| Version 4 |
Version 3 |
| \begin{DEF} |
\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 $\Map{f_i}A{A_i}$
|
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. |
are morphisms. |
|
|
| $A$ is called the \emph{domain of the source} and the family $(A_i)_{i\in I}$ is called |
$A$ is called the \emph{domain of the source} and the family $(A_i)_{i\in I}$ is called |
| codomain of the source. |
codomain of the source. |
| \end{DEF} |
\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 |
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. |
morphisms. |
|
|
| Sources can be composed with morphisms. If $\mathcal S=(A,(f_i)_{i\in I}$ is a source and |
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 denote the notation $(B,(f_i\circ f)_{i\in I})=\mathcal S\circ |
$\Map fBA$ is a morphism, we denote 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 |
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. |
$\mathcal S=((f_i)_{i\in I}, A)$ is a sink and $\Map fAB$ is a morphism. |
|
|
| \begin{DEF} |
\begin{DEF} |
| A source $\mathcal S=(A,(f_i)_{i\in I})$ is called a \emph{monosource} if for any pair |
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 |
$\Map{r,s}BA$ of morphisms from the equality $\mathcal S\circ r=\mathcal S\circ s$ follows |
| $r=s$. |
$r=s$. |
|
|
| A sink $\mathcal S=((f_i)_{i\in I},A)$ is called an \emph{episink} if for any pair |
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 $r\circ\mathcal S=s\circ\mathcal S$. |
$\Map{r,s}AB$ 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 |
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. |
$\mathcal S=\overline{\mathcal S}\circ e$ for an epimorphism $e$, then $e$ is an isomorphism. |
|
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| An episink $\mc S$ is called \emph{extremal episink} if the following holds: Whenever |
An episink $\mc S$ is called \emph{extremal episink} if the following holds: Whenever |
| $\mathcal S=m\circ\overline{\mathcal S}$ pre nejak\'y monomorphism $m$, tak $m$ je |
$\mathcal S=m\circ\overline{\mathcal S}$ pre nejak\'y monomorphism $m$, tak $m$ je |
| isomorphism. |
isomorphism. |
| \end{DEF} |
\end{DEF} |
|
|
| Every limit is an extremal monosource, a colimit is an extremal episink. |
Every limit is an extremal monosource, a colimit is an extremal episink. |
|
|
| \begin{thebibliography}{1} |
\begin{thebibliography}{1} |
|
|
| \bibitem{ahs} |
\bibitem{ahs} |
| J.~Ad\'amek, H.~Herrlich, and G.~Strecker. |
J.~Ad\'amek, H.~Herrlich, and G.~Strecker. |
| \newblock {\em Abstract and Concrete Categories}. |
\newblock {\em Abstract and Concrete Categories}. |
| \newblock Wiley, New York, 1990. |
\newblock Wiley, New York, 1990. |
|
|
| \end{thebibliography} |
\end{thebibliography} |
