On the homotopy theory of Grothendieck \infty-groupoids

TL;DR

This paper explores the homotopy theory of Grothendieck ty-groupoids, establishing independence of homotopy groups from choices, characterizing weak equivalences, and constructing a functor that preserves homotopy groups and weak equivalences.

Contribution

It introduces a variation of Grothendieck ty-groupoids, proves independence of homotopy groups from choices, and constructs a fundamental ty-groupoid functor compatible with weak equivalences.

Findings

Homotopy groups of Grothendieck ty-groupoids are well-defined independently of choices.

Weak equivalences of ty-groupoids are characterized by equivalent conditions.

The fundamental ty-groupoid functor ty preserves weak equivalences.

Abstract

We present a slight variation on a notion of weak \infty-groupoid introduced by Grothendieck in Pursuing Stacks and we study the homotopy theory of these \infty-groupoids. We prove that the obvious definition for homotopy groups of Grothendieck \infty-groupoids does not depend on any choice. This allows us to give equivalent characterizations of weak equivalences of Grothendieck \infty-groupoids, generalizing a well-known result for strict \infty-groupoids. On the other hand, given a model category M in which every object is fibrant, we construct, following Grothendieck, a fundamental \infty-groupoid functor \Pi_\infty from M to the category of Grothendieck \infty-groupoids. We show that if X is an object of M, then the homotopy groups of \Pi_\infty(X) and of X are canonically isomorphic. We deduce that the functor \Pi_\infty respects weak equivalences.