I recently got an email question from Sergey Arkhipov with a question, which I couldn’t answer to my own satisfaction, so I thought I would throw it open to the peanut gallery.
One construction I’ve used a lot in my recent work is the equivariant derived category for the action of a group G on a space X (in basically whatever category you like). This is basically the poor man’s way of understanding sheaves on the quotient stack of that space by the group.
But, of course, one could forget that there was ever a space there, and just remember that you have a category of sheaves on X, which the group G acts on. So, questions:
- Is there a construction of the equivariant derived category which makes no reference to the space and just uses the category of sheaves?
- If there a generalization of this construction where the action of G can be replaced by one of an arbitrary monoidal category?
The first question is in that class of things I’m sure I could do myself if I forced myself to sit down and do it: the answer is something like replacing the category with the category of locally constant sheaves on BG valued in your category. The second, I’m less sure about.