structures in a cohesive (∞,1)-topos
The immediate notion of cohomology of topological spaces which are equipped with a -action for some topological group (often: ”-spaces”, for short) is the ordinary equivariant cohomology of the Borel construction/homotopy quotient of by .
Hence where ordinary cohomology has as coefficient objects abelian groups, Bredon cohomology has as coefficients functors
For a more general abstract characterization of Bredon cohomology see at global equivariant homotopy theory.
There is an interesting (nontrivially) equivalent definition by Moerdijk and Svensson, using the Grothendieck construction for a certain -valued presheaf on the orbit category.
Further remarks on this are in
G. Mukherjee, N. Pandey, Equivariant cohomology with local coefficients (pdf)
H. Honkasalo, Sheaves on fixed point sets and equivariant cohomology, Math. Scand. 78 (1996), 37–55 (pdf)
H. Honkasalo, A sheaf-theoretic approach to the equivariant Serre spectral sequence, J. Math. Sci. Univ. Tokyo 4 (1997), 53–65 (pdf)
For orbifolds there is a generalization of -theory which is closely related to the Bredon cohomology (rather than usual equivariant cohomology):