The n-Category Café

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.)

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 $f_!$ 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.

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.

All atomic toposes $E$ are Boolean.

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

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

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.

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.

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 $\mathcal{Y} \to \mathcal{X}$ is an etale geometric morphism, then there is a surjective family $\mathcal{Z}_i\to \mathcal{X}$ of open embeddings such that $\mathcal{Y} \times_\mathcal{X} \mathcal{Z}_i \to \mathcal{Z}_i$ is an equivalence?

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.

Funny – I learned from you that openness of a geometric morphism sits at the bottom of the latter of locally $n$-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?

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.

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 $S$.

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?

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 $E$ 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 $E$ 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” $E$ is equivalently one such that there are no nontrivial embeddings into $E$. This means that the only Lawvere-Tierney topologies on $E$ are the trivial one and the inconsistent one. Equivalently, there is a canonical bijection between the Lawvere-Tierney topologies on $E$ and the Lawvere-Tierney topologies on $Set$. 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 $E$ and the pullback of the subobject classifier in $Set$, i.e. to saying the topos is atomic.

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

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 $T_0$ space can form a Boolean topos is if the space is discrete. My bad!

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).

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 $G$, under which conditions is the category of (equivariant) sheaves on $G$ a connected Grothendieck topos?

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

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

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

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!

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 $Set^G$ is connected for any (discrete) groupoid $G$ that is connected in the categorical sense; but the topos $Sh(X)$ is also connected for any space $X$ 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 $Set^X$ for any discrete set $X$ 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 $I$-indexed family of disjoint sets of full branches where $I$ is an (externally specified) set with cardinality greater than the continuum. This has no points because in a model of this theory in $Set$ there can be at most continuum-many such branches; but it has nontrivial models in the classifying topos of bijections $I\cong \mathbb{N}$ 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 $Sh(X)$ is atomic iff $X$ 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 $n\to \infty$: if $n$-atomic meant that the locales of $k$-morphisms were discrete for $k\le n$, 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 $\infty Gpd$.

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 $Loc$ (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 $G_1 \rightrightarrows G_0$ (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 $Sh(G_0) \to Sh(G_1\rightrightarrows G_0)$.

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).

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.

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.

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

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

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

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

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”.

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 $p$-groups are all nilpotent, and how finite nilpotent groups all split as a product of $p$-groups for various $p$ (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 $\mathrm{S}_6$. So far I don’t really understand that claim; I’m still struggling to understand the exceptional triple cover of $\mathrm{A}_6$, 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.