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October 14, 2018

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 GG as a source of GG-sets. Since the category of GG-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 GG the category of GG-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 GG-sets for some group GG?

  • Which are equivalent to the category of GG-sets for a finite group GG?

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

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

  • Is the group of natural automorphisms of UU just GG?

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:HGf \colon H \to G we get an obvious functor

f *:GSetHSet 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 essTopos_{ess} consisting of topoi, essential geometric morphisms and natural transformations. If I’m reading the nnLab correctly, this makes GSetG Set into a full sub-2-category of Topos essTopos_{ess}. This makes it all the more interesting to know which topoi are equivalent to categories of GG-sets.

  • What properties characterize essential geometric morphisms of the form i *:GSetHSeti^\ast \colon G \Set \to H \Set when i:HGi \colon H \to G is the inclusion of a subgroup?

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

Next:

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

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

Set(G/H)Setp *GSeti *HSetSet 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 *p^* for pp 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!

Posted at October 14, 2018 5:36 PM UTC

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Re: Topoi of G-sets

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

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

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:𝒳𝒴f\colon \mathcal{X}\to \mathcal{Y} is etale if and only if

(1) the functor f *:𝒴𝒳f^\ast\colon \mathcal{Y}\to \mathcal{X} admits a left adjoint f !f_!, which

(2) is conservative, and

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

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

Posted by: Charles Rezk on October 14, 2018 6:53 PM | Permalink | Reply to this

Re: Topoi of G-sets

Thanks! Believe it or not, I have vague dreams of extending certain tiny pieces of group theory to \infty-groups: not so much the general theory as the Klein geometry perspective on finite groups as ‘symmetry groups of incidence geometries’, with a special focus on certain interesting examples. Your immediate leap from topoi to \infty-topoi is a bit intimidating to me here, since my initial thoughts lie on a humbler plane. But these ideas may be really useful later on!

Posted by: John Baez on October 14, 2018 7:12 PM | Permalink | Reply to this

Re: Topoi of G-sets

The leap to \infty-topoi is immediate only because that is the thing I actually know about. I would guess that everything I said here works for 1-topoi. (Typically, anything you can say about \infty-topoi is either (i) something that has an exact analogue in 1-topoi, or (ii) doesn’t have an analogue (or has a poor, inadequate analogue), in which case you should just stick with \infty-topoi cause they are better.)

Posted by: Charles Rezk on October 14, 2018 8:16 PM | Permalink | Reply to this

Re: Topoi of G-sets

As David pointed out, this is related to my musings about \infty-atomic geometric morphisms.

A morphism between 1-topoi satisfying (1) and (3) is called locally connected, or locally 0-connected to be more precise, since the conditions (1) and (3) when stated for 1-topoi refer only to 0-truncated objects (since all objects of a 1-topos are 0-connected).

One way to decategorify (2) would be to simply assert that the left adjoint f !f_! is conservative. I suspect that this would indeed ensure that the domain is the category of internal diagrams on some internal groupoid in the codomain. (In the 1-categorical case, of course, this is not equivalent to being etale.)

Another way to decategorify (2) would be to first relativize it to slice categories, and then weaken the assumption to conservativity on subobjects. This isn’t as weird as it may sound, since everything reasonable in topos theory should localize to slice categories, and slice categories in \infty-topos theory often decategorify to subobject lattices in 1-topos theory since the object classifier decategorifies to a subobject classifier. With this second decategorification of (2), I suspect that we get a condition equivalent to atomicity of the geometric morphism. Thus we would get from another perspective that while “\infty-atomic” and “\infty-etale” coincide, they bifurcate when we decategorify.

Posted by: Mike Shulman on October 15, 2018 7:00 PM | Permalink | Reply to this

Re: Topoi of G-sets

I have vague dreams of extending certain tiny pieces of group theory to ∞-groups: not so much the general theory as the Klein geometry perspective on finite groups as ‘symmetry groups of incidence geometries’, with a special focus on certain interesting examples.

Back to Klein 2-geometry and higher!

Now we have some nice finite 2-groups lying about, see Platonic 2-group, it should be easy to think of the corresponding geometric figure (or groupoid) they act on.

For instance, in the A-series, there is the cyclic group of order 3, extended to a 2-group by the third roots of unity. It seems to act on a triangle with a kind of gauge at each vertex.

I was fishing about for something along these lines back here.

Posted by: David Corfield on October 18, 2018 12:25 PM | Permalink | Reply to this

Re: Topoi of G-sets

Thanks for pointing out that paper on Platonic 2-groups; this is exactly the sort of thing I’d like to think about more. I should post some of my ideas.

Posted by: John Baez on October 18, 2018 9:09 PM | Permalink | Reply to this

Re: Topoi of G-sets

Jacob Lurie explained this to me at MO.

Also Mike Shulman was telling us about things in this area in ∞-Atomic Geometric Morphisms. See the comparison with the 1-case later on.

Posted by: David Corfield on October 14, 2018 9:04 PM | Permalink | Reply to this

Re: Topoi of G-sets

  • Is the group of natural automorphisms of UU just GG?

Well UU is represented by the GG-set freely generated by one element (a.k.a. the GG torsor) so by Yoneda we just need to look at the morphisms from this GG-set to itself, which are indeed GG.

Personally, I love the topos-as-a-space perspective. Does anybody know what the points of GSetG\mathrm{Set} are?

Posted by: Oscar Cunningham on October 14, 2018 7:43 PM | Permalink | Reply to this

Re: Topoi of G-sets

GSetG\mathrm{Set} has only one point. However, this point has a symmetry group, which is GG.

(Because GSetG\mathrm{Set} is the “classifying topos for GG-torsors”.)

Posted by: Charles Rezk on October 14, 2018 8:08 PM | Permalink | Reply to this

Re: Topoi of G-sets

Relatedly, if TT is any Lawvere theory (say) and Set TSet^T is the category of algebras, with underlying-set functor U:Set TSetU: Set^T \to Set, then the set of natural transformations

U nUU^n \to U

carries a TT-algebra structure that is naturally isomorphic to the free algebra F(n)F(n) on nn generators. You can generalize this to monads and their algebras, and nn doesn’t have to be finite.

The idea is that U nhom(n,U)hom(F(n),):Set TSetU^n \cong \hom(n, U-) \cong \hom(F(n), -): Set^T \to Set, and by Yoneda the set of natural transformations

hom(F(n),)U\hom(F(n), -) \to U

is naturally isomorphic to the underlying set UF(n)U F(n).

Posted by: Todd Trimble on October 14, 2018 8:49 PM | Permalink | Reply to this

Re: Topoi of G-sets

The result is recorded at nLab here.

