Homomorphisms of Gray-categories as pseudo algebras

TL;DR

This paper establishes an equivalence between locally strict trihomomorphisms of Gray-categories and Gray-functors, using monad theory and coherence results, with implications for descent theory and a potential Yoneda lemma for tricategories.

Contribution

It shows that locally strict trihomomorphisms are biequivalent to Gray-functors, extending 2D monad theory to 3D and providing a foundation for tricategory theory.

Findings

Gray-category of trihomomorphisms is isomorphic to pseudo algebra Gray-category.

Inclusion of functor category has a left adjoint with biequivalence components.

Results apply to Gray-valued presheaves and suggest a tricategory Yoneda lemma.

Abstract

Given Gray-categories $P$ and $L$, there is a Gray-category $\mathrm{Tricat}_{\mathrm{ls}}(P,L)$ of locally strict trihomomorphisms with domain $P$ and codomain $L$, tritransformations, trimodifications, and perturbations. If the domain $P$ is small and the codomain $L$ is cocomplete, we show that this Gray-category is isomorphic as a Gray-category to the Gray-category $\mathrm{Ps}$-$T$-$\mathrm{Alg}$ of pseudo algebras, pseudo functors, transformations, and modifications for a Gray-monad $T$ derived from left Kan extension. Inspired by a similar situation in two-dimensional monad theory, we apply the coherence theory of three-dimensional monad theory and prove that the the inclusion of the functor category in the enriched sense into this Gray-category of locally strict trihomomorphisms has a left adjoint such that the components of the unit of the adjunction are internal biequivalences. This proves that any locally strict trihomomorphism between Gray-categories with small domain and cocomplete codomain is biequivalent to a Gray-functor. Moreover, the hom Gray-adjunction gives an isomorphism of the hom 2-categories of tritransformations between a locally strict trihomomorphism and a Gray-functor with the corresponding hom 2-categories in the functor Gray-category. A notable example is given by locally strict Gray-valued presheafs with small domain. Our results have applications in three-dimensional descent theory and point into the direction of a Yoneda lemma for tricategories.