Formula omitted]-groupoids, crossed modules and the fundamental groupoid of a topological group

Abstract

THEOREN 1. The categories of 8-groupoids and of crossed modules are equivalent. This result was, we understand, known to Verdier in 1965; it was then used by Duskin [6] ; it was discovered independently by us in 1972. The work of Verdier and Duskin is unpublished, we have found that Theorem 1 is little known, and so we hope that this account will prove useful. We shall also extend Theorem 1 to include in Theorem 2 a comparison of homotopy notions for the two cittegories. As an application of Theorem 1, we consider the fundamental groupoid nX of a topological group X. Clemly nX is a B-groupoid ; its associated crossed module has (as does any crossed module) an obstruction class or k-invariant which in this case lies in @(,X, zl(X, e)). We prove in Theorem 3 that this k-invariant is the fist Postnikov invariant of the classifying space Bsx of the singular complex 8X of X. An example of the use of Theorem 3 is the (possibly well known) result that the first Postnikov invariant of Bob) is zero. Duskin was led to his application of Theorem 1 in the theory of group extensions by an interest in Isbell’s principle (a 9’ in an Lsl is an &’ in a &Y). We were led to Theorem 1 as part of a programme for exploiting double groupoids (thab is, groupoid objects in the category of groupoids) in homotopy theory. Basic results on and applications of double groupoids are given in [3], [2] and [4].