| Version 10 |
Version 9 |
|
A morphism $f : A\to B$ in a category $\mathcal{C}$ is called {\em epi} if for any object $C$ and any morphisms $g_1,g_2 : B\to C$, if $g_1 f = g_2 f$ then $g_1 = g_2$. In other words, any diagram
|
A morphism $f : A\to B$ in a category $\mathcal{C}$ is called an {\em epi} if for any object $C$ and any morphisms $g_1,g_2 : B\to C$, if $g_1 f = g_2 f$ then $g_1 = g_2$. In other words, any diagram
|
| \begin{center} |
\begin{center} |
| $\xymatrix{A \ar[r]^f & B \ar[r]^{g_1} & C}=\xymatrix{A \ar[r]^f & B \ar[r]^{g_2} & C}$ |
$\xymatrix{A \ar[r]^f & B \ar[r]^{g_1} & C}=\xymatrix{A \ar[r]^f & B \ar[r]^{g_2} & C}$ |
| \end{center} |
\end{center} |
| reduces to the diagram $$\xymatrix{B \ar[r]^{g_1} & C}=\xymatrix{B \ar[r]^{g_2} & C}.$$ |
reduces to the diagram $$\xymatrix{B \ar[r]^{g_1} & C}=\xymatrix{B \ar[r]^{g_2} & C}.$$ |
|
|
|
An \emph{epimorphism} is just an epi morphism, and epi is also known as \emph{right cancellable}, \emph{epimorphic}, or simply \emph{epic}.
|
An epi is also called an \emph{epimorphism}, and a morphism that is an epi is said to be \emph{epimorphic}, or simply \emph{epic}.
|
|
|
| \textbf{Remarks.} |
\textbf{Remarks.} |
| \begin{enumerate} |
\begin{enumerate} |
| \item If $\mathcal{C}$ is an abelian category, then an epi has the property that $gf=0$ implies $g=0$ (surely, since $gf=0=0f$, and the result follows). |
\item If $\mathcal{C}$ is an abelian category, then an epi has the property that $gf=0$ implies $g=0$ (surely, since $gf=0=0f$, and the result follows). |
| \item The dual notion of epi is that of \PMlinkname{monic}{Monic}. |
\item The dual notion of epi is that of \PMlinkname{monic}{Monic}. |
| \end{enumerate} |
\end{enumerate} |
