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

Jonathan Chiche

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arXiv · 1309.0191

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

Jonathan Chiche

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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.

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Jonathan Chiche. 2013-09-01. Th\'eories homotopiques des 2-cat\'egories. https://arxiv.org/abs/1309.0191

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