Posted by: David Corfield on October 14, 2018 9:09 PM | Permalink | Reply to this

Re: Topoi of G-sets

David wrote:

The result is recorded at nLab here.

Nice! For those too busy to click every link that’s dangled before them, this a quick proof of the ‘Tannaka–Krein reconstruction theorem for GG-sets’ that I mentioned. I’m glad it’s on the nLab because it’s a nice warmup for the usual Tannaka–Krein theorem — and it’s presented in a way that makes it rather clear how it’s an extension of the famous Cayley theorem. The key idea behind both results is the Yoneda lemma.

I’m vaguely dreaming of a book that introduces group theory using a lot more more category theory than usual… sort of using group theory to motivate category theory and vice versa. This result deserves to be in such a book.

(I’m planning to start writing a lot more expository stuff on pure math in a few years. To keep from getting overburdened, the trick will be to sublimate my urge to write textbooks and instead write papers, which are a lot easier to finish.)

Posted by: John Baez on October 15, 2018 1:09 AM | Permalink | Reply to this

Re: Topoi of G-sets

Part of the lore here is SGA 1, where there is a lot of talk about connected atomic toposes. A connected atomic topos with a point (a geometric morphism from SetSet) is a classifying topos of a localic group GG (consisting of continuous discrete actions of GG). It seems you are interested in the case where GG is discrete.

It’s nice to know that the points of Set GSet^G (i.e., the set-theoretic models of the theory for which this is a classifying topos) are precisely GG-torsors.

Posted by: Todd Trimble on October 14, 2018 9:06 PM | Permalink | Reply to this

Re: Topoi of G-sets

Todd wrote:

Part of the lore here is SGA 1, where there is a lot of talk about connected atomic toposes.

Great! I just updated my post a minute ago, before reading this, to mention atoms.

A connected atomic topos with a point (a geometric morphism from Set) is a classifying topos of a localic group GG (consisting of continuous discrete actions of GG).

Excellent! When are such topoi Boolean? Does that happen if and only if GG is discrete?

By the way, the idea of ‘continuous discrete actions’ makes me nervous when GG is not discrete.

It seems you are interested in the case where GG is discrete.

I’m mainly interested in the case where GG is finite, but I’m just roaming around trying to understand the overall landscape, so I’m happy to consider more general cases.

Posted by: John Baez on October 14, 2018 10:03 PM | Permalink | Reply to this

Re: Topoi of G-sets

All atomic toposes EE are Boolean.

The notion of atomic topos is actually pretty interesting: it means the inverse image of the global sections functor, Γ *:SetE\Gamma^\ast: Set \to E, is logical, i.e., a logical morphism of toposes!! The consequence here is that it preserves the subobject classifier, which in SetSet is 2=1+12 = 1 + 1, so the subobject classifier in EE has to be 1+11 + 1, and this means EE is Boolean.

Such a confluence between the geometric and logical aspects of topos theory has some interesting effects. :-)

Posted by: Todd Trimble on October 14, 2018 11:42 PM | Permalink | Reply to this

Re: Topoi of G-sets

I can’t figure out from the links what an atomic topos is supposed to be. I am guessing it is related, for instance, to the notions of “orbital” and “atomic orbital” infinity category of (I think) Barwick and Glasman.

Posted by: Charles Rezk on October 14, 2018 11:25 PM | Permalink | Reply to this

Re: Topoi of G-sets

The nLab gives several equivalent characterizations of atomic topoi. The one that most appeals to me is the one I added to my blog article after first writing it. An object is an atom if its only subobjects are itself and the initial object; a Grothendieck topos is atomic if every object is a coproduct of atoms.

This appeals to me because it captures the idea that each object can be broken into pieces that can’t themselves be further broken down. For example the atomic GG-sets are just the transitive GG-sets, and every GG-set is a coproduct of atoms, namely its orbits.

(Note that we can still have an epimorphism from one atom to another atom that’s not an isomorphism, so while an atom doesn’t have any nontrivial parts, you may still be able to squash it down to a smaller atom.)

It could be that some of the other characterizations of atomic topoi are more illuminating for experts; in fact I feel that must be so. One of the most appealing to me is that a topos is atomic iff the subobject lattice of every object is a complete atomic Boolean algebra. And had I read this earlier, I wouldn’t have asked which atomic topoi are Boolean! Obviously they all are, given this.

Posted by: John Baez on October 15, 2018 12:32 AM | Permalink | Reply to this

Re: Topoi of G-sets

In case people want to follow up on what Charles is indicating, we have a page – nLab: Parametrized Higher Category Theory and Higher Algebra – which details the program.

Posted by: David Corfield on October 15, 2018 7:17 AM | Permalink | Reply to this

Re: Topoi of G-sets

Right – to be clear, the terminologies “atomic geometric morphism” and “atomic orbital category” are unrelated. On the other hand “paramterized category theory” is just a renaming of “indexed category theory”, which is another name for “fibered category theory”. In general, I tend not to expect the terminology used by Barwick et al to be related to “classical” terminology.

Posted by: Tim Campion on October 16, 2018 6:09 AM | Permalink | Reply to this

Re: Topoi of G-sets

Yes, we had a chat about that naming decision here, Mike in #38 arriving at a compromise suggestion to use ‘parametrized’ for indexing over EI-categories.

Posted by: David Corfield on October 16, 2018 7:58 AM | Permalink | Reply to this

Re: Topoi of G-sets

So we have atomic geometric morphism, one whose inverse image is a logical functor.

Then an atomic topos is one for which the global sections functor to SetSet is atomic.

Then Mike was looking for an \infty-version of the former, and claims to find it with ‘local homeomorphism’ or étale geometric morphism.

Presumably he could have taken the corresponding step to speak of an \infty-atomic (,1)(\infty,1)-topos as one with \infty-atomic global sections to \inftyGrpd.

What would this have to do, if anything, with Barwick et al.’s ‘atomic \infty-categories’, where every retraction is an equivalence?

Posted by: David Corfield on October 15, 2018 10:17 AM | Permalink | Reply to this

Re: Topoi of G-sets

I’ve been wondering: in the context of \infty-topoi, is it really appropriate to think of an etale geometric morphism as a kind of “local homeomorphism”? “Etale geomtric morphism” is a fancy word for “slice category of the base”. So in 1-topos theory, this is like a space over the base which has discrete fibers, which ends up being a local homeomorphism. But in the \infty-topos case, we’re really looking at a stack over the base topos – the fibers are arbitrary homotopy types.

