Definition 0.1. A category of Borel spaces 𝔹 has, as its objects, all Borel spaces (X_b; ℬ(X_b)), and as its morphisms the Borel morphisms f_b between Borel spaces; the Borel morphism composition is defined so that it preserves the Borel structure determined by the σ-algebra of Borel sets.

Remark 0.1. The category of standard Borel G-spaces 𝔹_G is defined in a similar manner to 𝔹, with the additional condition that Borel G-space morphisms commute with the Borel actions a : G × X → X defined as Borel functions (or Borel-measurable maps). Thus, 𝔹_G is a subcategory of 𝔹; in its turn, 𝔹 is a subcategory of 𝕋op–the category of topological spaces and continuous functions.

The category of rigid Borel spaces can be defined as above with the additional condition that the only automorphism f : X_b → X_b (bijection) is the identity 1_(Xb;ℬ(Xb)).