We develop the theory of equivariant sheaves over profinite spaces, where the
group is also taken to be profinite. We construct a good notion of equivariant
presheaves, with a suitable sheafification functor. Using these results on
equivariant presheaves, we give explicit constructions of products of
equivariant sheaves of R-modules. We introduce an equivariant analogue of
skyscraper sheaves, which allows us to show that the category of equivariant
sheaves of R-modules over a profinite space has enough injectives.
This paper also provides the basic theory for results by the authors on
giving an algebraic model for rational G-spectra in terms of equivariant
sheaves over profinite spaces. For those results, we need a notion of
Weyl-G-sheaves over the space of closed subgroups of G. We show that
Weyl-G-sheaves of R-modules form an abelian category, with enough injectives,
that is a full subcategory of equivariant sheaves of R-modules. Moreover, we
show that the inclusion functor has a right adjoint.Comment: 36 page
