|
|
|
Viewing Version
9
of
'epi'
|
[ view 'epi'
|
back to history
]
| Title of object: |
epi |
| Canonical Name: |
Epi |
| Type: |
Definition |
| Created on: |
2004-11-21 14:16:26 |
| Modified on: |
2007-06-16 11:23:43 |
| Classification: |
msc:18A05, msc:18A20 |
| Defines: |
epic |
| Synonyms: |
epi=epimorphism
epi=epimorphic |
Preamble:
% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb,amscd}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{xypic}
% used for TeXing text within eps files
\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
% making logically defined graphics
% there are many more packages, add them here as you need them
% define commands here |
Content:
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}
$\xymatrix{A \ar[r]^f & B \ar[r]^{g_1} & C}=\xymatrix{A \ar[r]^f & B \ar[r]^{g_2} & C}$
\end{center}
reduces to the diagram $$\xymatrix{B \ar[r]^{g_1} & C}=\xymatrix{B \ar[r]^{g_2} & C}.$$
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.}
\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 The dual notion of epi is that of \PMlinkname{monic}{Monic}.
\end{enumerate} |
|
|
|
|
|
