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$C_1$-category (Definition)
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



"$C_1$-category" is owned by bci1.

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See Also: Grothendieck category, $C_3$-category

Other names:  C1-category
Also defines:  C1-category
Keywords:  categories, Ab5 categories

Cross-references: morphism, monomorphisms, coproducts, category

This is version 5 of $C_1$-category, born on 2010-05-09, modified 2010-05-09.
Object id is 855, canonical name is C_1Category2.
Accessed 594 times total.

Classification:
Physics Classification: 00. (GENERAL)
Pending Errata and Addenda
None.
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