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Topics in Three-Dimensional Descent Theory

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Topics in Three-Dimensional Descent Theory TOPICSINTHREE-DIMENSIONAL DESCENTTHEORY Dissertation zur Erlangung des Doktorgrades an der Fakult¨atf¨urMathematik, Informatik und Naturwissenschaften Fachbereich Mathematik der Universit¨atHamburg vorgelegt von Lukas Buhn´e Hamburg, 2015 Gutachter waren: Prof. Dr. Christoph Schweigert Dr. Nick Gurski Prof. Dr. Ross Street Tag der Disputation: 29. April 2015 3 Contents Introduction 5 Summary of results . 22 Outline . 23 Acknowledgments . 28 1. Homomorphisms of Gray-categories as pseudo algebras 29 1.1. Preliminaries . 29 1.1.1. Conventions . 29 1.1.2. The category 2Cat ........................... 30 1.1.3. The symmetric monoidal closed category Gray ........... 30 1.1.4. Cubical functors . 34 1.1.5. V -enriched monad theory . 38 1.2. The Gray-category of pseudo algebras . 40 1.2.1. Definitions and two identities . 41 1.2.2. Coherence via codescent . 51 1.3. The monad of the Kan adjunction . 52 1.3.1. A V -monad on [obP ; L] ........................ 52 1.3.2. Explicit description of the monad . 56 1.3.3. Some properties of tensor products . 61 1.4. Identification of the V -category of algebras . 70 1.5. Identification of pseudo algebras . 74 1.5.1. Homomorphisms of Gray-categories . 74 1.5.2. The correspondence of Gray homomorphisms and pseudo algebras 81 1.5.3. The correspondence with locally strict trihomomorphisms . 87 2. A Yoneda lemma for tricategories 93 2.1. Codescent objects . 93 2.1.1. The codescent object of a free algebra . 106 2.2. The locally-strict Yoneda lemma . 112 2.2.1. Representables as free algebras . 112 2.2.2. The Gray-enriched Yoneda lemma. 113 2.2.3. The locally strict Yoneda lemma . 114 2.3. Change of local functors . 118 2.4. Representables and the cubical Yoneda embedding . 132 2.5. Preservation of representables under strictification . 139 2.6. The Yoneda lemma for tricategories . 147 3. A three-dimensional descent construction 149 3.1. The definition of the descent construction . 150 3.2. Strictification of the descent construction . 151 3.3. The codescent object of the weight of the descent construction . 152 3.4. Identification with descent data . 168 3.5. Outlook . 171 A. Functor tricategories 173 A.1. The functor tricategory for target a Gray-category . 175 A.2. Representables . 177 A.3. Composition of trihomomorphisms . 182 A.4. Whiskering with a trihomomorphism on the left . 184 A.5. Whiskering on the left with strictification . 189 A.6. Whiskering with a trihomomorphism on the right . 192 A.7. Biequivalences of trihomomorphisms . 201 A.8. Internal coherence for Bicat . 202 5 Introduction TOPICS IN THREE-DIMENSIONAL DESCENT THEORY Structures satisfying a property only up to a distinguished natural coherence isomor- phism appear frequently in mathematics and physics. Category theory provides a formal framework, in which to speak and reason about these structures. In particular, since many constructions are much easier carried out in the strict situation where the coher- ence isomorphisms are trivial, one wishes to know if any such structure is equivalent to a strict one. This is the concept of strictification. On the other hand, the definition of weak notions of higher categories and their homomorphisms is a rich source of examples for these situations. This is because the definition of weak higher categories involves categorification, the process where axioms and more generally equations are replaced by new coherence isomorphisms subject to new axioms. It is a highly nontrivial task to identify which equations have to be replaced, and what kind of and how many axioms have to be enforced in order to allow for strictification. In this thesis we are concerned with the concept of strictification in three-dimensional category theory. Specifically, our interest concerns strictification for the higher cells of weak 3-categories and is driven by fundamental questions in three-dimensional category theory arising from the construc- tion of bundle gerbes in terms of descent [61], which was motivated by Mathematical Physics. A strict 3-category can be defined as a category enriched in the symmetric monoidal closed category of small 2-categories equipped with the cartesian product and a suitable internal hom. The weak concept of a 3-category is the concept of a tricategory intro- duced by Gordon, Power, and Street [21]. In contrast to monoidal categories, and more generally bicategories, not every tricategory is triequivalent to a strict 3-category. For example, just as a monoidal category gives rise to a bicategory with one object, a braided monoidal category gives rise to a tricategory with one object and one 1-cell. If this tri- category were equivalent to a strict 3-category, every braided monoidal category would have to be strictly