$C_1$-category

Definition 0.1   A category $ \mathcal{C}_1$ with coproducts is called a $ C_1$ -category if for every family of of monomorphisms $ \left\{u_i: A_i \to B_i\right\}$ the morphism

$\displaystyle \iota := \oplus_i \, u_i: \oplus_i \, A_i \to \oplus_i \, B_i $

is also a monomorphism ([1]).

Remark 0.1   With certain additional conditions (as explained in ref. [1]) $ \mathcal{C}_1$ may satisfy the Grothendieck axiom $ \mathcal{A}b5$ , thus becoming a $ C_3$ -category (Ch. 11 in [1]).

Bibliography

1
See p.81 in 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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