  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.
