nLab
equivariant sheaf

Let GG be a group in some category CC of spaces. We assume that for every space XX in CC some category of sheaves F XF_X is given, in a way making a fibered category FF over CC. For example, we can consider sheaves of abelian groups over topological spaces.

Consider a GG-space XX with action ρ:G×XX\rho: G\times X\to X and projection p:G×XXp: G\times X\to X. These data give rise to an action groupoid in the category of spaces, denoted by [X/G][X/G] or sometimes X//GX//G. This groupoid can also be identified with its nerve, which is a simplicial object in spaces.

An equivariant sheaf over a GG-space XX is a GG-equivariant object in F XF_X. In other words, it is a sheaf xx over XX together with an isomorphism θ:p *xρ *x\theta : p^* x \to \rho^* x of sheaves over G×XG\times X satisfying the usual cocycle condition on G×G×XG\times G\times X. All equivariant sheaves form the equivariant fiber F X GF_X^G of FF over XX. The equivariant fiber over XX is thought of as a fiber over the simplicial object [X/G][X/G].

If the fibered category FF is a stack for some subcanonical Grothendieck topology on CC and XX a GG-torsor over some true base space X/GX/G in CC then there is a descent along torsor: the equivariant fiber F X GF_X^G is canonically equivalent to the usual fiber over X/GX/G. In other words the fibers over [X/G][X/G] and X/GX/G are equivalent.

If FF is not a stack one can instead of a Grothendieck topology in CC use the effective descent topology and include into the condition for a torsor instead of local triviality in a Grothendieck topology that PP over X/GX/G is an effective descent epimorphism relative to the fibered category FF over CC.

Notice that being an equivariant sheaf is an additional structure on a sheaf, rather than a property. Mumford introduced this notion in the geometric invariant theory under the name GG-linearization of a sheaf. GG-equivariant sheaves generalize (sheaves of sections of) GG-equivariant bundles, as studied for example in representation theory earlier by Borel, Weil and Bott.

Note also that if “space” means topological space and GG is a topological group acting on the point 11, then equivariant sheaves are just the same as continuous GG-sets. In particular, if GG is a discrete group, then equivariant sheaves are just the same as ordinary GG-sets.

Observe that the definition of equivariant sheaf only depends on the action groupoid, and thus can be generalized to equivariant sheaves on any internal groupoid in the category of spaces. If “space” means locale, then every Grothendieck topos can be presented as the category of equivariant sheaves on some localic groupoid. This is a theorem of André Joyal and Miles Tierney, and can be found in chapter C5 of the Elephant.

The appropriate notion of the “equivariant derived category” is in general not equal to the derived category of the abelian category of equivariant sheaves; one needs to resolve appropriately.

category: sheaf theory

Revised on March 6, 2013 19:44:56 by Zoran Škoda (161.53.130.104)