July 24, 2013

Higher Compactness

Posted by Mike Shulman

Higher category theorists enjoy categorifying stuff. That word can mean lots of things, but one thing it can mean is to take some familiar 1-category, embed it naturally into a higher category, and then try to extend some of the appropriate theory. And one of the threads in topos theory is to do this with topology, where we start with the 1-category of topological spaces.

So how do we categorify topological spaces? The first thing we do (normally) is to replace them by locales. A locale is a lattice that behaves like the open-set lattice of a topological space: it has finite intersections, arbitrary unions, and the former distribute over the latter. A big class of topological spaces are uniquely determined by their open-set lattice, so this doesn’t lose very much, and it makes it easier to categorify since a lattice is, after all, a (0,1)-category. (It’s possible to skip this step by using ionads, but let’s not get into that today.)

Now for lots of reasons that I also don’t want to get into, the appropriate categorification of a locale is a Grothendieck topos: a locally presentable category with finite limits and arbitrary colimits, where the former distribute over the latter in some well-defined way. Hopefully this is mostly intuitive, limits and colimits being categorifications of intersections and unions in lattice theory. This definition generalizes to higher categories as well, at least to the $(n,1)$-case, and we have full embeddings $Locale = (0,1)Topos \hookrightarrow (1,1)Topos \hookrightarrow (2,1)Topos \hookrightarrow \dots \hookrightarrow (\infty,1)Topos.$ However, these embeddings are not the identity, i.e. the lattice that is a locale is not itself a (1,1)-topos, but has to be made into one by passing to sheaves. Similarly, for $n\lt m$, an $(n,1)$-topos is not itself an $(m,1)$-topos, but has to be made into one by passing to $(m,1)$-sheaves.

The question I want to discuss today is, how do we categorify notions like compactness and local compactness from topology to higher toposes?

Compactness of a space $X$ is clearly a property of its open-set lattice $O(X)$: whenever we have a family of opens $(U_i)_i$ such that $\bigcup_i U_i = \top$ is the top element of the lattice, then there is a finite subfamily with that property. An equivalent way to say that is if we have a directed family of opens $(U_i)_i$ such that $\bigcup_i U_i = \top$, then there is some $i$ with $U_i = \top$. (To go from the first to the second, consider the directed family of finite unions.)

And another equivalent way to say that is that the map $r_\ast : O(X) \to O(1) = \{\bot,\top\}$, defined by $r_\ast(\top) = \top$ and $r_\ast(U)=\bot$ for all $U\neq \top$, preserves directed unions. This map $r_\ast$ is the right adjoint to the inverse image $r^\ast : O(1) \to O(X)$ induced by the continuous map $r:X\to 1$, so it’s actually a fairly canonical sort of beast. Moreover, this phrasing tells us that compactness is not really a property of the space $X$, but rather of the map $X\to 1$. For an arbitrary continuous map $f:X\to Y$, we can consider the property that $f_\ast$ (the right adjoint of the inverse image $f^\ast:O(Y) \to O(X)$) preserves directed unions, and this defines the property of $f$ being a proper map.

Now for an $(n,1)$-topos $X$, let’s write $O_n(X)$ for the $(n,1)$-category that it “is”. A morphism of such toposes $f:X\to Y$ is essentially by definition an adjoint pair of functors $f^\ast: O_n(Y) \rightleftarrows O_n(X) : f_\ast$, and so we have an obvious categorification of properness: we can ask that $f_\ast$ preserve filtered colimits. In particular, the terminal topos $1$ has $O_n(1)$ being the category of $(n-1)$-groupoids (0-groupoids being sets and $(-1)$-groupoids being truth values), and $r_\ast:O_n(X) \to O_n(1)$ is the “global sections” functor $Hom(1,-)$, so it is natural to consider “compactness” of $X$ to mean that this functor preserves filtered colimits.

