On homotopy types modelized by strict \infty-groupoids

TL;DR

This paper investigates which homotopy types can be modeled by strict -groupoids, showing that simply connected cases correspond to products of Eilenberg-Mac Lane spaces and relating them to derived categories.

Contribution

It establishes an equivalence between simply connected strict -groupoid homotopy types and derived categories of abelian groups, characterizing the modeled homotopy types.

Findings

Simply connected -groupoid homotopy types are equivalent to derived categories of abelian groups.

These homotopy types are exactly products of Eilenberg-Mac Lane spaces.

Brief exploration of 3-categories with weak inverses.

Abstract

The purpose of this text is the study of the class of homotopy types which are modelized by strict \infty-groupoids. We show that the homotopy category of simply connected \infty-groupoids is equivalent to the derived category in homological degree greater or equal to 2 of abelian groups. We deduce that the simply connected homotopy types modelized by strict \infty-groupoids are precisely the products of Eilenberg-Mac Lane spaces. We also briefly study 3-categories with weak inverses. We finish by two questions about the problem suggested by the title of this text.