Let be an Abelian cocomplete category, defined as the dual of an Abelian complete category.
A -category is defined as a cocomplete Abelian category
such that the following distributivity relation holds for any direct family and any subobject :
A -category is also called an -category.
The dual of the Cartesian closed category of finite Abelian quantum groups with exponential
elements (including Lie groups) and quantum group homomorphisms is a -category.