sourceSource42006-06-30 08:25:342006-07-01 10:51:59Definitionsourcemonosourceextremal monosourcesinkextremal episink% this is the default PlanetMath preamble. as your knowledge
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In the whole entry we suppose we are given a category $\Kat A$. By an object we always mean an object in $\Kat A$ and by a morphisms an $\Kat A$-morphism.
\begin{DEF}
A \emph{source} in a category $\Kat A$ is a pair $(A,(f_i)_{i\in I})$ where $A$ is an object and $\Map{f_i}A{A_i}$ are morphisms indexed by a class $I$.
The object $A$ is called the \emph{domain of the source} and the family $(A_i)_{i\in I}$ is called the codomain of the source.
\end{DEF}
A \emph{sink} is a pair $((f_i)_{i\in I},A)$ where $A$ is an object and $\Map{f_i}{A_i}A$ are
morphisms.
Sources can be composed with morphisms. If $\mathcal S=(A,(f_i)_{i\in I}$ is a source and
$\Map fBA$ is a morphism, we use the notation $(B,(f_i\circ f)_{i\in I})=\mathcal S\circ f$.
Similarly, for sinks, we use the notation
$f\circ\mathcal S=((f\circ f_i)_{i\in I},B)$ if
$\mathcal S=((f_i)_{i\in I}, A)$ is a sink and $\Map fAB$ is a morphism.
\begin{DEF}
A source $\mathcal S=(A,(f_i)_{i\in I})$ is called a \emph{monosource} if for any pair
$\Map{r,s}BA$ of morphisms from the equality $\mathcal S\circ r=\mathcal S\circ s$ follows
$r=s$.
A sink $\mathcal S=((f_i)_{i\in I},A)$ is called an \emph{episink} if for any pair
$\Map{r,s}AB$ of morphisms $r=s$ whenever $r\circ\mathcal S=s\circ\mathcal S$.
A monosource $\mc S$ is called \emph{extremal monosource}, if the following holds: Whenever
$\mathcal S=\overline{\mathcal S}\circ e$ for an epimorphism $e$, then $e$ is an isomorphism.
An episink $\mc S$ is called \emph{extremal episink} if the following holds: Whenever
$\mathcal S=m\circ\overline{\mathcal S}$ pre for a monomorphism $m$, tak $m$ is an isomorphism.
\end{DEF}
Every limit is an extremal monosource, a colimit is an extremal episink.
\begin{thebibliography}{1}
\bibitem{ahs}
J.~Ad\'amek, H.~Herrlich, and G.~Strecker.
\newblock {\em Abstract and Concrete Categories}.
\newblock Wiley, New York, 1990.
\end{thebibliography}