Is there, for example, some theorem saying that if 𝒴𝒳\mathcal{Y} \to \mathcal{X} is an etale geometric morphism, then there is a surjective family 𝒵 i𝒳\mathcal{Z}_i\to \mathcal{X} of open embeddings such that 𝒴× 𝒳𝒵 i𝒵 i\mathcal{Y} \times_\mathcal{X} \mathcal{Z}_i \to \mathcal{Z}_i is an equivalence?

Posted by: Tim Campion on October 16, 2018 5:57 AM | Permalink | Reply to this

Re: Topoi of G-sets

Arbitrary homotopy types, but discrete homotopy types, in that they have no topology, only homotopy. I wouldn’t expect a categorification of “local homeomorphism” to necessarily involve open embeddings, those being a rather 1-categorical phenomenon.

Posted by: Mike Shulman on October 16, 2018 5:11 PM | Permalink | Reply to this

Re: Topoi of G-sets

Funny – I learned from you that openness of a geometric morphism sits at the bottom of the latter of locally nn-connected geometric morphisms, making it rather 1-categorical in nature. But embeddings make perfect sense to me \infty-categorically. So I think I’d agree that there’s something funny about asking for open embeddings as I did, but because of the “open” part, not because of the “embedding” part. I take it from the placement of your italics that your thinking is different?

Posted by: Tim Campion on October 16, 2018 9:05 PM | Permalink | Reply to this

Re: Topoi of G-sets

It’s true, openness could also need categorification. But there’s no conflict between embeddings making sense \infty-categorically and their being a “1-categorical phenomenon” that sometimes have to be generalized when working higher-categorically. For instance, the notion of equivalence relation, meaning a monomorphism into a cartesian square that is reflexive, symmetric, and transitive, makes perfect sense \infty-categorically, but nevertheless for many purposes (e.g. exactness properties) it is not as interesting a notion as that of an internal groupoid object.

Posted by: Mike Shulman on October 16, 2018 9:55 PM | Permalink | Reply to this

Re: Topoi of G-sets

I have a deep intuitive understanding for the concept of an ‘atom’, an object with only nothing and itself as subobjects. The word ‘atom’ (which means ‘uncuttable’) really says it all.

I don’t have any such understanding for the concept of an ‘atomic geometric morphism’, a geometric morphism whose inverse image part is a logical functor.

Can anyone explain this latter concept in a hand-wavy poetic way that will help me grok it?

For example, why should all the objects of a topos be atoms iff its global sections functor is atomic? I’m not looking for a proof, but the intuition.

Posted by: John Baez on October 15, 2018 6:15 PM | Permalink | Reply to this

Re: Topoi of G-sets

FWIW, here’s Johnstone from the introduction to C3.5 of Sketches of an Elephant:

Traditionally, a geometric morphism with logical inverse image is said to be atomic. This is perhaps not a particularly good name… it derives from a characteristic property of atomic SetSet-toposes… as we shall see in 3.5.9 [sic] below, and is not really appropriate for morphisms between non-Boolean toposes.

I think the reference to 3.5.9 should be to 3.5.7, where he writes

…for any bounded atomic morphism p:ESp:E\to S we can choose a site of definition for EE consisting of objects AA which are connected, i.e. satisfy p !(A)1p_!(A)\cong 1. In the case when pp is atomic, connected objects of EE are generally called atoms… an atom AA has the property that… the map Sub S(1)Sub E(A)Sub_S(1) \to Sub_E(A)… is biective. (The name ‘atom’ is really appropriate only in the classical case S=SetS=Set, when this condition says that AA has no subobjects other than itself and 0 (and these two are distinct)…)

However, I’m not sure I agree with his position here. The condition Sub Set(1)Sub E(A)Sub_Set(1) \cong Sub_E(A) seems to me like a reasonable constructive version of “AA has precisely two subobjects, itself and 0”: internally, it says that the subobjects of AA are precisely those of the form {xAP}\{x\in A \mid P \} for all truth values PP. And the paper of Barr and Diaconescu shows that atomicity of an arbitrary geometric morphism p:ESp:E\to S is equivalent to saying that every object p *(Ω A)p_*(\Omega^A) — the “subobject lattice of AEA\in E to the eyes of SS” — is isomorphic to some power-object Ω B\Omega^{B} in SS, where in fact B=p !(A)B = p_!(A). They don’t seem to state this explicitly, but surely such an isomorphism decomposes AA as an SS-indexed coproduct of “atoms” in the constructive sense, with B=p !(A)B = p_!(A) as indexing object, and the existence of such decompositions should conversely imply atomicity.

So, with a suitable modification to make it constructive, I think the characterization that appeals to you is perfectly reasonable… at least for 1-toposes!

Posted by: Mike Shulman on October 15, 2018 6:36 PM | Permalink | Reply to this

Re: Topoi of G-sets

Heh, my post above was actually written before I saw your previous post above it, but it’s not too bad if read as an answer!

But more explicitly, suppose f:ESf:E\to S is atomic, so that f *:SEf^\ast:S\to E is logical. Then in particular, f *f^\ast preserves the subobject classifier, which means by adjunction that its left adjoint f !f_! preserves subobject lattices. In other words, the lattice of subobjects of AEA\in E coincides with the lattice of subobjects of f !ASf_! A \in S. Thus, from the perspective of SS, a subobject of AA is uniquely determined by which of the “atomic components” (elements of f !Af_! A) that it contains: each of these is “uncuttable” and AA is just a collection of these.

Posted by: Mike Shulman on October 15, 2018 9:04 PM | Permalink | Reply to this

Re: Topoi of G-sets

Great! I thought you were answering my question — and you sort of were, but not quite as clearly as if you’d known that’s what you were doing.

I now understand what ‘atomic’ about having a logical left adjoint; the concept of an ‘atom’ is getting relativized using intuitionistic logic, so instead of an atom being something that’s ‘either all there or not there at all, and not both’, we are allowing truth values from our topos SS.

Posted by: John Baez on October 15, 2018 9:20 PM | Permalink | Reply to this

Re: Topoi of G-sets

I’ve been wondering: what is the relationship between atomic toposes and what I’d call “complete toposes”. I’d say that a topos is “complete” if it classifies a complete geometric theory, i.e. a theory such that any extension of it is either conservative or inconsistent. So a “complete topos” would be a topos 𝒳\mathcal{X} such that any geometric morphism 𝒴𝒳\mathcal{Y} \to \mathcal{X} is surjective unless 𝒴\mathcal{Y} is the terminal category.

It seems to me that atomic toposes and complete toposes share the properties of being boolean and two-valued, but I’m not sure how much further the similarities go.

Posted by: Tim Campion on October 16, 2018 5:49 AM | Permalink | Reply to this

Re: Topoi of G-sets

It would be nice to find a different name than “complete”, since “complete topos” can also mean a topos that is complete as a category (as all Grothendieck toposes are, but not all complete elementary toposes are Grothendieck).