symmetric cf. [21, Rem. 8.8]. Rather, any tricategory is equivalent to a semi-strict 3-category: a Gray-category [21]. A Gray-category is a category enriched in the symmetric monoidal closed category Gray of small 2-categories and strict functors equipped with the strong Gray product [22] of 2-categories and a suitable internal hom cf. Section 1.1.3. The semi-strictness of Gray-categories has its roots in the definition of the monoidal structure of Gray: The Gray product encodes the non-trivial interchange corresponding to the braiding of a monoidal category. The other side of the story is that 6 the closed structure of Gray is given by pseudonatural transformations and modifications of strict functors of 2-categories, but there are two possibilities to compose pseudonat- ural transformations, which coincide only up to an invertible interchange modification. While the question of strictification for a tricategory has basically been answered by the coherence theorem of Gordon, Power, and Street, and later by Gurski [24] using free constructions with coherence theorems in the form of commuting diagrams and the explicit construction of a strictification GrT, a Gray-category, for a tricategory T, much less is known about the higher cells of tricategories. In enriched category theory the basic object of interest is a category enriched in a monoidal category. For enriched categories A and B, there are two basic notions of cells between them: that of a functor F : A ! B, and that of an enriched natural transfor- mation between two such functors. Under suitable assumptions, these organize into an enriched category themselves: the enriched functor category [A; B]. Its underlying cat- egory has objects the enriched functors between A and B and morphisms the enriched natural transformations. For example, for enrichment in the category Set of small sets with the cartesian product, an enriched category is a locally small ordinary category, and the functor category consists exactly of the ordinary functors between such categories and natural transformations. Hence, in three-dimensional category theory, there is the well-understood concept of the functor Gray-category [A; B] between Gray-categories A and B. By the definition of an enriched category, for two objects F; G in [A; B], that is, two Gray-functors F; G : A ! B, we have a hom object [A; B](F; G) in the category of enrichment Gray. Thus, [A; B](F; G) is a 2-category with objects the Gray-natural transformations between F and G, but there are also 1- and 2-cells in this 2-category, and these constitute the higher cells between Gray-categories. By the definition of the hom object of the enriched functor Gray-category in terms of an end|a basic form of a limit in enriched category theory|each of these higher cells will consist of component cells in the target Gray-category B and each of them will be subject to a naturality condition in the form of an equation of cells in the target B cf. Corollary 1.5. The weak notions of cells between Gray-categories arise if axioms are only enforced on cells of the highest dimension, that is, on 2-cells in the hom 2-categories of the target, which we call the 3-cells of the target Gray-category. We are in particular interested in the case, where an equation of each kind of cell is replaced by the appropriate notion of 'equivalence` between such cells and where axioms are only allowed for 3-cells. Since 3-cells between two fixed 2-cells form a set, the only notion of 'equivalence` is equality. Hence, 3-cells will be subject to axioms. The higher cells between two fixed 1-cells form a category, in which the appropriate notion of 'equivalence` is an isomorphism. Finally, the higher cells between two fixed objects form a 2-category|the hom object of the category enriched in Gray. The notion of equivalence in a 2-category that we use is that of an adjoint equivalence. Hence, an equation of 2-cells is replaced by an invertible 3-cell subject to new axioms that have to be specified. An equation of 1-cells is replaced by an adjoint equivalence 2-cell, but since equations of 2-cells are not allowed, further data in the form of invertible 3-cells subject to yet new axioms has to be specified. If the axioms of a Gray-category are themselves replaced in this fashion, one recovers the notion of a cubical tricategory. 7 On the other hand, we have the definitions of weak higher cells between two tricate- gories T and S . These are trihomomorphisms, tritransformations, trimodifications and perturbations, which are defined in a similar fashion involving bicategories, pseudofunc- tors, and the cartesian product instead of 2-categories, strict functors, and the Gray product. One expects that these organize into a functor tricategory Tricat(T; S ), but in fact, the precise structure of this tricategory has yet only been described in special cases.
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