Definition 0.1. Let 𝒜 be an Abelian cocomplete category, defined as the dual of an Abelian complete category.

A C₃-category is defined as a cocomplete Abelian category 𝒜 such that the following distributivity relation holds for any direct family and any subobject B:

([1])

Remark 0.1.

A C₃-category is also called an 𝒜b5-category.

Example 0.1. The dual of the Cartesian closed category of finite Abelian quantum groups with exponential elements (including Lie groups) and quantum group homomorphisms is a C₃-category.