  We study a category of probability spaces and measure-preserving Markov
kernels up to almost sure equality. This category contains, among its
isomorphisms, mod-zero isomorphisms of probability spaces. It also gives an
isomorphism between the space of values of a random variable and the
sigma-algebra that it generates on the outcome space, reflecting the standard
mathematical practice of using the two interchangeably, for example when taking
conditional expectations.
  We show that a number of constructions and results from classical probability
theory, mostly involving notions of equilibrium, can be expressed and proven in
terms of this category. In particular: - Given a stochastic dynamical system
acting on a standard Borel space, we show that the almost surely invariant
sigma-algebra can be obtained as a limit and as a colimit; - In the setting
above, the almost surely invariant sigma-algebra gives rise, up to isomorphism
of our category, to a standard Borel space; - As a corollary, we give a
categorical version of the ergodic decomposition theorem for stochastic
actions; - As an example, we show how de Finetti's theorem and the
Hewitt-Savage and Kolmogorov zero-one laws fit in this limit-colimit picture.
  This work uses the tools of categorical probability, in particular Markov
categories, as well as the theory of dagger categories.