Also, is there a more constructive way to phrase the definition?

Posted by: Mike Shulman on October 16, 2018 5:13 PM | Permalink | Reply to this

Re: Topoi of G-sets

Well, that’s sort of the thing. I think when you phrase the definition of a “complete topos” more constructively, you at least very nearly get the notion of an atomic topos.

The surjection / embedding factorization system can be thought of in logical terms. An embedding into a topos EE corresponds to localization at a Lawvere-Tierney topology, or in logical terms, to adding more axioms to the geometric theory one is classifying, without adding more function or relation symbols. On the other hand, a surjection into EE corresponds to adding functions and relations to your theory without adding new provable statements in the old language, i.e. to a conservative extension of your theory. The (surjection, embedding) factoization corresponds (contravariantly) to factoring an interpretation as the composite of two interpretations, the first of which adds new axioms, and the second of which adds new vocabulary.

Anyway, this factorization means that a “complete topos” EE is equivalently one such that there are no nontrivial embeddings into EE. This means that the only Lawvere-Tierney topologies on EE are the trivial one and the inconsistent one. Equivalently, there is a canonical bijection between the Lawvere-Tierney topologies on EE and the Lawvere-Tierney topologies on SetSet. So I think the constructive (or relative) statement would be something along these lines.

This sounds very close to saying that there is an isomorphism between the subobject classifier in EE and the pullback of the subobject classifier in SetSet, i.e. to saying the topos is atomic.

Here’s a stab at a more precise version of the statement: ESE \to S is “complete” if for every embedding FEF \to E over SS, the square E F,F SE^\to_\to F,F'{}^\to_\to S is a pullback, where FFSF \to F' \to S is the (surjection, embedding) factorization of FSF \to S. This says that “EE is as complete as SS is” – the only way to add axioms to EE is to add axioms to SS and induce up.

Posted by: Tim Campion on October 16, 2018 8:48 PM | Permalink | Reply to this

Re: Topoi of G-sets

Darn, that “square” in the last paragraph should be F E,F SF^\to_\to E,F' {}^\to_\to S. If I were less lazy I would figure out how to actually render it as a square, too!

Posted by: Tim Campion on October 16, 2018 9:09 PM | Permalink | Reply to this

Re: Topoi of G-sets

Ah, I see what you’re thinking. You’re right, they do look quite similar from that perspective. But I think they’re nevertheless distinct. On the one hand, atomicity speaks about subobjects of all objects, not just the terminal object, so that the inverse image functor preserves all powerobjects, not just the subobject classifier. On the other hand, the subobject classifier only classifies open subtoposes, while of course not all subtoposes are open.

Posted by: Mike Shulman on October 16, 2018 10:00 PM | Permalink | Reply to this

Re: Topoi of G-sets

Re: your questions about essential geometric morphisms between HSetH\mathrm{Set} and GSetG\mathrm{Set}. We can describe the whole 11-category of such morphisms: Map Topos(HSet,GSet)Fun(H,G). \mathrm{Map}_{\mathrm{Topos}}(H\mathrm{Set}, G\mathrm{Set}) \approx \Fun(H,G). (I’m writing geometric morphisms in the direction of the right adjoint.) That is, up to isomorphism, geometric morphisms correspond to group homomorphisms up to conjugation.

The etale geometric morphisms in this are the ones which can factor as HSet(GSet) /XGSet. H\mathrm{Set} \approx (G\mathrm{Set})_{/X} \to G\mathrm{Set}. The only slices of GSetG\mathrm{Set} which are equivalent to HSetH\mathrm{Set} are going to be those such that X=G/HX=G/H' for some subgroup HGH'\leq G which is isomorphic to HH. So I expect the etale morphisms HSetGSetH\mathrm{Set}\to G\mathrm{Set} to be those which are represented by injective group homomorphisms.

[For XGSetX\in G\mathrm{Set} such that X=G/H iX=\coprod G/H_i, we have GSet /XH iSetG\mathrm{Set}_{/X} \approx \prod H_i\mathrm{Set}.]

These are also going to be essential, since it is harder to be etale than essential. I would guess there are no other essential geometric morphisms in this case, but I’m not sure.

Posted by: Charles Rezk on October 15, 2018 1:03 AM | Permalink | Reply to this

Re: Topoi of G-sets

GGSet is both a (Grothendieck) topos and monadic over Set. How close does this come to a characterization?

Posted by: Tobias Fritz on October 15, 2018 12:04 PM | Permalink | Reply to this

Re: Topoi of G-sets

Another example is MM-sets for a monoid MM. A more interesting example is the Jónsson-Tarski topos. I think the latter might even be Boolean.

In the nLab article I just linked to, there’s a reference to the article by Johnstone When is a Variety a Topos?, which has more information on this question.

Posted by: Todd Trimble on October 15, 2018 2:36 PM | Permalink | Reply to this

Re: Topoi of G-sets

Thomas Holder kindly set me straight on the fact that the Jónsson-Tarski topos is definitely not Boolean. For some reason I jumped to the conclusion that one of its famous slices, which is the category of sheaves on Cantor space, was a Boolean topos, and that was giving me some slight hope. However, the only way that sheaves on a T 0T_0 space can form a Boolean topos is if the space is discrete. My bad!

Posted by: Todd Trimble on October 15, 2018 7:11 PM | Permalink | Reply to this

Re: Topoi of G-sets

The category of reflexive graphs is both a presheaf category and monadic over Set. However it’s also equivalent to MMSet for a suitable monoid MM, so it doesn’t really go beyond Todd’s examples — it’s just a really important example.

Posted by: John Baez on October 15, 2018 3:47 PM | Permalink | Reply to this

Re: Topoi of G-sets

Here are some more results from the Elephant that may be relevant:

  • C5.2.13 and C5.3.8: a Grothendieck topos is connected, atomic, and has a point if and only if it consists of the continuous actions of some localic group.

  • C5.2.14(c): The localic group in question can be characterized somewhat explicitly in terms of classifying topoi. If the topos EE classifies some theory TT, then the point s:SetEs:Set \to E classifies some TT-model MSetM\in Set, and the localic group GG is the automorphism group of this model MM with the “topology of pointwise convergence” (the scare quotes are because in general we’re talking about locales rather than spaces).

  • C5.2.10: For a discrete group GG, the topos GSetG Set admits a proper surjection from a localic topos if and only if GG is finite.

Posted by: Mike Shulman on October 15, 2018 7:07 PM | Permalink | Reply to this

Re: Topoi of G-sets

Mike wrote:

A Grothendieck topos is connected, atomic, and has a point if and only if it consists of the continuous actions of some localic group.

