The category of standard Borel spaces is the initial countably complete Boolean countably extensive category, meaning it is freely generated by countable products, disjoint unions, and complementation.
Borceux, Handbook of categorical algebra, Encyclopedia of Mathematics and its Applications, vol
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A universal characterization of standard Borel spaces
The category of standard Borel spaces is the initial countably complete Boolean countably extensive category, meaning it is freely generated by countable products, disjoint unions, and complementation.