cohomology and .H-Coextensions

148

Let N be a fixed group. Let SchZ2 (G, N ) ⊆ SchZ2 (G)

.

denote the subcategory whose objects’ first component is N and whose morphism’s first component is .idN : φ = (idN , φ) : (N, f, α) → (N, f ' , α ' ).

.

In other words, the set of objects of .SchZ2 (G, N ), denoted by .

2 (G, N ), Z˜ Sch

consists of pairs of maps .(f : G → Aut(N ), α : G × G → N) such that (4.25), (4.26) and (4.27) hold. A morphism φ : (f, α) → (f ' , α ' )

.

in .SchZ2 (G, N ) is a map .φ : G → N such that .φ(1) = 1 and α(x, y) = φ(xy) + α ' (x, y) − fy' φ(x) − φ(y),

.

fy (n) = φ(y) + fy' (n) − φ(y), for all .x, y ∈ G and .n ∈ N . This category .SchZ2 (G, N ) is a groupoid and, by 2 (G, N ), the second definition, its set of isomorphism classes of objects is .HSch nonabelian cohomology set of G with coefficients in N : 2 2 HSch (G, N ) = Z˜ Sch (G, N )/ ∼ =.

.

The restriction of the equivalence .ΔJ : SchZ2 (G) 2 .SchZ (G, N ) yields an equivalence of categories

≃

GrExt(G) to

ΔJ : SchZ2 (G, N ) ≃ Ext(G, N )

.

between the groupoid of Schreier 2-cocycles of G in N and the groupoid .Ext(G, N ) of group coextensions of G by N , i

p

N ͨ→ E ↠ G,

.

p

i

i'

p'

in which a morphism from .N ͨ→ E ↠ G to .N ͨ→ E ↠ G is an isomorphism of groups .f : E → E ' such that .f i = i ' and .p' f = p. Then, if Ext(G, N )

.

4.5 Nonabelian Group Coextensions

149

denotes the set of isomorphism classes of group coextensions of G by N, .ΔJ induces a bijection 2 HSch (G, N ) ∼ = Ext(G, N )

.



 [f, α] I→ [N ͨ→ G ⋊α N ↠ G] .

Hence, 2 (G, N ) classifies Theorem 4.14 (Schreier) For every pair of groups G and N , .HSch group coextensions of G by N. .□

For every group N , let .Out(N ) = Aut(N )/Inn(N ) be its group of outer automorphisms and let .p : Aut(N ) ↠ Out(N ) be the canonical projection. For every group G, there is a canonical commutative triangle of maps

.

2 (G, N ) to the homowhere .χ carries the class of a Schreier 2-cocycle .(f, α) ∈ Z˜ Sch op morphism .χ (f, α) = pf : G → Out(N ) (although f is not a homomorphism, pf is by (4.27)). This map .χ determines a partition of the Schreier cohomology set of G with coefficients in N

  2 2 G, (N, ρ) .HSch (G, N ) = HSch ρ

where .ρ varies in the set of homomorphisms .HomGr (Gop , Out(N )) and, for every homomorphism .ρ : Gop → Out(N ),   2 G, (N, ρ) = χ −1 (ρ) HSch

.

denotes the fibre over .ρ of .χ. Similarly, if   Ext G, (N, ρ) = χ¯ −1 (ρ),

.

then there is a partition of the set of isomorphic classes of coextensions of G by N Ext(G, N ) =



.

  Ext G, (N, ρ)

ρ 2 (G, N ) ∼ Ext(G, N ) such that, for each .ρ : Gop → Out(N ), the bijection .HSch = restricts to a bijection

    2 G, (N, ρ) ∼ HSch = Ext G, (N, ρ) .

.

4 Cohomology and .H-Coextensions

150

A pair .(N, ρ), where N is a group and .ρ : Gop → Out(N ) is a homomorphism, is what Eilenberg and Mac Lane term G-kernel in [2] (or what Mac Lane terms abstract kernel in [7]). Given a G-kernel .(N, ρ), select a map .f : G → Aut(N ) such that .pf = ρ, with .f1 = idN , and define the semifunctor Gf : D(G) → Gr,

.

as in the construction of the functor J in (4.28), by  .

Gf (x) = N, Gf 〈u, x, v〉 = fv : N → N,

for every .x, u, v ∈ G. Then,

  2 G, (N, ρ) . Lemma 4.9 There is a canonical bijection .H 2 (G, Gf ) ∼ = HSch

2 (G, (N, ρ)), .α I→ (f, α), clearly induces a Proof The map .Z˜ 2 (G, Gf ) → Z˜ Sch   2 2 injective map .H (G, Gf ) → HSch G, (N, ρ) , which is also surjective: 2 (G, (N, ρ)). Since .pf = ρ = pf ' , there is a map .φ : G → N, Let .(f ' , α ' ) ∈ Z˜ Sch with .φ(1) = 1 such that .fx (n) = −φ(x)+fx' (n)+φ(x) for every .x ∈ G and .n ∈ N . Defining, for every .x, y ∈ G,

α(x, y) = −φ(xy) + α ' (x, y) + fy' φ(x) + φ(y),

.

we find 2-cocycle .α ∈ Z˜ 2 (G, Gf ) with an isomorphism .φ : (f ' , α ' ) → (f, α) in 2 .SchZ (G, N ). ⨆ ⨅ The center .ZGf of the semifunctor .Gf is the .D(G)-module such that  .

ZGf (x) = ZN, ZGf 〈u, x, v〉 = fv : ZN → ZN,

which, by the equivalence of categories .ı ∗ : D(G)-Mod → G-Mod in Theorem 3.3, corresponds to the G-module defined by the center ZN of N with G-action xn = fx −1 (n)

.

for every .x ∈ G and .n ∈ N. This G-action on ZN does not depend on the choice of the representatives .fx ∈ ρ(x), since every inner automorphism of N leaves the elements of the center fixed. We refer to this structure of G-module on ZN as the obtained via .ρ : Gop → Out(N ), and denote by n HEM (G, (ZN, ρ)),

.

n = 0, 1, · · · ,

References

151

the corresponding cohomology groups. By Theorem 3.3, there are natural isomorphisms n H n (G, ZGf ) ∼ (G, (ZN, ρ)), = HEM

.

n = 0, 1, · · · .

Finally, notice that a 2-precocycle .α ∈ PZ˜ 2 (G, Gf ) is simply a normalized map .α : G × G → N such that fy fx (n) = −α(x, y) + fx (n) + α(x, y)

.

for every .x, y ∈ G and .n ∈ N . Such a 2-precocycle .α always exists, since both automorphisms .fy fx and .fxy represent the same outer automorphism, namely ˜ 3 (G, ZGf ) that represents the .ρ(xy) = ρ(y)ρ(x). Moreover, the 3-cocycle .hα ∈ Z 3 cohomology class .Obs(Gf ) in .H (G, ZGf ), hα (x, y, z) = α(xy, z) + fz α(x, y) − α(y, z) − α(x, yz),

.

3 (G, (ZN, ρ)) that represents the cohomoljust corresponds to the 3-cocycle in .Z˜ EM ogy class defined by Eilenberg-Mac Lane as the obstruction of the G-kernel .(N, ρ), 3 .Obs(N, ρ) ∈ HEM (G, (ZN, ρ)). All in all, Theorem 4.11 (or Theorem 4.12), for .S = G and .G = Gf : D(G) → Gr as above, yields

Theorem 4.15 (Eilenberg-Mac Lane) Let G be a group. Every G-kernel .(N, ρ) invariably determines a 3-cohomology class 3 Obs(N, ρ) ∈ HEM (G, (ZN, ρ))

.

called the obstruction of the   G-kernel. The set .Ext G, (N, ρ) of classes of extensions of G by the G-kernel .(N, ρ) is not empty if and only if its obstruction vanishes.   If .Obs(N, ρ) = 0, then the elements of .Ext G, (N, ρ) are in one-to one 2 (G, (ZN, ρ)). .□ correspondence with the elements of .HEM

References 1. Clifford, A.H., Preston, G.B.: The Algebraic Theory of Semigroups, vol. I. Mathematical Surveys, vol. 7. American Mathematical Society, Providence (1961) 2. Eilenberg, S., MacLane, S.: Cohomology theory in abstract groups II, group extensions with a non abelian kernel. Ann. Math. 48, 326–341 (1947) 3. Grillet, P.A.: Left coset extensions. Semigroup Forum 7, 200–263 (1974) 4. Grillet, P.A.: Semigroups: An Introduction to the Structure Theory. Monographs and Textbooks in Pure and Applied Mathematics, vol. 193. Marcel Dekker, Inc., New York (1995) 5. Leech, J.: H-coextensions of monoids. Mem. Am. Math. Soc. 1, 157(2), 1–66 (1975)

152

4 Cohomology and .H-Coextensions

6. Leech, J.: Extending groups by monoids. J. Algebra 74, 1–19 (1982) 7. Mac Lane, S.: Homology. Classics in Mathematics. Springer-Verlag, Berlin (1995) 8. Schreier, O.: Über die Erweiterung von Gruppen I. Monatsh. Math. Phys. 34, 165–180 (1926) 9. Wells, C.: Extension theories for categories. Preliminary Report (1980). Revised version (2001) available from https://www.academia.edu/4333242

Chapter 5

Cohomology of Monoids with Operators

This chapter deals with .𝚪-monoids, where .𝚪 is a fixed monoid of operators that acts on any given monoid S via a monoid homomorphism .𝚪 → End(S). A homomorphism of .𝚪-monoids is monoid homomorphism .f : S → T that preserves the action of .𝚪. That is, .f (γx) = γf (x). Based on the papers [3–5], we provide here a cohomology of .𝚪-monoids S, with cohomology groups denoted by .H𝚪n (S, A). This theory is extended in [2] to presheaves of monoids in view of other applications. When both .𝚪 and S are groups, this theory goes back to that first introduced by J.H.C. Whitehead in his seminal 1950 paper [8] on the cohomology of groups with operators. If .𝚪 = 1 is trivial, then the cohomology groups .H𝚪n (S, A) are simply the ordinary .D-cohomology ones .H n (S, A). In the first section we construct a small category .D𝚪 (S) of a .𝚪-monoid S. This category is the domain of the .D𝚪 (S)-modules that provide the cohomology groups of S with coefficients. We also define .𝚪-derivations and the functor .Der𝚪 (S, −) whose derived functors are the .𝚪-cohomology groups H𝚪n (S, A) = R n−1 Der𝚪 (S, −)(A).

.

In Sect. 5.2, we obtain certain long exact sequences that compare .𝚪-cohomology groups to other known cohomology groups. In Sect. 5.3 we construct a cochain complex .C𝚪• (S, A) that computes the cohomology groups of a .𝚪-monoid S. Finally, in Sect. 5.4 we establish bijections .Ext𝚪 (S, A) ∼ = H𝚪2 (S, A), and show that 2 .H (S, A) classifies .𝚪-coextensions of S by .A. 𝚪

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 A. M. Cegarra, J. Leech, The Cohomology of Monoids, RSME Springer Series 12, https://doi.org/10.1007/978-3-031-50258-3_5

153

154

5 Cohomology of Monoids with Operators

5.1 The Cohomology of 𝚪-Monoids Hereafter, .𝚪 is a fixed monoid of operators. A .𝚪-monoid is a monoid S together with a left .𝚪-action .(γ , x) I→ γx of .𝚪 on S such that .

γ(xy)

= γx γy,

τ(γx)

= τ γx,

1x

= x,

for all .x, y ∈ S and .γ ∈ 𝚪. Let 𝚪-Mon

.

be the category whose objects are .𝚪-monoids, in which a morphism .p : S → S ' is a monoid homomorphism p such that p(γx) = γp(x)

.

for all .x ∈ S and .γ ∈ 𝚪.

D𝚪(S)-Modules If S is a .𝚪-monoid S, let .D𝚪 (S) denote the category whose objects are the elements x ∈ S, with one morphism

.

〈u, x, v, γ 〉 : x → y,

.

for every .u, v ∈ S and .γ ∈ 𝚪 such that .u γx v = y. Composition is by '

'

〈u' , u γx v, v ' , γ ' 〉〈u, x, v, γ 〉 = 〈u' γ u, x, γ v v ' , γ ' γ 〉,

.

and .〈1, x, 1, 1〉 is the identity on x, for all .x ∈ S. A .D𝚪 (S)-module is a functor from .D𝚪 (S) to .Ab, the category of abelian groups. These are the natural coefficients for a cohomology of .𝚪-monoids. Indeed the category .D𝚪 (S)-.Mod of .D𝚪 (S)-modules is equivalent to the category of abelian group objects in the category of .𝚪-monoids over S (which provide the coefficients of Beck cohomology [1]); this is proved like Theorem 6 in Wells [7]. Let D𝚪 (S)-Mod

.

denote the category of .D𝚪 (S)-modules (whose morphisms are the natural transformations between them).

5.1 The Cohomology of 𝚪-Monoids

155

There is an embedding of categories D(S) ͨ→ D𝚪 (S),

.

〈u, x, v〉 I→ 〈u, x, v, 1〉,

so that every .D𝚪 (S)-module .A : D𝚪 (S) → Ab has an underlying .D(S)-module, also denoted by .A, for which we employ the notations in (2.1). We also denote γ .A〈1, x, 1, γ 〉a by . a, for every .x ∈ S, .γ ∈ 𝚪 and .a ∈ A(x). Hence A〈u, x, v, γ 〉(a) = uγa v

.

for all .u, x, v ∈ S, .γ ∈ 𝚪 and .a ∈ A(x). Then, for all .u, x, v ∈ S, .a, b ∈ A(x), and τ, γ ∈ 𝚪,

.

γ

.

(a + b) = γa + γb,

γ

(ua) = γuγa,

γ

(av) = γa γv,

( a) = τ γ a, 1a = a.

τ γ

If .A and .A' are .D𝚪 (S)-modules, then .F : A → A' is a morphism of .D𝚪 (S)modules if and only if .F is a morphism of .D(S)-modules and .F(γa) = γF(a), for any .x ∈ S, .a ∈ A(x), and .γ ∈ 𝚪. The monoid semidirect product .S ⋊ 𝚪 of S and .𝚪 is the cartesian product set .S × 𝚪 with multiplication '

(u' , γ ' )(u, γ ) = (u' γ u, γ ' γ ).

.

