Quantitative sheaves for probability [efr-5ZC5]

Given a topological space, its points are in bijection with a certain class of functors Sh(X) \to \mathsf {Set}. I have the idea that some quantitative version of this can encode information about a probability measure on X, thus giving us a categorical language for probability spaces.

The inspiration for this idea is that, for deterministic continuous-time dynamical systems, the family of "interval systems" gives a full subcategory equivalent to the interval category, and thus a functor from systems to temporal sets describing the "ordinary" trajectories. We would like to extend this idea to stochastic systems as in Trajectories of stochastic systems from the point of view of categorical systems theory.