nLab equivariant derived category

Idea

Given a locally compact group $G$ and a $G$-space $X$, one can consider the category of $G$-equivariant sheaves on $X$. It has been understood by Joseph Bernstein and Valery Lunts that the derived category of the abelian category of equivariant sheaves (or of any other abelian category) is not the most sensible choice, when the action is not free and $G$ is not discrete. Instead one needs to use $G$-resolutions, i.e. to effectively consider resolutions of the space $X$ itself while defining the derived category.

References

• J. Bernstein, V. Lunts, Equivariant sheaves and functors, Springer Lecture Notes in Math. 1578 (1994). MR95k:55012

• notes by Zhiwei Yun, pdf

• Masaki Kashiwara, Equivariant derived category and representation of real semisimple Lie groups, pdf

• Geoff Vooys, Equivariant Functors and Sheaves (arXiv:2107.01778)