The projection functor .j : 〈u, x, v, γ 〉 I→ (u, γ ), from .D𝚪(S) to .S ⋊ 𝚪 (regarded as a category with only one object), induces a full exact embedding j ∗ : (S ⋊ 𝚪)-Mod → D𝚪(S)-Mod

.

that carries any ordinary .(S ⋊ 𝚪)-module A, with action .((u, γ ), a) I→ (u, γ ) a, to the .D𝚪 (S)-module .j ∗A defined by .j ∗A(x) = A, for all .x ∈ S, and .u γ a v = (u, γ ) a, for all .u, v ∈ S, .γ ∈ 𝚪, and .a ∈ A. When S is a group, the projection functor .j : D𝚪(S) → S ⋊ 𝚪 is actually an equivalence of categories, so it induces an equivalence of categories (S ⋊ 𝚪)-Mod ≃ D𝚪(S)-Mod.

.

Via the projection homomorphism .S ⋊ 𝚪 → 𝚪, .(x, γ ) I→ γ , every .𝚪-module is also an .(S ⋊ 𝚪)-module, and thus we have also a full exact embedding 𝚪-Mod → D𝚪(S)-Mod.

.

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5 Cohomology of Monoids with Operators

The 𝚪-Cohomology Groups Let S be a .𝚪-monoid. A .𝚪-derivation of S in a .D𝚪(S)-module .A is a derivation of S in the underlying .D(S)-module .A, .d : S → A, such that d(γa) = γ(da)

.

for any .γ ∈ 𝚪, .x ∈ S, and .a ∈ A(x). We denote by Der𝚪 (S, A) ⊆ Der(S, A)

.

the abelian subgroup of all .𝚪-derivations .d : S → A. The functor Der𝚪 (S, −) : D𝚪(S)-Mod −→ Ab,

.

is representable. The ideal augmentation .IS is in fact a .D𝚪(S)-module in which IS〈1, x, 1, γ 〉 : IS(x) → IS(γx) acts on generators by

.

γ

.

((u0 , u1 ) − (x, 1)) = (γu0 , γu1 ) − (γx, 1).

A straightforward verification shows that, for any .D𝚪(S)-module .A, the isomorphism .Der(S, A) ∼ = HomD(S) (IS, A) in Proposition 2.2 induces an isomorphism Der𝚪 (S, A) ∼ = HomD𝚪(S) (IS, A).

.

(5.1)

which is natural in S and .A. For every .n ≥ 1, .H𝚪n (S, −) is the .(n − 1)th derived functor of .Der𝚪 (S, −), so that, for every .D𝚪(S)-module .A, H𝚪n (S, A) = R n−1 Der𝚪 (S, −)(A),

.

or equivalently, by (5.1), H𝚪n (S, A) = Extn−1 D𝚪(S) (IS, A).

.

(5.2)

When both .𝚪 and S are groups, the categories .(S ⋊ 𝚪)-.Mod and .D𝚪 (S)-Mod are canonically equivalent and, for any .(S ⋊ 𝚪)-module A, by Cegarra et al. [4, Theorem 2.6], the .𝚪-cohomology groups .H𝚪n (S, j ∗A) agree with the vector n (𝚪, S; A) as first defined by Whitehead in [8] for groups cohomology groups .Hn−1 with operators. (See also [6].) Some properties of the .𝚪-cohomology groups immediately follow from their definition: Each .H𝚪n (S, −) is an additive functor; .H𝚪1 (S, −) = Der𝚪 (S, −); for

5.2 The Linking Long Exact Sequences

157

every short exact sequence of .D𝚪(S)-modules .0 → A' → A → A'' → 0 , there is a long exact cohomology sequence .

· · · → H𝚪n (S, A' ) → H𝚪n (S, A) → H𝚪n (S, A'' ) → H𝚪n+1 (S, A' ) → · · · ;

if .A is an injective .D𝚪(S)-module, then .H𝚪n (S, A) = 0 for .n ≥ 2; etc. The following property is naturally expected. Proposition 5.1 If F is a free .𝚪-monoid, then .H𝚪n (F, −) = 0, for .n ≥ 2. Proof The free .𝚪-monoid on a set X is the free monoid on the set .𝚪 ×S = {γs | x ∈ X, γ ∈ 𝚪}, with .𝚪 acting on generators by .τ (γx) = τ γx. If .A is a .D𝚪(F )-module .A, by Corollary 2.1, there is a natural isomorphism 

Der(F, A) ∼ =

A(γx),

.

  d I→ d(γx) (γ ,x)∈𝚪×X ,

(γ ,x)∈𝚪×X

which restricts to a natural isomorphism Der𝚪 (F, A) ∼ =



.

A(x),

  d I→ dx x∈X .

x∈X

Hence .Der𝚪 (F, −) is exact. Therefore, its right-derived functors .R n−1 Der𝚪 (F, −) = H𝚪n (F, −) vanish for all .n ≥ 2. ⨆ ⨅

5.2 The Linking Long Exact Sequences The .𝚪-cohomology groups .H𝚪n (S, A), of any .𝚪-monoid S with coefficients in a .D𝚪 (S)-module .A, are closely related both to the cohomology groups n .H (D𝚪 (S), A), of the category .D𝚪 (S) with coefficients in .A, and the EilenbergMac Lane cohomology groups .H n (𝚪, A(1)), of the monoid of operators .𝚪 with coefficients in the .𝚪-module defined by the abelian group .A(1) on which .𝚪 acts by the homomorphisms .A〈1, 1, 1, γ 〉 : a I→ γa. This relationship is stated by means of the following long exact sequence that links these cohomology groups. Theorem 5.1 Let S be a .𝚪-monoid. For every .D𝚪 (S)-module .A, there is a natural long exact sequence .

n · · · → H𝚪n (S, A) → H n (D𝚪 (S), A) → HEM (𝚪, A(1)) → H𝚪n+1 (S, A) → · · ·

Proof The short exact sequence of .D(S)-modules in (2.22), 0 → IS → F0 S → Z → 0,

.

158

5 Cohomology of Monoids with Operators

is of .D𝚪(S)-modules, in which .F0 S〈1, x, 1, γ 〉 : F0 S(x) → F0 S(γx) acts on generators by .γ(u0 , u1 ) = (γu0 , γu1 ). Then, we have the induced long exact sequence

.

We have .ExtnD𝚪 (S) (IS, A) = H𝚪n+1 (S, A) and .ExtnD𝚪 (S) (Z, A) = H n (D𝚪 (S), A), and prove that there are natural isomorphisms ExtnD𝚪 (S) (F0 S, A) ∼ = H n (𝚪, A(1)).

.

For each .n ≥ 0, let .𝚪n : S = ObD𝚪 (S) → Set be the functor such that .𝚪n (1) = 𝚪 n and .𝚪n (x) = ∅ for all .x /= 1, and let .Pn be the free .D𝚪 (S)-module on .𝚪n . Then, for every .x ∈ S, Pn (x) = Z[{(γ0 , · · · , γn , u0 , u1 ) ∈ 𝚪 n+1 × S 2 | u0 u1 = x}]

.

and, for each arrow .〈u, x, v, γ 〉 : x → y in .D𝚪 (S), the induced homomorphism Pn 〈u, x, v, γ 〉 : Pn (x) → Pn (y), .a I→ u γa v, is defined by

.

u γ(γ0 , γ1 , . . . , γn , u0 , u1 ) v = (γ γ0 , γ1 , . . . , γn , uγu0 , γu1 v)

.

for every generator .(γ0 , · · · , γn , u0 , u1 ) of .Pn (x). These .Pn , .n ≥ 0, form an augmented chain complex of .D𝚪 (S)-modules, P• → F0 S,

.

whose differential operators, .∂ : Pn (x) → Pn−1 (x), at every .x ∈ S, are defined on generators by

.

∂(γ0 , . . . , γn , u0 , u1 ) =

n−1 

(−1)i (γ0 , . . . , γi γi+1 , . . . , γn , u0 , u1 )

i=0

+ (−1)n (γ0 , . . . , γn−1 , u0 , u1 ), and the augmentation .ϵ : P0 → F0 S is the morphism of .D𝚪(S)-modules such that, at every .x ∈ S, .ϵ : P0 (x) → F0 S(x) is the homomorphism such that ϵ(γ0 , u0 , u1 ) = (γu0 , γu1 ),

.

for any .γ0 ∈ 𝚪 and .u0 , u1 ∈ S with .u0 u1 = x.

5.2 The Linking Long Exact Sequences

159

Every .Pn is projective. For each .n ≥ 0, let .Фn : Pn (x) → Pn+1 (x) be the homomorphism such that Фn (γ0 , · · · , γn , u0 , u1 ) = (1, γ0 , · · · , γn , u0 , u1 ),

.

for every generator .(γ0 , · · · , γn , u0 , u1 ) of .Pn (x), and let .Ф−1 : F0 S(x) → P0 (x) be the homomorphism such that Ф−1 (u0 , u1 ) = (1, u0 , u1 )

.

for every generator .(u0 , u1 ) of .P0 (x). A straightforward verification on generators shows that these homomorphisms .Фn , .n ≥ −1 constitute a contracting homotopy of the augmented chain complex .P• (x) → F0 S(x): ϵФ−1 (u0 , u1 ) = ϵ(1, u0 , u1 ) = (u0 , u1 )

.

(∂Фn + Фn−1 ∂)(γ0 , . . . , γn , u0 , u1 ) = ∂(1, γ0 , . . . , γn , u0 , u1 ) n−1  + (−1)i (1, γ0 , . . . , γi γi+1 , . . . , γn , u0 , u1 ) i=0

+ (−1)n (1, γ0 , . . . , γn−1 , u0 , u1 ) = (γ0 , . . . , γn , u0 , u1 ) n  (−1)i (1, γ0 , . . . , γi−1 γi , . . . , γn , u0 , u1 )

+

i=1

+ (−1)n+1 (1, γ0 , . . . , γn−1 , u0 , u1 ) n−1  (−1)i (1, γ0 , . . . , γi γi+1 , . . . , γn , u0 , u1 )

+

i=0

+ (−1)n (1, γ0 , . . . , γn−1 , u0 , u1 ) = (γ0 , . . . , γn , u0 , u1 ). Therefore .P• → F0 S is a projective resolution of .F0 S and, for every .n ≥ 0, we have   ExtnD𝚪 (S) (F0 S, A) = H n HomD𝚪 (S) (P• , A) .

.

Theorem 1.1 now provides natural isomorphisms of abelian groups HomD𝚪(M) (Pn , A) ∼ =



.

𝚪n

A(1) = C n (𝚪, A(1)),

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5 Cohomology of Monoids with Operators

which constitute an isomorphism of cochain complexes between .HomD𝚪(M) (P• , A) and the standard cochain complex .C • (𝚪, A(1)) that computes the cohomology groups of the monoid .𝚪 with coefficients in the .𝚪-module .A(1). (See (3.2).) Hence,   n ExtnD𝚪 (S) (F0 S, A) ∼ (𝚪, A(1)). = H n C • (𝚪, A(1)) = HEM

.

⨆ ⨅ If S is a group, then we saw that the categories of .D𝚪 (S)-.Mod and .S ⋊ 𝚪-.Mod are equivalent. Therefore, for any .(S ⋊ 𝚪)-module A, we have isomorphisms n H n (D𝚪 (S), j ∗A) = ExtnD𝚪 (S) (Z, j ∗A) ∼ (S ⋊ 𝚪, A), = ExtnS⋊𝚪 (Z, A) = HEM

.

for every .(S⋊ 𝚪)-module A, where .j ∗ A is .D𝚪(S)-module via the projection functor .j : D𝚪 (S) → S ⋊ 𝚪 given by .〈u, x, v, γ 〉 I→ (u, γ ). Hence, the linking exact sequence becomes Corollary 5.1 Let S be a .𝚪-group, where .𝚪 is any monoid of operators. For any (S ⋊ 𝚪)-module A, there is a natural long exact sequence

.

.

n+1 n n · · · → HEM (S ⋊ 𝚪, A) → HEM (𝚪, A) → H𝚪n+1 (S, A) → HEM (S ⋊ 𝚪, A) → · · ·

which is natural in .A. If .𝚪 is a group, then: Proposition 5.2 If .𝚪 be a group, then there is an equivalence of categories D𝚪 (S) ≃ D(S ⋊ 𝚪),

.

for every .𝚪-monoid S. Proof Let .F : D𝚪 (S) → D(S ⋊𝚪) be the functor that sends .x ∈ S to .(x, 1) ∈ S ⋊𝚪 and sends .〈u, x, v, γ 〉 : x → y to 〈(u, γ ), (x, 1), (

.

γ −1

v, γ −1 )〉 : (x, 1) → (y, 1).

Then F is faithful. To show that it is also full, let 〈(u, γ ), (x, 1), (v, η)〉 : (x, 1) → (y, 1)

.

be a morphism in .D(S ⋊ 𝚪). Then, .(u, γ )(x, 1)(v, η) = (y, 1), .u γx γv = y and −1 ; hence .η = γ F 〈u, x, γv, γ 〉 = 〈(u, γ ), (x, 1), (v, η)〉.

.

5.3 The Whitehead Cochain Complex

161

Finally, for any object .(x, γ ) of .D(S ⋊ 𝚪), there is an isomorphism 〈(1, 1), (x, 1), (1, γ )〉 : (x, 1) → (x, γ ).

.

⨆ ⨅ When .𝚪 is a group, the induced equivalence .D𝚪 (S)-Mod ≃ D(S ⋊ 𝚪)-Mod sends the trivial .D𝚪 (S)-module .Z to the trivial .D(S ⋊ 𝚪)-module .Z. Hence there are natural isomorphisms H n (D𝚪 (S), A) ∼ = H n (S ⋊ 𝚪, A),

.

between the cohomology groups .H n (D𝚪(S), A) = ExtnD𝚪 (S) (Z, A) of the category n n .D𝚪(S), and the Leech cohomology groups .H (S ⋊ 𝚪, A) = Ext D(S⋊𝚪) (Z, A) of the monoid .S ⋊ 𝚪. Then, the linking cohomology exact sequence becomes: Corollary 5.2 Let .𝚪 be a group and let S be a .𝚪-monoid. For every .D(S ⋊ 𝚪)module .A, there is a natural long exact sequence .

n · · · → H𝚪n (S, A) → H n (S ⋊ 𝚪, A) → HEM (𝚪, A(1)) → H𝚪n+1 (S, A) → · · ·

which is natural in .A.

