### Topoi of G-sets

#### Posted by John Baez

I’m thinking about finite groups these days, from a Klein geometry perspective where we think of a group $G$ as a source of $G$-sets. Since the category of $G$-sets is a topos, this lets us translate concepts, facts and questions about groups into concepts, facts and questions about topoi. I’m not at all good at this, so here are a bunch of basic questions.

For any group $G$ the category of $G$-sets is a Boolean topos, which means basically that its internal logic obeys the principle of excluded middle.

Which Boolean topoi are equivalent to the category of $G$-sets for some group $G$?

Which are equivalent to the category of $G$-sets for a

*finite*group $G$?

It might be easiest to start by characterizing the categories of $G$-sets where $G$ is a *groupoid*, and then add an extra condition to force $G$ to be a group.

The category $G Set$ comes with a forgetful functor $U: G Set \to Set$.

- Is the group of natural automorphisms of $U$ just $G$?

This should be easy to check, I’m just feeling lazy. If some result like this is true, how come people talk so much about the Tannaka–Krein reconstruction theorem and not so much about this simpler thing? (Maybe it’s just too obvious.)

Whenever we have a homomorphism $f \colon H \to G$ we get an obvious functor

$f^\ast \colon G Set \to H Set$

This is part of an essential geometric morphism, which means that it has both a right and left adjoint. By this means we can actually get a 2-functor from the 2-category of groups (yeah, it’s a 2-category since groups can be seen as one-object categories) to the 2-category $Topos_{ess}$ consisting of topoi, essential geometric morphisms and natural transformations. If I’m reading the $n$Lab correctly, this makes $G Set$ into a full sub-2-category of $Topos_{ess}$. This makes it all the more interesting to know which topoi are equivalent to categories of $G$-sets.

- What properties characterize essential geometric morphisms of the form $i^\ast \colon G \Set \to H \Set$ when $i \colon H \to G$ is the inclusion of a subgroup?

Whenever we have this, we get a transitive $G$-set $G/H$, which is thus a special object in $G Set$. These objects are just the **atoms** in $G Set$: that is, the objects whose only subobjects are themselves and the initial object. Indeed $G Set$ is an **atomic topos**, meaning that every object is a coproduct of atoms. That’s just a fancy way of saying that every $G$-set can be broken into orbits, which are transitive $G$-sets.

Next:

- What properties characterize essential geometric morphisms of the form $i^\ast \colon G \Set \to H \Set$ when $i \colon H \to G$ is the inclusion of a
*normal*subgroup?

In this case $G/H$ is a group with a surjection $p \colon G \to G/H$, so we get another topos $(G/H)Set$ and essential geometric morphisms

$Set \longrightarrow (G/H)Set \stackrel{p^\ast}{\longrightarrow} G Set \stackrel{i^\ast}{\longrightarrow} H Set \longrightarrow Set$

What properties characterize essential geometric morphisms of the form $p^*$ for $p$ a surjective homomorphism of groups?

Is there a concept of ‘short exact sequence’ of essential geometric morphisms such that the above sequence is an example?

Well, my questions could go on all day, but this is enough for now!

## Re: Topoi of G-sets

If you let me talk about $\infty$-topoi, then I know a characterization. I’ll write $\mathcal{S}$ for the $\infty$-topos of $\infty$-groupoids; then any $\infty$-groupoid $X$ determines a slice category $\mathcal{S}_{/X}$, which comes with a unique geometric morphisms $\pi\colon \mathcal{S}_{/X}\to \mathcal{S}$.

Of course then $\mathcal{S}_{X}\approx \mathrm{Fun}(X,\mathcal{S})$ (thinking of $X$ as an $\infty$-groupoid; e.g., we could take $X=B G$ for some group $G$).

Any geometric morphism which is equivalent to the projection of a slice $\infty$-topos to its base is called

etale, and there’s an intrinsic characterization of these [HTT 6.3.5.11]: an $f\colon \mathcal{X}\to \mathcal{Y}$ is etale if and only if(1) the functor $f^\ast\colon \mathcal{Y}\to \mathcal{X}$ admits a left adjoint $f_!$, which

(2) is conservative, and

(3) has a “push-pull formula”, i.e., the “obvious” map $f_!(f^\ast\times_{f^\ast Y} Z) \to X\times_Y f_!Z$ is an equivalence.

Applied to the unique geometric morphism $\pi\colon \mathcal{X}\to \mathcal{S}$, this gives a criterion for $\mathcal{X}$ to be equivalent to $\Fun(X, \mathcal{S})$. Presumably there is a “classical” 1-topos analogue of this theory, but I don’t know what it is.