nLab
Bredon cohomology

Idea

The usual cohomology for $G$-spaces is the equivariant cohomology $H_G^{*}(X)$ (in classical case just the cohomology of the Borel construction $EG{\times }_{G}X$, sometimes called in this context the homotopy orbit space).

Bredon cohomology is another cohomology of $G$-spaces, defined with the help of the orbit category and its usual coefficients are presheaves of abelian groups on the orbit category.

Where ordinary cohomology has as coefficient objects abelian groups, Bredon cohomology has as coefficients functors

$$O(G)\to \mathrm{Ab}$$O(G) \to Ab

from the orbit category of $G$ into Ab.

More details are for the moment at equivariant cohomology in the section Bredon equivariant cohomology.

References

There is an interesting (nontrivially) equivalent definition by Moerdijk and Svensson, using the Grothendieck construction for a certain $\mathrm{Cat}$-valued presheaf on the orbit category.

• I. Moerdijk, J-A. Svensson, The equivariant Serre spectral sequence, Proc. Amer. Math. Soc. 118 (1993), no. 1, 263–278 doi:10.2307/2160037

Further remarks on this are in

• G. Mukherjee, N. Pandey, Equivariant cohomology with local coefficients (pdf)

• H. Honkasalo, Sheaves on fixed point sets and quivariant 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 $K$-theory which is closely related to the Bredon cohomology (rather than usual equivariant cohomology):

• Alejandro Adem, Yongbin Ruan, Twisted Orbifold K-Theory, Commun.Math.Phys. 237 (2003) 533-556, doi:10.1007/s00220-003-0849-x arXiv