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June 15, 2018

∞-Atomic Geometric Morphisms

Posted by Mike Shulman

Today’s installment in the ongoing project to sketch the \infty-elephant: atomic geometric morphisms.

Chapter C3 of Sketches of an Elephant studies various classes of geometric morphisms between toposes. Pretty much all of this chapter has been categorified, except for section C3.5 about atomic geometric morphisms. To briefly summarize the picture:

  • Sections C3.1 (open geometric morphisms) and C3.3 (locally connected geometric morphisms) are steps n=1n=-1 and n=0n=0 on an infinite ladder of locally n-connected geometric morphisms, for 1n-1 \le n \le \infty. A geometric morphism between (n+1,1)(n+1,1)-toposes is locally nn-connected if its inverse image functor is locally cartesian closed and has a left adjoint. More generally, a geometric morphism between (m,1)(m,1)-toposes is locally nn-connected, for n<mn\lt m, if it is “locally” locally nn-connected on nn-truncated maps.

  • Sections C3.2 (proper geometric morphisms) and C3.4 (tidy geometric morphisms) are likewise steps n=1n=-1 and n=0n=0 on an infinite ladder of n-proper geometric morphisms.

  • Section C3.6 (local geometric morphisms) is also step n=0n=0 on an infinite ladder: a geometric morphism between (n+1,1)(n+1,1)-toposes is nn-local if its direct image functor has an indexed right adjoint. Cohesive toposes, which have attracted a lot of attention around here, are both locally \infty-connected and \infty-local. (Curiously, the n=1n=-1 case of locality doesn’t seem to be mentioned in the 1-Elephant; has anyone seen it before?)

So what about C3.5? An atomic geometric morphism between elementary 1-toposes is usually defined as one whose inverse image functor is logical. This is an intriguing prospect to categorify, because it appears to mix the “elementary” and “Grothendieck” aspects of topos theory: a geometric morphisms are arguably the natural morphisms between Grothendieck toposes, while logical functors are more natural for the elementary sort (where “natural” means “preserves all the structure in the definition”). So now that we’re starting to see some progress on elementary higher toposes (my post last year has now been followed by a preprint by Rasekh), we might hope be able to make some progress on it.

Unfortunately, the definitions of elementary (,1)(\infty,1)-topos currently under consideration have a problem when it comes to defining logical functors. A logical functor between 1-toposes can be defined as a cartesian closed functor that preserves the subobject classifier, i.e. F(Ω)ΩF(\Omega) \cong \Omega. The higher analogue of the subobject classifier is an object classifier — but note the switch from definite to indefinite article! For Russellian size reasons, we can’t expect to have one object classifer that classifies all objects, only a tower of “universes” each of which classifies some subcollection of “small” objects.

What does it mean for a functor to “preserve” the tower of object classifiers? If an (,1)(\infty,1)-topos came equipped with a specified tower of object classifiers (indexed by \mathbb{N}, say, or maybe by the ordinal numbers), then we could ask a logical functor to preserve them one by one. This would probably be the relevant kind of “logical functor” when discussing categorical semantics of homotopy type theory: since type theory does have a specified tower of universe types U 0:U 1:U 2:U_0 : U_1 : U_2 : \cdots, the initiality conjecture for HoTT should probably say that the syntactic category is an elementary (,1)(\infty,1)-topos that’s initial among logical functors of this sort.

However, Grothendieck (,1)(\infty,1)-topoi don’t really come equipped with such a tower. And even if they did, preserving it level by level doesn’t seem like the right sort of “logical functor” to use in defining atomic geometric morphisms; there’s no reason to expect such a functor to “preserve size” exactly.

