$C_1$-categoryC_1Category2010-05-09 15:21:412010-05-09 16:23:19DefinitionC1-categorycategoriesAb5 categories\begin{definition}
A category $\mathcal{C}_1$ with coproducts is called a \emph{$C_1$-category} if for every family of
of monomorphisms $\left\{u_i: A_i \to B_i\right\}$ the morphism
$$\iota := \oplus_i \, u_i: \oplus_i \, A_i \to \oplus_i \, B_i $$
is also a monomorphism (\cite{BM266}).
\end{definition}
\begin{remark}
With certain additional conditions (as explained in ref. \cite{BM266}) $\mathcal{C}_1$ may satisfy the Grothendieck axiom $\mathcal{A}b5$, thus becoming a
$C_3$-category (Ch. 11 in \cite{BM266}).
\end{remark}
\begin{thebibliography}{9}
\bibitem{BM266}
See p.81 in ref. $[266]$ in the
\PMlinkname{Bibliography for categories and algebraic topology}{BibliographyForCategoryTheoryAndAlgebraicTopologyApplicationsInTheoreticalPhysics}
\bibitem{NP288}
Ref. $[288]$ in the \PMlinkname{Bibliography for categories and algebraic topology}{BibliographyForCategoryTheoryAndAlgebraicTopologyApplicationsInTheoreticalPhysics}
\end{thebibliography}