A universal characterization of standard Borel spaces

TL;DR

This paper establishes that the category of standard Borel spaces is the universal initial object in a 2-category of categories with specific algebraic and completeness properties, providing a foundational characterization.

Contribution

It proves the universal initiality of the category of standard Borel spaces within a broad class of algebraic and categorical structures, extending to $oldsymbol{ ext{kappa}}$-complete Boolean categories.

Findings

Standard Borel spaces form a universal initial object in a 2-category of algebraic categories.

The dual of $oldsymbol{ ext{kappa}}$-presented $oldsymbol{ ext{kappa}}$-complete Boolean algebras is initial in a related 2-category.

This characterization unifies various algebraic and categorical properties of Borel spaces and Boolean algebras.

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.