Infinite products and zero-one laws in categorical probability

Markov categories are a recent category-theoretic approach to the foundations of probability and statistics. Here we develop this approach further by treating infinite products and the Kolmogorov extension theorem. This is relevant for all aspects of probability theory in which infinitely many random variables appear at a time. These infinite tensor products ⨂i∈JXi come in two versions: a weaker but more general one for families of objects (Xi)i∈J in semicartesian symmetric monoidal categories, and a stronger but more specific one for families of objects in Markov categories.As a first application, we state and prove versions of the zero--one laws of Kolmogorov and Hewitt--Savage for Markov categories. This gives general versions of these results which can be instantiated not only in measure-theoretic probability, where they specialize to the standard ones in the setting of standard Borel spaces, but also in other contexts.

Higher arity self-distributive operations in Cascades and their cohomology

We investigate constructions of higher arity self-distributive operations, and give relations between cohomology groups corresponding to operations of different arities. For this purpose we introduce the notion of mutually distributive [Formula: see text]-ary operations generalizing those for the binary case, and define a cohomology theory labeled by these operations. A geometric interpretation in terms of framed links is described, with the scope of providing algebraic background of constructing [Formula: see text]-cocycles for framed link invariants. This theory is also studied in the context of symmetric monoidal categories. Examples from Lie algebras, coalgebras and Hopf algebras are given.