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sarahzrf · 3 years ago

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First portion of an incomplete mathpost

I've tweeted a couple of times now that "forming a filterquotient is just taking the stalk of a stack". Let me explain what that means, for people who know what stalks, stacks, and elementary topoi are, but not filterquotients. You should probably also know what a filter is, though.

Before anything else, let me give a "direct" definition of what a filterquotient of a topos is, so that once the rest of the post works its way back around, you'll be able to see the equivalence. Let $\mathcal{E}$ be an elementary topos and $\Phi \subseteq \operatorname{Sub}(1)$ be a filter in the poset of subobjects of $\mathcal{E}$'s terminal object. For $U$ a subobject of $1$ and objects $X$, $Y$ we define the set $\operatorname{Hom}_U(X, Y)$ of "$U$-partial morphisms" from $X$ to $Y$ to be $\operatorname{Hom}(U \times X, Y)$; if $V \subseteq U$, then there is an evident "restriction map" from $\operatorname{Hom}_U(X, Y)$ to $\operatorname{Hom}_V(X, Y)$. We define the filterquotient $\mathcal{E}_\Phi$ to be the category with objects the same as those of $\mathcal{E}$ and where the hom-set from $X$ to $Y$ is $\varinjlim_{U \in \Phi} \operatorname{Hom}_U(X, Y)$, with composition given by composition in $\mathcal{E}$ on representatives. This new category will still be an elementary topos, although it need not be a Grothendieck topos even if $\mathcal{E}$ was one, and there is an evident functor $\mathcal{E} \to \mathcal{E}_\Phi$ which is a logical morphism. Exercise to understand this definition: Work out $\mathcal{E}_\Phi$ looks like in the case where $\mathcal{E}$ is $\mathrm{Set}^{\mathbb{N}}$ and $\Phi$ is the filter of cofinite subsets of $\mathbb{N}$.

Now we can work our way to seeing how this is a case of taking the stalk of a stack.

First of all, I'm using "stalk" in a slightly generalized sense, and I'm not completely sure whether any of its standard meanings actually apply literally to the case of filterquotients. To be a little more precise about the kind of generalization in question: traditionally, a stalk is something that you take of a sheaf and at a point, and while it's obvious that I'm generalizing along the first axis already by talking about stacks instead of just sheaves, I'm actually also generalizing along the second axis, by talking about stalks at filters instead of just stalks at points. Before I touch on filterquotients or stacks at all, let me begin by explaining what this latter generalization means.

The traditional first definition of a stalk goes something like this: you start with a space $X$, a sheaf $F$ on $X$ valued in some category $C$ of algebraic structures, and a point $p \in X$. Then you build the "stalk of $F$ at $p$" by first collecting all local sections of $F$ defined on all open neighborhoods of $p$, and then modding out by a "defines the same germ at $p$" relation. This is a very nuts-and-bolts definition that makes the content of the stalk very clear. However, it is not very categorypilled, which makes me sad. We can reformulate this definition to make it clear that the stalk is a certain colimit; this also allows generalization of the notion of stalk to [pre]sheaves valued in non-concrete categories. Let's be precise: let $N_p \subseteq O(X)$ be the collection of open neighborhoods of $p$, considered as a full subcategory of $O(X)$. Then we can restrict $F : O(X)^{\mathrm{op}} \to C$ to have domain $N_p^{\mathrm{op}}$. Considering this functor as a diagram in $C$ and taking its colimit, we get the stalk of $F$ at $p$.

