Functors between categories of equivariant sheaves are equivariant sheaves on the product?

Question

This is a follow up question to this question which remained unanswered (satisfactorily) even after a large bounty. I have made a litlle progress and I have no a more specific question which might be easier to answer. I hope that in writing this follow up question I'm not violating any rules. If I am I apologize.

Let $G$ and $H$ be finite groups and $X$ and $Y$, $G$ and $H$-sets respectively. The main question is still the following:

Question: What kind of natural functors are there between categories of equivariant sheaves $Sh_G(X) \to Sh_H(Y)$?

Let's work with sheaves of sets and suppose for the moment $X$ and $Y$ points (Let's work with right actions throughout). In this case we're looking for functors: $G$-Sets $\to H$-Sets. The general yoga of functors = bi-modules gave me the idea to look inside the category of $G^{op} \times H$-Sets. Here's a nice step forward:

There's always a canonical functor

$$G \text{-Sets} \times (G^{op} \times H) \text{ -Sets} \to H \text{-Sets}$$

Which on objects acts by taking the "tensor product" $(A,P) \mapsto A \times_G P$ (note that we modded out by the right action of $G$ on $A$ and the left action on $P$ so now we only have a right $H$ action).

In the case of sheaves of vector spaces ($X$ and $Y$ still points) this is in fact precisely the familiar identification of functors with bimodules if we take the corresponding group algebras.

Suppose $X$ and $Y$ are arbitrary $G$ and $H$ sets now. I'd very much like to generalize the above to this situation but am still confused as to how to do so. Ideally I would like the following to hold and to have a concrete description as in the case where everything is a point:

Let $X$ and $Y$ as before. Every $G \times H$ equivariant sheaf on the product $X \times Y$ gives a pair of natural functors $Sh_G(X) \to Sh_H(Y)$ and $Sh_H(Y) \to Sh_G(X)$. And any interesting functors between these categories is in fact of this type.

Is there a way to make this true?

Answer 1

Let $G$ be a discrete group and $X$ a $G$-set. The equivariant category $X//G$ has elements of $X$ as objects, each triple $(g,x,g\cdot x)$ defines a morphism $x \to g\cdot x$, the composition corresponds to the composition of these actions. So e.g. if $X$ is a point, then $X//G$ has a single point $*$ as objects and $Hom(*,*) = G$. If $X=G $ with left action, then $X//G$ has $G$ objects and each pair $x,x' : Ob(X//G)$ is connected by a unique morphism.

Theorem: the category of $G$-equivariant sheaves over $X$ is equivalent to the category of presheaves $P(X//G) = [(X//G)^{op}, Set]$.

The category of equivariant sheaves on $X\times Y$ is thus equivalent to $$\begin{eqnarray} [(X\times Y // G\times H)^{op}, Set] & = & [(X//G)^{op} \times (Y//H)^{op}, Set] \\ & = & \left[(X//G)^{op}, P(Y//H)\right] \end{eqnarray}$$

Thus if your nice functors are given by $G\times H$-equivariant sets, then their action on $P(X//G)$ is uniquely determined by the action on subcategory $X//G \xrightarrow{Y} P(X//G)$, where $Y$ is the Yoneda embedding. Any presheaf is canonically a colimit of representables, and in fact $P(C)$ is the free cocompletion of $C$ w.r.t all colimits. Thus the interesting functors in your question are really functors preserving all colimits.

Theorem: any colimit preserving functor $Sh_G(X) \to Sh_H(Y)$ is represented by $G\times H$-equivariant sheaf on $X \times Y$.

Of course, not all functors preserve all colimits. For example, the right adjoint of any functor defined above will preserve all limits but no colimits in general. So the claim that all interesting functors are of this type is a bit thin, but they certainly are the most accessible.