I like this sort of result a lot. I’m a bit worried by how this relates to Todd’s remark:

A connected atomic topos with a point (a geometric morphism from SetSet) is a classifying topos of a localic group GG (consisting of continuous discrete actions of GG).

Why is he saying ‘discrete’ and you’re not? My guess is that you’re both talking about a localic group acting continuously on a discrete set.

I’m also a bit puzzled by how this is related to these other results on the nLab:

Proposition [Joyal-Tierney 84]. For every Grothendieck topos \mathcal{E} there is a localic groupoid 𝒢\mathcal{G} such that Sh(𝒢)\mathcal{E} \simeq Sh(\mathcal{G}).

and

Proposition [Butz-Moerdijk 98]. If \mathcal{E} has enough points, then there exists a topological groupoid 𝒢\mathcal{G} such that Sh(𝒢)\mathcal{E} \simeq Sh(\mathcal{G}).

The first one makes it sound like we add the adjectives ‘connected’, ‘atomic’ and ‘with a point’ to get from localic groupoids to localic groups. But the second one makes it sound like we add the adjective ‘have enough points’ to get from localic groupoids to topological groupoids. I would be happiest if one set of adjectives took us from localic groupoids to localic groups, another disjoint set of adjectives took us from localic groupoids to topological groupoids, the combination of these sets of adjectives took us to topological groups, and one extra adjective took us to discrete groups.

Posted by: John Baez on October 15, 2018 10:19 PM | Permalink | Reply to this

Re: Topoi of G-sets

Yes, ‘continuous discrete actions’ means continuous actions of a topological on a discrete space (ie a set). This is called Cont(G)Cont(G) in the Elephant (Example A.1.2.6). Note also that a sheaf on the one-object localic (or topological) groupoid arising from a given group GG is exactly a continuous action. This is because it is a sheaf XX on the one-point space (i.e. a set) together with the appropriate action by the arrows, namely an automorphism of the constant sheaf with fibre XX on GG (really, it’s s *Xt *Xs^\ast X \to t^\ast X, but s=t:G*s=t\colon G\to \ast…) satisfying the usual conditions.

I think of ‘has a point’ and ‘has enough points’ as different adjectives. If a topos doesn’t have a point, then there’s no fibre functor to look at the automorphisms thereof. And one needs enough points even in the localic topos case to know that the locale YY, such that the topos is Sh(Y)Sh(Y), is spatial.

Perhaps you want to look at étendus

Posted by: David Roberts on October 16, 2018 12:30 AM | Permalink | Reply to this

Re: Topoi of G-sets

Yes, as David said, your guess is right.

I’m not quite sure what is bothering you about the sets of adjectives {\{connected, atomic, has a point}\} and {\{has enough points}\}. As sets of adjectives they are certainly disjoint; in particular, as David says, having a point and having enough points are two different properties. Maybe you want the adjectives in the two sets to be “pairwise independent”, in that no adjective in one set implies any adjective in the other set, whereas it seems that “having enough points” implies “having a point”? But in fact this seeming is an illusion… the trivial topos (sheaves on the empty space) has no points, but that’s enough! (-:

Posted by: Mike Shulman on October 16, 2018 12:54 AM | Permalink | Reply to this

Re: Topoi of G-sets

Okay, maybe I’m getting the point… or at least, enough points.

I’d like to take this:

A Grothendieck topos is connected, atomic, and has a point iff it’s equivalent to the category of all continuous actions of some localic group on discrete sets.

and try to see what each one of the three assumptions gives us. If we remove all three we get this:

Any Grothendieck topos is equivalent to the category of sheaves on a localic groupoid.

Is something like the following true?

“A Grothendieck topos is connected iff it’s equivalent to the category of sheaves on a localic groupoid whose locale of objects is connected.” (???)

What if we only know the Grothendieck topos has a point? What if we only know it’s atomic?

Posted by: John Baez on October 16, 2018 6:46 AM | Permalink | Reply to this

Re: Topoi of G-sets

Joyal and Tierney did show that every Grothendieck topos can be presented as the topos of equivariant sheaves on some localic groupoid in the paper An extension of the Galois theory of Grothendieck Mem. Amer. Math. Soc. no 309 (1984).

Posted by: lt on October 16, 2018 5:27 PM | Permalink | Reply to this

Re: Topoi of G-sets

Yes, that’s the result I stated in the comment to which you replied. Let me spell out my questions more precisely:

  • Given a localic groupoid GG, under which conditions is the category of (equivariant) sheaves on GG a connected Grothendieck topos?

  • Given a localic groupoid GG, under which conditions is the category of (equivariant) sheaves on GG an atomic Grothendieck topos?

  • Given a localic groupoid GG, under which conditions is the category of (equivariant) sheaves on GG a Grothendieck topos with a point?

Posted by: John Baez on October 16, 2018 8:30 PM | Permalink | Reply to this

Re: Topoi of G-sets

I don’t know the answers to those questions, and I don’t immediately see anything helpful in the Elephant.

Posted by: Mike Shulman on October 16, 2018 10:03 PM | Permalink | Reply to this

Re: Topoi of G-sets

Thanks for trying!

These questions seems interesting to me because we’re going from ‘a category of sheaves on localic groupoid’ to ‘a category of sheaves on a localic group’ by imposing three extra properties (connected, atomic and ‘with a point’, the last perhaps better thought of as an extra structure)… and I’m having a bit of trouble thinking of a localic group as a localic groupoid with three extra things. I easily can see two — ‘connected’ and ‘with a point’ — since a group is a connected groupoid equipped with a chosen point. But three?

My best guess is that the third extra thing is somehow connected to the fact that we’re talking about localic groupoids. I also fear I may be mixing up two different concepts of ‘connected’: the concept of connected for groupoids saying there’s a morphism between any pair of objects, and some concept of connected for localic groupoids where we demand that the locale of objects (or maybe the locale morphisms) be connected. Maybe this puzzlement is connected to my confusion about the ‘third extra thing’.

The nLab says:

An example of a connected atomic topos without a point is given in (Johnstone, example D3.4.14).

This might help, since it would give an example of a localic groupoid whose category of sheaves has just two of the three properties I’m talking about!

Posted by: John Baez on October 17, 2018 6:07 AM | Permalink | Reply to this

Re: Topoi of G-sets

Yes, I believe there are two “concepts” of connectedness going on here, plus a third that combines the two. (I say “concepts” because I don’t know precise statements of them all in this context.)