5.3 The Whitehead Cochain Complex For every .𝚪-monoid S we now construct, for every .D𝚪 (S)-module .A, a cochain complex .C𝚪• (S, A) whose homology groups are the .𝚪-cohomology groups n .H (S, A). 𝚪 p,q For every .p ≥ 0 and .q > 0, let .C𝚪 (S, A) be the abelian group, under pointwise addition, whose elements are all mappings .ϕ that assign to each .γ1 , . . . , γp ∈ 𝚪 and .x1 , . . . , xq ∈ S an element ϕ(γ1 , . . . , γp ; x1 , . . . , xq ) ∈ A(γ1 ···γp(x1 · · · xq ))

.

where .γ1 , . . . , γp = 1 if .p = 0. Let C𝚪n (S, A) =

.



p,q

C𝚪 (S, A)

p+q =n p ≥ 0, q > 0

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5 Cohomology of Monoids with Operators

if .n > 0, with .C𝚪0 (S, A) = 0. An n-cochain on S with values in .A is an element .ϕ of .C𝚪n (S, A) regarded as a mapping of the disjoint union  .

𝚪p × S q ,

p+q =n p ≥ 0, q > 0

where .𝚪 p = {1} if .p = 0, into the disjoint union .



A(x), such that

x∈S

ϕ(γ1 , . . . , γp ; x1 , . . . , xq ) ∈ A(γ1 ···γp(x1 · · · xq ))

.

whenever .p ≥ 0, .q > 0, .γ1 , . . . , γp ∈ 𝚪 and .x1 , . . . , xq ∈ S; this last condition ensures that n-cochains can be added pointwise. The coboundary .δϕ of an n-cochain .ϕ ∈ C𝚪n (S, A) is the .(n+1)-cochain defined as follows. Let .p ≥ 0, .q > 0, .p + q = n + 1, .γ1 , . . . , γp ∈ 𝚪 and .x1 , . . . , xq ∈ S. If .p = 0, then .q = n + 1 and δϕ(1; x1 , . . . , xq+1 ) = x1 ϕ(1; x2 , . . . xq+1 )

.

+

q  (−1)j ϕ(1; x1 , . . . , xj xj +1 , . . . , xq+1 ) j =1

+ (−1)q+1 ϕ(1; x1 , . . . , xq )xq+1 If .p > 0, then δϕ(γ1 , . . . , γp ; x1 , . . . , xq ) = γ1ϕ(γ2 , . . . , γp ; x1 , . . . , xq )

.

+

p−1 

(−1)i ϕ(γ1 , . . . , γi γi+1 , . . . , γp ; x1 , . . . , xq )

i=1

+ (−1)p ϕ(γ1 , . . . , γp−1 ; γp x1 , . . . , γp xq ) + (−1)p +

q−1 

γ1 ···γp

x1 ϕ(γ1 , . . . , γp ; x2 , . . . xq )

(−1)p+j ϕ(γ1 , . . . , γp ; x1 , . . . , xj xj +1 , . . . , xq)

j =1

+ (−1)p+q ϕ(γ1 , . . . , γp ; x1 , . . . , xq ) γ1 ···γpxq . When both monoids .𝚪 and S are groups and A is any .(S ⋊ 𝚪)-module, the cochain complex .C𝚪• (S, A) was introduced by Whitehead in [8] to define the aforen (𝚪, S; A). mentioned vector cohomology groups .Hn−1

5.3 The Whitehead Cochain Complex

163

Theorem 5.2 Let S be a .𝚪-monoid. For any .D𝚪(S)-module .A, .C𝚪• (S, A) is a cochain complex, and there are natural isomorphisms H𝚪n (S, A) ∼ = H n C𝚪• (S, A),

.

n = 1, 2, . . . .

Proof We shall start by constructing a double chain complex of free .D𝚪(S)modules, denoted by .F•,• as follows. For every integers .p, q ≥ 0, let .Xp,q : ObD𝚪 (S) → Set be the functor, from the discrete category of objects of .D𝚪 (S) to the category of sets, which carries each .x ∈ S to the set

 Xp,q (x) = (γ1 , . . . , γp ; x1 , . . . , xq+1 ) ∈ 𝚪 p × S q+1 | γ1 ···γp (x1 · · · xq+1 ) = x ,

.

where .γ1 , . . . , γp = 1 if .p = 0, and let .Fp,q = FXp,q be the free .D𝚪 (S)-module on .Xp,q . Then, for every .x ∈ S, Fp,q (x) = Z[{(γ0 , . . . , γp ; x0 , . . . , xq+2 ) | x0 γ0 ···γp(x1 · · · xq+1 ) xq+2 = x}],

.

and, for .〈u, x, v, γ 〉 : x → y any arrow in .D𝚪 (S), the induced homomorphism Fp,q (x) → Fp,q (y), .a I→ uγa v, is defined by

.

uγ(γ0 , . . . , γp ; x0 , . . . , xq+2 ) v

.

= (γ γ0 , γ1 , . . . , γp ; uγx0 , x1 , . . . , xq+1 , γxq+2 v). for every generator .(γ0 , . . . , γp ; x0 , . . . , xq+2 ) of .Fp,q (x). These .Fp,q , .p, q ≥ 0, form a double chain complex of .D𝚪 (S)-modules, .F•,• , whose vertical and horizontal differentials .

∂ h : Fp,q → Fp−1,q ,

∂ v : Fp,q → Fp,q−1 ,

are the unique morphisms of .D𝚪 (S) modules such that ∂ h (γ1 , . . . , γp ; x1 , . . . , xq+1 ) = γ1 (γ2 , . . . , γp ; x1 , . . . , xq+1 )

.

p−1 

(−1)i (γ1 , . . . , γi γi+1 , . . . , γp ; x1 , . . . , xq+1 )

i=0

+ (−1)p (γ1 , . . . , γp−1 ; γpx1 , . . . , γpxq+1 ), ∂ v (γ1 , . . . , γp ; x1 , . . . , xq+1 ) = (−1)p (γ1 , . . . , γp ; γ0 ···γpx1 , . . . , xq+1 ) q  + (−1)q (γ1 , . . . , γp ; x1 , . . . , xj xj +1 , . . . ,xq+1 ) j =1

+ (−1)q (γ1 , . . . , γp ; x1 , . . . , xq ,

xq+1 ) .

γ0 ···γp

164

5 Cohomology of Monoids with Operators

So that, for every object .x ∈ S of .D𝚪(S), .

∂ h : Fp,q (x) → Fp−1,q (x), ∂ v : Fp,q (x) → Fp,q−1 (x),

are respectively defined on generators by p−1 

∂ h (γ0 , . . . , γp ; x0 , . . . , xq+2 ) =

.

(−1)i (γ0 , . . . , γi γi+1 , . . . , γp ; x0 , . . . , xq+2 )

i=0

+(−1)p (γ0 , . . . , γp−1 ; x0 ,γpx1 , . . . ,γpxq+1 , xq+2 ), ∂ v (γ0 , . . . , γp ; x0 , . . . , xq+2 ) = (−1)p (γ0 , . . . , γp ; x0 γ0 ···γpx1 , x2 , . . . , xq+2 )

.

+

q 

(−1)q (γ0 , . . . , γp , x0 , . . . , xj xj +1 , . . . , xq+2 )

j =1

+ (−1)q+1 (γ0 , . . . , γp , x0 , . . . , xq ,

xq+1 xq+2 ) .

γ0 ···γp

Let .TotF•,• be its total complex: 

Totn F•,• =

.

∂ = ∂ h + ∂ h : Totn F•,• → Totn−1 F•,• .

Fp,q ,

p+q=n

Recall now the standard resolution .F• S of the trivial .D(S)-module .Z in (2.4). This is in fact a complex of .D𝚪(S)-modules, where, for each .γ ∈ 𝚪 and .x ∈ S, the structure action homomorphisms .Fn S〈1, x, 1, γ 〉 : Fn S(x) → Fn S(γx), .a I→ γx, are defined on generators by γ

.

(x0 , x1 , . . . , xn , xn+1 ) = (γx0 ,γ x1 , . . . ,γ xn ,γ xn+1 ).

Since .IS = Ker(F0 S → Z), we have an exact complex of .D𝚪 (S)-modules ∂

∂

F•+1 S = · · · → F3 S −→ F2 S −→ F1 S,

.

∂

with .Coker(F2 S → F1 S) = IS. Regarding .F•+1 S as a double complex concentrated in the degree zero in the horizontal direction, there is a morphism of double complexes ϵ : F•,• → F•+1 S,

.

5.3 The Whitehead Cochain Complex

165

determined by the morphisms .ϵ : F0,n → Fn+1 S which, at any element .x ∈ S, consist of the abelian group homomorphisms ϵ : F0,n (x) → Fn+1 S(x)

.

such that ϵ(γ0 ; x0 , . . . , xn+2 ) = (x0 , γ0x1 , . . . , γ0xn+1 , xn+2 ).

.

for every generator .(γ0 ; x0 , . . . , xn+2 ) of .F0,n (x). For every .q ≥ 0 and .x ∈ S, the homomorphisms .Ф−1 : Fq+1 S(x) → F0,n (x) and .Фp : Fp,q (x) → Fp+1,q (x), for .p ≥ 0, such that .

Ф−1 (x0 , . . . , xn+2 ) = (1; x0 , . . . , xn+2 ), Фp (γ0 , . . . , γp ; x0 , . . . , xq+2 ) = (1, γ0 , . . . , γp ; x0 , . . . , xn+2 ),

constitute a contracting homotopy of the augmented complex .F•,q (x) → Fq+1 S(x). Hence, the augmented complex of .D𝚪 (S)-modules .F•,n → Fn+1 S → 0 is exact, so that, by Dold-Puppe Theorem, the augmentation .ϵ induces a homology equivalence between the associated total complexes Tot F•,• → Tot F•+1 S = F•+1 S.

.

Therefore  .Hn Tot F•,• ) = Hn (F•+1 S) =



0

n > 0,

IS i = 0.

Indeed, .Tot F•,• S is a projective resolution of the .D𝚪(S)-module .IS, since, for  every .n ≥ 0, .Totn F•,• = p+q=n Fp,q is a free .D𝚪(S)-module on the the functor .Xn : ObD𝚪(S) → Set defined by Xn (x) =



.

Xp,q (x),

p+q=n

for every .x ∈ S. Then, by (5.2), for any .D𝚪(S)-module .A,   n−1 HomD𝚪(S) (Tot F•,• , A) . H𝚪n (S, A) = Extn−1 D𝚪 (S) (IS, A) = H

.

Theorem 1.1 now yields natural isomorphisms of abelian groups HomD𝚪(S) (Totn F•,• , A) ∼ = C𝚪n+1 (S, A),

.

(n ≥ 0),

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5 Cohomology of Monoids with Operators

which commute with coboundaries. Therefore, .C𝚪• (S, A) is a cochain complex and we are in the presence of an isomorphism of cochain complexes HomD𝚪(S) (Tot F•,• S, A) ∼ = C𝚪•+1 (S, A).

.

  Hence .H𝚪n (S, A) = H n−1 HomD𝚪(S) (Tot F•,• , A) ∼ = H n C𝚪• (S, A).

⨆ ⨅

5.4 Cohomology and 𝚪-Coextensions Let .𝚪 be a given monoid of operators. A .𝚪-coextension of a .𝚪-monoid S by a D𝚪 (S)-module .A is a coextension .(A, E, p, +) of the underlying monoid S by the underlying .D(S) module .A, as defined in Sect. 2.4, in which E is a .𝚪-monoid, .p : E ↠ S is a homomorphism of .𝚪-monoids (that preserves the action of .𝚪), and .

γ

.

(w + a) = γ w + γ a

(5.3)

for all .γ ∈ 𝚪, .x ∈ S , .w ∈ p−1 (x), and .a ∈ A(x). Two such .𝚪-coextensions .(A, E, p, +) and .(A, E ' , p' , +) are equivalent if there exists a .𝚪-isomorphism .F : E ∼ = E ' such that .p' F = p and .F (w + a) = F (w) + a, for .w ∈ E and .a ∈ A(p(w)). Let Ext𝚪 (S, A)

.

denote the class (actually a set, by next Theorem 5.3) of equivalence classes [A, E, p, +] of .𝚪-coextensions .(A, E, p, +)) of S by .A.

.

Theorem 5.3 (Classification) For every .𝚪-monoid S and .D𝚪 (S)-module .A, there is a natural bijection Ext𝚪 (S, A) ∼ = H𝚪2 (S, A).

.

Proof Recall from Sect. 5.3 that the coboundary of a 1-cochain .φ ∈ C𝚪1 (S, A) is the 2-cochain .δφ ∈ C𝚪2 (S, A) defined by δφ(x, y) = x φ(y) − φ(xy) + φ(x) y, .

(5.4)

δφ(γ ; x) = φ(x) − φ( x).

(5.5)

.

γ

γ

for all for all .x, y ∈ S and .γ ∈ 𝚪. Similarly, the coboundary of a 2-cochain .α ∈ C𝚪2 (S, A) is the 3-cochain .δα ∈ C𝚪3 (S, A) defined by δα(x, y, z) = xα(y, z) − α(xy, z) + α(x, yz) − α(x, y)z, .

.

(5.6)

5.4 Cohomology and 𝚪-Coextensions

167

δα(γ ; x, y) = γα(x, y) − α(γ x, γ y) − γx α(γ ; y).

(5.7)

+ α(γ ; xy) − α(γ ; x) γy, '

δα(γ ' , γ ; x) = γ α(γ ; x) − α(γ ' γ ; x) + α(γ ' , γx).

(5.8)

for all for all .x, y, z ∈ S and .γ , γ ' ∈ 𝚪. Two 2-cocycles .α, α ' ∈ Z𝚪2 (S, A) are cohomologous if .α ' = α + δφ for some 1-cochain .φ ∈ C𝚪1 (S, A). To each .𝚪-coextension .(A, E, p, +) of S by .A, assign a cocycle .αs as follows. For each .x ∈ S choose .s(x) ∈ p−1 (x). For every .x, y ∈ S and .γ ∈ 𝚪 there exist unique .αs (x, y) ∈ A(xy) and .αs (γ ; x) ∈ A(γx) such that s(xy) = s(x)s(y) + αs (x, y),

.

s(γx) = γs(x) + αs (γ ; x). Then .δαs ∈ Z𝚪2 (S, A): The cocycle condition .δαs (x, y, z) = 0 holds since s((xy)z) = s(xy)s(z) + αs (xy, z) = (s(x)s(y) + αs (x, y))s(z) + αs (xy, z)

.

(2.23)

= s(x)s(y)s(z) + αs (x, y) z + αs (xy, z),

s(x(yz)) = s(x)s(yz) + αs (x, yz) = s(x)(s(y)s(z) + αs (y, z)) + αs (x, yz) (2.23)

= s(x)s(y)s(z) + x αs (y, z) + αs (x, yz),

and the result follows by comparison. Similarly, .δαs (γ ; x, y) = 0 since s(γx γy) = s(γx)s(γy) + αs (γx, γ y)

.

