Double groupoids and homotopy 2-types

TL;DR

This paper establishes an equivalence between a category of double groupoids satisfying a filling condition and the homotopy category of 2-types, providing a new algebraic model for certain topological spaces.

Contribution

It introduces a new 'homotopy double groupoid' construction and proves an equivalence with the homotopy category of 2-types, linking higher categorical structures to homotopy theory.

Findings

The geometric realization functor induces an equivalence between double groupoids and homotopy 2-types.

A new 'homotopy double groupoid' construction for topological spaces is explicitly defined.

Double groupoids with a filling condition characterize homotopy 2-types.

Abstract

This work contributes to clarifying several relationships between certain higher categorical structures and the homotopy types of their classifying spaces. Double categories (Ehresmann, 1963) have well-understood geometric realizations, and here we deal with homotopy types represented by double groupoids satisfying a natural `filling condition'. Any such double groupoid characteristically has associated to it `homotopy groups', which are defined using only its algebraic structure. Thus arises the notion of `weak equivalence' between such double groupoids, and a corresponding `homotopy category' is defined. Our main result in the paper states that the geometric realization functor induces an equivalence between the homotopy category of double groupoids with filling condition and the category of homotopy 2-types (that is, the homotopy category of all topological spaces with the property that the $n^{\text{th}}$ homotopy group at any base point vanishes for $n\geq 3$). A quasi-inverse functor is explicitly given by means of a new `homotopy double groupoid' construction for topological spaces.