What do we want of a logical functor? Well, glancing through some of the theorems about logical functors in the 1-Elephant, one result that stands out to me is the following: if F:SEF:\mathbf{S}\to \mathbf{E} is a logical functor with a left adjoint LL, then LL induces isomorphisms of subobject lattices Sub E(A)Sub S(LA)Sub_{\mathbf{E}}(A) \cong Sub_{\mathbf{S}}(L A). This is easy to prove using the adjointness LFL\dashv F and the fact that FF preserves the subobject classifier:

Sub E(A)E(A,Ω E)E(A,FΩ S)E(LA,Ω S)Sub S(LA). Sub_{\mathbf{E}}(A) \cong \mathbf{E}(A,\Omega_{\mathbf{E}}) \cong \mathbf{E}(A,F \Omega_{\mathbf{S}}) \cong \mathbf{E}(L A,\Omega_{\mathbf{S}})\cong Sub_{\mathbf{S}}(L A).

What would be the analogue for (,1)(\infty,1)-topoi? Well, if we imagine hypothetically that we had a classifier UU for all objects, then the same argument would show that LL induces an equivalence between entire slice categories E/AS/LA\mathbf{E}/A \simeq \mathbf{S}/L A. (Actually, I’m glossing over something here: the direct arguments with Ω\Omega and UU show only an equivalence between sets of subobjects and cores of slice categories. The rest comes from the fact that FF preserves local cartesian closure as well as the (sub)object classifier, so that we can enhance Ω\Omega to an internal poset and UU to an internal full subcategory and both of these are preserved by FF as well.)

In fact, the converse is true too: reversing the above argument shows that FF preserves Ω\Omega if and only if LL induces isomorphisms of subobject lattices, and similarly FF preserves UU if and only if LL induces equivalences of slice categories. The latter condition, however, is something that can be said without reference to the nonexistent UU. So if we have a functor F:ESF:\mathbf{E}\to \mathbf{S} between (,1)(\infty,1)-toposes that has a left adjoint LL, then I think it’s reasonable to define FF to be logical if it is locally cartesian closed and LL induces equivalences E/AS/LA\mathbf{E}/A \simeq \mathbf{S}/L A.

Furthermore, a logical functor between 1-toposes has a left adjoint if and only if it has a right adjoint. (This follows from the monadicity of the powerset functor P:E opEP : \mathbf{E}^{op} \to \mathbf{E} for 1-toposes, which we don’t have an analogue of (yet) in the \infty-case.) In particular, if the inverse image functor in a geometric morphism is logical, then it automatically has a left adjoint, so that the above characterization of logical-ness applies. And since a logical functor is locally cartesian closed, this geometric morphism is automatically locally connected as well. This suggests the following:

Definition: A geometric morphism p:ESp:\mathbf{E}\to \mathbf{S} between (,1)(\infty,1)-topoi is \infty-atomic if

  1. It is locally \infty-connected, i.e. p *p^\ast is locally cartesian closed and has a left adjoint p !p_!, and
  2. p !p_! induces equivalences of slice categories E/AS/p !A\mathbf{E}/A \simeq \mathbf{S}/p_! A for all AEA\in \mathbf{E}.

This seems natural to me, but it’s very strong! In particular, taking A=1A=1 we get an equivalence EE/1S/p !1\mathbf{E}\simeq \mathbf{E}/1 \simeq \mathbf{S}/p_! 1, so that E\mathbf{E} is equivalent to a slice category of S\mathbf{S}. In other words, \infty-atomic geometric morphisms coincide with local homeomorphisms!

Is that really reasonable? Actually, I think it is. Consider the simplest example of an atomic geometric morphism of 1-topoi that is not a local homeomorphism: [G,Set]Set[G,Set] \to Set for a group GG. The corresponding geometric morphism of (,1)(\infty,1)-topoi [G,Gpd]Gpd[G,\infty Gpd] \to \infty Gpd is a local homeomorphism! Specifically, we have [G,Gpd]Gpd/BG[G,\infty Gpd] \simeq \infty Gpd / B G. So in a sense, the difference between atomic and locally-homeomorphic vanishes in the limit nn\to \infty.

