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Stream: theory: category theory

Topic: slices of G-Set

Mike Shulman (Feb 12 2022 at 06:07):

If $H$ is a subgroup of $G$, with corresponding orbit $X := G/H \in G\mathrm{Set}$, then I believe there is an equivalence of categories $H\mathrm{Set} \simeq G\mathrm{Set}/X$. Put differently, the geometric morphism $H \mathrm{Set} \to G\mathrm{Set}$ induced by the inclusion is etale. If this is true, does anyone know a reference for it?

Morgan Rogers (he/him) (Feb 12 2022 at 07:46):

All geometric morphisms between toposes of the form G-Set are atomic and induced by group homs, and injective homomorphisms induce etale surjections. The former fact is an exercise in Mac Lane & Moerdijk which I can dig up if you need, the latter thing is in the first chapter of my thesis but a better reference will eventually be my joint work with @Jens Hemelaer on factorizing geometric morphisms between toposes of M-sets, assuming I devote some time to finishing it within the next few months.

Morgan Rogers (he/him) (Feb 12 2022 at 07:52):

(A quick way to see that G/H is the correct thing to slice over to recover the gm induced by the inclusion $H \hookrightarrow G$ is to observe that another expression for this quotient is $G \otimes_H 1$, which is the image of 1 under the left adjoint to the restriction functor (the latter can be expressed as $\mathrm{Hom}(G,-)$).

Mike Shulman (Feb 12 2022 at 08:02):

Thanks! I went back and looked at your thesis, but wasn't able to find the statement about etale-ness, can you point to it specifically?

Morgan Rogers (he/him) (Feb 12 2022 at 11:26):

Ah sorry that's because the part about all morphisms being atomic stops being true for monoids which aren't groups. You need to combine the fact that the hyperconnected-localic factorization of a gm coming from a semigroup homomorphism corresponds to the surjective--injective factorization of that morphism, with the fact that atomic+localic=etale

Morgan Rogers (he/him) (Feb 12 2022 at 11:26):

The former of the two is what you'll find in my thesis :sweat_smile:

John Baez (Feb 12 2022 at 17:09):

Mike Shulman said:

If $H$ is a subgroup of $G$, with corresponding orbit $X := G/H \in G\mathrm{Set}$, then I believe there is an equivalence of categories $H\mathrm{Set} \simeq G\mathrm{Set}/X$.

If it's true there should be a pretty simple explicit 'formula' for the equivalence, right?

I'm reminded of 'Mackey theory', which relates representations of the subgroup $H \subseteq G$ and equivariant vector bundles over $X := G/H$, meaning vector bundles $p: E \to X$ with a $G$-action lifting the $G$-action on $X$. This is often studied in a context where everything is a manifold or topological space, which is making it hard for me to find references to how it works in the simpler case of sets. Also, I'm having trouble finding any reference to an equivalence, because people are often in a rush to exploit what's going on, and don't slow down to examine the equivalence that I believe is lurking here.

John Baez (Feb 12 2022 at 17:16):

Anyway:

Given an $H$-set $S$ you can form the $G$-set

$\displaystyle{ F(S) = \frac{G \times S}{(gh, s) \sim (g,hs)} }$

This functor $F$ is the left adjoint to restricting $G$-actions to $H$-actions. But $F(S)$ is a $G$-set over $X := G/H$ via $[g,s] \mapsto [g]$, where $[g]$ is the right coset of $g$.

John Baez (Feb 12 2022 at 17:19):

Conversely, given a $G$-set $Y$ over $G/H$ we can look at the fiber over the coset $[1]$. Since $H$ fixes $[1]$ it acts on this fiber, so this fiber becomes an $H$-set.

John Baez (Feb 12 2022 at 17:20):

I think this is the equivalence you want - but yeah, I'm not proving it's an equivalence.

Anyway, this is the essential idea behind 'Mackey theory', translated from the usual Vect context to Set. People call $F(S)$, as $G$-set over $X$, an 'induced bundle'. This stuff is widely used in group representation theory.

Jens Hemelaer (Feb 13 2022 at 13:55):

It is indeed the right equivalence. If $X$ is an object in a topos $\mathbf{Sh}(\mathcal{C},J)$, then you can explicitly write down a site for the corresponding slice topos $\mathbf{Sh}(\mathcal{C},J)/X$:

• as category you take the slice category $\mathcal{C}/X$, i.e. the category of elements of $X$;
• as covering sieves you take the sieves that are covering sieves after projecting to $\mathcal{C}$.

For the topos $\mathbf{PSh}(G)$ and $X=G/H$, the category of elements of $X$ is a connected groupoid such that each object has automorphism group isomorphic to $H$, so it is equivalent as a category to $H$. The Grothendieck topology is the presheaf topology. This shows that $\mathbf{PSh}(G)/X \simeq \mathbf{PSh}(H)$. Explicitly writing down the equivalence takes some more work, but eventually you get the equivalence that @John Baez wrote down above.

Jens Hemelaer (Feb 13 2022 at 14:14):

More generally, the étale morphisms to a presheaf topos $\mathbf{PSh}(\mathcal{C})$ are precisely the geometric morphisms $\mathbf{PSh}(\mathcal{D}) \to \mathbf{PSh}(\mathcal{C})$, induced by a discrete fibration $\mathcal{D} \to \mathcal{C}$. This is a bit similar to the correspondence between sheaves and étalé spaces. Maybe it is easier to find a reference for this more general statement? I have a feeling that it is a special case of a result in Caramello–Zanfa, but I haven't found a precise reference.

Zhen Lin Low (Feb 13 2022 at 14:32):

It must be in the Elephant somewhere, and was probably known to Grothendieck... (only half joking)

Jens Hemelaer (Feb 13 2022 at 15:13):

It was known to Grothendieck! It is in SGA4, Exposé IV, Section 5.8 :grinning_face_with_smiling_eyes: Thanks for the suggestion, I don't have the reflex to check SGA4 yet because the pdf wasn't always searchable.

Reid Barton (Feb 13 2022 at 15:57):

In fact this is essentially "the" example of an "etale map", with $k \subseteq K$ a finite separable field extension, $G = \mathrm{Gal}(k)$ and $H = \mathrm{Gal}(K)$. Then $\mathrm{Spec}\, K \to \mathrm{Spec}\, k$ is etale, the etale topos of $k$ is equivalent to $G$-sets with $\mathrm{Spec}\, K \to \mathrm{Spec}\, k$ corresponding to $G/H$, and the etale topos of $K$ is $H$-sets. (Though here $G$ is really a profinite group and the sets are supposed to have continuous $G$-action. Hopefully an actual algebraic geometer can correct anything I got wrong.)

Mike Shulman (Feb 13 2022 at 19:43):

Thank you everyone, this is great!