$C_2$-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]).

Bibliography

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

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



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