The first thing to notice here is that we only use $p$ in order to form $N_p$. In particular, for any collection of opens $S \subseteq O(X)$, we can carry out the construction of restricting $F$ to have domain $S^{\mathrm{op}}$ and taking the colimit of the resulting diagram, so long as $C$ is cocomplete enough; nothing about these latter steps is reliant on us having an actual point in hand. The second thing to notice is that while we can do this for any collection of opens, $N_p$ does have some special properties that make taking stalks particularly nice. Most importantly, it is a filter in $O(X)$---it's called the "neighborhood filter" of $p$, to be precise. I actually can't speak off the top of my head to the significance, in this context, of the "upward closed" half of the definition of filters, but the "downward directed" half is crucial: it means that the category $N_p^{\mathrm{op}}$ is filtered, so that a colimit indexed by it is a filtered colimit. Filtered colimits are lovely for many reasons; one very very important one is that while a general colimit of algebraic structures need not be computed on the underlying sets as the same colimit---think of how a coproduct of groups is not computed on underlying sets as a coproduct of sets---a filtered colimit of algebraic structures is computed on underlying sets as the same colimit and then simply re-equipped with operations. Indeed, this is what allows the nuts-and-bolts description of stalks as a certain disjoint-union-and-quotient to work for sheaves valued in any category of algebraic structures, without needing to speak directly of abstract colimits. For this reason among others, I claim that being a filter is a desirable condition to impose on a collection of opens for it to be worth thinking about "stalks" there. So: given a sheaf (or presheaf) $F : O(X)^{\mathrm{op}} \to C$, for any filter $\Phi \subseteq O(X)$, let's call the colimit of the diagram $F|_{\Phi^{\mathrm{op}}}$ the stalk of $F$ at $\Phi$, or stalk of $F$ on $\Phi$. In the case that $C$ is a category of algebraic structures, this looks pretty much the same as usual, thanks to $\Phi$ being directed: we gather up all of the sections of $F$ on opens belonging to $\Phi$, then quotient by "agreeing on something in the intersection".

Now let me give two key sources of filters in $O(X)$ not of the form $N_p$ that are worth caring about, both to justify this generalization and to give an intuition for what it can look like.

We can form a neighborhood filter of any subset of $X$, not just of a point: if $A \subseteq X$ is any subset, not necessarily open or closed, let $N_A$ be the collection of opens that are supersets of $A$. This will always be a filter. This is a pretty natural object. For example, we can rephrase one classic result about paracompact spaces as follows: For any sheaf of sets, there is a map from the stalk on $N_A$ to the set of sections on $A$ of the étalé space, and for closed $A$ and paracompact $X$, this map is a surjection.

In many spaces, typically non-compact ones, we can form various filters of "neighborhoods of infinity" of one kind or another. Taking the stalk at such a filter gives us the "germs at infinity" of the corresponding kind. For example, if $X$ is the real line, we can let $\Phi$ be the collection of all opens which contain $(a, \infty)$ for some $a$. Then the stalk at $\Phi$ of, say, the sheaf of real-valued functions, really is the collection of germs at positive infinity.

(An aside on filters: In general, as in both of these examples, one can think of a filter as the collection of everything that contains a given point, or more generally, a given region. This makes it a kind of "formal intersection"; for example, a point is part of the region described by a filter (and we can evaluate germs on the filter at that point) iff it is an element of the literal intersection of all of the opens of the filter. However, the fact that the intersection is only formal means that a filter generally carries more information, and is a "larger region" than, the literal intersection of all of the opens in it. For example, the neighborhood filter of a point, or more generally a subset, can be seen as an "infinitesimal neighborhood" of that point or subset, while in a space satisfying mild separation axioms, the intersection of all of the opens in this filter is just the point or subset itself. And a filter of neighborhoods of infinity can be seen as a point at infinity, or really an infinitesimal neighborhood of such, while the intersection of all of the opens in such a filter is generally empty. In this view on filters, the stalk on a filter reveals itself as "local sections defined on the region described by the filter"; it is a colimit because we are taking a formal intersection (limit), passing it through a contravariant functor, and then evaluating.)

Much more generally than just the case of topological spaces, if $(D, J)$ is any site where $D$ is a poset, we can take the stalk of a sheaf on $(D, J)$ at a filter in $D$. We can generalize this further still, to non-poset sites, with a definition motivated by topos theory, but it will not be necessary for what I want to talk about and would require a pretty lengthy detour.

Having explained what it means to take a stalk at a filter, let me now talk about taking stalks of stacks. The definition that follows is just what seems to me to be what taking the stalk of a [pre]stack should clearly mean; I have not seen it in any sources. That said, I would not be surprised if it is out there, given that I haven't read a ton of stuff that uses stacks!