A topos, regarded as a generalized space, is kind of like a topological groupoid (or more precisely a localic groupoid) in that it has higher categorical structure but also “topological” structure at each level. But I don’t understand how those two structures interact for toposes as well as I’d like; it seems that they’re not completely independent as they are for (say) topological groupoids in this sense, plus the categorical structure is at least partially directed (some toposes have a category of points rather than just a groupoid of them).

But I do know that the notion of “connected” topos combines both “categorical” connectedness and “topological” connectedness. For instance, the topos Set GSet^G is connected for any (discrete) groupoid GG that is connected in the categorical sense; but the topos Sh(X)Sh(X) is also connected for any space XX that is connected in the topological sense.

In particular, the latter gives lots of examples of connected toposes with a point that are not atomic. On the other hand, an example of an atomic topos with a point (indeed, enough points) that is not connected is given by Set XSet^X for any discrete set XX with more than one element.

The example in D3.4.14 of a connected atomic topos without a point is very syntactic: it’s the classifying topos of the geometric theory of a decidable complete rooted binary tree equipped with an II-indexed family of disjoint sets of full branches where II is an (externally specified) set with cardinality greater than the continuum. This has no points because in a model of this theory in SetSet there can be at most continuum-many such branches; but it has nontrivial models in the classifying topos of bijections II\cong \mathbb{N} which is known to be consistent. I don’t think this example helps my intuition any, but maybe it will help you.

I am tempted to conjecture that atomicity of the topos of sheaves on a localic groupoid corresponds to some kind of “discreteness” property of its locale of objects. For instance, C3.5.3 says that a localic topos Sh(X)Sh(X) is atomic iff XX is a discrete locale. And if the locale of objects is discrete, then connectedness ought to reduce somehow to “categorical” connectedness, hence imply that our localic groupoid is equivalent to one with one object, i.e. a localic group. This would also explain how atomicity “vanishes in the limit nn\to \infty: if nn-atomic meant that the locales of kk-morphisms were discrete for knk\le n, then in the limit \infty-atomic would mean that our localic \infty-groupoid is just a discrete one, i.e. that we are etale over Gpd\infty Gpd.

But D3.4.14 means that this idea can’t be quite right: if that topos were presentable by a localic groupoid with a discrete locale of objects, then its lack of a point would mean that that locale would be empty, hence the topos would be trivial. But maybe there is some slightly weaker notion of “object-discreteness” for localic groupoids for which this could be true. (It seems that it can’t be simply a condition on the locale of objects, by C3.5.3.)

By the way, in understanding how properties of toposes are related to those of localic groupoids, a relevant question to ask is when two localic groupoids give rise to equivalent toposes. This is studied in C5.3 of the Elephant, but there doesn’t seem to be a complete answer given. Three classes of functors between localic groupoids are shown to induce equivalences of toposes:

  • open weak equivalences: functors that are internally full-and-faithful in LocLoc (in the usual sense for internal categories), and essentially “surjective” where the relevant notion of surjection is an open surjection.
  • dually, proper weak equivalences.
  • the map from any open or proper localic groupoid G 1G 0G_1 \rightrightarrows G_0 (i.e. one for which the source and target maps are open or proper) to its “étale-completion”, which is the kernel pair of the geometric morphism Sh(G 0)Sh(G 1G 0)Sh(G_0) \to Sh(G_1\rightrightarrows G_0).

and it is shown that any geometric morphism between toposes of sheaves on étale-complete open (or proper) localic groupoids can be represented by a span of functors between these étale-complete open (or proper) localic groupoids in which the backwards-pointing arrow is an open (or proper) weak equivalence. This gives a sort of representation of the 2-category of toposes as a 2-category of fractions of that of étale-complete open localic groupoids (but not proper ones, since not every topos can be presented by a proper localic groupoid).

Posted by: Mike Shulman on October 18, 2018 5:27 PM | Permalink | Reply to this

Re: Topoi of G-sets

Mike wrote:

A topos, regarded as a generalized space, is kind of like a topological groupoid (or more precisely a localic groupoid) in that it has higher categorical structure but also “topological” structure at each level. But I don’t understand how those two structures interact for toposes as well as I’d like…

Thanks for admitting this, and thanks for tackling my puzzles even if the answers are inconclusive — because it reassures me that there’s something interesting going on here, not just my own ignorance.

James Dolan used to complain about how topological groupoids have both a topology on the space of objects and also spaces of isomorphisms between objects, which together yield a double \infty-groupoid — one which happens to be just a 1-groupoid in the ‘algebraic’ direction, but a true \infty-groupoid in the ‘topological’ direction.

There’s nothing bad about double structures, but it pays to know when you’re dealing with one, and why. As you say, when we turn a localic groupoid into a topos we seem to be combining the algebraic and topological directions in a funny way.

Posted by: John Baez on October 18, 2018 8:28 PM | Permalink | Reply to this

Re: Topoi of G-sets

Yeah. The category of localic groupoids is naturally enriched over double categories, with two kinds of 2-cells: one (invertible) induced by the groupoid structure and the other (thin) induced by the poset structure of a locale/frame. Somehow the “sheaves” construction combines these two kinds of 2-cells to produce the one kind of 2-cell that appears in the 2-category of toposes, while inverting some hard-to-describe-in-general class of “weak equivalences”.

It might clarify things to go to localic \infty-groupoids and \infty-toposes. Or it might not. But I think that chapter of the \infty-Elephant hasn’t been written yet.

Posted by: Mike Shulman on October 18, 2018 11:48 PM | Permalink | Reply to this

Re: Topoi of G-sets

Another avenue of approach to these things begins with Bunge’s characterization of presheaf toposes, which speaks again of “atoms” but in a slightly stronger sense. One version of her theorem (from her 1966 thesis) states that presheaf categories are precisely (cocomplete) atomic categories. I should explain the latter terminology, since it means something different from a Grothendieck topos being atomic.

An object aa of a cocomplete category EE is atomic if E(a,):ESetE(a, -): E \to Set preserves colimits. A cocomplete category is said to be atomic if it the full subcategory Atom(E)Atom(E) of atomic objects is small and dense. Here density means that for every object ee of EE is canonically a colimit of atoms, i.e., the canonical map

a:Atom(E)E(a,e)ae\int^{a: Atom(E)} E(a, e) \cdot a \to e

is an isomorphism.

(This can be seen a doctrinal variation on Gabriel-Ulmer duality, which starts by defining a finitely presentable object aa as one where E(a,):ESetE(a, -): E \to Set preserves filtered colimits, and defines locally finite presentable categories as cocomplete categories where every object is canonically a filtered colimit of finitely presented objects. The analogous theorem is that every locally finitely presentable category is of the form Lex(C op,Set)Lex(C^{op}, Set) where CC, the (small) category of finitely presentable objects, is finitely cocomplete. Whereas in the present case, the duality is between presheaf categories [C op,Set][C^{op}, Set] and small Cauchy complete categories CC.)

