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We study a category of probability spaces and measure-preserving Markovkernels up to almost sure equality. This category contains, among itsisomorphisms, mod-zero isomorphisms of probability spaces. It also gives anisomorphism between the space of values of a random variable and thesigma-algebra that it generates on the outcome space, reflecting the standardmathematical practice of using the two interchangeably, for example when takingconditional expectations.  We show that a number of constructions and results from classical probabilitytheory, mostly involving notions of equilibrium, can be expressed and proven interms of this category. In particular: - Given a stochastic dynamical systemacting on a standard Borel space, we show that the almost surely invariantsigma-algebra can be obtained as a limit and as a colimit; - In the settingabove, the almost surely invariant sigma-algebra gives rise, up to isomorphismof our category, to a standard Borel space; - As a corollary, we give acategorical version of the ergodic decomposition theorem for stochasticactions; - As an example, we show how de Finetti's theorem and theHewitt-Savage and Kolmogorov zero-one laws fit in this limit-colimit picture.  This work uses the tools of categorical probability, in particular Markovcategories, as well as the theory of dagger categories.

Categorical probability spaces, ergodic decompositions, and transitions to equilibrium