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

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 and 2- are equivalent in a remarkably strong sense: there is an isomorphism, compatible with localization, between the ordered classes of basic localizers of and 2-. It follows that weak homotopy equivalences in 2- can be characterised in an internal way, without mentioning topological spaces or simplicial sets.
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Taxonomy
TopicsHomotopy and Cohomology in Algebraic Topology · Algebraic structures and combinatorial models · Advanced Topology and Set Theory
