Definition 0.1
A
category of Borel spaces

has, as its objects, all
Borel spaces

, and as its morphisms the
Borel morphisms

between Borel spaces; the Borel morphism composition is defined so that it preserves the Borel structure determined by the

-algebra of Borel sets.
Remark 0.1
The
category of standard Borel G-spaces

is defined in a similar manner to

, with the additional condition that Borel G-space morphisms commute with
the
Borel actions

defined as
Borel functions
(or Borel-measurable maps). Thus,

is a subcategory of

; in its turn,

is a subcategory of

-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
(bijection) is the identity
.