There is some flexibility in interpreting precisely what this should mean, and accordingly there are various different flavors of equivariant cohomology theory, all closely related, but different.
Borel equivariant cohomology We can form the action groupoid of the action of on : the weak quotient of the action. Then for any coefficient object we may think of the cohomology of as being the -equivariant cohomology of . This definition is known Borel equivariant cohomology
This notion of equivariant cohomology captures notably the standard notion of equivariant -principal bundles for the delooping of the structure group . If is the point, then this reduces to the definition of group cohomology of the group . Group cohomology is the Borel-equivariant cohomology of the point.
This is described below in the section Borel equivariant cohomology.
Equivariant multiplicative cohomology For cohomology in with coefficient objects components of an E-∞-ring spectrum – a multiplicative cohomology theory – one finds that there is a refinement of Borel equivariant cohomology that is of interest.
This is described below in the section Multiplicative equivariant cohomology.
of (∞,1)-presheaves on has as objects topological spaces/simplicial sets with a -action, whose homotopy type is determined on the subspaces of fixed points under the action of all closed subgroups . Integer-graded Bredon cohomology is the (abelian) cohomology inside this (∞,1)-topos. It can also be generalized to -graded Bredon cohomology. This is the kind of equivariant cohomology used mostly in equivariant stable homotopy theory.
This is described below in the section Bredon equivariant cohomology.
We first state the general abstract definition of equivariant cohomology and then derive from it the more concrete formulations that are traditionally given in the literature.
Recall from the discussion at cohomology that in full generality we have a notion of cohomology of an object with coefficients in an object whenever and are objects of some (∞,1)-topos . The cohomology set is the set of connected components in the hom-object ∞-groupoid of maps from to : .
Recall moreover from the discussion at space and quantity that objects of an (∞,1)-topos of (∞,1)-sheaves have the interpretation of ∞-groupoids with extra structure. For instance for -sheaves on a site of smooth test spaces such as Diff these objects have the interpretation of Lie ∞-groupoids.
In this case, for some such ∞-groupoid with structure, let be its 0-truncation, which is the space of objects of , the categorically discrete groupoid underlying . We think of the morphisms in as determining which points of are related under some kind of action on , the 2-morphisms as relating these relations on some higher action, and so on. Equivariance means, roughly: functorial transformation behaviour of objects on with respect to this “action” encoded in the morphisms in . This is the intuition that is made precise in the following
In the simple special case that one should keep in mind, is for instance the action groupoid of the action, in the ordinary sense, of a group on : its morphisms connect those objects of that are related by the action by some group element .
It is natural to consider the relative cohomology of the inclusion . Equivariant cohomology is essentially just another term for relative cohomology with respect to an inclusion of a space into a (-)groupoid.
In some (∞,1)-topos the equivariant cohomology with coefficient in an object of a 0-truncated object with respect to an action encoded in an inclusion is simply the -valued cohomology of .
More specifically, an equivariant structure on an -cocycle on is a choice of extension
i.e. a lift of through the projection .
By comparing the definition of equivariant cohomology with that of group cohomology one sees that group cohomology can be equivalently thought of as being equivariant cohomology of the point.
For some group let be the stack of -principal bundles. Let be some finite group (just for the sake of simplicity of the example) and let be an action of on a space . Let be the corresponding action groupoid.
Then a cocycle in the -equivariant cohomology is
Consider for definiteness , the path ∞-groupoid of . (All other cases are in principle obtaind from this by truncation and/or strictification).
Then for some coefficient -groupoid, a morphism can be thought of as classifying a -principal ∞-bundle on the space .
On the other hand, a morphism out of is something like a flat connection (see there for more details) on this principal -bundle, also called an -local system. (More details on this are at differential cohomology).
Accordingly, an extension of through the inclusion is the process of equipping a principal -bundle with a flat connection.
Comparing with the above definition of eqivariant cohomology, we see that flat connections on bundles may be regarded as path-equivariant structures on these bundles.
