Coherence in Three-Dimensional Category Theory by Nick Gurski

By Nick Gurski

Size 3 is a crucial test-bed for hypotheses in better classification concept and occupies anything of a different place within the express panorama. on the center of issues is the coherence theorem, of which this publication offers a definitive remedy, in addition to protecting comparable effects. alongside the best way the writer treats such fabric because the grey tensor product and provides a building of the elemental 3-groupoid of an area. The booklet serves as a accomplished advent, overlaying crucial fabric for any scholar of coherence and assuming just a easy figuring out of upper class idea. it's also a reference element for lots of key strategies within the box and hence a necessary source for researchers wishing to use larger different types or coherence ends up in fields equivalent to algebraic topology or theoretical laptop technological know-how.

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9 Let Hom(A, B) denote the full sub-bicategory of Bicat(A, B) with objects the strict functors. Recall that if B is a 2-category, then Hom(A, B) is a 2-category for any bicategory A. 10 Let A, B be 2-categories. 1 The evaluation map e : Hom(A, B) × A → B is a cubical functor. 2 The function which sends a 2-functor F : A1 → Hom(A2 , B) to the composite F×1 e A1 × A2 → Hom(A2 , B) × A2 → B is a natural isomorphism between 2-functors A1 → Hom(A2 , B) and cubical functors A1 × A2 → B. Proof For the first part, the evaluation map e is defined by the following formulas.

Then there is a cubical functor F : A × B → C such that (1) F agrees with G on objects and (2) there is an invertible icon ν : G ⇒ F. Proof Define F to agree with G on objects. For a 1-cell ( f, g) : (a, b) → (a , b ), define F to be the composite G( f,1) G(1,g) F(a, b) −→ F(a , b) −→ F(a , b ), where we have already used that F(a, b) = G(a, b). For a 2-cell (α, β) : ( f, g) → ( f , g ), define F(α, β) to be the horizontal composite G( f,1) F(a, b)  G(α,1) G( f ,1) C Q F(a , b) G(1,g)  G(1,β) G(1,g ) C Q F(a , b ).

Finally, we quotient out by the equivalence relation generated by naturality of the 2-cells a f gh , l f , r f , the middle-four interchange law, the rule that the composition α ◦ β in FG agrees with that of G if α, β are arrows in some G(a, b), and the two bicategory axioms. Note that there is an obvious inclusion i : G → FG of categoryenriched graphs. 10 1. The data above satisfy the necessary axioms to constitute a bicategory. 2. Let B be a bicategory. Then given a map f : G → B of category-enriched graphs, there is a unique strict functor of bicategories f˜ : FG → B such that f˜i = f in Gr (Cat).

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