Coherence in Three-Dimensional Category Theory

286 Pages·2013·4.742 MB·English

by  Nick Gurski

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Coherence in Three-Dimensional
Category Theory
NICK GURSKI
UniversityofSheffield
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pages cm.–(Cambridgetractsinmathematics;201)
ISBN978-1-107-03489-1(hardback)
1. Tricategories. I. Title.
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Contents
Introduction page1
1 Tricategories 1
2 Gray-monads 3
3 Anoutline 5
Acknowledgements 12
Part I Background 13
1 Bicategoricalbackground 15
1.1 Bicategoricalconventions 15
1.2 Matesinbicategories 17
2 Coherenceforbicategories 21
2.1 TheYonedaembedding 21
2.2 Coherenceforbicategories 22
2.3 Coherenceforfunctors 29
3 Gray-categories 35
3.1 TheGraytensorproduct 36
3.2 Cubicalfunctors 38
3.3 ThemonoidalcategoryGray 45
3.4 Afactorization 48
Part II Tricategories 57
4 Thealgebraicdefinitionoftricategory 59
4.1 Basicdefinition 59
4.2 Adjointequivalencesandtricategoryaxioms 65
4.3 Trihomomorphismsandotherhighercells 66
4.4 Unpackedversions 78
v
vi Contents
4.5 Calculationsintricategories 83
4.6 Comparingdefinitions 85
5 Examples 86
5.1 Primaryexample:Bicat 86
5.2 Fundamental3-groupoids 89
6 Freeconstructions 97
6.1 Graphs 97
6.2 Thecategoryoftricategories 99
6.3 FreeGray-categories 103
7 Basicstructure 106
7.1 Structureoffunctors 107
7.2 Structureoftransformations 110
7.3 Pseudo-icons 114
7.4 Changeofstructure 123
7.5 Triequivalences 127
8 Gray-categoriesandtricategories 129
8.1 Cubicaltricategories 129
8.2 Gray-categories 133
9 CoherenceviaYoneda 138
9.1 Localstructure 139
9.2 Globalresults 141
9.3 ThecubicalYonedalemma 144
9.4 Coherencefortricategories 154
10 Coherenceviafreeconstructions 156
10.1 Coherencefortricategories 157
10.2 Coherenceanddiagramsofconstraints 160
10.3 Anon-commutingdiagram 161
10.4 Strictifyingtricategories 162
10.5 Coherenceforfunctors 171
10.6 Strictifyingfunctors 177
Part III Gray-monads 181
11 CodescentinGray-categories 183
11.1 Laxcodescentdiagrams 184
11.2 Codescentdiagrams 188
11.3 Codescentobjects 190
Contents vii
12 Codescentasaweightedcolimit 196
12.1 WeightedcolimitsinGray-categories 197
12.2 Examples:coinsertersandcoequifiers 200
12.3 Codescent 207
13 Gray-monadsandtheiralgebras 209
13.1 Enrichedmonadsandalgebras 210
13.2 Laxalgebrasandtheirhighercells 213
13.3 Totalstructures 219
14 Thereflectionoflaxalgebrasintostrictalgebras 227
14.1 Thecanonicalcodescentdiagramofalaxalgebra 228
14.2 Theleftadjoint,laxcase 230
14.3 Theleftadjoint,pseudocase 242
15 Ageneralcoherenceresult 244
15.1 Weakcodescentobjects 245
15.2 Coherenceforpseudo-algebras 266
Bibliography 273
Index 277

