The classifying space of a crossed complex

The aim of this paper is to show one more facet of the role of crossed complexes as generalisations of both groups (or groupoids) and of chain complexes. We do this by defining and establishing the main properties of a classifying space functor B: mia -> SToft from the category of crossed complexes to the category of spaces. The basic example of a crossed complex is the fundamental crossed complex n\X of a filtered space X. Here 77XX is the fundamental groupoid n1(X1,X0) and 77nX for n ^ 2 is the family of relative homotopy groups nn(Xn,Xn_1,p) for all peX0. These come equipped with the standard operations of vx X on 77n X and boundary maps S: 7TnX->-77ra_1X. The axioms for crossed complexes are those universally satisfied for this example. Every crossed complex is of this form for a suitable filtered space X ([17], corollary 93, see also Section 2 below). Thus a crossed complex C is like a chain complex of modules with a groupoid G as operators, but with non-Abelian features in dimensions 1 and 2, in the sense that the part C2->Cl is a crossed module with cokernel G. So crossed complexes have the virtues of chain complexes, in the sense of having a familiar homological algebra, at the same time as being able to carry non-Abelian information, such as that involved in a presentation of a group G. Their convenience is also shown by results of [2], which give them as the first level or linear approximation to combinatorial homotopy theory. Further details of the history of their use are given in [18]. Our main result is the following classification theorem.