# nLab Bredon cohomology

cohomology

### Theorems

#### $\left(\infty ,1\right)$-Topos Theory

(∞,1)-topos theory

## Constructions

structures in a cohesive (∞,1)-topos

# Contents

## Idea

The immediate notion of cohomology of topological spaces which are equipped with a $G$-action for some topological group $G$ (often: ”$G$-spaces”, for short) is the ordinary equivariant cohomology ${H}_{G}^{*}\left(X\right)≔{H}^{•}\left(X//G\right)$ of the Borel construction/homotopy quotient of $X$ by $G$.

In contrast to this, Bredon cohomology is another notion of cohomology of $G$-spaces, defined via the orbit category ${\mathrm{Orb}}_{G}$ of $G$: its coefficients are presheaves (of abelian groups) on the orbit category.

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

${\mathrm{Orb}}_{G}^{\mathrm{op}}\to \mathrm{Ab}$Orb_G^{op} \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.

For a more general abstract characterization of Bredon cohomology see at global equivariant homotopy theory.

## 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.

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 $K$-theory which is closely related to the Bredon cohomology (rather than usual equivariant cohomology):

Revised on October 7, 2013 08:56:56 by Urs Schreiber (89.204.154.179)