When pairing equivariant cohomology with other variants of cohomology such as twisted cohomology or differential cohomology one has to exercise a bit of care as to what it really is that one wants to consider. A discussion of this is (beginning to appear) at differential equivariant cohomology.
A refinement of Borel equivariant cohomology. For the moment see
According to the nPOV on cohomology, if and are objects in an (∞,1)-topos, the 0th cohomology is , while if is a group object, then . More generally, if is times deloopable, then . In Top, this gives you the usual notions if is a (discrete) group, and in general, classifies principal ∞-bundles in whatever (∞,1)-topos.
Now consider the -topos of -equivariant spaces, which can also be described as the (∞,1)-presheaves on the orbit category of . For any other group there is a notion of a principal -bundle (where is the group of equivariance, and is the structure group of the bundle), and these are classified by maps into a classifying -space . So the principal -bundles over can be called . If we had something of which was a delooping, we could call the principal -bundles ””, but there does not seem to be such a thing. It seems that is not connected, in the sense that is not an effective epimorphism and thus is not the quotient of a group object in .
If we have an object of our -topos that can be delooped infinitely many times, then we can define for any integer by looking at all the spaces . These integer-graded cohomology groups are closely connected to each other, e.g. they often have cup products or Steenrod squares or Poincare duality, so it makes sense to consider them all together as a cohomology theory . We then are motivated to put together all of the objects into a spectrum object, a single object which encodes all of the cohomology groups of the theory. A general spectrum is a sequence of objects such that ; the stronger requirement that restricts us to “connective” spectra, those that can be produced by successively delooping a single object of the -topos. In Top, the most “basic” spectra are the Eilenberg-MacLane spectra produced from the input of an ordinary abelian group.
Now we can do all of this in , and the resulting notion of spectrum is called a naive -spectrum: a sequence of -spaces with . Any naive -spectrum represents a cohomology theory on -spaces. The most “basic” of these are “Eilenberg-Mac Lane -spectra” produced from coefficient systems, i.e. abelian-group-valued presheaves on the orbit category. The cohomology theory represented by such an Eilenberg-Mac Lane -spectrum is called an (integer-graded) Bredon cohomology theory.
It turns out, though, that the cohomology theories arising in this way are kind of weird. For instance, when one calculates with them, one sees torsion popping up in odd places where one wouldn’t expect it. It would also be nice to have a Poincare duality theorem for -manifolds, but that fails with these theories. The solution people have come up with is to widen the notion of “looping” and “delooping” and thereby the grading:
instead of just looking at , we look at , where is a finite-dimensional representation of and is its one-point compactification. Now if is a -space that can be delooped ” times,” we can define . If can be delooped times for all representations , then our integer-graded cohomology theory can be expanded to an -graded cohomology theory, with cohomology groups for all formal differences of representations . The corresponding notion of spectrum is a genuine -spectrum, which consists of spaces for all representations such that . A naive Eilenberg-Mac Lane -spectrum can be extended to a genuine one precisely when the coefficient system it came from can be extended to a Mackey functor, and in this case we get an -graded Bredon cohomology theory .
-graded Bredon cohomology has lots of formal advantages over the integer-graded theory. For instance, the torsion that popped up in odd places before can now be seen as arising by “shifting” of something in the cohomology of a point in an “off-integer dimension,” which was invisible to the integer-graded theory. Also there is a Poincare duality for -manifolds: if is a -manifold, then we can embed it in a representation (generally not a trivial one!) and by Thom space arguments, obtain a Poincare duality theorem involving a dimension shift of , where is generally not an integer (and, apparently, not even uniquely determined by !). Unfortunately, however, -graded Bredon cohomology is kind of hard to compute.
Matvei Libine, Lecture Notes on Equivariant Cohomology (arXiv)
B. Guillou, A short note on models for equivariant homotopy theory (pdf)
Equivariant degree-2 -Lie group cohomology is discussed in