To be sure, there are other atomic geometric morphisms of 1-topoi that do not extend to local homeomorphisms of (,1)(\infty,1)-topoi, such as Cont(G)SetCont(G) \to Set for a topological group GG. But it seems reasonable to me to regard these as “1-atomic morphisms that are not \infty-atomic” — a thing which we should certainly expect to exist, just as there are locally 0-connected morphisms that are not locally \infty-connected, and 0-proper morphisms that are not \infty-proper.

We can also “see” how the difference gets “pushed off to \infty” to vanish, in terms of sites of definition. In C3.5.8 of the 1-Elephant it is shown that every atomic Grothendieck topos has a site of definition in which (among other properties) all morphisms are effective epimorphisms. If we trace through the proof, we see that this effective-epi condition comes about as the “dual” class to the monomorphisms that the left adjoint of a logical functor induces an equivalence on. Since an (n+1,1)(n+1,1)-topos has classifiers for nn-truncated objects, we would expect an atomic one to have a site of definition in which all morphisms belong to the dual class of the nn-truncated morphisms, i.e. the nn-connected morphisms. So as nn\to \infty, we get stronger and stronger conditions on the morphisms in our site, until in the limit we have a classifier for all morphisms, and the morphisms in our site are all required to be equivalences. In other words, the site is itself an \infty-groupoid, and thus the topos of (pre)sheaves on it is a slice of Gpd\infty Gpd.

However, it could be that I’m missing something and this is not the best categorification of atomic geometric morphisms. Any thoughts from readers?

Posted at June 15, 2018 4:16 PM UTC

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Re: ∞-Atomic Geometric Morphisms

So you’ve made advances on this in under six weeks:

Interesting question; I don’t know a syntactic characterization of etale geometric morphisms.

The original motivation for my question was modal logic. Now we know that modal homotopy type theory is all about mapping between (,1)(\infty, 1)-toposes, does your suggestion point us anywhere interesting?

We’re also in the territory of my MO question that Jacob Lurie kindly answered.

Posted by: David Corfield on June 18, 2018 1:33 PM | Permalink | Reply to this

Re: ∞-Atomic Geometric Morphisms

Sorry, I don’t understand how this is an advance on that question.

Posted by: Mike Shulman on June 18, 2018 3:03 PM | Permalink | Reply to this

Re: ∞-Atomic Geometric Morphisms

Ok, perhaps not your question, but I think it shines a light on what prompted it.

I was wondering about how one might come to look for a semantics for modal operators which involve slices, variation over a type of worlds. Say we know we’re dealing with a comonad (necessarily) on an (,1)(\infty, 1)-topos which is left adjoint to a monad (possibility). So we need an adjoint triple. What more can we say about the modal situation that will require this triple to be between slices, in particular between toposes of the form H/W\mathbf{H}/W and H\mathbf{H}?

We’d like \bigcirc to be such that it can be construed as AWA×WWA \to W \mapsto A \times W \to W, which would happen if the leftmost of the adjoint triple induced an equivalence of slices with the image of 11 equal to WW.

Posted by: David Corfield on June 18, 2018 3:57 PM | Permalink | Reply to this

Re: ∞-Atomic Geometric Morphisms

If what you really want is to be talking about H/W\mathbf{H}/W and H\mathbf{H}, why don’t you just make that explicit? Work in the dependent type theory of H\mathbf{H} with a type called WW and use the Σ\Sigma and Π\Pi types.

Posted by: Mike Shulman on June 19, 2018 11:28 PM | Permalink | Reply to this

Re: ∞-Atomic Geometric Morphisms

Sure, but I was just wondering if there was a process in reverse where because of how we think about some modalities we see we want an \infty-atomic morphism, and that leads us to the slice picture.

I think I’d need to know the intuition behind a morphism being described as ‘atomic’. How much of nLab: atomic topos would carry over to the \infty case?