The interesting thing is that if $X$ is an ordinary topological space (or locale), then compactness in the usual sense (i.e. $r_\ast : O_0(X) \to O_0(1)$ preserving directed unions) does not imply that its corresponding $(n,1)$-topos of sheaves is compact in this sense (i.e. $r_\ast : O_n(X) \to O_n(1)$ need not preserve filtered colimits for $n\gt 0$)! The following counterexample can be found in Sketches of an Elephant: let $X$ be the quotient of $[0,1] + [0,1]$ with both copies of $0$ identified and both copies of $1$ identified. Then $X$ is a quotient of a compact space, hence compact, but it is not “strongly compact”: if $Y_i$ is the quotient of $[0,1] + [0,1]$ with both copies of $x$ identified for $x\lt 1/2^i$ or $x \gt 1-1/2^i$, then the colimit of the diagram $Y_0 \to Y_1 \to Y_2 \to \dots$ is the terminal object, which has a global section, but none of the $Y_i$ has a global section.

In 1-topos theory, a morphism $f:X\to Y$ of 1-toposes such that $f_\ast$ preserves filtered colimits is said to be tidy, and if $Y=1$ then $X$ is said to be strongly compact. I found this very mysterious until I learned about $(n,1)$-toposes, and realized that compactness/properness and strong-compactness/tidiness were just the cases $n=0$ and $n=1$ of a tower of conditions on $(n,1)$-topoi for $0 \le n \le \infty$.

Presumably there are toposes, even toposes of sheaves on topological spaces, which are $n$-compact but not $(n+1)$-compact for all $n$. It would be interesting to try to find some. However, if a space is not only (0-)compact but also Hausdorff (which is equivalently to say that its diagonal $X\to X\times X$ is also proper), then it is automatically $n$-compact for all $n$, including $n=\infty$. So this hierarchy is only nontrivial for “ill-behaved” spaces (for some meaning of “ill-behaved”).

There is also a “dual” hierarchy which yields a tower of more familiar conditions on topological spaces. It turns out that $n$-compactness is equivalent to asking that the functor $f_\ast:O_n(X) \to O_n(Y)$ is an $O_n(Y)$-indexed functor. If we instead ask that the inverse image functor $f^\ast$ have a left adjoint $f_!$ which is $O_n(Y)$-indexed, then we get the definition of when $f$ is locally $(n-1)$-connected. For 1-toposes, we get ordinary local connectedness, while for 0-toposes we get local inhabitedness. The latter is also called openness of $f$. Again, I found the connection between openness and local connectedness in 1-topos theory very confusing until I realized that they were the cases $n=0,1$ of a hierarchy.

Now we come to something that is more conjectural: what about local compactness? It’s a bit trickier to characterize local compactness of a space in terms of its open set lattice, but it can be done. In any lattice with directed unions, define $U\ll V$ (pronounced $U$ is way below $V$) to mean that whenever $V = \bigcup_i W_i$ is the union of a directed family $(W_i)_i$, then there is some $i$ with $U \le W_i$. In $O(X)$, this holds if there is a compact set $K$ with $U\subseteq K \subseteq V$ (and the converse holds if $X$ is locally compact). Then local compactness of $X$ can be phrased as the requirement that every $V$ is the join of all opens that are way below it.

This gives the definition of a locally compact locale. A more categorification-friendly characterization involves the lattice $Idl(O(X))$ of order-ideals in $O(X)$, i.e. directed lower sets. The “principal ideal” map $O(X) \to Idl(O(X))$ is the free cocompletion of $O(X)$ under directed unions, and so since $O(X)$ has directed unions, this map has a left adjoint $\bigcup : Idl(O(X)) \to O(X)$. It turns out that $X$ is locally compact, in the sense described above, just when $\bigcup$ has a left adjoint.

