group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
geometric representation theory
representation, 2-representation, ∞-representation
Grothendieck group, lambda-ring, symmetric function, formal group
principal bundle, torsor, vector bundle, Atiyah Lie algebroid
Eilenberg-Moore category, algebra over an operad, actegory, crossed module
Be?linson-Bernstein localization?
For $X$ a space and $G$ a group acting on it, then the Borel equivariant cohomology of $X$ is the cohomology of the homotopy quotient $X//G$.
Since a standard way to model the homotopy quotient is the Borel construction, this is called Borel equivariant cohomology.
This is the special case of genuine equivariant cohomology where the action on the coefficients is trivial.
In the more general context of (global) equivariant stable homotopy theory, Borel-equivariant spectra are those which are right induced from plain spectra, hence which are in the essential image of the right adjoint to the forgetful functor from equivariant spectra to plain spectra.
cohomology in the presence of ∞-group $G$ ∞-action:
Borel equivariant cohomology | $\phantom{AAA}\leftarrow\phantom{AAA}$ | general (Bredon) equivariant cohomology | $\phantom{AAA}\rightarrow\phantom{AAA}$ | non-equivariant cohomology with homotopy fixed point coefficients |
---|---|---|---|---|
$\phantom{AA}\mathbf{H}(X_G, A)\phantom{AA}$ | trivial action on coefficients $A$ | $\phantom{AA}[X,A]^G\phantom{AA}$ | trivial action on domain space $X$ | $\phantom{AA}\mathbf{H}(X, A^G)\phantom{AA}$ |
Loring Tu, What is… Equivariant Cohomology?, Notices of the AMS, Volume 85, Number 3, March 2011 (pdf, pdf)
Stefan Schwede, Example 4.5.19 of Global homotopy theory 2018 (arXiv:1802.09382)
Last revised on June 27, 2019 at 09:16:30. See the history of this page for a list of all contributions to it.