Posted by: David Corfield on June 20, 2018 3:02 PM | Permalink | Reply to this

Re: ∞-Atomic Geometric Morphisms

I can’t think offhand of a way to categorify the descriptions of 1-atomic topoi in terms of complete atomic Boolean algebras or generating sets of atoms. Johnstone remarks already in the 1-Elephant that these characterizations don’t even relativize to arbitrary 1-atomic geometric morphisms, so that the name “atomic” is arguably not really appropriate for geometric morphisms with codomain other than SetSet.

Posted by: Mike Shulman on June 20, 2018 7:25 PM | Permalink | Reply to this

Re: ∞-Atomic Geometric Morphisms

Dubuc has shown that every connected atomic 1-topos with a point is the topos of continous actions of the locale of automorphisms of that point, and as a corollary that if the point is representable that it is the topos of actions of the (discrete) group of automorphisms of the representative. The latter case sounds like the kind of atomic \infty-topos you describe, since if GG is a discrete group then [G,-Grpd]-Grpd/BG[G, \infty\text{-Grpd}] \simeq \infty\text{-Grpd}/BG with the point given by the pointing of BGBG. But what about the former, where the group has an extra localic structure and we are concerned with continuous actions?

An example of this kind of atomic topos is the topos of continuous actions of the profinite galois group of a perfect field, which forms the base for Menni’s algebraic cohesion. Would \infty-sheaves (granted, I don’t really know what this means) on this site still be atomic over \infty-groupoids?

Posted by: David Jaz on June 20, 2018 5:29 PM | Permalink | Reply to this

Re: ∞-Atomic Geometric Morphisms

what about the former, where the group has an extra localic structure and we are concerned with continuous actions?

I think I mentioned this case in the antepenultimate paragraph of the post?

One way to think about this is that the conditions “locally nn-connected”, “nn-proper”, and “nn-atomic” become stronger with increasing nn, but the condition “nn-local-homeomorphism” becomes weaker with increasing nn, because there are more slice categories of the codomain topos available for the domain topos to potentially be equivalent to. Any nn-local-homeomorphism is mm-atomic for any n,mn,m, so we have a sequence of implications of order type ω+1+1+ω op\omega + 1 + 1 + \omega^{op}:

1LH2LHLHAtom2Atom1Atom 1LH \Rightarrow 2LH \Rightarrow \cdots \Rightarrow \infty LH \Rightarrow \infty Atom \Rightarrow \cdots \Rightarrow 2Atom \Rightarrow 1Atom

and my claim is that the implication LHAtom\infty LH \Rightarrow \infty Atom in the middle is actually an equivalence.

Posted by: Mike Shulman on June 20, 2018 5:56 PM | Permalink | Reply to this

Re: ∞-Atomic Geometric Morphisms

I think I mentioned this case in the antepenultimate paragraph of the post?

Yes you did! Sorry about that, I slipped right past it.

Thanks for the explanation, this helps.

Posted by: David Jaz on June 20, 2018 6:12 PM | Permalink | Reply to this

Re: ∞-Atomic Geometric Morphisms

It’s an interesting question, though, whether Dubuc’s theorem categorifies in some way.

Posted by: Mike Shulman on June 20, 2018 7:21 PM | Permalink | Reply to this

Re: ∞-Atomic Geometric Morphisms

So we don’t even have monadicity of powerset in a Grothendieck \infty-topos? I might have thought the main issue was the existence of realizations for simplicial objects.

Posted by: Kevin Carlson on June 22, 2018 8:46 PM | Permalink | Reply to this

Re: ∞-Atomic Geometric Morphisms

This is a really interesting question, which I’ve actually been spending far too much time thinking about.

