PlanetPhysics/C 3Category

Abelian C₃-category

Let ${\mathcal {A}}$ be an Abelian cocomplete category, defined as the dual of an Abelian complete category.

A $C_{3}$-category is defined as a cocomplete Abelian category ${\mathcal {A}}$ such that the following distributivity relation holds for any direct family $\left\{A_{i}\right\}$ and any subobject $B$:

$(\bigcup A_{i})\bigcap B=\bigcup (A_{i}\bigcap B),$ ()

A $C_{3}$-category is also called an ${\mathcal {A}}b5$-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 $C_{3}$-category.