Introduction
In the study of higher categories, dimension three occupies an interesting
positiononthelandscapeofhigherdimensionalcategorytheory.Fromtheper-
spectiveofa“hands-on”approachtodefiningweakn-categories,tricategories
representthemostcomplicatedkindofhighercategorythatthecommunityat
largeseemscomfortableworkingwith.Ontheotherhand,dimensionthreeis
the lowest dimension in which strict n-categories are genuinely more restric-
tive than fully weak ones, so tricategories should be a sort of jumping off
point for understanding general higher dimensional phenomena. This work
is intended to provide an accessible introduction to coherence problems in
three-dimensionalcategory.
1 Tricategories
Tricategories were first studied by Gordon, Power, and Street in their 1995
AMSMemoir.Theywereawarethatstrict3-groupoidsdonotmodelhomotopy
3-types, and thus the aim of their work was to create an explicit defini-
tion of a weak 3-category which would not be equivalent (in the appropriate
three-dimensional sense) to that of a strict 3-category. The main theorem of
Gordon et al. (1995) is often stated: every tricategory is triequivalent to a
Gray-category.Triequivalenceisastraightforwardgeneralizationoftheusual
notionthattwocategoriesareequivalentwhenthereisafunctorbetweenthem
whichisessentiallysurjective,full,andfaithful.Thenewandinterestingfea-
ture of this result is the appearance of Gray-categories. These are categories
whichareenrichedoverthemonoidalcategoryGray;thismonoidalcategory
has the category of 2-categories and strict 2-functors as its underlying cat-
egory, but its monoidal structure is not the Cartesian one. Gray-categories
can thus be viewed as a maximally strict yet still completely general form
ofweak3-category,anditisknown,forinstance,thatGray-groupoidsmodel
allhomotopy3-types.
1
2 Introduction
My interest in tricategories began while carrying out joint work with
Eugenia Cheng on the Stabilization Hypothesis of Baez and Dolan. The
Stabilization Hypothesis roughly states that k-degenerate weak (n + k)-
categoriescorrespondtowhattheycalledk-tuplymonoidaln-categories.Here,
k-degeneratemeansthatthe(n+k)-categoriesonlyhaveasingle0-cell,sin-
gle1-cell,andsoon,uptohavingonlyasingle(k −1)-cell:thusthebottom
k dimensions are degenerate. A k-tuply monoidal n-category is one which is
monoidal,andask increasesthatmonoidalstructurebecomesmoreandmore
commutative until it stabilizes when k = n +2. Some relevant examples to
keepinmindare
• the case k = 1,n = 0 gives 1-degenerate categories (categories with a
single object) on the one hand or 1-tuply monoidal 0-categories (sets with
anassociativeandunitalmultiplication)ontheotherhand;and
• fixing n = 1 we get weak 2-categories with a single object, weak
3-categories with a single object and single 1-cell, and weak 4-categories
withasingleobject,1-cell,and2-cellontheonehandandmonoidalcate-
gories,braidedmonoidalcategories,andsymmetricmonoidalcategorieson
theotherhand.
The Stabilization Hypothesis is a guiding principle of higher category the-
ory,yetwefoundthatnosystematicstudyoflowdimensionalcaseshadbeen
carriedout.
As had already been discovered by Tom Leinster, k-degenerate (n + k)-
categories and k-tuply monoidal n-categories were not precisely the same
structures, at least when using the explicit, algebraic notions of weak
n-category.Asanexample,abicategorywithasingleobjectandsingle1-cell
is not only a commutative monoid given by the set of 2-cells I ⇒ I under
composition(whereI isthesingle1-cell),butisinfactacommutativemonoid
equippedwithadistinguishedinvertibleelement.Thiselementcorrespondsto
the left (or right, they are equal) unit isomorphism, and satisfies no axioms.
Soinfactitisthealgebraicnatureofthedefinitionofbicategorythatcreates
thisextrapieceofdata.Tocarryoutthesameanalysisindimensionthree,we
neededafullyalgebraicdefinitionoftricategory,andthedefinitionofGordon,
Power,andStreetwasonlypartiallyalgebraic.
Theoriginaldefinitionwaspartiallyalgebraicbecauseitincludeddatahav-
ingcertainpropertiesbutnotthedatanecessarytocheckthoseproperties.In
particular,theassociativityequivalencefor1-cellcompositionisa2-cell
ah,g,f :(h⊗g)⊗ f ⇒h⊗(g⊗ f),