The n-Category Café

So in what sort of site can we replace covering families by single covers? Obviously our site has to have sufficiently many coproducts, to start with. And in this case, if $\{u_i \to x\}_{i\in I}$ is a covering family and the coproduct $\coprod_i u_i$ exists, then $\coprod_i u_i \to x$ is factored through by our given covering family, and hence is indeed, itself, a singleton covering family.

But if covering families are to be determined by singleton covers, we should also have the converse: if $\coprod_i u_i \to x$ is a singleton cover, then $\{u_i \to x\}_{i\in I}$ should be a covering family. As a particular case of this, the family of coprojections into a coproduct $\{u_i \to \coprod_j u_j\}_{i\in I}$ should be a covering family — and this case implies the general one, since $\{u_i \to x\}_{i\in I}$ is a composite of $\coprod_i u_i \to x$ with $\{u_i \to \coprod_j u_j\}_{i\in I}$.

Now recall from my last post that when the family of coprojections into a colimit in a site are covering, and some other condition holds, then that colimit is called postulated. For coproducts, the extra condition makes them stable and disjoint. In other words, the following are equivalent:

1. A subcanonical site with postulated coproducts.

2. A subcanonical site which is an extensive category, such that all families $\{u_i \to \coprod_j u_j\}_{i\in I}$ are covering.

We call a site (not necessarily subcanonical) with the latter property superextensive. The prefix “super-” is dual to the prefix “sub-” occurring in “subcanonical”: the latter means “contained in the canonical topology”, while the former means “containing the extensive topology”. The extensive topology on an extensive category is, of course, that generated by the families of coprojections $\{u_i \to \coprod_j u_j\}_{i\in I}$. (I guess it was Toby Bartels and me who came up with the terminology “superextensive”.)

So a superextensive site is a very natural context in which to talk about the relationship of covering families to singleton covers. There are also lots of examples: $Top$ or $Diff$ or $Sch$ with their usual topologies, or the canonical topology on any Grothendieck topos. This answers the first question.

Let me next answer the third question: what can’t we say, even in a superextensive site? The main thing we can’t say, but which occasionally people are misled into believing to be true, is that the category of sheaves (or stacks) on a superextensive site is equivalent to the category of sheaves (or stacks) for its singleton covers alone.

When you phrase things in the way that I have, this seems fairly clear: you also need the sheaf condition for the coproduct injections! Of course, since coproducts are disjoint, the sheaf condition for these is pretty simple: it just says that $X(\coprod_i u_i) \cong \prod_i X(u_i)$. (For the nullary coproduct, this means $X(\emptyset) \cong 1$.)

However, there are things we can say about superextensive sites, which in some cases does justify restricting attention to the singleton covers. One thing we can say is proven on the nlab page superextensive site: if $X$ is a sheaf for the extensive topology (that is, we have $X(\coprod_i u_i) \cong \prod_i X(u_i)$), then its sheafification with respect to singleton covers only is still a sheaf for the extensive topology, and therefore also a sheaf with respect to the whole superextensive topology. (Note that in general, it is not true that sheafification for one topology preserves sheaves for another topology.)

This is interesting because sheaves and stacks for the extensive topology are generally much easier to come by. In particular, any internal category in an extensive category is automatically a stack for the extensive topology. This obviously means that it a stack for the whole superextensive topology if and only if it is so for the singleton covers. But the theorem above (or more precisely, its presumed analogue for stacks) means that when stackifying such an internal category, it suffices to consider only singleton covers as well.

There’s another interesting thing we can say, which requires a bit of technology from my exact completions paper. First of all, there’s a size issue; so far I’ve been talking about “coproducts” quite loosely, but of course an extensive category with all coproducts is rarely small, whereas toposes of sheaves make the most sense for small sites. To be precise, let $\kappa$ be what I call an “arity class”, i.e. a class of small cardinalities closed under indexed sums and decompositions. For instance, $\kappa$ could be the set $\omega$ of finite cardinals, or the class “$\infty$” of all small cardinalities, or the set $\{1\}$. We then have notions of $\kappa$-extensive category and $\kappa$-superextensive site, which involve only coproducts of cardinalities belonging to $\kappa$.

Now we also have the notion of $\kappa$-ary site, which is a site whose covering families are (generated by) those of sizes belonging to $\kappa$. For instance, in a $\{1\}$-ary (or “unary”) site, all covering families are generated by singleton covers. There’s also a solution-set condition for finite limits in the definition of $\kappa$-ary site, but let’s ignore that by assuming all our sites have finite limits.

Obviously, it’s most natural to consider $\kappa$-superextensive sites which are also $\kappa$-ary. In fact, if $C$ is a $\kappa$-extensive category with finite limits, then there is a bijection between (1) unary topologies on $C$ and (2) $\kappa$-superextensive $\kappa$-ary topologies on $C$. From (2) to (1), we restrict to the singleton covers, while from (1) to (2) we add the families $\{u_i \to \coprod_j u_j\}_{i\in I}$.

Now every $\kappa$-ary site has a $\kappa$-ary exact completion: a universal morphism of sites into a category that is both Barr-exact and $\kappa$-extensive. For instance, the $\infty$-ary exact completion of a small site is precisely its topos of sheaves.

Finally, the theorem is the following: for any $\kappa$-superextensive $\kappa$-ary site $C$, the following two categories are equivalent:

1. The $\kappa$-ary exact completion of $C$.

2. The unary exact completion of the corresponding unary topology on $C$.

Roughly, this means that while the category of sheaves (the $\infty$-ary exact completion) of a superextensive site is not the same as the category of sheaves for its singleton covers alone, we can still recover that category of sheaves from the singleton cover topology by a unary exact completion. (I say “roughly” because if we take $\kappa=\infty$, then $C$ will not be small, which means that its $\infty$-ary exact completion is not the entire category of sheaves but only the “small sheaves”. But arguably, for large sites one should only be interested in the small sheaves anyway.)

Assuming that something analogous is true for stacks, this roughly justifies the use of internal categories and anafunctors (defined as usual using only singleton covers) in a superextensive site. Internal categories and anafunctors look very much like a construction of the “unary 2-exact completion”; thus the stack-version of this theorem would tell us that they also construct the $\infty$-ary 2-exact completion (roughly, the 2-topos of 2-sheaves) with respect to the whole superextensive topology. (I should mention that David Roberts’ thesis also includes some remarks on the value of anafunctors in superextensive sites.)

I think this is a nice application of the general theory of exact completion (and of its hypothetical 2-categorical analogue). In particular, even stating this theorem depends on having a notion of (unary) exact completion of sites, rather than merely of left-exact, weakly left-exact, or regular categories.