Alex SIMPSON - Probability sheaves
Channel: Institut des Hautes Études Scientifiques (IHÉS)
Date: 2015-12-03

In [2], Tao observes that the probability theory concerns itself with properties that are \preserved with respect to extension of the underlying sample space", in much the same way that modern geometry concerns itself with properties that are invariant with respect to underlying symmetries. Reformulating this in category-theoretic language, probabilistic concepts organise themselves into presheaves over a category of sample spaces. In this talk, I observe that they further form sheaves, and I consider ramications of this observation.
As a suitable category of sample spaces, I take the category of measure-preserving measurable maps (modulo almost sure equality) between standard (a.k.a. Lebesgue-Rokhlin) probability
spaces. In this category, every cospan completes to a commutative square enjoying a universal conditional independence property. As a consequence, the category carries an atomic Grothendieck topology, whose sheaves can themselves be characterised in terms of conditional independence. Examples of such probability sheaves include sheaf representations of standard
probability spaces (given by representables), sheaves of random variables, sheaves of probability measures (given by a general coend construction), and sheaves of orbits of ergodic group
actions.
In general, I argue that the resulting atomic topos of probability sheaves is a natural category of generalised probabilistic concepts. Moreover, as a boolean topos, it models a mathematical
universe in which random variable occurs as a primitive rather than derived mathematical notion. I believe this model has the potential to inform the development of an alternative approach to probability theory founded on primitive random variables, somewhat along the lines envisaged by Mumford in [1].

References
[1] D. Mumford. The dawning of the age of stochasticity. In
Mathematics: Frontiers and Perspectives
. AMS, 2000.
[2] T. Tao. A review of probability theory. Note 0 in 254A { Random Matrices. Lecture notes on blog at terrytao.wordpress.com, 2010.