# PlanetPhysics/C 3Category

## Abelian ${\displaystyle C_{3}}$-category

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

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

${\displaystyle (\bigcup A_{i})\bigcap B=\bigcup (A_{i}\bigcap B),}$  ([1])

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

## References

1. See p.82 and eq. (1) in ref. ${\displaystyle [266]}$  in the Bibliography for categories and algebraic topology
2. Ref. ${\displaystyle [288]}$  in the Bibliography for categories and algebraic topology