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# $C_{2}$-category

In general, a *$C_{2}$-category* is an $\mathcal{A}b4$-category, or, alternatively, an $\mathcal{A}b3$- and $\mathcal{A}b3^{*}$ -category $C^{{\ast}}$ with certain additional conditions for the canonical morphism from direct sums to products of any family of objects in $\mathcal{C}$ [2]).

###### Definition 0.1.

A *$C_{2}$-category* is defined as a category $\mathcal{C}$ that has products, coproducts and a zero object, and if the morphism $\iota:\oplus A_{i}\to\mathbf{X}A_{i}$ is a monomorphism for any family of objects $\left\{A_{i}\right\}$ in $\mathcal{C}$ (p. 81 in [1]).

###### Remark 0.1.

One readily obtains the result that a $C_{2}$-category is $C_{1}$ ([1]).

# References

- 1 Ref. $[266]$ in the Bibliography for categories and algebraic topology
- 2 Ref. $[288]$ in the Bibliography for categories and algebraic topology

Defines:

$C_2$

Keywords:

$C_2$-category, Ab3-, Ab3*- and Ab4- categories, canonical monomorphism

Related:

Category,GrothendieckCategory, C_1Category, C_3Category, IndexOfCategories

Synonym:

$Ab4$-category

Type of Math Object:

Definition

Major Section:

Reference

Groups audience:

## Mathematics Subject Classification

18-00*no label found*18E05

*no label found*

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