Consider a [pre]stack on a site $(D, J)$ in the form of a pseudofunctor $\mathcal{S} : D^{\mathrm{op}} \to \mathrm{Cat}$ (or more generally, we could consider stacks valued in a bicategory other than $\mathrm{Cat}$). If $D$ is a poset and $\Phi$ is a filter in $D$, then the same construction as above makes perfect sense: we have a cofiltered diagram in $D$ which we send through $\mathcal{S}$ to get a filtered diagram in $\mathrm{Cat}$. The stalk of $\mathcal{S}$ on $\Phi$ should be the 2-colimit of this filtered diagram. Filteredness allows us to give a pleasantly explicit formula for this 2-colimit for the case of values in $\mathrm{Cat}$, and for the case of values in any "concrete" bicategory where filtered 2-colimits are computed on underlying categories: The objects of $\mathcal{S}_\Phi$ are pairs $(d \in D, A \in \mathcal{S}(d))$, and the hom-sets are given by $\operatorname{Hom}((c, A), (d, B)) = \varinjlim_{e \leq c, d} \operatorname{Hom}(A|_e, B|_e)$. Composition is defined on representatives by... [TODO finish definition, double check, check coherence issues, justify the construction]

Let's do a simple example computation. Let $X$ be your favorite topological space, $\mathcal{S}$ be the stack of locally trivial real vector bundles on it (as a stack of groupoids), and $p$ be any point of $X$; we compute $\mathcal{S}_p$. We begin by establishing the connected components. Let $V_n = (X, X \times \mathbb{R}^n) \in \mathcal{S}_p$; I claim that every object of $\mathcal{S}_p$ is isomorphic to exactly one $V_n$. [TODO ...] Then, we establish the automorphism groups of each $V_n$. Since a pullback of a trivial bundle is trivial [TODO _fucking_ coherences] we have $\operatorname{Hom}(V_n, V_n) \cong \varinjlim_{p \in U} \operatorname{Hom}(U \times \mathbb{R}^n, U \times \mathbb{R}^n) \cong \varinjlim_{p \in U} \operatorname{Hom}(U, \operatorname{GL}(n))$. Of course, this is just the group of germs at $p$ of $\operatorname{GL}(n)$-valued functions.

Now we can start talking about which stack we're going to be taking a stalk of.

If $C$ is any category with pullbacks, there is a prestack on it of fundamental interest, called the "codomain fibration" when viewed as a category fibered in categories and the "self-indexing" when viewed as a contravariant pseudofunctor to $\mathrm{Cat}$. In its guise as the codomain fibration, it is $\operatorname{cod} : \operatorname{Arr}(C) \to C$. In its guise as the self-indexing, it is the pseudofunctor defined on objects by $S(A) = C/A$ and on morphisms by $S(f) = f^*$. For $C$ a Grothendieck topos, the self-indexing is in fact a stack with regard to the canonical topology on $C$---which, for a Grothendieck topos, is the one where coverings are families of morphisms such that the induced map from a coproduct is an epi. More generally, for $C$ an elementary topos (in fact, much weaker conditions suffice), the self-indexing is still a stack with respect to a "finitary" version of this topology, where the coverings are those finite families of morphisms such that the induced map from a coproduct is an epi. Finally, for the purposes we are interested in, we want to consider the restriction of the self-indexing to the poset of subobjects of $1$ (it is legitimate to call this "restriction" because this poset is equivalent to the full subcategory of $C$ on objects whose unique map to $1$ is monic). In the case of a Grothendieck topos, the resulting prestack continues to be a stack on this poset if we give it the Grothendieck topology where a family of elements covers another element just when that element is their sup (equivalently, is leq their sup). In the case of an elementary topos, arbitrary joins may not exist; but we will still be a stack for finite covering families.

As an example, let $X$ be a topological space and consider the Grothendieck topos $\operatorname{Sh}(X)$. Under the equivalence of $\operatorname{Sh}(X)$ with the category of étalé spaces over $X$, we can view the self-indexing of $\operatorname{Sh}(X)$ as a stack on that category instead. And given the pseudonatural-in-$F$ equivalence $\operatorname{Sh}(X)/F \simeq \operatorname{Sh}(\operatorname{Et}(F))$, we see that the resulting stack is the one which assigns each étalé space the category of sheaves on it. The terminal étalé space is $X$ itself; subobjects are just opens of $X$, so the subobject poset is $O(X)$, and the sup-based Grothendieck topology described above is the usual one on $O(X)$, so the restricted stack is a stack on $X$ in the usual sense. In particular, it is the stack $U \mapsto \operatorname{Sh}(U)$.

Finally, we can put the pieces together!

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