= (γs(x) + αs (γ ; x))(γs(y) + αs (γ ; y)) + αs (γx, γy) (2.23) γ

=

s(x) γs(y) + αs (γ ; x) γy + γx αs (γ ; y) + αs (γx, γy),

s(γ(xy)) = γs(xy) + αs (γ ; xy) = γ (s(x)s(y) + αs (x, y)) + αs (γ ; xy) (5.3) γ

=

s(x) γs(y) + γαs (x, y) + αs (γ ; xy),

and also .δαs (γ ' , γ ; x) = 0, since '

'

'

s(γ (γx)) = γ s(γx) + αs (γ ' ; γx) = γ (γs(x) + αs (γ ; x)) + αs (γ ' , γx)

.

(5.3) γ ' γ

=

'

'

s(x) + γ αs (γ ; x) + αs (γ ' ; γx),

'

s(γ γx) = γ γs(x) + αs (γ ' γ ; x). Hence .αs = αs (A, E, p, +) is a 2-cocycle.

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5 Cohomology of Monoids with Operators

Moreover, the cohomology class .[αs ] ∈ H𝚪2 (S, A) does not depend on the choice of the section s of .p : E ↠ S. Indeed, if .s ' (x) ∈ p−1 (x) for all .x ∈ S, then there is a unique 1-cochain .φ ∈ C𝚪1 (S, A) such that .s(x) = s ' (x) + φ(x) for all .x ∈ S. Since s(xy) = s(x)s(y) + αs (x, y) = (s ' (x) + φ(x))(s ' (x) + φ(x)) + αs (x, y)

.

(2.23) '

= s (x) s ' (y) + x φ(y) + φ(x) y + αs (x, y),

s(xy) = s ' (xy) + φ(xy) = s ' (x) s ' (y) + αs ' (x, y) + φ(xy), we have .αs ' (x, y) = αs (x, y) + δφ(x, y), for all .x, y ∈ S. Similarly, since s(γ x) = γs(x) + αs (γ ; x) = γ(s ' (x) + φ(x)) + αs (γ ; x)

.

(5.3) γ '

=

s (x) + γφ(x) + αs (γ ; x),

s(γ x) = s ' (γx) + φ(γx) = γs ' (x) + αs ' (γ ; x) + φ(γx), it follows that .αs ' (γ ; x) = αs (γ ; x) + δφ(γ ; x) for all .γ ∈ 𝚪 and .x ∈ S. Thus, αs ' = αs + ∂φ and therefore .[αs ' ] = [αs ] in .H𝚪2 (S, A). In addition, if .(A, E, p, +) and .(A, E ' , p' , +) are two .𝚪-coextensions of S by .A and .F : E ∼ = E ' is an equivalence of .𝚪-coextensions, then .

αFs (A, E ' , p' , +) = αs (A, E, p, +).

.

Hence, we have a well-defined map Ψ : Ext𝚪 (S, A) → H𝚪2 (S, A)

.

given by .Ψ[A, E, p, +] = [αs (A, E, p, +)]. To show that .Ψ is surjective we construct, for each .α ∈ Z𝚪2 (S, A) a .𝚪-extension of S by .A, (A, S ⋊α A, π, +),

.

such that .Ψ[A, S⋊α A, π, +] = [α]. The .𝚪-monoid, .S⋊α A consists of pairs .(x, a) where .x ∈ S and .a ∈ A(x), with multiplication and action of .𝚪 (x, a)(y, b) = (xy, x b + a y − α(x, y)),

.

γ

(x, a) = (γx, γa − α(γ ; x)).

5.4 Cohomology and 𝚪-Coextensions

169

The projection .π : S ⋊α A ↠ S sends .(x, a) to x, and the actions + : π −1 (x) × A(x) → π −1 (x) are defined by

.

(x, a) + a ' = (x, a + a ' ).

.

(5.9)

A straightforward verification shows that the equalities (x, a)((y, b)(z, c)) = ((x, a)(y, b))(z, c),

.

γ

((x, a)(y, b)) = γ (x, a) γ (y, b), γ 'γ

'

(x, a) = γ (γ (x, a)),

follow from the cocycle conditions .δα(x, y, z) = 0, .δα(γ ; x, y) = 0 and δα(γ ' , γ ; x) = 0, respectively. Furthermore, .δα(x, 1, 1) = 0 and .δα(1, 1, x) = 0 imply .xα(x, 1) = α(1, 1) and .α(1, 1)x = α(1, x), so that .(1, α(1, 1)) is the identity element of .S ⋊α A. The equalities .γ(1, α(1, 1)) = (1, α(1, 1)) and .1(x, a) = (x, a) similarly follow from .δα(γ ; 1, 1) = 0 and .δα(1, 1; x) = 0. Therefore, .S ⋊α A is a .𝚪-monoid. Finally, the actions (5.9) are simply transitive and the conditions (2.23) and (5.3) hold. Thus, .(A, S ⋊α A, π, +) is a .𝚪-coextension of the .𝚪-monoid S by the .D𝚪 (S)-module .A. Moreover, if we choose the obvious section map .s : S → S⋊αA with .s(x) = (x, 0), the equalities .

(xy, 0) = (x, 0)(y, 0) + α(x, y),

.

(γ x, 0) = γ (x, 0) + α(γ , x),

show that .αs (A, S ⋊α A, π, +) = α, and thus .Ψ[A, S ⋊α A, π, +] = [α]. Finally, we show that .Ψ is injective. If .(A, E, p, +) is any .𝚪-coextension of S by .A, .s : S → E is any section map of p, and .αs = αs (A, E, p, +), then (x, a) I→ s(x) + a

.

is an equivalence of .𝚪-coextensions .(A, S ⋊α A, π, +) ≃ (A, E, p, +). Furthermore, if .α, α ' ∈ Z𝚪2 (S, A) are cohomologous, so that .α ' = α + δφ for some 1 .φ ∈ C (S, A), then 𝚪 (x, a) I→ (x, a + φ(x))

.

is an equivalence of .𝚪-coextensions .(A, S ⋊α A, π, +) ≃ (A, S ⋊α ' A, π, +). Therefore, .Ψ is injective. ⨆ ⨅

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References 1. Beck, J.: Triples, Algebras and Cohomology. Thesis (Ph.D.) Columbia University, 1967 (Repr. Theory Appl. Categ. 2 (2003), 1–59) 2. Carrasco, P., Cegarra, A.M.: Cohomology of presheaves of monoids. Mathematics 8(1), 116, 1–35 (2020) 3. Cegarra, A.M.: Cohomology of monoids with operators. Semigroup Forum 99, 67–105 (2019) 4. Cegarra, A.M., Garía-Calcines, J.M., Ortega, J.A.: Cohomology of groups with operators. Homology Homotopy Appl. 4, 1–23 (2002) 5. Cegarra, A.M., García-Calcines, J.M., Ortega, J.A.: On graded categorical groups and equivariant group extensions. Can. J. Math. 54, 970–997 (2002) 6. Cockcroft, W.H.: Interpretation of Vector Cohomology Groups. Am. J. Math. 76, 599–619 (1954) 7. Wells, C.: Extension theories for monoids. Semigroup Forum 16, 13–35 (1978) 8. Whitehead, J.H.C.: On Group extensions with operators. Q. J. Math. Oxford 2, 219–228 (1950)

Chapter 6

Cohomology and Monoidal Groupoids

In this chapter we analyze monoidal abelian groupoids and classify them by third .Dcohomology groups of monoids, using a 3-dimensional Schreier-Grothendieck-like theory of factor sets. We also consider monoidal abelian groupoids with a coherent monoid .𝚪-action. The particular case of categorical groups was dealt with by Grothendieck and Sinh in 1975, [31]. Categorical groups (monoidal groupoids whose objects are all quasi-invertible [24]) are also called Gr-categories [7, 31], weak 2-groups [1] and group-like categories [18]) and have been applied in several contexts, including homotopy theory, topological quantum field theory or theory of gerbes. (See [1] and the references therein.) A fundamental result about categorical groups is their classification by Eilenberg-Mac Lane 3-cohomology classes of groups. The appearance of monoidal groupoids in several branches of mathematics makes them interesting in their own right. Arbitrary monoidal groupoids are analyzed in [11], where a 3-dimensional Schreier-Grothendieck theory of nonabelian factor sets for their classification is established. Here, we focus on the special case of monoidal abelian groupoids (whose automorphism groups are all abelian).

6.1 Monoidal Groupoids In this section, we fix notation, recall basic terminology, and present the needed background for discussing monoidal groupoids. A groupoid is a small category in which every morphism is an isomorphism. We write the composition of morphisms in a groupoid additively: the composite of morphisms .a : X → Y and .b : Y → Z is .b + a : X → Z; the identity morphism of an object X is .0X or 0; the inverse of a (iso)morphism .a : X → Y is .−a : Y → X.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 A. M. Cegarra, J. Leech, The Cohomology of Monoids, RSME Springer Series 12, https://doi.org/10.1007/978-3-031-50258-3_6

171

172

6 Cohomology and Monoidal Groupoids

A monoidal groupoid G = (G, ⊗, I, a, r, l)

.

consists of a groupoid .G, a functor .⊗ : G × G → G (the tensor product), an object I (the unit object), and natural morphisms

.

a X,Y,Z : (X ⊗ Y ) ⊗ Z → X ⊗ (Y ⊗ Z),

.

r X : X ⊗ I → X, l X : I ⊗ X → X, called the associativity, right unit, and left unit constraints, respectively, often denoted by just by .a, .l, .r, such that the associativity pentagon and unit triangle diagrams below commute for every objects .X, Y, Z, T of .G:

(6.1)

.

(6.2)

.

For later reference, note the conditions above imply that the equality l = r I : I ⊗ I → I,

. I

(6.3)

and the commutativity of the following two triangles [24, Proposition 1.1]

(6.4)

.

¯ be monoidal groupoids. A ¯ = (G, ¯ ⊗, ¯ a, ¯ I, ¯ r¯ , l) Let .G = (G, ⊗, I, a, r, l) and .G monoidal functor ¯ F = (F, φ, φ ∗ ) : G → G

.

6.1 Monoidal Groupoids

173

¯ endowed with consists of a functor between the underlying groupoids .F : G → G a family .φ of morphisms ¯ Y, φ X,Y : F (X ⊗ Y ) → F X⊗F

.

that are natural in X and Y , and a morphism .φ ∗ : F I → I¯ such that the following diagrams commute for every objects .X, Y, Z:

(6.5)

.

(6.6)

.

¯ are monoidal functors, then a monoidal transformation If .F, F ' : G → G θ : F → F'

.

is a natural transformation such that the following diagrams commute for every objects .X, Y :

.

(6.7)

¯¯ are two monoidal functors, the composite .F '' = ¯ and .F ' : G ¯ →G If .F : G → G ¯ ¯ is also monoidal, with structure data given by the compositions :G→G

F 'F

.

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6 Cohomology and Monoidal Groupoids

This composition is associative and unitary, with identity of a monoidal groupoid .G the monoidal functor     idG = idG , idX⊗Y X,Y ∈ObG , idI : G → G.

.

Hence monoidal groupoids and monoidal functors constitute a category. Actually, this is the underlying category of the 2-category of monoidal groupoids, whose 2cells are the monoidal transformations. In this 2-category, both the horizontal and vertical composition of 2-cells are defined as the ordinary vertical and horizontal composition of natural transformations. Note that every 2-cell is invertible, so that this 2-category is a track category [4] (also known as a .(2, 1)-category). A monoidal equivalence between monoidal groupoids is an equivalence in this 2-category, that is, a monoidal functor .F : G → G' for which there exist a monoidal functor ' ' ' ' .F : G → G and monoidal transformations .idG → F F and .F F → idG' . Two monoidal groupoids are monoidal equivalent if they are connected by a monoidal equivalence. From Saavedra [30, I, Proposition 4.4.2], it is known that a monoidal functor ' .F : G → G is a monoidal equivalence if and only if the underlying functor is an equivalence of categories; that is, if and only if the functor F is full, faithful, and each object of .G' is isomorphic to an object F X for some .X ∈ ObG; equivalently, by Higgings [21, Chapter 6, Corollary 2], if and only if the induced map on the sets of isomorphism classes of objects ' ObG/∼ = → ObG /∼ =,

.

[X] I→ [F X],

is a bijection, and the induced homomorphisms on the automorphism groups AutG (X) → AutG' (F X),

.

a I→ F a,

are all isomorphisms. Some groups of automorphisms in any monoidal groupoid .G are always abelian. Recall that an object X of .G is invertible (also called quasi-invertible) if there exists an object .X' with morphisms .X ⊗ X' → I and .X' ⊗ X → I. Lemma 6.1 Let .G be a monoidal groupoid. If .X ∈ ObG is invertible, then the group .AutG (X) is abelian and isomorphic to .AutG (I). Proof First, by the Eckmann-Hilton argument [17], the group .AutG (I) is abelian since the multiplication AutG (I) × AutG (I) → AutG (I),

.

(a, b) I→ r I + (a ⊗ b) − r I ,

is a group homomorphism. Now, if X is an invertible object of .G, the endofunctor X ⊗ − : G → G, is actually an equivalence since the associativity and unit

.

6.2 Monoidal Groupoids with Operators

175

constraints yield natural isomorphisms

.

(X ⊗ −)(X' ⊗ −) ∼ = ((X ⊗ X' ) ⊗ −) ∼ =I⊗−∼ = idG , ' ' ∼ ((X ⊗ X) ⊗ −) = ∼I⊗−= ∼ idG . (X ⊗ −)(X ⊗ −) =

Hence, AutG (I) ∼ = AutG (X ⊗ I) ∼ = AutG (X).

.

a I→

0X ⊗ a

I→ r X + (0X ⊗ a) − rX ⨆ ⨅

and the group .AutG (X) is also abelian.

Recall that a groupoid .G is abelian if all its automorphism groups .AutG (X), .X ∈ ObG, are abelian. A categorical group is a monoidal groupoid whose objects are all invertible. By Lemma 6.1 above, all the automorphism groups of any categorical group are abelian. Hence Proposition 6.1 Every categorical group is a monoidal abelian groupoid.

□

.

