Notes on 2-groupoids, 2-groups and crossed modules

Abstract

This paper contains some basic results on 2-groupoids, with special emphasis on computing derived mapping 2-groupoids between 2-groupoids and proving their invariance under strictification. Some of the results proven here are presumably folklore (but do not appear in the literature to the author's knowledge) and some of the results seem to be new. The main technical tool used throughout the paper is the Quillen model structure on the category of 2-groupoids introduced by Moerdijk and Svensson.

Stacky abelianization of algebraic groups

“…Furthermore, a morphism of crossed modules defines a strict 1-morphism between the corresponding gr-categories. These facts remain true if we replace crossed modules by strictly stable crossed modules and gr-categories by strictly commutative Picard groupoids; see, for instance, [Con84], [Bre92], [Bre94], and [Noo05].…”

“…Furthermore, G acts on H by conjugation. One can show that H → G is a crossed module and [G/H] is equivalent to M; see, for instance, [Noo05].…”

Non-Hausdorff symmetries of C*-algebras

“…For any connected CW-complex X , there is an associated crossed module C and a map X → BC that induces an isomorphism on π 1 and π 2 and thus is a homotopy equivalence if π n (X ) = 0 for n ≥ 3. Thus any homotopy 2-type is represented by the classifying space of some crossed module (see also [8,9]). …”

“…Crossed modules of discrete groups were introduced in homotopy theory to classify 2-connected spaces up to homotopy equivalence. They are equivalent to strict 2-groups, which are central objects of study in higher category theory (see [1,9]). The crossed modules of topological groupoids that we introduce below are equivalent to strict topological 2-groupoids.…”

n-Groupoids and Stacky Groupoids

“…An n-group is an n-groupoid for which X 0 is a point. When n = 2, they are different from the various kinds of 2-group(oid)s or double groupoids in [5,8] (see [20,Appendix] for an explanation of the relation between our 2-group and the one in [5]), and are not exactly the same as in [30], as he requires a choice of composition and strict units; however, they are the same as in [14]. A usual groupoid (category with only isomorphisms) is equivalent to a 1-groupoid in the sense of Def.…”