Representing Bredon cohomology with local coefficients

TL;DR

This paper develops a new homotopical framework for Bredon cohomology with local coefficients using equivariant crossed complexes and constructs a parametrized G-spectrum that captures this cohomology theory.

Contribution

It introduces a novel representation of Bredon cohomology with local coefficients via equivariant crossed complexes and constructs a corresponding naive parametrized G-spectrum.

Findings

Bredon cohomology with local coefficients is represented as homotopy classes in equivariant crossed complexes.

A naive parametrized G-spectrum is constructed to model the cohomology theory.

The spectrum reduces to Bredon cohomology with local coefficients on suspension spectra.

Abstract

For a discrete group G, we represent the Bredon cohomology with local coefficients as the homotopy classes of maps in the category of equivaraint crossed complexes. Subsequently, we construct a naive parametrized G-spectrum, such that the cohomology theory defined by it reduces to the Bredon cohomology with local coefficients when restricted to suspension spectra.