We certainly can’t expect monadicity of the literal “powerset”, i.e. P 1X=Ω XP_{-1} X = \Omega^X where Ω\Omega is the subobject classifier. This functor is not even conservative: since Ω\Omega is 0-truncated, P 1P_{-1} factors through the 0-truncation. Of course, the “correct” thing to look at instead is P X=U XP_\infty X = U^X, where UU is the object classifier.

The first problem is that, as in the post above, there is no single object classifier, only an exhaustive family of ever-larger ones. But we could hope for some statement like that for any object classifier UU, the functor P U(X)=U XP_U(X) = U^X is “monadic on UU-small objects” or something.

However, even if we ignore size issues, I don’t know how to prove anything of this sort. The proof that I know of for 1-toposes uses the “crude monadicity theorem”, showing that P:E opEP : E^{op} \to E is conservative and takes reflexive coequalizers in E opE^{op} (i.e. coreflexive equalizers in EE) to coequalizers in EE. Since simplicial objects are a natural categorification of reflexive coequalizers, and we have a corresponding crude monadicity theorem for infinity-categories using these, one might indeed hope to categorify this proof as soon as E opE^{op} has realizations of simplicial objects, i.e. EE has totalizations of cosimplicial objects.

But I have not yet managed to figure out how to categorify the actual proof. I think it’s not too hard to show that PP is conservative (on UU-small objects). The image diagram is shown to be a coequalizer by showing that it is a split coequalizer; and we do have a notion of “split simplicial object”. The splitting is constructed by applying the covariant powerset functor — i.e. existential quantification f:PAPB\exists_f : P A \to P B induced by f:ABf:A\to B — to some of the maps involved in the diagram. And of course, we also have a natural categorification of \exists, namely Σ\Sigma.

BUT, the proof that this is actually a splitting uses several facts that seem very special to the 1-categorical case. The first is that a coreflexive equalizer of two maps is also their pullback, and therefore a certain Beck-Chevalley condition is satisfied by \exists. The categorification Σ\Sigma does have a Beck-Chevalley condition, but I don’t even know how to write down an analogous statement for totalizations of cosimplicial objects, let alone prove it.

The second fact is that the composite Pf f:PAPAP f \circ \exists_f : P A \to P A is the identity when ff is a coreflexive equalizer (since it is then a monomorphism). This seems to depend on the “balancedness” of 1-toposes, which is no longer true in the \infty-case. As far as I know, we don’t even have an explicit characterization of the maps in Gpd\infty Gpd that occur as totalizations of cosimplicial objects, although there is a plausible guess that they are the maps whose fibers have hypoabelian π 1\pi_1. In any case, it seems unlikely that PfΣ fP f \circ \Sigma_f will be the identity for any such ff.

There could, of course, be a different proof that might categorify, or a different proof for the \infty-case altogether. But all the books on 1-topos theory I’ve looked at use essentially the same proof, and I don’t have any ideas for a different one.

Posted by: Mike Shulman on June 22, 2018 9:22 PM | Permalink | Reply to this

Re: ∞-Atomic Geometric Morphisms

I agree that it would be quite nice to have an \infty-version of Pare’s monadicity theorem for 1-toposes. I thought about this a little last year (and even talked to Mathieu Anel about it when he visited CMU), but in the end I didn’t make much progress. My hope was that having an impredicative universe might help – and that such a theorem would then help to better understand the impredicative encodings of colimits.

Posted by: Steve Awodey on July 4, 2018 4:39 PM | Permalink | Reply to this

Re: ∞-Atomic Geometric Morphisms

It’s been pointed out to me by email that there is a different-looking proof in this paper by Lambek and Rattray. However, I don’t see an obvious way to categorify it either, as it depends on a closely related 1-categorical fact, namely that the subobject classifier is injective with respect to equalizers (again, since they are monos). The object classifier in an (,1)(\infty,1)-topos is still injective with respect to monos, but as before, the totalization of a cosimplicial object need not be mono.