When we categorify, the role of $Idl(O(X))$ is played by the Ind-completion of $O_n(X)$, which is again its free cocompletion under filtered colimits. Now we can ask whether $colim : Ind(O_n(X)) \to O_n(X)$ has a left adjoint, and call this something like local $n$-compactness of $X$.

As before, it turns out that local (0-)compactness does not imply local 1-compactness: essentially the same counterexample applies. And again as before, there is a stronger notion which does suffice for local 1-compactness, called stable local compactness. This means that the way-below relation is preserved by finite intersections (including the empty intersection, which implies that the space is compact). In the topos-theoretic literature, local 1-compactness of a locale is called meta-stable local compactness.

As far as I know, local $n$-compactness for $n\gt 1$ has not been studied (but I would love to be wrong!). Here are some basic questions about it that would be nice to answer:

• Are there examples of locally $n$-compact spaces that are not locally $(n+1)$-compact, for all $n$?
• Can local $n$-compactness, at least for spaces or locales, be expressed in a way that looks more evidently like a “local” sort of $n$-compactness?
• Are stably locally compact locales automatically locally $n$-compact for all $n$, just as compact Hausdorff locales are automatically $n$-compact for all $n$?
• In the cases $n=0,1$, the locally $n$-compact $(n,1)$-toposes are exactly the exponentiable ones. Is this true for $n\gt 1$ as well?
Posted at July 24, 2013 9:30 PM UTC

TrackBack URL for this Entry:   http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/2635

Re: Higher Compactness

Does anything interesting happen to the compactness theorem in logic when you increase $n$?

In view of the compactness-ultrafilter connection, perhaps we need the answer to your query to Tom.

Posted by: David Corfield on July 25, 2013 9:40 PM | Permalink | Reply to this

Re: Higher Compactness

That’s a good question! Wikipedia says that the compactness theorem interpreted topologically is about Stone spaces, which are compact Hausdorff. If that’s right, then they are automatically also $n$-compact for all $n$. So we might be using $n$-compactness already without realizing it. (-:

Posted by: Mike Shulman on July 25, 2013 9:45 PM | Permalink | Reply to this

Re: Higher Compactness

Wonderful! This makes me wonder about categorifying downward Löwenheim–Skolem. With $n$-compactness for all $n$ this could make possible a version of Lindström’s theorem in an $(\infty,1)$-topos. I’m out of my league here so just spitballing, but something like this would certainly help shed some light on the internal logic of $(\infty,1)$-topoi.

Posted by: Patrick Durkin on July 25, 2013 10:58 PM | Permalink | Reply to this

Re: Higher Compactness

So there’s a compactness theorem for homotopy type theory? Could it give non-standard models?

Posted by: David Corfield on July 26, 2013 4:48 PM | Permalink | Reply to this

Re: Higher Compactness

No, I don’t know of any such theorem. One obvious approach to it would be to try to construct filterquotients of $(\infty,1)$-topoi, but I don’t know that anyone’s done that.

Posted by: Mike Shulman on July 26, 2013 5:20 PM | Permalink | Reply to this

Re: Higher Compactness

Since we have a model by Kan complexes and Kan fibrations in the topos of simplicial sets, another approach might be show you can always cook up a submodel for a finite set of types (I’m fuzzy on the logic here. Is this what we want?). Does the obvious restriction of the singular functor $S:$Top$\to$ sSet (objects in the image are Kan complexes) achieve this?

Posted by: Patrick Durkin on July 26, 2013 6:36 PM | Permalink | Reply to this

Re: Higher Compactness

I don’t know what you mean.

Posted by: Mike Shulman on July 26, 2013 6:55 PM | Permalink | Reply to this

Re: Higher Compactness

Haha yes, oops. I see now why this makes no sense. I need to read up more before I open my mouth again :x

Posted by: Patrick Durkin on July 26, 2013 7:09 PM | Permalink | Reply to this

Re: Higher Compactness

I like this question.

