[Paper Review] La théorie de l'homotopie des 2-catégories

Jonathan Chiche|arXiv (Cornell University)|Nov 25, 2014

Homotopy and Cohomology in Algebraic Topology10 references3 citations

TL;DR

This thesis establishes a homotopy theory for 2-categories by constructing a Quillen model structure on 2-Cat, proving that the homotopy category of 2-categories is equivalent to that of small categories via the nerve functor. It generalizes Grothendieck's homotopy theory from categories to 2-categories using simplicial methods, fundamental localizers, and a 2-categorical version of Quillen's Theorem A, showing that the nerve functor induces an equivalence between the homotopy categories Ho(Cat) and Ho(2-Cat).

ABSTRACT

On développe une théorie de l'homotopie des 2-catégories analogue à la théorie de l'homotopie des catégories développée par Grothendieck dans "À la poursuite des champs". Il s'agit de la thèse de doctorat de l'auteur. We develop a homotopy theory of 2-categories analogous to Grothendieck's homotopy theory of categories developed in "Pursuing Stacks." This is the author's PhD thesis.

Motivation & Objective

• To extend Grothendieck's homotopy theory from categories to 2-categories.
• To define and study fundamental localizers in the 2-categorical setting.
• To establish a Quillen equivalence between the homotopy categories of small categories and 2-categories.
• To generalize Quillen's Theorem A to 2-categories and lax morphisms.
• To construct a model structure on 2-Cat whose weak equivalences are detected by the nerve functor.

Proposed method

• Uses the nerve functor N: Cat → b∆ and its 2-categorical analogue to relate 2-categories to simplicial sets.
• Applies the theory of fundamental localizers to define weak equivalences in 2-Cat.
• Constructs a 2-categorical version of Quillen's Theorem A for lax morphisms and 2-functors.
• Introduces twisted cylinders and 2-truncations to analyze homotopy types in 2-Cat.
• Uses the adjunction between c: b∆ → Cat and N: Cat → b∆, and extends it to 2-categories via the Grothendieck construction.
• Applies the Sd and Ex functors on simplicial sets to stabilize the nerve and realize a Quillen equivalence between Cat and 2-Cat.

Experimental results

Research questions

• RQ1 Is there a Quillen model structure on 2-Cat whose weak equivalences are detected by the nerve functor?
• RQ2 Can Quillen's Theorem A be generalized to the 2-categorical setting for lax functors?
• RQ3 What is the relationship between the homotopy category of 2-categories and that of small categories?
• RQ4 How do fundamental localizers behave in the 2-categorical context?
• RQ5 Can the nerve functor induce an equivalence between Ho(Cat) and Ho(2-Cat)?

Key findings

• The nerve functor N: Cat → b∆ induces a Quillen equivalence between the model categories of small categories and simplicial sets.
• The homotopy category of 2-categories Ho(2-Cat) is equivalent to Ho(Cat), the homotopy category of small categories.
• A 2-categorical version of Quillen's Theorem A holds for lax functors, providing a criterion for weak equivalences in 2-Cat.
• The fundamental localizer for 2-Cat is characterized via the nerve functor and the 2-skeleton truncation.
• The adjunction (c, N) does not induce a Quillen equivalence on 2-Cat, but the composition cSd² and Ex²N does.
• The construction of the Grothendieck 2-category ∆/X provides a homotopy inverse to the nerve functor at the level of homotopy categories.