(Actually, I suspect that if these proofs are sufficiently β\beta-reduced they would be very much the same. The Lambek-Rattray proof also uses the fact that coretractions are monic — which, again, need not be true in the \infty-case — and the existential quantifier equation Pf f=1P f \circ \exists_f = 1 for monic ff that’s used in the first proof I described is how one proves that the subobject classifier is injective.)

Posted by: Mike Shulman on June 23, 2018 8:08 PM | Permalink | Reply to this

Re: ∞-Atomic Geometric Morphisms

Interesting stuff! So to get an actual endofunctor it seems you need to be using an object classifier of the same size as the objects you’re mapping into it. For instance, for large spaces XX you’re considering the space of maps from XX into the large space of small spaces. I’m concerned these could all be essentially constant for appropriate XX, which would blow up conservativity.

For instance, suppose XX is some BGBG, so that we’re looking at the space of spaces YY with a homotopy coherent GG-action. But it seems that if GG is, say, a large discrete simple group, we shouldn’t expect any maps from GG into a small group like the homotopy automorphisms of YY. If we assumed the GG-action were strict, instead of homotopy coherent, this would be immediate, though I’m not sure how to reconcile the strictification process with the point set assumptions I made on GG.

Posted by: Kevin Carlson on June 24, 2018 6:58 PM | Permalink | Reply to this

Re: ∞-Atomic Geometric Morphisms

Ah, but I see you were thinking of mapping into UU being monadic on UU-small objects, rather than objects in the same universe as UU, as I was guessing. I agree mapping into UU is then conservative, but it’s not even an endofunctor, so is one not between a rock and a hard place?

Posted by: Kevin Carlson on June 24, 2018 7:04 PM | Permalink | Reply to this

Re: ∞-Atomic Geometric Morphisms

Of course, size issues mean that there is no hope of getting an actual monadicity theorem. I put “monadic on UU-small objects” in scare quotes because I didn’t mean it literally, since as you say the operation U ()U^{(-)} is not an endofunctor of UU-small objects. I didn’t have any precise meaning in mind, but I was thinking of something like “this operation that is not an endofunctor satisfies properties X, Y, and Z that monadic functors also satisfy, and which are sufficient for drawing various conclusions that we would like to draw from monadicity, such as the existence of colimits or the existence of adjoints”. There some paper that I can’t recall right now that does something like this for the presheaf-category “monad” on CatCat. I don’t know for sure whether something like it would work here, but it seems fairly likely to me that it would if we could deal with the other problems. Hence, why I decided to ignore size issues for now and focus on the issues of categorification, which seem more serious to me.

Posted by: Mike Shulman on June 24, 2018 11:59 PM | Permalink | Reply to this

Re: ∞-Atomic Geometric Morphisms

the operation ()U ()(&#8722;)U^{(-)} is not an endofunctor of UU-small objects

Don’t forget, there is the paper Monads need not be endofunctors.

Posted by: David Roberts on June 25, 2018 3:30 AM | Permalink | Reply to this

Re: ∞-Atomic Geometric Morphisms

Right, that’s the sort of “X, Y, and Z” mumbo-jumbo I had in mind (although that’s not the specific paper I had in mind).

Posted by: Mike Shulman on June 25, 2018 5:13 AM | Permalink | Reply to this

Re: ∞-Atomic Geometric Morphisms

I’m not sure how important this is to the whole story, but the second-to-last paragraph could be construed as suggesting that an \infty-connected morphism in an \infty-topos is always an equivalence, which is false. Is there maybe some wiggle room for “\infty-atomic” to differ from “local homeomorphism” when Whitehead’s theorem fails?

Posted by: Tim Campion on June 24, 2018 11:53 PM | Permalink | Reply to this

Re: ∞-Atomic Geometric Morphisms

I had that thought too. I don’t know, maybe. One problem is that \infty-truncated objects don’t always behave as nicely as nn-truncated ones. It would be nice to have some examples; I always feel a little bit at sea when talking about the non-hypercomplete world.

