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$C_2$-category (Definition)

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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See Also: Grothendieck category

Also defines:  $C_2$-category
Keywords:  Grothendieck category

Cross-references: monomorphism, coproducts, category, objects, morphism
There is 1 reference to this object.

This is version 4 of $C_2$-category, born on 2010-05-09, modified 2010-05-09.
Object id is 853, canonical name is C2Category.
Accessed 482 times total.

Classification:
Physics Classification: 00. (GENERAL)
 02. (Mathematical methods in physics)
 02.70.-cxx (Computational techniques )
 02.90.+p (Other topics in mathematical methods in physics )
Pending Errata and Addenda
None.
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