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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.
Definition 1 A 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 domain of the source and the family $(A_i)_{i\in I}$ is called the codomain of the source.
A 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.
Definition 2 A source $\mathcal S=(A,(f_i)_{i\in I})$ is called a 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 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 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 extremal episink if the following holds: Whenever $\mathcal S=m\circ\overline{\mathcal S}$ pre for a monomorphism $m$ , tak $m$ is an isomorphism.
Every limit is an extremal monosource, a colimit is an extremal episink.
- 1
- J. Adámek, H. Herrlich, and G. Strecker.
Abstract and Concrete Categories.
Wiley, New York, 1990.
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"source" is owned by kompik.
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Cross-references: colimit, limit, monomorphism, isomorphism, epimorphism, equality, codomain, domain, class, indexed by, morphisms, mean, object, category
There are 33 references to this entry.
This is version 7 of source, born on 2006-06-30, modified 2006-07-01.
Object id is 8109, canonical name is Source4.
Accessed 5467 times total.
Classification:
| AMS MSC: | 18A05 (Category theory; homological algebra :: General theory of categories and functors :: Definitions, generalizations) |
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Pending Errata and Addenda
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