In any abelian groupoid the following useful property holds. Lemma 6.2 If the following squares in an abelian groupoid commute,

.

then .b = b' . Proof .b = −f + a + f = −f + g + b' − g + f = −f + g − g + f + b' = b' . ⨅ ⨆

6.2 Monoidal Groupoids with Operators Let .𝚪 be a given monoid of operators. A .𝚪-monoidal groupoid is a pseudo-functor (see [32], for example) from the category .𝚪 to the 2-category of monoidal groupoids, that is, a monoidal groupoid on which .𝚪 acts coherently by monoidal endofunctors. .𝚪-monoidal groupoids are the objects of a 2-category, whose arrows, called .𝚪monoidal functors, are pseudo-transformations between these, and whose 2-cells, called .𝚪-monoidal transformations are modifications between these. For later use, we unpack below the ingredients of this 2-category that will be used in this chapter.

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6 Cohomology and Monoidal Groupoids

A .𝚪-monoidal groupoid, G = (G, F, θ ),

.

consists of − − .− .− . .

a monoidal groupoid .(G, ⊗, I, a, r, l), γ a family .F of monoidal endofunctors .Fγ = (Fγ , φ γ , φ ⋆ ) : G → G, .γ ∈ 𝚪, γ ,λ γ λ a family .θ of monoidal transformations .θ : F F → Fγ λ , .γ , λ ∈ 𝚪, 1 1 a monoidal transformation .θ : idG → F ,

such that the following two coherence conditions hold: − for every .γ , λ, δ ∈ 𝚪, the following square commutes.

.

(6.8)

.

− for every .γ ∈ 𝚪 and .X ∈ ObG, both inner triangles in the square below commute.

.

(6.9)

.

¯ are two such .𝚪-monoidal groupoids, then a If .G and .G equivalence

𝚪-monoidal

.

¯ (F, ψ) : G → G

.

is a pseudo-natural equivalence; in other words it consists of the following data ¯ − a monoidal equivalence .F = (F, φ, φ ∗ ) : G → G, γ γ ¯ .− a family .ψ of monoidal transformations .ψ : F F → F Fγ , .

.

6.3 Realizing 3-Cohomology Classes

177

one for each .γ ∈ 𝚪, such that the following two axioms hold: − for every .γ , λ ∈ 𝚪, the diagram below commutes,

.

(6.10)

.

− the following triangle commutes.

.

(6.11)

.

If .G = (G, F, θ ) is a .𝚪-monoidal groupoid, then [G] = [G, F, θ ]

.

denotes its .𝚪-monoidal equivalence class.

6.3 Realizing 3-Cohomology Classes In [31, II, Prop. 13] (See also [1, Theorem.8.3.7].) Grothendieck and Sinh proved that, for any group G, any G-module A, and every cohomology class .k ∈ H 3 (G, A), there exists a categorical group .G, unique up to monoidal equivalence, such that G is the group of isomorphism classes of objects of .G, .A ∼ = AutG (I), and the action of G on A and the cohomology class .k are canonically deduced from the functoriality of the tensor and the naturality and coherence of the constraints of .G. This fact was historically relevant since it pointed out the utility of categorical groups, for example, in homotopy theory: as .H 3 (G, A) = H 3 (BG, A) is the 3rd cohomology group of the classifying space BG of the group G with local coefficients in A, for any triplet of data .(G, A, k) as above, there exists a path-connected CW-complex X, unique up to homotopy equivalence, such that .πi X = 0 if .i /= 1, 2, .π1 X = G, 3 .π2 X = A (as G-modules), and .k ∈ H (G, A) is the unique non-trivial Postnikov invariant of X. Therefore, categorical groups arise as algebraic homotopy 2-types of path-connected spaces. Indeed, strict categorical groups (that is, categorical groups in which all the structure constraints are identities) are the same thing as crossed modules, whose use in homotopy theory goes back to Whitehead (1949). (See [8] for

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6 Cohomology and Monoidal Groupoids

the history) In presence of a group of operators .𝚪, the cohomological classification of .𝚪-categorical groups was discussed in [15]. With the Schreier-Grothendieck theory in mind, we construct a 3-dimensional cohomological theory for (.𝚪-) monoidal abelian groupoids to provide a minimal data set for the construction of all (.𝚪-) monoidal abelian groupoids up to (.𝚪-) monoidal equivalences. Let .𝚪 be a given monoid of operators. As in Sect. 5.3, for every .𝚪-monoid S and .D𝚪 (S)-module .A, the abelian group 3 .H (S, A) can be regarded as the group of cohomology classes of 3-cocycles, where 𝚪 a 3-cocycle .h ∈ Z𝚪3 (S, A) is a mapping .

3 from  the disjoint union .S . x∈S A(x), such that



h(x, y, z) ∈ A(xyz),

(𝚪 × S 2 )



(𝚪 2 × S) into the disjoint union

h(γ , τ ; x) ∈ A(γ τ x),

.

h(γ , τ ; x) ∈ A(γ τ x)

for all .x, y, z ∈ S and .γ , τ ∈ 𝚪, and such that .δh = 0, where .δh ∈ C𝚪4 (S, A) is the 4-cochain .

defined by δh(x, y, z, t) = xh(y, z, t) − h(xy, z, t) + h(x, yz, t).

(6.12)

.

− h(x, y, zt) + h(x, y, z)t, δh(γ ; x, y, z) = γh(x, y, z) − h(γx, γy, γz) − γxh(γ ; y, z).

(6.13)

+ h(γ ; xy, z) − h(γ ; x, yz) + h(γ ; x, y) z, γ

δh(γ , τ ; x, y) = γh(τ ; x, y) − h(γ τ ; x, y) + h(γ ; τx, τy). +

(6.14)

xh(γ , τ ; y) − h(γ , τ ; xy) + h(γ , τ ; x) y,

γτ

γτ

δh(γ , τ, υ; x) = γh(τ, υ; x) − h(γ τ, υ; x)

(6.15)

+ h(γ , τ υ; x) − h(γ , τ ; x), v

for all .x, y, z, t ∈ S and .γ , τ, υ ∈ 𝚪. Two such .h, h' ∈ Z𝚪3 (S, A) are cohomologous if .h' = h + δφ for some 2cochain .φ ∈ C𝚪2 (S, A), where the coboundary .δφ is explicitly shown in (5.6), (5.7) and (5.8).

6.3 Realizing 3-Cohomology Classes

179

A 3-cohomology object H = (S, A, k)

.

consists of a .𝚪-monoid S, a .D𝚪 (S)-module .A, and a cohomology class .k ∈ H𝚪3 (S, A). These are the objects of the category of 3-cohomology objects, denoted by H3𝚪 .

.

If .H = (S, A, k) and .H' = (S ' , A' , k ' ) are 3-cohomology objects, then a morphism (F, f ) : H → H'

.

in .H3𝚪 consists of a morphism of .𝚪-monoids .f : S → S ' and a morphism of .D𝚪 (S)modules .F : A → f ∗A' such that .f ∗ k ' = F∗ k, where f ∗ : H𝚪3 (S ' , A' ) → H𝚪3 (S, f ∗A' ),

.

F∗ : H𝚪3 (S, A) → H𝚪3 (S, f ∗A' )

are the corresponding induced homomorphisms on cohomology. The composition of a morphism .(F, f ) : H → H' with a morphism .(F' , f ' ) : H' → H'' is defined by (F' , f ' )(F, f ) = (f ∗F' F, f ' f ),

.

and the identity of a 3-cohomology object .H = (S, A, k) is .idH = (idS , idA ). A morphism .(F, f ) : H → H' in .H3𝚪 is an isomorphism if and only if both f and .F are isomorphisms. If .H = (S, A, k) is a 3-cohomology object, then [H] = [S, A, k]

.

denotes its isomorphism class in .H3𝚪 . For each .𝚪-monoidal abelian groupoid .G = (G, F, θ ) we construct a 3cohomology object HG = (π0 G, π1 G, k G )

.

as follows.

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6 Cohomology and Monoidal Groupoids

The 𝚪-Monoid π0 G The .𝚪-monoid .π0 G consists of the isomorphism classes .[X] of objects X of the underlying groupoid .G, with multiplication .[X][Y ] = [X ⊗ Y ], identity .1 = [I], and action .γ [X] = [Fγ X] of .𝚪. The multiplication in .π0 G is associative and with an identity element thanks to the associativity and unit constraints .a X,Y,Z : (X ⊗ Y ) ⊗ Z ∼ = X ⊗ (Y ⊗ Z), .r X : X ⊗ I ∼ = X and .l X : I ⊗ X ∼ = X. The equalities .γ ([X][Y ]) = γ [X] γ [Y ] and .γ [I] = [I] follow from the γ γ isomorphisms .φ X,Y : Fγ (X ⊗ Y ) ∼ = Fγ X ⊗ Fγ Y and .φ ∗ : Fγ I ∼ = I. 1 γ λ γ λ Similarly, the equalities . [X] = [X] and . ( [X]) = [X] follows from the γ ,λ structure isomorphisms .θ 1X : X ∼ = F1 X and .θ X : Fγ Fλ X ∼ = Fγ λ X.

The D𝚪 (π0 G)-Module π1 G For each .x ∈ π0 G, choose an object sx of .G such that [sx] = x,

(6.16)

π1 G(x) = AutG (sx),

(6.17)

.

and choose .s1 = I. Let .

the abelian group of automorphisms of .sx in .G. For every .x, y ∈ π0 G and .γ ∈ 𝚪, choose morphisms in .G ϒx,y : s(xy) → sx ⊗ sy and ϒγ' ,y : s(γy) → Fγ sy.

.

(6.18)

It is possible to normalize .ϒ by choosing ⎧ ⎪ ⎪ ϒ1,x = −l sx : sx → I ⊗ sx, ⎪ ⎨ ϒ = −r : sx → sx ⊗ I, x,1 sx . γ ⎪ ϒγ' ,1 = −φ ∗ : I → Fγ I, ⎪ ⎪ ⎩ ϒ ' = θ 1 : sx → F1sx. sx 1,x

(6.19)

6.3 Realizing 3-Cohomology Classes

181

(Note that .l I = r I , by (6.3), and .θ 1I = −φ 1∗ , by the commutativity of the coherence triangle in (6.7)) For every .x, y ∈ π0 G and .γ ∈ 𝚪, homomorphisms

.

π1 G〈x, y, 1, 1〉 : π1 G(y) → π1 G(xy),

a I→ xa,

π1 G〈1, y, x, 1〉 : π1 G(y) → π1 G(yx),

a I→ ax,

π1 G〈1, y, 1, γ 〉 : π1 G(y) → π1 G(γ y),

a I→ γa,

are then defined by the commutative squares

.

(6.20) By Lemma 6.2, these homomorphisms do not depend on the choice of .ϒx,y and ϒγ' ,y . For every .x ∈ π0 G and .a ∈ π1 G(x), the equalities

.

1a = a = a1 = 1 a

.

hold by Lemma 6.2, since the morphisms .l sx , .r sx and .θ 1sx are natural. For every .x, y, z ∈ π0 G and .a ∈ π1 G(z), the equalities (xy)a = x(ya), x(ay) = (xa)y, a(xy) = (ax)y,

.

also follow from Lemma 6.2 and the following pairs of commutative diagrams in G, in which the squares .(A) commute since .a is natural, and the remaining squares by (6.20) since .⊗ is a functor:

.

182

6 Cohomology and Monoidal Groupoids

.

Similarly, for every .x, y ∈ π0 G, .a, a ' ∈ π1 G(y) and .γ , λ ∈ 𝚪, the equalities γ

.

(a + a ' ) = γ a + γ a ' , γ (xa) = γ x γ a, γ (ax) = γ a γ x, γ (λ a) = γ λ a

follows from Lemma 6.2 and the following pairs of commutative diagrams in .G, in which the squares .(B) commute since .φ γ is natural, the square .(C) commutes since γ ,λ .θ is natural, and the remaining squares commute by (6.20), since .⊗ and .Fγ are

6.3 Realizing 3-Cohomology Classes

functors:

.

183

184

6 Cohomology and Monoidal Groupoids

.

Thus, .π1 G is a .D𝚪 (π0 G)-module.

The Cohomology Class k G ∈ H𝚪3 (π0 G, π1 G) We now construct a 3-cocycle h ∈ Z𝚪3 (π0 G, π1 G)

. G

whose values .hG (x, y, z) ∈ π1 G(xyz), .hG (γ ; x, y) ∈ π1 G(γ x γ y) and h (γ , λ; x) ∈ π1 G(γ λ x), for each .x, y, z ∈ π0 G and .γ , λ ∈ 𝚪, are the automorphisms of .G such that the following diagrams commute:

. G

.

.

(6.21)

(6.22)

6.3 Realizing 3-Cohomology Classes

.

185

(6.23)

The four parts of the cocycle condition .δhG = 0 follow from the commutativity of the following diagrams, in which cells .(A) commute since .a is natural, cells γ .(B) commute since the morphisms .φ are natural, cells .(C) commute since the γ ,δ morphisms .θ are natural, and the remaining cells commute as indicated since .⊗ and .F γ are functors. It follows from (6.1), (6.5), (6.7), and (6.8) respectively that the vertical arrows on the right of each diagram compose to identity morphisms. Hence the vertical arrows on the left also compose to identity morphisms, by Lemma 6.2. This yields each part of the condition .δhG = 0.

.

186

.

where .(∗) =

.

6 Cohomology and Monoidal Groupoids

6.3 Realizing 3-Cohomology Classes

and .(∗∗) =

.

∂hG (γ , λ; x, y) = 0:

.

.

187

188

6 Cohomology and Monoidal Groupoids

∂hG (x, y, z, u) = 0:

.

.

If .ϒ is normalized, as in (6.19), then so is .hG : ⎧ ⎨ hG (x, y, 1) = hG (x, 1, y) = hG (1, x, y) = 0, . h (1, ; x, y) = 0, ⎩ G hG (γ , 1; x) = hG (1, γ ; x) = 0.

(6.24)

6.3 Realizing 3-Cohomology Classes

189

This follow from Lemma 6.1 and the following commutative diagrams, in which cells .(D) commute since .l and .r are natural, cells .(E) commute since .θ 1 is natural, and the remaining cells commute as indicated. .∂hG (1, x, y) = 0:

.

∂hG (x, 1, y) = 0:

.

.

∂hG (x, y, 1) = 0:

.

.

∂hG (1; x, y) = 0:

.

.

190

6 Cohomology and Monoidal Groupoids

∂hG (γ , 1; x) = 0:

.

.

∂hG (1, γ : x) = 0:

.

.

Finally, .k G = [hG ] is the cohomology class of .hG in .H𝚪3 (π0 G, π1 G).

