If we're to think of a (grothendieck) topos as a generalized space, then it makes sense to ask for a group action on that space (and more to the point: a quotient of our space by that group action).
Concretely, we should have a geometric morphism $\mathcal{E} \to \mathcal{E}$ for each element of our group, compatible with the group operation. Said more quickly, it should be a functor $BG \to \mathfrak{Top}$ from $G$ (viewed as a category) into the (2-)category of topoi.
Now, in classical geometric settings we have good machinery for taking quotients by (free) group actions. In the case of a nonfree action, our quotient spaces are orbifolds, which are well modeled by groupoids.
My question is whether there's a good reference for groups acting on a topos. In particular, whether there's a notion of the quotient topos. We know that, given a (localic) groupoid $H$, we can build a topos of (continuous) $H$-sets, and I would be curious to know if the quotient of a topos $Sh(X)$ by a group action $G$ can be identified with (continuous) $X//G$-sets, where $X//G$ is the action groupoid of $G$ on $X$, which is frequently used as the stand-in for the orbifold $X/G$.
This seems believable, but I don't have the time to work on it myself, and it seems like the kind of thing that someone may have thought about some decades ago.
Thanks in advance!
Best Answer
I don't have a reference, but: given a group $G$ acting on a category $C$ (or more generally an object in a $2$-category) we can construct another category $C^G$ (or $C^{hG}$ maybe) called its category of homotopy fixed points, which is a $2$-limit; see e.g. this blog post for a discussion. If $G$ acts on a topological space $X$ then the category $\text{Sh}(X)^G$ can be identified with sheaves on the action groupoid $X/G$. The reason we're taking fixed points and not a quotient here is because $\text{Sh}(X)$ is being thought of as "functions" so the relationship to spaces is contravariant; I think this is equivalent to considering topoi with algebraic morphisms, or "logoi."
Another interesting example which motivates the blog post above is that if $L/K$ is a finite Galois extension with Galois group $G$ and $C$ is a suitable category of objects over $L$ (e.g. $L$-vector spaces, $L$-algebras) then $C^G$ recovers the category of objects over $K$; this is an abstract way of describing Galois descent.
Also, it's worth keeping in mind that there are extra coherence conditions for describing a group action on a category; the condition $g(hx) = (gh)x$ becomes a family of isomorphisms which itself satisfies a condition generalizing a $2$-cocycle condition. This is necessary if you want the notion of a group action on a category to be invariant under equivalence of categories.