Mackey 2-Functors and Mackey 2-Motives

237 Pages·2020·2.645 MB·English

by  Paul Balmer, Ivo Dell'Ambrogio

Paul Balmer P Monographs in Mathematics
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Ivo Dell’Ambrogio
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Mackey 2-Functors and l
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Mackey 2-Motives r
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I
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o Paul Balmer

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This book is dedicated to equivariant mathematics, specifically the study of additive b
r
categories of objects with actions of finite groups. The framework of Mackey 2-functors o
g Mackey 2-Functors and
axiomatizes the variance of such categories as a function of the group. In other words, i
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it provides a categorification of the widely used notion of Mackey functor, familiar to

representation theorists and topologists. Mackey 2-Motives

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The book contains an extended catalogue of examples of such Mackey 2-functors that
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are already in use in many mathematical fields from algebra to topology, from geometry c
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to KK-theory. Among the first results of the theory, the ambidexterity theorem gives e
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a way to construct further examples and the separable monadicity theorem explains
2
how the value of a Mackey 2-functor at a subgroup can be carved out of the value at -
F
a larger group, by a construction that generalizes ordinary localization in the same way u
that the étale topology generalizes the Zariski topology. The second part of the book n
c
provides a motivic approach to Mackey 2-functors, 2-categorifying the well-known span t
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construction of Dress and Lindner. This motivic theory culminates with the following r
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application: The idempotents of Yoshida’s crossed Burnside ring are the universal source a
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of block decompositions.
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The book is self-contained, with appendices providing extensive background and
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terminology. It is written for graduate students and more advanced researchers interested c
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in category theory, representation theory and topology. e
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ISBN 978-3-03719-209-2
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Monographs / Balmer/Dell‘Ambrogio | Font: Rotis Semi Sans | Farben: Pantone 116, Pantone 287 | RB 16 mm

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Paul Balmer
Ivo Dell’Ambrogio
Mackey 2-Functors and
Mackey 2-Motives
Authors:
Paul Balmer Ivo Dell’Ambrogio
Mathematics Department Laboratoire Paul Painlevé
University of California Los Angeles Université de Lille, CNRS, UMR 8524
Los Angeles, CA 90095-1555 59000 Lille
USA France
e-mail: [email protected] e-mail: [email protected]
2010 Mathematics Subject Classification: Primary: 20J05, 18B40, 55P91; secondary: 18M30, 18N10, 18N25,
18N40, 19A22, 20C99
Key words: Groupoids, Mackey formula, equivariant, 2-functors, derivators, ambidexterity, separable
monadicity, spans, string diagrams, motivic decompositions, Burnside algebras
ISBN 978-3-03719-209-2
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fortheirloveandsupport,andforpatientlyindulgingusduringmonths
ofgroupoid-jugglingandstring-untangling.