I don’t know many answers, but I can give you one data point: I believe that a locally compact Hausdorff (topological) space satisfies your condition for every $n$, and is also exponentiable in the world of $\infty$-topoi.

Does the proof of exponentiability in the cases $n \leq 1$ proceed directly from your hypothesis, or do you first use it to construct “individual” compact sets? The argument for exponentiability that I know in the case $n = \infty$ seems to need the latter (coded as Pro-objects of your locale, say).

Posted by: Jacob Lurie on July 25, 2013 11:14 PM | Permalink | Reply to this

Re: Higher Compactness

The proofs that I’ve seen proceed by first constructing the exponential into the object classifier, and then other exponentials out of that. This is how it’s done in B4.3 and C4.4 of Sketches of an Elephant, for instance. There are probably other proofs, however.

Posted by: Mike Shulman on July 26, 2013 12:04 AM | Permalink | Reply to this

Re: Higher Compactness

Let me just spitball, focusing on classifying maps into the object classifier (I think the reduction to that case is pretty formal). Let me use the term “geometric morphism” to mean a functor which preserves small colimits and finite limits (which is the opposite of the usual convention). To address your fourth question, we might want to show that if $\mathcal{X}$ is an $\infty$-topos satisfying your hypothesis, then there is another $\infty$-topos $\mathcal{X}'$ having the following universal property: objects of $\mathcal{X} \times \mathcal{Y}$ (product formed in the world of $\infty$-topoi) are classified by geometric morphisms from $\mathcal{X}'$ to $\mathcal{Y}$.

The product can be described concretely as the $\infty$-category of $\mathcal{Y}$-valued sheaves on $\mathcal{X}$: that is, functors $\mathcal{X}^{op} \rightarrow \mathcal{Y}$ which preserve small limits. Our obstacle is that this notion is “infinitary”, but geometric morphisms into $\mathcal{Y}$ are only “allowed” to mention the preservation of finite limits.

In the case where $\mathcal{X}$ is the $\infty$-topos of sheaves on a locally compact (Hausdorff) topological space $X$, you can argue as follows: instead of describing $\mathcal{Y}$-valued sheaves on $X$ by their values on open subsets of $X$, you can instead describe them in terms of their values on compact subsets $K \subseteq X$. And the axiomatization of sheaves in terms of compact sets is in some sense simpler: you need to say that the value of a sheaf on a union $K \cup K'$ is the fiber product of the values on $K$ and $K'$ over $K \cap K'$, that the value on $\emptyset$ is final, and that the value on any compact set $K$ is given by the (filtered) direct limit of its values on all compact neighborhoods of $K$. The virtue of this description is that it references only finite limits in $\mathcal{Y}$: the infinite limits have been traded for infinite (filtered) colimits. And this is the sort of thing that can be classified by geometric morphisms from an $\infty$-topos into $\mathcal{Y}$.

Is there any hope of translating this sketch into an argument which only uses your hypothesis on $\mathcal{X}$? The trouble I see is that I would want to reference some poset (or category) of “compact pieces” of $\mathcal{X}$, but your condition doesn’t seem to tell me that any individual “compact pieces” exist: only that there are lots of filtered colimits of “open pieces” that behave as if they could be rewritten as colimits of “compact pieces” instead.

Posted by: Jacob Lurie on July 27, 2013 4:28 PM | Permalink | Reply to this

Re: Higher Compactness

Ah, I see what you’re getting at. The proof for 0- and 1-topoi does go sort of along those lines, finding a suitable small replacement for $\mathcal{X}$ determined by its local compactness. If I’m beta-reducing the proof correctly, then what you get in the case of a locally compact locale $X$ is the site $O(X)^{\mathrm{op}}$, with the topology in which a cosieve on an open $U\in O(X)$ is covering iff it contains all $V$ such that $U\ll V$.

In other words, in your description you can also use open sets rather than compact sets, making use of the way-below relation to define the appropriate infinite filtered-colimit condition.

