  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 $\bigotimes_{i \in
J} X_i$ come in two versions: a weaker but more general one for families of
objects $(X_i)_{i \in J}$ 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.
