  Motivated by advances in categorical probability, we introduce
non-commutative almost everywhere (a.e.) equivalence and disintegrations in the
setting of C*-algebras. We show that C*-algebras (resp. W*-algebras) and a.e.
equivalence classes of 2-positive (resp. positive) unital maps form a category.
We prove non-commutative disintegrations are a.e. unique whenever they exist.
We provide an explicit characterization for when disintegrations exist in the
setting of finite-dimensional C*-algebras, and we give formulas for the
associated disintegrations.
