Equivariant Derived Categories via their properties. - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T19:19:32Z http://mathoverflow.net/feeds/question/3340 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/3340/equivariant-derived-categories-via-their-properties Equivariant Derived Categories via their properties. Garlef Wegart 2009-10-29T21:28:00Z 2009-10-30T06:29:33Z <p>There are some ways to define equivariant derived categories of all sorts. But all the ways i know of involve giving a concrete construction. Is the other way around possible? Is there some universal property that characterizes them? More explicitly i'm thinking of a bifibration D -> T where T is for example some nice subcategory of Top and D are the derived k-sheaves. For every group object G in T there should be a equivariant bifibration D_G -> T^G and these fibrations should again be bifibered over the category of group objects in T.</p> <p><strong>So here's the question</strong>: What properties do we actually want in such a situation?</p> <p>For example i have in mind the "induction equivalence" and "quotient equivalence" as described in the book by Bernstein and Lunts.</p> http://mathoverflow.net/questions/3340/equivariant-derived-categories-via-their-properties/3414#3414 Answer by David Treumann for Equivariant Derived Categories via their properties. David Treumann 2009-10-30T06:29:33Z 2009-10-30T06:29:33Z <p>A bifibration is presumably equivalent to the data of a functor from T to categories, with the property that the map between categories induced by a map between spaces has an adjoint. So we can understand your bifibration D -> T as a contravariant functor that takes a topological space to its category of sheaves, and a continuous map to the pullback operation. </p> <p>If by "category of sheaves" we mean the 1960's style triangulated category, this functor has bad properties--e.g. there is no simple way to recover D(X) from the categories D(U), where U runs through an open cover of X. But by now there is better technology: we can replace the triangulated category of sheaves by a suitable infinity-category. In this setting, D(X) will be an inverse limit in the infinity-categorical sense of the categories D(U) belonging to an open cover. This is a nice way of formulating what is sometimes called cohomological descent. We actually have descent for more general kinds of covers, for instance coming from free group actions. If Y is a principal G-bundle over X, then D(X) is the inverse limit of the cosimplicial diagram of infinity-categories D(Y x G x ... x G). This is the kind of thing that makes the equivariant theory of sheaves work.</p> <p>So we can try to answer your question like this. A candidate theory of equivariant sheaves is a contravariant functor D_G from T^G to infinity-categories. It must agree with the usual theory of sheaves on T, which we identify with the full subcategory of T^G consisting of free G-spaces. And it must satisfy a strong enough kind of descent. After pinning down that last condition, I am pretty sure this will uniquely determine D_G.</p> <p>You could translate all this back into the language of bifibrations (well, infinity bifibrations), but that point of view isn't so easy for me.</p> <p>(Another way to put this is that D is itself a sheaf of categories on T in a suitable Grothendieck topology, and that this extends in a unique way to sheaf of categories on T^G. But I'm worried that there is some technical problem with this statement.)</p>