Contents
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1 Surveyofresults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.1 ThedefinitionofMackey2-functors . . . . . . . . . . . . . . . . . . 7
1.2 Rectification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3 Separablemonadicity . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.4 Mackey2-functorsandGrothendieckderivators . . . . . . . . . . . . 15
1.5 Mackey2-motives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.6 Pointerstorelatedworks . . . . . . . . . . . . . . . . . . . . . . . . 18
2 Mackey2-functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.1 Commaandiso-commasquares . . . . . . . . . . . . . . . . . . . . 19
2.2 Mackeysquares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3 GeneralMackey2-functors . . . . . . . . . . . . . . . . . . . . . . . 25
2.4 Separablemonadicity . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.5 Decategorification . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3 Rectificationandambidexterity. . . . . . . . . . . . . . . . . . . . . . . 33
3.1 Selfiso-commas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.2 Comparingthelegsofaselfiso-comma . . . . . . . . . . . . . . . . 39
3.3 Thecanonicalmorphism‚fromlefttorightadjoint . . . . . . . . . . 41
3.4 RectificationofMackey2-functors . . . . . . . . . . . . . . . . . . . 58
4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.1 Examplesfromadditivederivators . . . . . . . . . . . . . . . . . . . 63
4.2 Mackeysub-2-functorsandquotients . . . . . . . . . . . . . . . . . . 66
4.3 Extendingexamplesfromgroupstogroupoids . . . . . . . . . . . . . 68
4.4 Mackey2-functorsofequivariantobjects . . . . . . . . . . . . . . . . 72
5 Bicategoriesofspans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.1 Spansina(2,1)-category . . . . . . . . . . . . . . . . . . . . . . . . 81
5.2 Theuniversalpropertyofspans . . . . . . . . . . . . . . . . . . . . . 95
5.3 Bicategoricalupgradesoftheuniversalproperty . . . . . . . . . . . . 106
5.4 Pullbackof2-cellsinthebicategoryofspans. . . . . . . . . . . . . . 113
5.5 Heuristicaccountandthe2-dualversion . . . . . . . . . . . . . . . . 122
6 Mackey2-motives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.1 Mackey2-motivesandtheiruniversalproperty . . . . . . . . . . . . . 131
6.2 Astrictpresentationandacalculusofstringdiagrams . . . . . . . . . 143
6.3 ThebicategoryofMackey2-functors . . . . . . . . . . . . . . . . . . 150
viii Contents
7 AdditiveMackey2-motivesanddecompositions . . . . . . . . . . . . . 155
7.1 AdditiveMackey2-motives . . . . . . . . . . . . . . . . . . . . . . . 155
7.2 TheYoneda2-embedding . . . . . . . . . . . . . . . . . . . . . . . . 159
7.3 PresheavesoveraMackey2-functor . . . . . . . . . . . . . . . . . . 161
7.4 CrossedBurnsideringsandMackey2-motives . . . . . . . . . . . . . 163
7.5 MotivicdecompositionsofMackey2-functors . . . . . . . . . . . . . 171
AppendixA:Categoricalreminders . . . . . . . . . . . . . . . . . . . . . . 175
A.1 Bicategoriesand2-categories . . . . . . . . . . . . . . . . . . . . . . 175
A.2 Mates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
A.3 Stringdiagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
A.4 Howtoreadstringdiagramsinthisbook . . . . . . . . . . . . . . . . 194
A.5 Theordinarycategoryofspans . . . . . . . . . . . . . . . . . . . . . 197
A.6 Additivityforcategories. . . . . . . . . . . . . . . . . . . . . . . . . 199
A.7 Additivityforbicategories . . . . . . . . . . . . . . . . . . . . . . . 203
AppendixB:OrdinaryMackeyfunctorsonagivengroup . . . . . . . . . . 215
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
Introduction
In order to study a given group G, it is natural to look for mathematical objects on
whichG actsbyautomorphisms.Forinstance,inordinaryrepresentationtheory,one
considersvectorspacesonwhichG actslinearly. Intopology,onemightprefertopo-
logicalspacesandcontinuousG-actions. Infunctionalanalysis,itmightbeoperator
algebrasonwhichGisexpectedtoact. Andsoon,andsoforth. Those‘G-equivariant
objects’usuallyassembleintoacategory,thatweshalldenoteM.G/. Constructing
such categories M.G/ of G-equivariant objects in order to study the group G is
a simple but powerful idea. It is used in all corners of what we shall loosely call
‘equivariantmathematics’.
Inthiswork,wefocusonfinitegroupsG andadditivecategoriesM.G/,i.e.cat-
egories in which one can add objects and add morphisms. Although topological or
analyticalexamplesmaynotseemveryadditiveatfirstsight,theycanbeincludedin
our discussion by passing to stable categories. Thus, to name a few explicit exam-
ples ofsuch categoriesM.G/, let us mention categoriesof kG-modulesM.G/ D
Mod.kG/ or their derived categories M.G/ D D.kG/ in classical representation
theoryoverafieldk,homotopycategoriesofG-spectraM.G/DSH.G/inequivari-
anthomotopytheory, andKasparovcategoriesM.G/ D KK.G/ of G-C(cid:2)-algebras
in noncommutative geometry. As the reader surely realizes at this point, the list of
suchexamplesis virtuallyendless: justletG actwhereverit can! In fact, the entire
Chapter4ofthisbookisdevotedtoareviewofexamples.
Let us try to isolate the properties that such categoriesM.G/ havein common.
First of all, it is clear that in all situations we can easily construct a similar cate-
goryM.H/ foranyothergroup H, in particularfor subgroupsH (cid:2) G. The vari-
ance of M.G/ in the group G, through restriction, induction, conjugation, etc, is
the bread and butter of equivariantmathematics. It is then a natural question to ax-
iomatizewhatitmeanstohaveareasonablecollectionofadditivecategoriesM.G/
indexedbyfinitegroupsG,withalltheselinksbetweenthem. Inviewoftheubiquity
ofsuchstructures,itissomewhatsurprisingthatsuchanaxiomatictreatmentdidnot
appearearlier.
In fact, a lot of attention has been devoted to a similar but simpler structure, in-
volvingabeliangroupsinsteadofadditivecategories.Thesearetheso-calledMackey
functors.Letusquicklyremindthereaderofthisstandardnotion,goingbacktowork
ofGreen[Gre71]andDress[Dre73]almosthalfacenturyago.
AnordinaryMackeyfunctorM involvesthedataofabeliangroupsM.G/indexed
byfinitegroupsG. TheseM.G/comewithrestrictionhomomorphismsRGWM.G/
H
! M.H/, inductionortransferhomomorphismsIGWM.H/ ! M.G/, and conju-
H
gationhomomorphismsc WM.H/ ! M.xH/,forH (cid:2) G andx 2 G. Thisdatais
x
subjecttoacertainnumberofrules,mostofthemratherintuitive. Amongthem,the
critical rule is the Mackey double-cosetformula, which says that for all H;K (cid:2) G