# 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 $$x$$

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

1. This determines the point uniquely, i.e if $$(-)_x \simeq (-)_y$$ then $$x = y$$
2. This is a left exact left adjoint $$Sh(X) \to Set$$ - i.e part of a geometric morphism $$Set \to Sh(X)$$.
3. 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.