A universal characterization of standard Borel spaces

Abstract

We prove that the category $\mathsf{SBor}$ of standard Borel spaces is the (bi-)initial object in the 2-category of countably complete Boolean (countably) extensive categories. This means that $\mathsf{SBor}$ is the universal category admitting some familiar algebraic operations of countable arity (e.g., countable products, unions) obeying some simple compatibility conditions (e.g., products distribute over disjoint unions). More generally, for any infinite regular cardinal $\kappa$, the dual of the category $\kappa\mathsf{Bool}_\kappa$ of $\kappa$-presented $\kappa$-complete Boolean algebras is (bi-)initial in the 2-category of $\kappa$-complete Boolean ($\kappa$-)extensive categories.