6.4 The Classification Theorem Let .𝚪 be a given monoid of operators. ¯ be .𝚪-monoidal abelian groupoids. If .G and .G ¯ are .𝚪Theorem 6.1 Let .G and .G monoidal equivalent, then .HG and .HG¯ are isomorphic. Proof First, we show that the isomorphism class of the 3-cohomology object .HG does not depend on the choice of s, .ϒ and .ϒ ' . ˜ G = (π0 G, π˜ 1 G, k˜ G ) be constructed from other choices of objects .s˜ x Let .H in (6.16) and morphisms .ϒ˜ x,y : s˜ (xy) → s˜ x ⊗ s˜ y and .ϒ˜ γ' ,y : s˜ (γy) → Fγ s˜ y in (6.18). Construct an isomorphism of .D𝚪 (π0 G)-modules .F : π1 G ∼ = π˜ 1 G as follows. For each .x ∈ π0 G, choose a morphism ψx : sx → s˜ x;

.

6.4 The Classification Theorem

191

choose .ψ1 = 0I : I → I. For every .a ∈ π1 G(x), define .Fa ∈ π˜ 1 G(x) by the commutative square:

(6.25)

.

That .F : π1 G → π˜ 1 G is a morphism of .D𝚪 (π0 G)-modules follows from Lemma 6.2, applied to the following pairs of commutative diagrams. For every .x, y, z ∈ π0 G, .γ ∈ 𝚪 and .a ∈ π1 G(y). .F(xa) = xF(a):

.

F(az) = F(a)z:

.

.

F(γ a) = γ F(a):

.

.

Hence, .F : π1 G ∼ = π˜ 1 G is an isomorphism of .D𝚪 (π0 G)-modules. Next, construct a 2-cochain .g ∈ C𝚪2 (π0 G, π˜ 1 G), such that F∗ hG = h˜ G + ∂g.

.

For every .x, y ∈ π0 G and .γ ∈ 𝚪, define .g(x, y) ∈ π˜ 1 G(xy) and .g(γ ; x) ∈ π˜ 1 G(γx) by the commutative squares:

192

6 Cohomology and Monoidal Groupoids

.

(6.26) That .h˜ G = F∗ hG − ∂g follows from the commutative diagrams below, in which cells .(A), .(B) and .(C) commute since .a, .φ γ and .θ γ ,δ are natural and the remaining cells commute as indicated since .⊗ and .Fγ are functors. In these diagrams, the vertical arrows on the left compose respectively to .F∗ hG (x, y, z) − ∂g(x, y, z), .F∗ hG (γ , λ; x) − ∂g(γ , λ; x) and .F∗ hG (γ ; x, y) − ∂g(γ ; x, y), and also match the definition of .h˜ G (x, y, z), .h˜ G (γ ; x, y) and .h˜ G (γ , λ; x); this proves the required equalities. ˜ G (x, y, z) = FhG (x, y, z) − ∂g(x, y, z): .h

.

h˜ (γ , δ; x) = FhG (γ , δ; x) − ∂g(γ , δ; x):

. G

6.4 The Classification Theorem

.

h˜ (γ ; x, y) = FhG (γ ; x, y) − ∂g(γ ; x, y):

. G

.

193

194

6 Cohomology and Monoidal Groupoids

˜ G is an Then, .F∗ k G = [F∗ hG ] = [h˜ G ] = k˜ G , so that .(F, idπ0 G ) : HG ∼ = H 3 ˜ isomorphism in .H𝚪 , and therefore .[HG ] = [HG ]. ¯ be .𝚪-monoidal abelian groupoids and let .(F, ψ) : G → G ¯ be a Now let .G and .G .𝚪-monoidal equivalence. Let .HG = (π0 G, π1 G, k G ) be constructed as in Sect. 6.3 using representatives objects sx and morphisms .ϒx,y : s(xy) → sx ⊗ sy and ' γ γ .ϒγ ,y : s( y) → F sy of .G. The bijection ¯ f : π0 G ∼ = π0 G,

.

[X] I→ [F X],

is an isomorphism of .𝚪-monoids: indeed, for every objects .X, Y of .G and every ¯ Y , .ψ γ : F ¯ γ F X → F Fγ X and γ ∈ 𝚪, the isomorphisms .φ : F (X ⊗ Y ) → F X⊗F ¯ .φ ∗ : F I → I yield .

¯ Y ] = [F X][F Y ] f ([X][Y ]) = f [X ⊗ Y ] = [F (X ⊗ Y )] = [F X⊗F

.

= f [X] f [Y ], ¯ = 1, f (1) = f [I] = [F I] = [I] ¯ γ F X] = γ [F X] = γf [X]. f (γ [X]) = f [Fγ X] = [F Fγ X] = [F Next, since f is a bijection, we may choose, for every .x, y ∈ π0 G and .γ ∈ 𝚪, s¯ f (x) = F sx

.

as the representative object of .f (x) in (6.16) and

.

¯ π1 G, ¯ as representative morphisms in (6.18) for the construction of .HG¯ = (π0 G, k G¯ ). For every .x ∈ π0 G, the equivalence F induces an isomorphism of abelian groups ¯ x), F : π1 G(x) → π1 G(f

.

a I→ Fa = F a,

¯ is an isomorphism of .D𝚪 (π0 G)-modules: this follows In fact, .F : π1 G ∼ = f ∗ π1 G from the commutative diagrams below, where cells .(B) and .(D) commute since .φ and .ψ γ are natural and the remaining cells commute by (6.20) since F is a functor. The required equalities then follow from the definitions and Lemma 6.2. .f (x)F(a) = F(xa):

6.4 The Classification Theorem

195

.

F(a) f (z) = F(az):

.

.

γ F(a)

.

= F(γ a):

.

Finally, f and .F induce isomorphisms of cocycle groups ¯ π1 G) ¯ ∼ ¯ f ∗ : Z𝚪3 (π0 G, = Z𝚪3 (π0 G, f ∗ π1 G),

.

¯ F∗ : Z𝚪3 (π0 G, π1 G) ∼ = Z𝚪3 (π0 G, f ∗ π1 G), and the equality F∗ hG = f ∗ hG¯ ,

.

follows from the commutative diagrams below, where cells .(B) and .(D) commute since .φ and .ψ γ are natural, cell .(H ) commutes since .ψ γ is monoidal, and the remaining cells commute as indicated since F is a functor. Then it follows from the definitions and Lemma 6.2 that the top horizontal arrows .F hG (x, y, z), .−F hG (γ ; x, y) and .−F hG (γ , λ; x) must equal .hG ¯ (x, y, z), .−hG ¯ (γ ; x, y) and .−hG ¯ (γ , λ; x), as required.

196

h (f (x), f (y), f (z)) = FhG (x, y, z):

. G ¯

.

h (γ ; f (x), f (y)) = FhG (γ ; x, y):

. G ¯

.

6 Cohomology and Monoidal Groupoids

6.4 The Classification Theorem

197

h (γ , λ; f (x)) = FhG (γ , λ; x):

. G ¯

.

Then, F∗ k G = [F∗ hG ] = [f ∗ hG¯ ] = f ∗ [k G¯ ],

.

so that .(F, f ) : HG ∼ = HG¯ is an isomorphism in .H3𝚪 , and therefore .[HG ] = [HG¯ ].

⨆ ⨅

We refer to the isomorphism class [HG ] = [π0 G, π1 G, k G ]

.

as the classifying 3-cohomology object of the .𝚪-monoidal abelian groupoid .G. Theorem 6.2 (Classification Theorem) .[G] → I [HG ] is a one-to-one correspondence between .𝚪-monoidal equivalence classes of .𝚪-monoidal abelian groupoids and isomorphism classes of 3-cohomology objects. We begin the proof of Theorem 6.2 by constructing, for every 3-cohomology object .H = (S, A, k) a .𝚪-monoidal abelian groupoid GH = (GH , F, θ ).

.

Choose a 3-cocycle .h ∈ Z3 (S, A) such that .[h] = k. Then,

198

6 Cohomology and Monoidal Groupoids

− The objects of .GH are the elements of S. For every .x, y ∈ S,

.

GH (x, y) = ∅

.

if .x /= y, otherwise GH (x, y) = A(x).

.

The tensor product of .GH is x ⊗ y = xy

.

for every .x, y ∈ S, and .

for every morphisms .a, b of .GH . The unit object is .I = 1 ∈ S. The associativity and unit constraints are a x,y,z = h(x, y, z) : (xy)z → x(yz),

.

l x = −h(1, 1, x) : 1x = x → x, r x = h(x, 1, 1) : x1 = x → x. and are natural since the groups .A(x) are abelian. The coherence pentagon (6.1) commutes since .δh(x, y, z, t) = 0. (See (6.12).) Finally, .δh(x, 1, 1, y) = 0 yields xh(1, 1, y) − h(x, 1, y) + h(x, 1, 1)y = 0,

.

so that triangles (6.2) commute. Thus GH = (GH , ⊗, a, I, l, r)

.

is a monoidal abelian groupoid. − For each .γ ∈ 𝚪, the functor .Fγ : GH → GH is defined by

.

Fγ x = γx.

.

for every .x ∈ S, and a

γa

Fγ (x −→ x) = (γx −→ γx)

.

6.4 The Classification Theorem

199

for every morphism a of .GH . Its monoidal structure constraints are defined by γ

φ x,y = −h(γ ; x, y) : γ (xy) → γx γy,

.

γ

φ ⋆ = h(γ ; 1, 1) : γ 1 → 1. The diagrams (6.5) commute since .δh(γ ; x, y, z) = 0 (see (6.13)). Furthermore, the equalities .δh(γ ; x, 1, 1) = 0 and .δh(γ ; 1, 1, x) yield: .

γ

h(x, 1, 1) = h(γx, 1, 1) + γx h(γ ; 1, 1) − h(γ ; x, 1),

γ

h(1, 1, x) = h(1, 1, γx) + h(γ ; 1, 1) γx − h(γ ; 1, x),

so that the triangles (6.6) commute. Thus, every γ

Fγ = (Fγ , φ γ , φ ⋆ ) : GH → GH

.

is a monoidal functor. γ ,λ .− For every .γ , λ ∈ 𝚪, the monoidal transformation .θ : Fγ Fλ → Fγ λ is defined, at each .x ∈ S, by γ ,λ

= −h(γ , λ; x) : γ (λx) → γ λx.

θx

.

In (6.7), the left diagram commutes since .δh(γ , λ; x, y) = 0 (see (6.14)), and the right diagram commutes since .δh(γ , λ; 1, 1) = 0 yields γ

.

h(λ; 1, 1) + h(γ ; 1, 1) = h(γ λ; 1, 1) − h(γ , λ; 1).

Thus, every .θ γ ,λ is a monoidal transformation. 1 1 .− The natural transformation .θ : id GH → F , at each .x ∈ S, is defined by θ 1x = h(1, 1; x) : x → 1 x.

.

The equalities .δh(1, 1; x, y) = 0 and .δh(1, 1; 1, 1) = 0 in (6.14) yield h(1, 1; xy) − h(1; x, y) = h(1, 1; x) y + x h(1, 1; y),

.

h(1, 1; 1) = −h(1; 1, 1), so that both diagrams (6.7) commute. Thus .θ 1 is a monoidal transformation. Finally, the equality .δh(γ , λ, τ ; x) = 0 in (6.15) yields γ

.

h(λ, τ ; x) + h(γ , λτ ; x) = h(γ , λ; τx) + h(γ λ, τ ; x),

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6 Cohomology and Monoidal Groupoids

so that the diagram (6.8) commutes. Similarly, .δh(γ , 1, 1; x) = 0 and δh(1, γ , 1; x) = 0 yield .h(γ , 1; x) = γ h(1, 1; x) and .h(γ , 1; x) = γ h(1, 1; x), so that the diagram (6.9) commutes. Thus .GH = (GH , F, θ ) is .𝚪-monoidal abelian groupoid.

.

Proposition 6.2 Let .H and .H' be 3-cohomology objects. If .H and .H' are isomorphic, then .GH and .GH' are .𝚪-monoidal equivalent. Proof Suppose .H = (S, A, k) and .H' = (S ' , A' , k ' ) and let .GH and .GH' be constructed from representative cocycles .h ∈ Z𝚪3 (S, A) and .h' ∈ Z𝚪3 (S ' , A' ) of ' .k and .k respectively. Assume that .H ∼ = H' , so that there exist an isomorphism of .𝚪-monoids .f : ' ∼ S = S , an isomorphism of .D𝚪 (S)-modules .F : A ∼ = f ∗A' and a 2-cochain .g ∈ ' 2 ∗ ∗ ' C𝚪 (S, f A ) such that .f h = F∗ h + δg. Define .F = (F, φ, Ф∗ ) : GH → GH' by F (x) = f (x)

.

for every .x ∈ S, .

for every .a ∈ A(x), and .

φ x,y = g(x, y) : f (xy) −→ f (x)f (y), φ ∗ = −g(1, 1) : f (1) −→ 1.

for every .x, y ∈ S. Then, .φ x,y is natural in x and y, since .F : A → f ∗A' is a morphism of .D(S)-modules. The diagram (6.5) commutes since h' (f (x), f (y), f (z)) = Fh(x, y, z) + δg(x, y, z).

.

The equalities h' (f (x), 1, 1) = Fh(1, 1, x) + δg(x, 1),

.

h' (1, 1, f (x)) = Fh(1, 1, x) + δg(1, x), imply h' (f (x), 1, 1) = Fh(x, 1, 1) − g(x, 1) + f (x)g(1, 1),

.

h' (1, 1, f (x)) = Fh(1, 1, x) + +g(1, x) − g(1, 1)f (x),

6.4 The Classification Theorem

201

so that the coherence squares in (6.6) commute. Thus .F = (F, φ, φ ∗ ) is a monoidal functor. In fact, it is a monoidal equivalence since .f : π0 GH = S ∼ = S ' = π0 GH' is an isomorphism of monoids and, for every .x ∈ S, F : π1 GH (x) = A(x) ∼ = A' (f (x)) = π1 GH' (x)

.

is an isomorphism of abelian groups. For every .γ ∈ 𝚪, define .ψ γ : Fγ F → F Fγ at every .x ∈ S by γ

ψ x = −g(γ ; x) : γf (x) → f (γx).

.

These are natural since .F : A → f ∗A' is a morphism of .D𝚪 (S)-modules. The squares (6.7) commute since .h' (γ ; f (x), f (y)) = Fh(γ ; x, y) + δg(γ ; x, y). The equality .h' (γ ; 1, 1) = Fh(γ ; 1, 1) + δg(γ ; 1, 1) yields h' (γ ; 1, 1) = Fh(γ ; 1, 1) + γ g(1, 1) − g(1, 1) − g(γ ; 1)

.

so that the coherence triangle (6.7) commutes. The diagrams (6.10) commute since h' (γ , λ; f (x)) = h(γ , λ; x) + δg(γ , λ; x).

