C2-category
C2Category
2010-05-09 15:16:21
2010-05-09 15:16:21
Definition
$C_2$-category
Grothendieck category
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In general, a \emph{$C_2$-category} is an $\mathcal{A}b4$-category, or, alternatively, an $\mathcal{A}b3$- and $\mathcal{A}b3^*$ -category $\C$ with certain additional conditions for the canonical morphism from direct sums to products of any family of objects in $\mathcal{C}$ \cite{NP288}).
\begin{definition}
A \emph{$C_2$-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 \cite{BM266}).
\end{definition}
\begin{remark}
One readily obtains the result that a $C_2$-category is $C_1$ (\cite{BM266}).
\end{remark}
\begin{thebibliography}{9}
\bibitem{BM266}
Ref. $[266]$ in the
\PMlinkname{Bibliography for categories and algebraic topology}{CategoricalOntologyABibliographyOfCategoryTheory}
\bibitem{NP288}
Ref. $[288]$ in the \PMlinkname{Bibliography for categories and algebraic topology}{CategoricalOntologyABibliographyOfCategoryTheory}
\end{thebibliography}