Coherence in Three-Dimensional Category Theory by Nick Gurski

By Nick Gurski

Size 3 is a vital test-bed for hypotheses in better class concept and occupies whatever of a special place within the express panorama. on the middle of concerns is the coherence theorem, of which this booklet offers a definitive remedy, in addition to protecting comparable effects. alongside the way in which the writer treats such fabric because the grey tensor product and provides a building of the basic 3-groupoid of an area. The ebook serves as a entire creation, overlaying crucial fabric for any scholar of coherence and assuming just a easy knowing of upper type idea. it's also a reference aspect for lots of key techniques within the box and accordingly an essential source for researchers wishing to use better different types or coherence ends up in fields comparable to algebraic topology or theoretical desktop technological know-how.

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Here we have written F(1, −), F(−, 1), for Fa1 (−), resp. Fa2 (−). The constraint cells are given by or are identities as necessitated by the definition of cubical functor, and it is simple to check that the axioms above give the axioms for a weak functor. 3 A cubical functor F : A1 × A2 × A3 → B determines, and is uniquely determined by (1) For each object a1 ∈ A1 , a cubical functor of two variables Fa1 : A2 × A3 → B, and similarly for objects a2 ∈ A2 , a3 ∈ A3 ; (2) For each pair of objects a1 , a2 in A1 , A2 , respectively, the equation Fa1 (a2 , −) = Fa2 (a1 , −) holds, and similarly for pairs a1 , a3 and a2 , a3 ; such that the following axiom holds: Given a 1-cell ( f 1 , f 2 , f 3 ) : (a1 , a2 , a3 ) → (a1 , a2 , a3 ) in A1 × A2 × A3 , the equation below holds.

16 The category 2Cat of 2-categories and 2-functors has the structure of a closed symmetric monoidal category when equipped with • • • • the Gray tensor product, A ⊗ B, unit object the terminal 2-category, the internal hom-functor Hom(A, B), and symmetry either given by the construction given in Section 1 or by the procedure above. 17 Note that this is a different closed symmetric monoidal structure than the one given by Cartesian product and the usual hom-2-category having 0-cells 2-functors, 1-cells 2-natural transformations, and 2-cells modifications.

Thus the statement of the coherence theorem for bicategories becomes the following. 13 (Coherence for bicategories) The functor induced by j : G → Fs G is a strict biequivalence. 14 Let G, G be category-enriched graphs, and let S, T : G → G be maps between them. The category-enriched graph Eq(S, T ) is defined to have objects those a ∈ G 0 such that S0 a = T0 a. The category Eq(S, T )(a, b) has objects pairs (h, α) where h : a → b in G and α : Sh → T h is an isomorphism in G (S0 a, S0 b). The morphisms β : (h, α) → (h , α ) are those β : h → h in G such that α ◦ S(β) = T (β) ◦ α.

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