| Version 5 |
Version 4 |
| \begin{theorem} (Proposition 1.6. in ref. \cite{BM266}) |
\begin{theorem} (Proposition 1.6. in ref. \cite{BM266}) |
| A cocomplete Abelian category $\mathcal{A}$ is $C_3$ if and only if for every direct family of subobjects $\left\{A_i\right\}$ of an object $A$ , and every morphism $g: B \to A$, one has the following equation: |
A cocomplete Abelian category $\mathcal{A}$ is $C_3$ if and only if for every direct family of subobjects $\left\{A_i\right\}$ of an object $A$ , and every morphism $g: B \to A$, one has the following equation: |
|
|
| $$g^{-1}(\bigcup A_i) = \bigcup g^{-1}(A_i).$$ |
$$g^{-1}(\bigcup A_i) = \bigcup g^{-1}(A_i).$$ |
| \end{theorem} |
\end{theorem} |
|
|
| \textbf{Remark:} |
\textbf{Remark:} |
| The proof involves the exact sequence: |
The proof involves the exact sequence: |
| $$ 0 \to g^{-1}(A_i) \to B \to Im / A_i \bigcap Im \to 0 ,$$ |
$$ 0 \to g^{-1}(A_i) \to B \to Im / A_i \bigcap Im \to 0 ,$$ |
|
|
| where $Im$ is the image of the morphism $g$. |
where $Im$ is the image of the morphism $g$. |
|
|
|
|
| \begin{thebibliography}{9} |
\begin{thebibliography}{9} |
| \bibitem{BM266} |
\bibitem{BM266} |
| See p.83 and eq. (3) in ref. $[266]$ in the |
See p.83 and eq. (3) in ref. $[266]$ in the |
| \PMlinkname{Bibliography for categories and algebraic topology}{CategoricalOntologyABibliographyOfCategoryTheory} |
\PMlinkname{Bibliography for categories and algebraic topology}{CategoricalOntologyABibliographyOfCategoryTheory} |
|
|
| \end{thebibliography} |
\end{thebibliography} |
