Stochastic Stalks

Stalks and points

Recall that a point of a topos \(\mathcal{E}\) is a geometric morphism from the topos \(Set\). My preferred way to think about this is to consider the sheaf topos \(Sh(X)\) on some (sober) topological space \(X\). Then given \(x \in X\) and a sheaf \(S\), we can form the stalk at \(x\)

\(S_x := \operatorname{colim}_{x \in U \subseteq X \text{ open}} S(U)\)

This determines the point uniquely, i.e if \((-)_x \simeq (-)_y\) then \(x = y\)
This is a left exact left adjoint \(Sh(X) \to Set\) - i.e part of a geometric morphism \(Set \to Sh(X)\).
All left exact left adjoints \(Sh(X) \to Set\) have this form.

Hence it makes sense to identify literal points of \(X\) with points of \(Sh(X)\) in the above sense.

Random points

We can think of a probability measure on a space \(X\) as a sort of “generalized point”, which has been “smeared out”. How can we lift this intuition to the level of \(Sh(X)\)? It seems sort of obvious that we shouldn’t expect this to work on the level of sets - they are somehow too “discrete” to capture quantitative information about probabilities. It’s actually worth mentioning here that any point of \(Sh(X)\) is determined by its action on sheaves represented by open sets, which must each be sent to either \(\emptyset\) or \(*\) (this follows from the “left exact left adjoint” assumptions). The fact that it’s a left exact left adjoint furthermore implies that it must be a sort of infinitely-additive \(\{0,1\}\)-valued probability measure defined on the open sets, and it follows from this that it’s a “dirac measure”. This is how to prove that all points are really represented by a point. It seems that one way of considering “random points” would be to let the functor take values in a category with objects that can reasonably represent more complicated probability measures. It also seems unlikely that we can rely completely on universal properties to carry the day here. If \(U, V \subset X\) are open sets, then \(U \cap V\) is their product, both in \(O(X)\) and in \(Sh(X)\). So if a functor \(P: Sh(X) \to C\) preserves products, \(P(U \cap V)\) depends only on \(P(U)\) and \(P(V)\) - so this clearly can’t capture all possible probability measures.

One attempt: Stochastic stalks

I haven’t solved this problem completely (I’m not convinced a good general solution exists). One approach is to think about integration of metric spaces over a measure, which I will now explain. A sheaf of metric spaces is a functor \(O(X)^{op} \to Met\), where \(Met\) is the category of metric spaces and short, i.e distance-nonincreasing, maps, which satisfies the sheaf axiom. We denote the category of such sheaves by \(Sh(X,Met)\). By “the sheaf axiom”, I mean it preserves limits. Since \(Met\) does not have all limits, this is a bit subtler than it may appear. However, since \(Met\) does have finite limits, this difficulty disappears if we assume \(X\) is compact.

Let \(M\) be a sheaf of metric spaces and let \(P\) be a Radon probability measure on \(X\). Then we define \(M_X\) to be the product \(\prod_{x \in X}M_x\) of all the stalks (just considered as a set). Equip \(M_X\) with a pseudometric \(d\) by setting \(d(a,b) = \int d(a_x,b_x)P(dx)\). In other words, we integrate the distances according to the given probability measure. If we quotient out with the relation \(a \sim b\) if \(d(a,b) = 0\), this gives a proper metric space, \(M_X/\sim\).

Now for each \(U \subset X\) with \(P(U) = 1\), we consider the map \(M(U) \to M_X/\sim\) given by taking all the germs. We let \(M_P\) be the metric space consisting of the images of all these maps.

This defines a functor \(Sh(X,Met) \to Met\). We can recover the probability measure on an open set \(A\) by considering a sheaf \(M\) given by two points at distance \(1\) if \(U \subset A\) and the singleton otherwise. Then \(M_P\) consists of two points at distance \(P(A)\). At least in the case of compact Hausdorff spaces, this determines the underlying measure uniquely. (In general, the “measure defined on open sets” that we recover in this way is called a valuation, and you can argue that we shouldn’t expect to tell the difference between different measures with the same valuation).

Questions

What useful properties characterize functors of the above form?
Is there a good way of doing this for \(\sigma\)-algebras instead of topologies?
Is there a good way of doing this for a general topos?

Example: Random variables

Let \(M\) be any metric space. Then we can form a sheaf of metric spaces where \(M(U)\) is the set of continuous functions \(U \to M\) in the sup metric. Then the metric space \(M_P\) is the set of \(M\)-valued random variables, metrized by letting \(d(A,B) := \mathbb{E}_P(d(A,B))\) - i.e metrized by expected distance.