Interesting Grothendieck topologies or coverages on the category Prob

Question

I am currently trying to understand Grothendieck Topologies and coverages and want to endow the category Prob, consisting of finite probability spaces and measure preserving maps, with a Grothendieck Topology. I have already tried endowing it with the trivial topology and dense topology but got no interesting results (yet). Is there an interesting topology that can be put onto Prob, or is there a general way to endow a given category with an interesting Grothendieck Topology?

Answer 1

This answer is motivated by the related question here.

I think it can help to write $\mathbf{Prob}$ as a disjoint union of the categories $\mathbf{Prop}_\lambda$, with $\mathbf{Prop}_\lambda$ the full subcategory of spaces with total measure $\lambda$, with $\lambda \geq 0$. This works because for any measure-preserving map $f: X \to Y$, we have that $\mu(X)=\mu(Y)$.

You can also use rescaling to show that all categories $\mathbf{Prop}_\lambda$ with $\lambda\neq 0$ are isomorphic to each other. This doesn't work for $\lambda=0$, instead $\mathbf{Prop}_0$ is isomorphic (or at least equivalent) to the category of finite sets.

Defining a Grothendieck topology on $\mathbf{Prop}$ amounts to defining a Grothendieck topology on $\mathbf{Prop}_\lambda$ for each $\lambda \geq 0$. In this way, in order to classify the Grothendieck topologies on $\mathbf{Prop}$, it is enough to classify them on $\mathbf{Prop}_1$ and on $\mathbf{Prop}_0$.

I'm not sure which one of these two will be the most difficult. On $\mathbf{Prop}_1$ you have for example the chaotic, trivial and dense topologies, and since you already computed that the dense topology agrees with the atomic topology, they all have a concrete description.

Other than that, maybe you can use that any full subcategory $\mathcal{C} \subseteq \mathbf{Prop}_1$ defines a Grothendieck topology, see the example on slide 9 here (and more generally any Grothendieck topology on $\mathcal{C}$ then extends to one on $\mathbf{Prop}_1$). A full subcategory of $\mathbf{Prop}_1$ that looks particularly interesting to me is the one consisting of the spaces in which all points have measure $>0$.