Spaces of maps into classifying spaces for equivariant crossed complexes
A crossed complex is an algebraic structure which can be used in homotopy theory; roughly speaking, it is a chain complex of groupoids. A filtered topological space \(X_*\) has a fundamental crossed complex \(\pi X_*\), and a crossed complex \(C\) has a classifying space \(BC\). If \(X_*\) is the skeletal filtration of a CW-complex \(X\) and if \(C\) is a crossed complex, then there is a weak equivalence \[ B\bigl( {\mathcal CRS} (\pi X_*,C) \bigr) \to {\mathcal TOP} (X,BC), \] where \({\mathcal CRS}\) is an internal hom for crossed complexes and \({\mathcal TOP}\) is a space of maps. The main result in this paper is a similar equivariant result for a group \(G\) which is discrete, or at least close to being discrete. The CW-complex becomes a \(G\)-CW-complex, and the crossed complex becomes a diagram of crossed complexes indexed by the orbit category of \(G\). Applications include results on equivariant Eilenberg-MacLane spaces, on local systems, and on equivariant \(2\)-types.
scientific article; zbMATH DE number 1052948
scientific article