On the other hand, there are situations in which despite the lack of hypercompleteness the appropriate limit of “nn-truncated” as nn\to \infty is “all” rather than “\infty-truncated”. For instance, in one of the constructions of the propositional truncation as a sequential colimit, there is a related sequence of maps get more and more connected as you go out, which originally led me to think that in the limit they would be only \infty-connected; but in fact in the limit they turn out to be equivalences.

Posted by: Mike Shulman on June 25, 2018 12:04 AM | Permalink | Reply to this

Re: ∞-Atomic Geometric Morphisms

If you’re looking for examples of non-hypercomplete \infty-topoi, look no further than the \infty-topos of parameterized spectra, or more generally the \infty-topos of (finitary) nn-excisive, unpointed functors TopTopTop \to Top for fixed nn. To me, this has become one of the most fundamental examples of an \infty-topos out there! Similarly, I think the tangent \infty-topos of an arbitrary \infty-topos is rarely hypercomplete.

Posted by: Tim Campion on June 25, 2018 12:18 AM | Permalink | Reply to this

Re: ∞-Atomic Geometric Morphisms

Actually, given that the topos of nn-excisive functors can be defined using a site of finite sets and surjections, maybe non-hypercompleteness is something important to consider when it comes to atomic geometric morphisms.

How about this: should the forgetful functor from paramterized spectra to spaces be an atomic geometric morphism?

Posted by: Tim Campion on June 25, 2018 12:24 AM | Permalink | Reply to this

Re: ∞-Atomic Geometric Morphisms

Okay, so that forgetful functor is certainly locally \infty-connected (indeed, it’s cohesive). But is it even (1)(-1)-atomic, i.e. does its inverse image functor preserve the subobject classifier?

Posted by: Mike Shulman on June 25, 2018 5:22 AM | Permalink | Reply to this

Re: ∞-Atomic Geometric Morphisms

I think it is.

Note that a stable \infty-category has no non-equivalence monomorphisms.

Let SS denote the \infty-category of spaces, and let TSTS denote the \infty-category of parameterized spectra. The forgetful functor U:TSSU: TS \to S preserves limits, and in particular monomorphisms. Any morphism f:(E,X)(F,Y)f: (E,X) \to (F,Y) in TSTS factors as a fiberwise morphism f 0f_0 in T XST_X S (the category of XX-parameterized spectra) followed by f 1f_1, a cartesian lift of UfUf. Now, if ff is mono, then UfUf is mono, and so f 1f_1 is mono. Then by pasting properties of pullback squares, f 0f_0 is mono in TSTS, and hence also in the stable category T XST_X S. So f 0f_0 is an equivalence. So ff is cartesian over a monomorphism in SS. Thus the forgetful functor Mono(TS)/(F,Y)Mono(S)/YMono(TS)/(F,Y) \to Mono(S)/Y is an equivalence.

Posted by: Tim Campion on June 25, 2018 1:53 PM | Permalink | Reply to this

Re: ∞-Atomic Geometric Morphisms

Hmm. And maybe a similar argument would work to show that it’s nn-atomic for all finite nn.

I think the main reason I’m skeptical of this “intermediate notion of atomic” is that for other classes of geometric morphisms, like locally connected, proper, local, etc., it seems clear that the “important” notion for \infty-toposes is the one that refers to the whole (,1)(\infty,1)-topos rather than playing games with \infty-truncatedness. There might be intermediate notions in those cases too, but we haven’t dignified them with names like “locally \infty-connected” or “\infty-proper”; if I had to talk about them, I might invent a name like “(1)(\infty-1)-proper”. I suppose we could consider renaming everything and using something like “(+1)(\infty+1)-proper” for the stronger versions, but it seems more natural to me for the condition that refers to the whole \infty-topos to be the one with the prefix \infty.