The characterization is not hard to establish. (We’ve discussed this at the Café before, so I’ll hold off from repeating the argument. It’s not a bad exercise though.)

Once you have characterized presheaf toposes, you can add on more hypotheses to get to GG-SetSet for a group GG. A Boolean presheaf topos is of the form Set GSet^G where GG is a groupoid, and then if you add a connectedness hypothesis, GG becomes a group (as we were saying). The thing that Mike said about getting finite groups was something new to me.

Posted by: Todd Trimble on October 17, 2018 3:00 AM | Permalink | Reply to this

Re: Topoi of G-sets

Thanks! I’m just blundering around, so I’m open to all sorts of nice ways to characterize categories like GSetG Set and the maps between them that arise from homomorphisms HGH \to G. This seems like a good one.

My interest in the case of GG finite comes from the rich theory of finite groups, which will have a reflection in topos theory. Topoi of the form GSetG Set for GG finite should be a kind of ‘dream world’ inside topos theory, sort of like how group algebras [G]\mathbb{C}[G] are incredibly nice algebras: finite-dimensional, semisimple, naturally extendable to Hopf algebras, etc.

By the way: I think in the case of categories of the form GSetG Set the two concepts of ‘atom’ agree. Right?

Posted by: John Baez on October 17, 2018 6:35 AM | Permalink | Reply to this

Re: Topoi of G-sets

I think they don’t agree. That is, if the atoms we were discussing earlier are the same as inhabited transitive GG-sets, then no: for atoms in the sense I’m mentioning here, there’s just one atomic object up to isomorphism, namely GG acting on itself.

To put it in the context of my prior remark: given a presheaf category Set C opSet^{C^{op}} with CC Cauchy complete, CC will be the full subcategory of atoms. So for Set G opSet^{G^{op}} you get GG back, since groups seen as one-object categories are Cauchy complete. (Same as for Set GSet^G since GG opG \simeq G^{op}.)

More explicitly, the reason G/HG/H is not an atom is that Set G(G/H,):Set GSetSet^G(G/H, -): Set^G \to Set does not preserve coequalizers. For example, there’s a coequalizer map GG/HG \to G/H, but the induced map

Set G(G/H,G)Set G(G/H,G/H)Set^G(G/H, G) \to Set^G(G/H, G/H)

is not a coequalizer.

Other names for atomic object: tiny object, Lawvere’s “a.t.o.m.” (“amazing tiny object model” – although there may be an internal/external distinction that I’m sloughing over there), and the more unwieldy “indecomposable projective object”.

Posted by: Todd Trimble on October 17, 2018 12:59 PM | Permalink | Reply to this

Re: Topoi of G-sets

Thanks. For some stupid reason I noticed that Set G(G/H,)Set^G(G/H,-) preserves coproducts and jumped to the conclusion that G/HG/H is an atom in this other sense, while failing to check coequalizers. It’s sort of funny, taking the ‘atom’ metaphor seriously, that squashing down the atom GG produces something that’s ‘too small to be an atom’ in this other sense… though Jim would say G/HG/H is cosmaller than GG.

Posted by: John Baez on October 17, 2018 3:23 PM | Permalink | Reply to this

Re: Topoi of G-sets

Incidentally, John: good luck in your quest to learn more about finite group theory! That’s an area where personally I feel woefully lacking. The place where I went to graduate school, Rutgers, was a real powerhouse in finite simple group theory: Gorenstein was there, and I think Lyons and O’Nan and Sims might all still be there, and then there are experts on the Monster like Lepowsky. And yet, I never learned much about group theory in all the long time I was there.

Posted by: Todd Trimble on October 17, 2018 1:18 PM | Permalink | Reply to this

Re: Topoi of G-sets

I’m having lots of fun slowly working through Robert Wilson’s The Finite Simple Groups, allowing myself plenty of time to stop and smell the roses, and take elaborate side-hikes. For example, I’ve become quite enamored with how pp-groups are all nilpotent, and how finite nilpotent groups all split as a product of pp-groups for various pp (a special case of the ‘fracture theorem’). This zone is a kind of halo around the niceness of finite abelian groups: a complete classification seems utterly hopeless, but the general structure is reminiscent of the abelian case.

I also like Wilson’s claim that a lot of peculiarities of sporadic finite simple groups trace their origin to the outer automorphism of S 6\mathrm{S}_6. So far I don’t really understand that claim; I’m still struggling to understand the exceptional triple cover of A 6\mathrm{A}_6, called the ‘Valentiner group’.

In short, there’s a great blend of grand vistas and intriguing specific objects, sort of like you see in the theory of simple Lie groups, but more complicated. The hard part is not getting put off by the fact that I’ll never understand more than a tiny bit of the subject.

Posted by: John Baez on October 17, 2018 3:42 PM | Permalink | Reply to this

Re: Topoi of G-sets

Here’s a bunch of thoughts about your Tannaka-Krein question, which may not meaningfully address your actual question. I was hoping to write a blog post about this at some point but never got around to it, among other things because I couldn’t tell whether everybody knows this stuff already.

First, I want to draw your attention to how cute the structure of the proof of “Tannaka-Krein reconstruction over Set” is: its content consists of exactly two applications of the Yoneda lemma, one to show that the group of automorphisms of the fiber functor is the group of automorphisms of the GG-set GG, and the other to show that the group of automorphisms of the GG-set GG is GG (Cayley’s theorem).

Second, here’s a context for relating this result to the usual Tannaka-Krein theorem. Let’s consider the 2-category of symmetric monoidal cocomplete categories, or SMCCs for short (where part of the requirement is that the monoidal structure distributes over colimits), and symmetric monoidal cocontinuous functors. Probably I should also require presentability but that won’t really be relevant to anything I’m about to say. Two important classes of examples to keep in mind are Grothendieck topoi and quasicoherent sheaves on stacks. The nLab term here is “symmetric 2-rig” but I really dislike that terminology, among other things because it’s actually incompatible with e.g. the standard notion of a 2-group (a 2-ring doesn’t have an underlying additive 2-group). But I do want to think of SMCCs as categorified commutative rings, so as a setup for doing “2-affine algebraic geometry,” where Grothendieck topoi live “over Spec(Set)” and quasicoherent sheaves live “over Spec(Ab).”

(Here’s a subrant of the rant: there should be two different words for Grothendieck-topoi-where-the-morphisms-are-geometric-morphisms and Grothendieck-topoi-where-the-morphisms-are-algebraic-morphisms, because the former look like locales / affine schemes but the latter look like Heyting algebras / commutative rings. This is important for e.g. clarifying things like what it means to perform free constructions of topoi; free often means in the second sense, not the first.)