The generalization of the way-below relation to 1-categories is “the profunctor of wavy arrows”. If $W:\mathcal{X} \to Ind(\mathcal{X})$ is the left adjoint of $colim$ (which is itself left adjoint to the Yoneda embedding $Y$), we define a profunctor $\mathcal{W}: \mathcal{X} ⇸ \mathcal{X}$ by $\mathcal{W}(a,b) = Ind(\mathcal{X})(Y(a),W(b)).$ An element of $\mathcal{W}(a,b)$ is called a wavy arrow from $a$ to $b$. The profunctor $\mathcal{W}$ is an idempotent comonad in the bicategory of profunctors, and in particular, it comes with a map of profunctors $\mathcal{W} \to Hom_{\mathcal{X}}$, so that every wavy arrow has an underlying straight arrow. We then define a topology on $\mathcal{X}^{\mathrm{op}}$ (or a small subcategory thereof) by declaring a cosieve on $U$ to be covering if it contains the underlying straight arrows of all wavy arrows with domain $U$, and proceed analogously.

Posted by: Mike Shulman on July 28, 2013 6:41 AM | Permalink | Reply to this

Re: Higher Compactness

So, if I’m understanding you correctly, in the locally compact case what you’re saying can be paraphrased as “we don’t need to use all compact sets, it suffices to use those which arise as closures of open sets”, and suggesting a way of “using” them which might generalize to other contexts.

So, in the context I was describing before, we want to describe the limit-preserving functors $\mathcal{X}^{op} \rightarrow \mathcal{Y}$. Given such a functor $\mathcal{F}$, we can define a new functor $\mathcal{F}': \mathcal{X}^{op} \rightarrow \mathcal{Y}$ by performing a left Kan extension along your profunctor of wavy arrows. We might then hope that the construction $\mathcal{F} \mapsto \mathcal{F}'$ is fully faithful, and that its essential image can be characterized by conditions involving the preservation of small colimits and finite limits (which will guarantee the existence of a classifying $\infty$-topos, at least modulo set-theoretic technicalities; it seems unreasonable in the $n=\infty$-case to require that it be handed to us as a sheaves on a Grothendieck site).

I tried playing around with this a little, and it isn’t obvious to me that the construction is fully faithful. One can attempt to mimic the proof that “sheaves on locally compact spaces are determined by their values on compact sets”, but the argument that I know uses in an essential way the fact that given a compact set $K$, the collection of open neighborhoods of $K$ is directed (under reverse inclusion). I’d then want the analogous statement “given $U$, the category of wavy arrows with domain $U$ is (op)-filtered”, but this doesn’t seem likely to be true (in the locally compact case, we could have open sets that aren’t “way-below” anything).

Posted by: Jacob Lurie on July 28, 2013 11:55 PM | Permalink | Reply to this

Re: Higher Compactness

it seems unreasonable in the $n=\infty$-case to require that it be handed to us as a sheaves on a Grothendieck site.

Yeah. Once again it seems that we could really use a more general theory of “$\infty$-sites” that suffices to present all $(\infty,1)$-topoi.

I don’t have any particular ideas myself for how to proceed without such a thing in this case.

Posted by: Mike Shulman on July 31, 2013 8:33 PM | Permalink | Reply to this

Re: Higher Compactness

With a categorified notion of local compactness, could we have a categorified Pontrjagin duality?

If $\mathbb{Z}$’s role is played by the sphere spectrum, what could play that of the circle group?

nLab’s entry points to Duality for topological abelian group stacks and T-duality as a categorification, which extends

Pontrjagin duality for locally compact groups to abelian group stacks whose sheaves of objects and automorphisms are represented by locally compact groups.

But I guess I’m wondering if there is a duality for some kind of connective spectra.

Posted by: David Corfield on August 9, 2013 8:32 AM | Permalink | Reply to this

Post a New Comment