Classifying spaces and Bredon (co)homology for transitive groupoids

TL;DR

This paper extends equivariant Bredon (co)homology to transitive topological groupoids by defining an orbit category and analyzing actions via isotropy groups, with applications to principal bundles and classifying spaces.

Contribution

It introduces a new framework for equivariant (co)homology of transitive groupoids, generalizing classical equivariant theories for groups.

Findings

Defined the orbit category for transitive topological groupoids.

Extended equivariant Bredon (co)homology to groupoid actions.

Applied the theory to principal bundles and classifying spaces.

Abstract

We define the orbit category for transitive topological groupoids and their equivariant CW-complexes. By using these constructions we define equivariant Bredon homology and cohomology for actions of transitive topological groupoids. We show how these theories can be obtained by looking at the action of a single isotropy group on a fiber of the anchor map, extending equivariant results for compact group actions. We also show how this extension from a single isotropy group to the entire groupoid action can be applied to the structure of principal bundles and classifying spaces.