David Spivak: Internal probability valuations

Talk at the Categorical Probability and Statistics workshop 2020: http://perimeterinstitute.ca/personal/tfritz/2019/cps_workshop/ Title: Internal probability valuations Speaker: David Spivak Chair: Peter McCullagh Date: June 7th, 2020 Abstract: There are many competing approaches to probability theory. A particularly nice one from a categorical viewpoint is that of (probability) valuations, which monotonically assign an element in [0,1] to each open set of a space, satisfying an inclusion-exclusion formula and a continuity requirement. Valuations are known to coincide with probability measures in many non-pathological cases, like on Polish spaces. The data and axioms of a topological space and a valuation on it can be stated constructively, and hence in the internal logic of any topos with a natural numbers object. In the topos of sets, for example, the result is just a valuation in the usual sense; however, understanding the semantics of valuations in more general toposes is an open problem. In this talk, I'll discuss work-in-progress with Tobias Fritz, in which we investigate the semantics of valuations on spatial toposes. We believe that the result will in particular provide a new constructive and synthetic approach to stochastic processes.