General forms of the Tannaka-Krein theorem due to Lurie and others assert that if XX and YY are stacks satisfying some hypotheses, then symmetric monoidal cocontinuous functors QC(Y)QC(X)QC(Y) \to QC(X), possibly satisfying some hypotheses, correspond to morphisms XYX \to Y of stacks. The specialization to Tannaka-Krein for finite groups occurs when Y=BGY = BG for a finite group GG and XX is a point (and we’re working over an algebraically closed field of characteristic zero, say): morphisms XYX \to Y then correspond to GG-torsors on a point, and there’s one isomorphism class of these with automorphism group GG, and correspondingly there’s a unique fiber functor from QC(BG)Rep(G)QC(BG) \cong \text{Rep}(G) to QC(pt)VectQC(\text{pt}) \cong \text{Vect}, with automorphism group GG.

So what these theorems say is that some stacks are 2-affine in the sense that their theory fully faithfully embeds into the theory of SMCCs.

Tannaka-Krein over Set is the same story about BGBG and a point, only instead of working over a field we work over Set, and “symmetric monoidal cocontinuous functor” is upgraded slightly to “geometric morphism” (it’s a bit annoying that these don’t quite coincide). The more general claim about torsors is then a special case of Diaonescu’s theorem, which further implies that the fiber functor G-SetSetG\text{-Set} \to \text{Set} is uniquely determined up to isomorphism by being geometric. Probably it’s even determined up to isomorphism by being symmetric monoidal cocontinuous.

Now, there are some fun SMCCs you can cook up, and then you can try to think about what their Spec’s looks like. For example, the free SMCC on an object is the category of combinatorial species. Its Spec is something like “the classifying space of sheaves”; it’s the object classifier in this setting. The free SMCC on a dualizable object is (I think?) the category of presheaves on the 1-dimensional (oriented) cobordism category. Its Spec is something like the “classifying space of vector bundles,” so it deserves to be called something like BGLBGL, although possibly only after “base changing from Set to Vect.” It should have some relationship to Deligne’s stuff about Rep(GL t)\text{Rep}(GL_t), along the lines of Rep(GL t)\text{Rep}(GL_t) being the free SMCC over Vect on a dualizable object of dimension tt.

Posted by: Qiaochu Yuan on October 19, 2018 10:14 AM | Permalink | Reply to this

Re: Topoi of G-sets

Nice stuff! I’ll only make a trivial comment:

The nLab term here is “symmetric 2-rig” but I really dislike that terminology, among other things because it’s actually incompatible with e.g. the standard notion of a 2-group (a 2-ring doesn’t have an underlying additive 2-group).

Having introduced the term ‘symmetric 2-rig’ to the world in a paper with James Dolan, I feel obliged to defend it. A ring has an underlying additive group but a rig does not, so there’s no reason to suspect that a 2-rig has an underlying additive 2-group — so you shouldn’t be annoyed that it doesn’t!

Somehow you slipped from talking about 2-rigs to talking about ‘2-rings’. These could be interesting, but they’re a special case of ring spectra. SetSet and VectVect are 2-rigs, not 2-rings.

Posted by: John Baez on October 19, 2018 3:31 PM | Permalink | Reply to this

Re: Topoi of G-sets

In regards to Qiaochu’s subrant, it’s worth mentioning that at least in the \infty-topos context, Joyal advocates using “topos” for “toposes with geometric morphisms as morphisms” and his term “logos” for “toposes with left exact left adjoints as morphisms”.

Posted by: Tim Campion on October 22, 2018 12:45 PM | Permalink | Reply to this

Re: Topoi of G-sets

I asked a question on the category theory mailing list and got a nice reply from Christopher Townsend. I might as well quote it here since while many of you are on the category theory mailing list, it’s good to have all the info in one place.

I wrote:

Joyal and Tierney proved that any Grothendieck topos is equivalent to the category of sheaves on a localic groupoid. I gather that we can take this localic groupoid to have a single object iff the Grothendieck topos is connected, atomic, and has a point. In this case the topos can also be seen as the category of continuous actions of a localic group on (discrete) sets.

I’m curious about how these three conditions combine to get the job done. So suppose 𝒢\mathcal{G} is a localic groupoid.

Under which conditions is the category of sheaves on 𝒢\mathcal{G} a connected Grothendieck topos?

Under which conditions is the category of sheaves on 𝒢\mathcal{G} an atomic Grothendieck topos?

Under which conditions is the category of sheaves on 𝒢\mathcal{G} a Grothendieck topos with a point?

(Maybe we should interpret “with a point” as an extra structure on 𝒢\mathcal{G} rather than a mere extra property; I don’t know how much this matters.)

Christopher Townsend wrote:

Thanks for the question, it has made me re-read some stuff.

I’m really just reading Elephant C3.5.6 and C5.2.13 for my answer.

I think the essence of your question is given a Grothendieck topos F, how does having a point, being atomic and being connected conspire to produce an (open) localic group G such that F is equivalent (over our base topos, say Set) to the category of G-equivariant sheaves?

If you can find a topological space/locale X and a cover p:Sh(X)\toF that is an effective descent morphism (e.g. an open surjection) then [by easy-ish abstract nonsense] there is a localic groupoid 𝒢\mathcal{G} such that F is the category of 𝒢\mathcal{G}-equivariant sheaves. The locale of objects of 𝒢\mathcal{G} is X. Therefore in the case of F having a point we have a cover with X=1; i.e. should give a localic group. But this does not complete the proof because for an arbitrary p:Set\toF over Set we don’t know if p is an effective descent morphism. However if F is atomic then any discrete cover (in particular Set; X=1 is discrete[!]) is open and if further F is connected then the cover p is a surjection. Hence p is of effective descent. (C3.5.6)

So, in summary, to make Joyal and Tierney work you need an open surjection. The atomicity gives you openness and connectedness gives you a surjection in the case you are looking at (though these two implications do not appear to be independent). Having the cover being a point forces the localic groupoid object to be 1; i.e. you have an open localic group.

Posted by: John Baez on October 20, 2018 3:27 PM | Permalink | Reply to this

Re: Topoi of G-sets

Another question I’m curious about: Is there a characterization of toposes which are localizations (i.e. embedded subtoposes) of GSetG-Set for a discrete groupoid GG?

I’d really like to know this \infty-categorically: is there a characterization of \infty-toposes which are localizations of Spaces/XSpaces/X for some XX?

Posted by: Tim Campion on October 22, 2018 12:47 PM | Permalink | Reply to this

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