Th\'eories homotopiques des 2-cat\'egories

TL;DR

This paper develops a homotopy theory for 2-categories, establishing an equivalence with the homotopy theory of categories and enabling internal characterizations of weak equivalences.

Contribution

It introduces the notion of basic localizer for 2-Cat and proves the equivalence of homotopy theories between Cat and 2-Cat, generalizing Grothendieck's framework.

Findings

Homotopy theories of Cat and 2-Cat are equivalent.

Weak homotopy equivalences in 2-Cat can be characterized internally.

Defined basic localizers for 2-Cat analogous to those for Cat.

Abstract

This text develops a homotopy theory of 2-categories analogous to Grothendieck's homotopy theory of categories developed in "Pursuing Stacks". We define the notion of "basic localizer of 2-Cat", 2-categorical generalization of Grothendieck's notion of basic localizer, and we show that the homotopy theories of $Cat$ and 2-$Cat$ are equivalent in a remarkably strong sense: there is an isomorphism, compatible with localization, between the ordered classes of basic localizers of $Cat$ and 2-$Cat$. It follows that weak homotopy equivalences in 2-$Cat$ can be characterised in an internal way, without mentioning topological spaces or simplicial sets.