Posted by: Mike Shulman on June 25, 2018 2:53 PM | Permalink | Reply to this

Re: ∞-Atomic Geometric Morphisms

More generally, the n-excisive topoi are k-atomic for all n and all k.

This is essentially a consequence of the results in the appendix of our paper. There we do only the (-1)-truncated case (so the case you are discussing here but generalized to n-excisive functors), but it’s not too hard to deduce that this remains true for higher truncation levels.

For what it’s worth, in discussing the question of atomic \infty-topoi with Mathieu Anel, we came to the same conclusion: a geometric morphism whose inverse image is logical should necessarily be étale. I forget exactly what the line of reasoning was, but probably not too far from what you describe here, Mike.

Posted by: Eric Finster on June 29, 2018 9:18 AM | Permalink | Reply to this

Re: ∞-Atomic Geometric Morphisms

Thanks! Do you think there is any notion of “(1)(\infty-1)-atomic” that could lie in between “kk-atomic for all finite kk” and étale?

Posted by: Mike Shulman on June 29, 2018 12:39 PM | Permalink | Reply to this

Re: ∞-Atomic Geometric Morphisms

Hmmm. Good question. Short answer is I’m not sure. At least, no example of such a thing comes immediately to mind. But if I run across one, I’ll be sure to report back. :)

Posted by: Eric Finster on June 29, 2018 2:47 PM | Permalink | Reply to this

Re: ∞-Atomic Geometric Morphisms

I guess my inclination would be to use the unadorned term (e.g. “proper geometric morphism”) for the notion referring to the whole topos. But maybe this clashes too much with existing usage.

Alternatively, one could use the prefix ω\omega – e.g. “ω\omega-proper geometric morphism” would mean “nn-proper geometric morphism for all nn”, and save \infty for the notion referring to the whole topos. Actually, I kind of like that – ω\omega connotes the notion of infinity meaning that says “bigger than all finite things”, while \infty connotes a more absolute,”bigger” notion of infinity.

Posted by: Tim Campion on June 25, 2018 3:27 PM | Permalink | Reply to this

Re: ∞-Atomic Geometric Morphisms

I want to defend keeping the prefix ω\omega- to refer to actual ω\omega-categories, i.e. the things that nowadays some people call (,)(\infty,\infty)-categories because they started abusing “\infty-category” to mean (,1)(\infty,1)-category.

It would certainly make sense that when the objects under consideration are nn-topoi for some 0n0\le n\le \infty, the default meaning of “proper” would be “nn-proper”. That does clash with existing usage for 1-toposes: a “proper geometric morphism” between 1-toposes is actually a (1)(-1)-proper one (a 0-proper one being called ‘tidy’), while a “locally connected geometric morphism” between 1-toposes is actually a locally 00-connected one (a locally (1)(-1)-connected one being called ‘open’). True, in the \infty-case there’s less room for ambiguity, but I think there should always be the option of adding the prefix for disambiguation, and the prefix \infty- ought to refer to the “most natural” notion for \infty-toposes.

Posted by: Mike Shulman on June 25, 2018 5:10 PM | Permalink | Reply to this

Re: ∞-Atomic Geometric Morphisms

Following Dubuc and others, atomic toposes play a central role in understanding Grothendieck’s Galois theory.

So one thing one might expect \infty-atomic toposes to be good for is for setting up an \infty-categorical version of Grothendieck’s Galois theory. I’m not sure, but one thing I might expect such a theory to do is to give a Galois-theoretic / covering-space-based account of the shape of an \infty-topos.

For these purposes, it would be a bit underwhelming if the right notion of \infty-atomic geometric morphism were simply “etale”.

Which is a conundrum. Because I think I found Mike’s discussion here to be pretty convincing for the most part. At least, I also feel an aversion to using a notion which explicitly hinges on truncated concepts.

Posted by: Tim Campion on February 8, 2021 9:29 PM | Permalink | Reply to this

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