.

Finally, .h' (1, 1; f (x)) = h(1, 1; x) + δg(1, 1; x) also yields h' (1, 1; f (x)) = h(1, 1; x) + g(1; x),

.

so that the coherence triangle (6.11) commutes. Thus F : GH → GH'

.

is a .𝚪-monoidal equivalence.

⨆ ⨅

Proposition 6.3 Every .𝚪-monoidal abelian groupoid .G is .𝚪-monoidal equivalent to .GHG . Proof Let .HG = (π0 G, π1 G, k G ) be constructed from representative objects sx and morphisms .ϒx,y , ϒγ' ,y , as in (6.16), (6.18). We may assume that .ϒx,1 , .ϒ1,x , ' ' .ϒ γ ,1 and .ϒ1,y were chosen as in (6.19); then .hG is normalized as in (6.24). Define a functor .F : GHG → G by .

202

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and define

.

Then .φ x,y is natural, since the diagram

.

commutes for every .a ∈ AutG (sx) and .b ∈ AutG (sy). Moreover, the coherence diagrams (6.5) commute since the diagrams (6.21) commute, and the squares (6.6) commute trivially. Thus, .F = (F, φ, φ ∗ ) is a monoidal equivalence. For every .γ ∈ 𝚪, define .ψ γ : Fγ F → Fγ F , at each .x ∈ π0 G, by ψ x = −ϒγ' ,x : Fγ sx → s(γx).

.

γ

Then .ψ γ is natural and monoidal since the diagrams (6.20) and (6.22) commute. Moreover, the coherence triangles (6.7) and (6.9) (trivially) commute, whereas the diagrams (6.8) commute since the diagrams (6.23) do. Thus, .F = (F, φ, φ ∗ ) : GHG → G is a .𝚪-monoidal equivalence. ⨆ ⨅ Proposition 6.4 Every 3-cohomology object .H is isomorphic to .HGH . Proof Let .H be a 3-cohomology object. By Proposition 6.3, .GH is .𝚪-monoidal ⨆ ⨅ equivalent to .GHG whence, by Theorem 6.1, .H is isomorphic to .HGH . H

This completes the proof of Theorem 6.2, which now follows from Theorem 6.1 and Propositions 6.3 and 6.4. In particular, if .𝚪 = 1 is trivial, then Theorem 6.2 yields Theorem 6.3 The mapping .[G] I→ [π0 G, π1 G, k G ] establishes a one to one correspondence between monoidal equivalence classes of monoidal abelian groupoids and isomorphism classes of 3-cohomology objects, that consist of a monoid S, a 3 .D(S)-module .A, and a cohomology class .k ∈ H (S, A). Remark 6.1 Recall that a categorical group is a monoidal abelian groupoid .G where .π0 G is a group. If G is a group, then every 3-cohomology object .(G, A, k) is isomorphic to a 3-cohomology system .(G, A, k) in which A is a G-module

References

203

3 (G, A) is an Eilenberg-Mac Lane 3-cohomology class of G with and .k ∈ HEM coefficients in A. Hence the classification of categorical groups by third cohomology classes in [31] is a particular case of Theorem 6.2.

Remark 6.2 Recall from Theorem 3.30 that, with a dimension shift, both the Barr-Beck cotriple cohomology theory [2, 6] and the Leech cohomology theory of monoids are the same. Hence, for any monoid S and any .D(S)-module .A, the Duskin [16] and Glenn [19] general interpretation theorem for cotriple cohomology classes shows that .H 3 (S, A) classifies 2-torsors over S under .A. A very similar result follows from the general result by Pirashvili [28, 29] and Baues-Dreckmann [3] about the classification of track categories: namely, 3 .H (S, A) classifies linear track extensions of S by the .D(S)-module .A (called “natural systems” in [5]). In fact, the concepts of “2-torsor over S under .A”, “linear track extension of S by .A”, and “ strict monoidal abelian groupoid .G with fixed isomorphisms .f : π0 G ∼ =S and .F : π1 G ∼ = f ∗ A” are equivalent: this follows from Lemmas 2.2 and 2.3 in [13] (or [14, Theorem 3.3]), since an internal groupoid in the category of monoids is the same as a strict monoidal groupoid. However, the strict version of the monoidal groupoid concept is not optimal for all applications. By the Mac Lane Coherence Theorem [24, 26], every monoidal abelian groupoid is equivalent to a strict one; but this does not extend to homomorphisms. In fact, there exist strict monoidal abelian groupoids .G and .G' that are related by a monoidal equivalence between them but there is no strict monoidal equivalence either from .G to .G' nor from .G' to .G. To apply the Duskin or Pirashvili classification results we must use only strict monoidal abelian groupoids and strict monoidal equivalences between them, so that we must define two strict monoidal abelian groupoids .G and .G' as “monoidal equivalent” if there is a zig-zag chain of strict equivalences such as .G ← G1 → · · · ← Gn → G' , thereby unnecessarily obscuring the conclusions. Remark 6.3 The cohomological classification of braided and symmetric abelian groupoids by means of higher 3-cohomology groups of commutative monoids can be found in [9] and [10], respectively. (Cf. [20, Chapter 5]). Also, the classification of strictly commutative monoidal abelian groupoids by means of Grillet’s symmetric third cohomology classes is shown in [12].

References 1. Baez J.C., Lauda, A.D.: Higher dimensional algebra V: 2-groups. Theory Appl. Categ. 12, 423–492 (2004) 2. Barr, M., Beck, J.: Homology and standard constructions. In: Seminar on Triples and Categorical Homology. Lecture Notes in Mathematics, vol. 80, pp. 245–335. Springer, Berlin (1969) 3. Baues, H.J., Dreckmann, W.: The cohomology of homotopy categories and the general linear group. K-Theory 3, 307–338 (1989)

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4. Baues, H.J., Jibladze, M.: Classification of abelian track categories. K-Theory 25, 299–311 (2002) 5. Baues, H.J., Wirsching, G.: Cohomology of small categories. J. Pure Appl. Algebra 38, 187– 211 (1985) 6. Beck, J.: Triples, algebras and cohomology. Thesis Ph.D. Columbia University (1967). (Repr. Theory Appl. Categ. 2 (2003), 1–59) 7. Breen, L.: textitTheorie de Schreier superieure. Ann. Sci. Ec. Norm. Sup. 25, 465–514 (1992) 8. Brown, R., Spencer, C.B.: G-groupoids, crossed modules and the fundamental groupoid of a topological group. Proc. Kon. Nederl. Acad. Wetensch. 79, 296–302 (1976) 9. Calvo-Cervera, M., Cegarra, A.M.: A cohomology theory for commutative monoids. Mathematics 3, 1001–1031 (2015) 10. Calvo-Cervera, M., Cegarra, A.M.: Higher cohomologies of commutative monoids. J. Pure Appl. Alg. 223, 131–174 (2019) 11. Calvo, M., Cegarra, A.M., Heredia, B.A.: Structure and classification of monoidal groupoids. Semigroup Forum 87, 35–79 (2013) 12. Calvo-Cervera, M., Cegarra, A.M., Heredia, B.A.: On the third cohomology group of commutative monoids. Semigroup Forum 92, 511–533 (2016) 13. Cegarra, A.M., Aznar, E.: An exact sequence in the first variable for torsor cohomology: the 2-dimensional theory of obstructions. J. Pure Appl. Alg. 39, 197–250 (1986) 14. Cegarra, A.M., Bullejos, M., Garzón, A.R.: Higher dimensional obstruction theory in algebraic categories. J. Pure Appl. Alg. 49, 43–102 (1987) 15. Cegarra, A.M., García-Calcines, J.M., Ortega, J.A.: On graded categorical groups and equivariant group extensions. Can. J. Math. 54, 970–997 (2002) 16. Duskin, J.: Simplicial Methods and the Interpretation of Triple Cohomology, vol. 163. Memoirs of the American Mathematical Society (1975) 17. Eckmann, B., Hilton, P.J.: Group-like structures in general categories, I. Multiplications and comultiplications. Math. Ann. 145, 227–255 (1962) 18. Fröhlich, A., Wall, C.T.C.: Graded monoidal categories. Compos. Math. 28, 229–285 (1974) 19. Glenn, P.: Realization of cohomology classes in arbitrary exact categories. J. Pure Appl. Alg. 25, 33–105 (1982) 20. Grillet, P.A.: The Cohomology of Commutative Semigroups. Lecture Notes in Mathematics, vol. 2037. Springer, Berlin (2022) 21. Higgins, PJ.: Categories and groupoids. Reprints Theory Appl. Categ. 7, 1–178 (2005) 22. Husainov, A.A.: On the Leech dimension of a free partially commutative monoid. Tbil. Math. J. 1, 71–87 (2008) 23. Husainov, A.A.: The homology of partial monoid actions and Petri nets. Appl. Categ. Struct. 21, 587–615 (2013) 24. Joyal, A., Street, R.: Braided tensor categories. Adv. Math. 82, 20–78 (1991) 25. Leech, J.: The structure of a band of groups. Mem. Am. Math. Soc. 1(2), 67–95 (1975) 26. Mac Lane, S.: Natural associativity and commutativity. Rice Univ. Stud. 49, 28–46 (1963) 27. Norman, M.: Two cohomology theories for structured spaces (2020). ArXiv 2004.1152v2 28. Pirashvili, T.: Models for the homotopy theory and cohomology of small categories. Soobshch. Akad. Nauk Gruzin. SSR 129, 261–264 (1988) 29. Pirashvili, T.: Cohomology of small categories in homotopical algebra. In: K-Theory and Homological Algebra (Tbilisi, 1987–88). Lecture Notes in Mathematics, vol. 1437, pp. 268– 302. Springer, Berlin (1990) 30. Saavedra, N.: Catégories Tannakiennes. Lecture Notes in Mathematics, vol. 265. Springer, Berlin (1972) 31. Sinh, H.X.: Gr-catégories. Thése de Doctorat, Université Paris VII (1975). https://pnp. mathematik.uni-stuttgart.de/lexmath/kuenzer/sinh.html 32. Street, R.: Categorical structures. In: Handbook of Algebra, vol. 1, pp. 529–577. NorthHolland, Amsterdam (1996) 33. Wells, C.: Extension theories for monoids. Semigroup Forum 16, 13–35 (1978)

Concluding Remarks

In this monograph we have sought to present a coherent account of the cohomology of monoids beginning with developments in the 1970s and 1980s and moving on to more recent results in the last 10 years. In particular we have augmented the earlier work by presenting a cohomology based on a free resolution of a .Z-object in the appropriate abelian category (as in the classical cohomology of groups). We have shown how other cohomology theories for monoids can be viewed as special cases of this theory. In particular, we have shown that for inverse monoids and regular monoids this cohomology agrees with those developed for inverse semigroups and more generally for regular semigroups by Lausch and Loganathan in the 1970s and 1980. Of more recent vintage, e.g., is Sect. 2.6 on the cohomology of cyclic monoids based on a 2016 publication by Calvo-Cervera and Cegarra. Section 4.4 on obstruction theory is also rather recent. Chapter 5 on the cohomology of .𝚪-monoids is based on a 2019 publication by Cegarra. The material on monoidal groupoids in Chap. 6, also due to Cegarra, is new. This monograph, however, is not exhaustive. .D-cohomology is discussed in a 2020 arXiv publication by Manual Norman on cohomology theories for structured spaces [27]. And then there are Ahmet A. Husainov’s publications on applications of “Leech homology” of monoids to problems arising in computer science [22, 23]. Here one is on the homological side of things using a dual .D-homology. A concluding reference brings us back to the publication of Leech’s AMS Memoir. Besides the .H-coextension paper, it also contained The Structure of a Band of Groups [25]. Here a semigroup variant of .D-cohomology is used to study a band of groups, that is, a semigroup that is a (necessarily disjoint) union of its maximal subgroups that has the added property that the pointwise product of any two maximal subgroups must be contained in a maximal subgroup. Such a “bouquet” of groups is an .Hcoextension of a band, that is, of a semigroup consisting of just idempotents. Again, there are the cohomology theories in the first sections of Chap. 3. Though seemingly more elementary, calculating their cohomology groups can be a challenge, as in the cohomology of groups. Understandably, Eilenberg-MacLane cohomology for © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 A. M. Cegarra, J. Leech, The Cohomology of Monoids, RSME Springer Series 12, https://doi.org/10.1007/978-3-031-50258-3

205

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Concluding Remarks

monoids and for semigroups continues to have its adherents. (See, e.g. the papers by William R. Nico, Boris V. Novikov, Ahmet A. Husainov, and, more recently, by Paul Fau and others.) So, it is safe to assume that, yes, there is more to come. Indeed, we trust this monograph will serve as an added catalyst for this to happen.

