The n-Category Café

A Characterization of Standard Borel Spaces

Posted by John Baez

People in measure theory find it best to work with, not arbitrary measurable spaces, but certain nice ones called standard Borel spaces. I’ve used them myself.

The usual definition of these looks kind of clunky: a standard Borel space is a set $X$ equipped with a $\sigma$-algebra $\Sigma$ for which there exists a complete metric on $X$ such that $\Sigma$ is the $\sigma$-algebra of Borel sets. But the results are good. For example, every standard Borel space is isomorphic to one of these:

• a finite or countably infinite set with its $\sigma$-algebra of all subsets,
• the real line with its sigma-algebra of Borel subsets.

So standard Borel spaces are a good candidate for Tom Leinster’s program of revealing the mathematical inevitability of certain traditionally popular concepts. I forget exactly how he put it, but it’s a great program and I remember some examples: he found nice category-theoretic characterizations of Lebesgue integration, entropy, and the nerve of a category.

Now someone has done this for standard Borel spaces!

This paper did it:

• Ruiyuan Chen, A universal characterization of standard Borel spaces, Journal of Symbolic Logic 88(2) (2023), 510–539.

Here is his result:

Theorem. The category $\mathsf{SBor}$ of standard Borel spaces and Borel maps is the (bi)initial object in the 2-category of countably complete countably extensive Boolean categories.

Here a category is countably complete if it has countable limits. It’s countably extensive if it has countable coproducts and a map

$$X \to \sum_i Y_i$$

into a countable coproduct is the same thing as a decomposition $X \cong \sum_i X_i$ together with maps $X_i \to Y_i$ for each $i$. It’s Boolean if the poset of subobjects of any object is a Boolean algebra.

So, believe it or not, these axioms are sufficient to develop everything we do with standard Borel spaces!