## You are here

Home$C_3$-category is $ Ab5$

## Primary tabs

# $C_{3}$-category is $Ab5$

###### Definition 0.1.

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 $B$:

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

[1]

###### Remark 0.1.

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

# References

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

Defines:

Abelian cocomplete category,

Keywords:

$C_1$-category, $C_2$-category, monomorphisms family, products, coproducts and zero objects, Ab5 category, abelian category with generators

Related:

C_1Category, C_2Category,AbelianCategory,GrothendieckCategory, C_3CategoryTheorem, ExactSequenceTheoremInC_3Category, GabrielPopescuTheoremForAb5Categories,GrothendiecksTheorem, IndexOfCategories

Synonym:

Ab5 category, cocomplete abelian category

Type of Math Object:

Definition

Major Section:

Reference

Groups audience:

## Mathematics Subject Classification

18-00*no label found*18A99

*no label found*18E15

*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff
- Corrections