The n-Category Café

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?