Symbols & Notations

A (A, i, E, p) (Schreier coextension of a monoid), 61 (A, E, P , +) (coextension of a category), 99 ≈1 , ≈2 (relations on D(S)), 75 a (associativity constraint), 172 A (module over a category), 1 Ab (category of abelian groups), 1 AbHExt(S) (category of H-coextensions of S with abelian kernel), 120 AbZ2 (S) (category of abelian 2-cocycles of S), 34 (A, E, p, +) (coextension of a monoid), 32

B BS

(classifying simplicial set of a monoid S), 92 BC (classifying simplicial set of a category C), 95 B n (S, A) (abelian group of n-coboundaries of S in A), 19 B• S (S-module Bar resolution), 57 B•e S (S-bimodule Bar resolution), 64 B˜ n (S, A) (abelian group of normalized n-coboundaries of S in A), 20 C C • (S, A) C𝚪• (S, A)

(Leech cochain complex), 18 (Whitehead cochain complex), 161 • (S, A) CEM (Eilenberg-Mac Lane cochain complex), 58

• (X, A) CGZ (Gabriel-Zisman cochain complex), 90 • (S, B) CHM (Hochschild-Mitchell cochain complex), 65 • (C, A) CW (Wells cochain complex), 96 C∞ (infinite cyclic monoid), 44 Cm,q (cyclic monoid of index m and period q), 44 ≡ (congruence on D(S)), 75 cd(C) (cohomological dimension of C), 9 Cat (category of small categories), 12 C (small category), 1 C ⋊α A (twisted crossed product category), 100 C-Mod (category of C-modules), 1 C˜ • (S, A) (Leech normalized cochain complex), 20 C˜ n (S, G) (group of nonabelian n-cochains of S in G, n = 1, 2, 3), 126, 137

D Δ(X)

(category of simplices of a simplicial set X), 90 Δ (simplicial category), 90 D' (S) (D' -category of an inverse monoid S), 83 D(S) (D-category of a monoid S), 72 D(S)-Mod (category of D(S)-modules), 76 Der(S, A) (abelian group of derivations of S in A), 26 Der𝚪 (S, A) (abelian group of 𝚪derivations of S in A), 156 DiffS (functor of S-differentials), 31

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 A. M. Cegarra, J. Leech, The Cohomology of Monoids, RSME Springer Series 12, https://doi.org/10.1007/978-3-031-50258-3

207

208 Dim(S)

(cohomological D-dimension of a monoid S), 42 D(S) (division D-category of a monoid S), 12 D(S)-Mod (category of D(S)-modules), 14 D(C) (division D-category of a category C), 94 (division D-category of a 𝚪-monoid D𝚪 (S) S), 154 (category of D𝚪 (S)-modules), D𝚪 (S)-Mod 154 Δ (functor from 2-cocycles to coextensions), 36, 130 (face map of a simplicial set), 90 di dimEM (S) (Eilenberg-Mac Lane cohomological dimension of S), 60 (Hochschild-Mitchell dimHM (S) cohomological bidimension of S), 69

E Ext(G, N ) (set of classes of group coextensions of a group G by a group N ), 148 Ext(S, A) (set of classes of coextensions of S by A), 33 Ext(C, A) (set of classes of coextensions of C by A), 101  (set of classes of Ext G, (N, ρ) coextensions of a group G by a G-kernel), 149 (set of classes of 𝚪Ext𝚪 (S, A) coextensions of S by A), 166 Ext(G, N ) (category of group coextensions of a group G by a group N), 148 Ext(S, A) (category of coextensions of S by A), 33 (category of coextensions of S by ExtD (S) D(S)-modules), 120 (category of coextensions of S by ExtD (S) D(S)-modules), 33

F (F, ψ) (𝚪-monoidal equivalence), 176 (monoidal functor), 172 F = (F, φ, φ ∗ ) F (free module functor), 5 (D(S)-module resolution of Z), 16 F• S (normalized D(S)-resolution of Z), 19 F˜ • S (change of monoid functor f∗ D(T )-Mod → D(S)-Mod), 21

Symbols & Notations f∗ f∗

(change of monoid functors H n (T , A) → H n (S, f ∗ A)), 21 (left adjoint functor to f ∗ ), 22

G G (group), 60 G(S) (group of units of S), 12 G = (G, ⊗, I, a, r, l) (monoidal groupoid), 172 G = (G, F, θ ) (𝚪-monoidal groupoid), 176 (monoidal groupoid defined by a GH 3-cohomology object), 197 G (semifunctor), 126 Gr (quotient category of groups), 131 GrExt(G) (category of group coextensions of a group G), 145 Gr (category of groups), 115 G (semifunctor induced by G to Gr), 131 𝚪-Mon (category of 𝚪-monoids), 154

H H 2 (S, G)

(nonabelian 2-cohomology set of S), 129 2 (G, N ) (Schreier cohomology set of HSch G), 148 (Leech cohomology group of H n (S, A) S), 16 (cohomology group of C), 7 H n (C, A) (D𝚪 -cohomology group of S), H𝚪n (S, A) 156 (Grillet cohomology group of HGn (S, A) S), 53 n (S, B) (D-cohomology group of S), 87 HD n (S, A) HBB (Barr-Beck cohomology group of S), 108 n (S, A) (Eilenberg-Mac Lane HEM cohomology group of S), 57 n (S, A) (Grothendieck cohomology HGT group of S), 104 n (X, A) (Gabriel-Zisman cohomology HGZ of X), 90 n (S, B) (Hochschild-Mitchell HHM cohomology group of S), 64 (Wells cohomology group of HWn (C, A) C), 95 H = (S, A, k) (3-cohomology object), 179 (classifying HG = (π0 G, π1 G, k G ) 3-cohomology object of G), 179 HExt(S) (category of H-coextensions of a monoid S), 112

Symbols & Notations HExtD (S) (category of H-coextensions of S by D-modules), 120 HZ2 (S) (category of nonabelian H-2-cocycles of S), 132 HomC (A, A' ) (abelian group of morphisms of C-modules), 2 H3𝚪 (category of 3-cohomology objects), 179

I ı (functor G → D(G)), 60 ı∗ (functor D(G)-Mod → G-Mod), 61 I (unit object), 172 IS (augmentation ideal of S), 29 IDer(S, A) (abelian group of inner derivations of S in A), 26 J J j j j∗

(functor SchZ2 (G) → Z2 (G)), 146 (functor D(S) → S), 59 (functor D𝚪 (S) → S ⋊ 𝚪), 155 (functor (S ⋊ 𝚪)-Mod → D𝚪 (S)-Mod), 155 j∗ (functor S-Mod → D(S)-Mod), 59 J, L, R, H, D (Green’s relations), 78 K kG κ κ∗ κ∗

(classifying 3-cohomology class), 190 (functor D(S) → S e ), 67 (functor S e -Mod → D(S)-Mod), 67 (left adjoint functor to κ ∗ ), 67

L L♯ , R ♯ (relations on D(S)), 72 l (left unit constraint), 172 𝓁 (functor S e → S), 66 𝓁∗ (functor S-Mod → S e -Mod), 66 L(S) (Green’s left preorder of S), 73 L(S) (division L-category of a monoid S), 13 lim (inverse limit functor), 7 ←

M Mon (category of monoids), 12 Mon ↓S (category of monoids over S), 27 N (N, ρ)

(abstract G-kernel), 150

209 n

(ordered set {0, 1, . . . , n}), 90

O Obs(N, ρ) (obstruction of the G-kernel (N, ρ)), 151 Obs(G) (obstruction of the semifunctor G), 138 ObC (set of objects of C), 2, 5 OrPZ˜ 2 (S, G) (set of orbits of 2-precocycles of S in G), 137 P P (C) (underlaying preorder of C), 73 PZ˜ 2 (S, G) (set of nonabelian 2-precocycles of S in G), 136 π (functor Δ(BS) → D(S)), 92 π (functor Δ(BC) → D(C)), 96 π∗ (functor D(S)-Mod → Δ(BS)-Mod), 93 π∗ (functor D(C)-Mod → Δ(BC)-Mod), 96 π0 G (classifying monoid of G), 180 π1 G (classifying module of G), 180 R Rn F (nth right derived functor of F ), 7 r (right unit constraint), 172 R(S) (Green’s right preorder of S), 73 R(S) (division R-category of a monoid S), 13 ϱ (functor D(S) ↠ D(S)), 72 ϱ∗ (functor D(S)-Mod → D(S)-Mod), 76 ϱ∗ (left adjoint functor to f ∗ ), 84 S S⋊A (semidirect product), 27 S⋊𝚪 (semidirect product), 155 S ⋊α A (twisted semidirect product), 36 S ⋊α G (twisted semidirect product), 130 S-Mod (category of S-modules), 57 Se (enveloping monoid), 64 S e -Mod (category of S-bimodules), 64 S (map A(t k ) → A(t k1 )), 45 SchZ2 (G) (category of Schreier 2-cocycles of a group G), 145 SchZ2 (G, N ) (category of Schreier 2-cocycles of G in N ), 148 Set (category of sets), 3 Σ(H ) (Schützenberger group of H ), 113 Σ(x) (Schützenberger group of x), 113

210 Σp si

Symbols & Notations (kernel of an H-coextension), 116 (degeneracy map of a simplicial set), 90

T T H n (Kp , p ∗ A) (congruence cohomology groups), 25 T (trace map), 45 ⊗ (tensor product functor)), 172 U [u, x, v] (morphism of D(S)), 72 〈u, x, v, γ 〉 (morphism of D𝚪 (S)), 154 〈u, x, v〉 (morphism of D(S)), 12 Z ZG

(center of a semifunctor G), 136

Z n (S, A)

(abelian group of n-cocycles of S in A), 19 ZS (monoid ring), 57 Z (trivial module of integers), 7 Z[−] (free abelian group functor Set → Ab), 3 Z2 (S) (category of nonabelian 2-cocycles of S), 127 Z2 (S, A) (category of 2-cocycles of S in A), 35 Z2 (S, G) (category of nonabelian 2-cocycles of S in G), 129 Z˜ 2 (S, G) (set of nonabelian 2-cocycles of S in G), 127 2 (G, N ) Z˜ Sch (set of Schreier 2-cocycles of G in N ), 148 Z˜ n (S, A) (abelian group of normalized n-cocycles of S in A), 20

Subject Index

A Abelian group coextension, 33 object over S, 28 groupoid, 175 kernel functor, 119 Abstract kernel, 150 Associativity constraint, 172 pentagon, 172 Augmentation ideal, 29

B Beck cohomology groups, 108 cotriple resolution, 107 Bimodule over a monoid, 64

C Cancellative monoid, 70 Categorical group, 175 Category of abelian group coextensions, 33 groups, 1 2-cocycles, 34 .C-modules, 1 .D(S)-modules, 76 .D(S)-modules, 14 .D𝚪 (S)-modules, 154 .𝚪-monoids, 154 group coextensions

by .D-modules, 120 of groups, 145 groups, 115 .H-coextensions, 112, 120 with abelian kernels, 120 by .D-modules, 120 monoids, 12 monoids over S, 27 nonabelian 2-cocycles, 127 .H-2-cocycles, 132 S-bimodules, 64 Schreier 2-cocycles, 145 simplices, 90 small categories, 12 S-modules, 57 3-cohomology objects, 179 Center of a semifunctor, 136 Change of monoid functor, 21 left adjoint to, 22 Classifying module, 180 monoid, 180 simplicial set of a category, 95 monoid, 92 3-cohomology class, 190 3-cohomology object, 179, 197 Coefficient system, 90 Coextension of a category, 99 monoid, 32 Cohomological .D-dimension of a monoid, 42 dimension of a category, 9

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 A. M. Cegarra, J. Leech, The Cohomology of Monoids, RSME Springer Series 12, https://doi.org/10.1007/978-3-031-50258-3

211

212

Subject Index

Cohomology of a category, 7 Congruence cohomology groups, 25 on .D(S), 75 Conjugation, 113 Contracting homotopy, 17 Coprime set of morphisms, 79 Cotriple, 106 Cyclic monoid, 44 Czech cochain complex, 104

bimodule, 65 5 .Δ(X)-module, 91 .D(S)-module, 15 .𝚪-monoid, 157 monoid, 29 resolution, 17, 57, 58, 65, 98 S-bimodule, 64 S-module, 57 .C-module,

G D .Δ(X)-module, 90 .D-cohomology of a monoid, 87 .D-morphism, 78 ' .D -category of an inverse monoid, 83 .D(S)-module, 76 .D-cohomology of a category, 95 .𝚪-monoid, 156 monoid, 16 .D(S)-module, 14 .D(C)-module, 95 .D𝚪 (S)-module, 154 Degeneracy map, 90 Derivations, 26 Differentials, 31 Division .D-category of a category, 94 .𝚪-monoid, 154 monoid, 12 .L-category of a monoid, 13 .R-category of a monoid, 13

.𝚪-

coextension, 166 derivation, 156 monoid, 154 monoidal equivalence, 176 monoidal groupoid, 176 Gabriel-Zisman cochain complex, 90 cohomology groups, 90 G-kernel, 150 Global dimension, 70 Green’s preorders, 73 relations, 78 Grillet cohomology groups, 53 Grothendieck cohomology groups, 104 flask resolution, 104 topology, 103 Group of units of a monoid, 12 Groupoid, 171

H .H-coextension,

E Eilenberg-Mac Lane Bar resolution, 57 Classification Theorem, 151 cochain complex, 58 cohomological dimension, 60 cohomology groups, 57, 60 Enveloping monoid, 64

F Face map, 90 Five term exact sequence, 25 Fixed point functor, 59 Flask sheaf, 104 Free abelian group functor, 3

112 with abelian kernel, 119 Hochschild-Mitchell Bar resolution, 64 cochain complex, 65 cohomological bidimension, 69 cohomology groups, 64

I Identity of a monoidal groupoid, 174 Index of a cyclic monoid, 44 Initial object, 9 pair, 10 Injective .C-module, 5 resolution, 8

Subject Index

213

Inner derivation, 26 Inverse limit functor, 7 Inverse monoid, 81 coextension, 89 Invertible object, 174

2-cocycle, 127 2-cohomology set, 129 3-cochain, 137 2-precocycle, 136 Normal morphism of .D(S), 82

K Kernel functor, 117–119 Künneth tensor formula, 69

O Obstruction of a G-kernel, 151 of an orbit of 2-precocycles, 137 of a semifunctor, 138 Orbits of 2-precocycles, 137

L Lallement’s Theorem, 111 Lausch-Loganathan cohomology, 89 Leech coboundaries, 19 cochain complex, 18 cochains, 19 cocycles, 19 cohomology groups, 16 normalized coboundaries, 20 normalized cochain complex, 20 normalized cocycles, 20 normalized resolution of .Z, 19 resolution of .Z, 16 Left invariance, 113, 114 unit constraint, 172 Linear extension of a category, 99 track extension, 203

M Module over a category, 1 monoid, 57 Monoidal equivalence, 174 functor, 172 groupoid, 172 transformation, 173 Monoid ring, 57

N Natural system, 95 Nerve of a category, 95 Nonabelian 1-cochain, 126 2-cochain, 126

P Period of a cyclic monoid, 44 Postnikov invariant, 177 Preorder of a category, 73 Projective .C-module, 4 resolution, 8 Pseudo-functor, 175

R Regular monoid, 122 Relations on .D(S), 72, 75 Right derived functors, 7 Right unit constraint, 172

S Schreier Classification Theorem, 149 coextension of a monoid, 61 2-cocycle, 145 2-cohomology set, 148 Schützenberger group, 113 Semidirect product, 27, 155 Semifree monoid, 43 Semifunctor, 126 Simplicial category, 90 Simplicial set, 90 Simply transitive action, 32 Smooth functor, 78 Symmetric .D(S)-module, 52

T 2-category of .𝚪-monoidal groupoids, 175 monoidal groupoids, 174 2-torsor, 203

214 3-cohomology object, 179 Tensor product, 172 Transitive cochains, 24 Twisted crossed product, 100 semidirect product, 36, 130

U Unit object, 172 Unit triangle, 172 Universal group, 64

Subject Index V Vector cohomology groups, 156

W Wells cochain complex, 96 cohomology groups, 95 Whitehead cochain complex, 161

Y Yoneda Lemma, 3