2014/08/15 by Lukas Buhné, Buhné, Lukas
Mathematics · Medicine · #Advanced Topics in Algebra #Homotopy and Cohomology in Algebraic Topology #Ophthalmology and Eye Disorders #math.CT
paper · pdf · doi:10.48550/arxiv.1408.3481
62 pages
arxiv created 2014/08/15 · arxiv updated 2014/08/18
Given Gray-categories P and L, there is a Gray-category Tricatls(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 Ps-T-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.