PlanetMath: source

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source

(Definition)

In the whole entry we suppose we are given a category $\mathbf{A}$ . By an object we always mean an object in $\mathbf{A}$ and by a morphisms an $\mathbf{A}$ -morphism.

Definition 1 A source in a category $\mathbf{A}$ is a pair $(A,(f_i)_{i\in I})$ where is an object and $f_i:A\to A_i$ are morphisms indexed by a class .

The object 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 is an object and $f_i:A_i\to A$ are morphisms.

Sources can be composed with morphisms. If $\mathcal S=(A,(f_i)_{i\in I}$ is a source and $f:B\to A$ 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 $f:A\to B$ is a morphism.

Definition 2 A source $\mathcal S=(A,(f_i)_{i\in I})$ is called a monosource if for any pair $r,s:B\to A$ of morphisms from the equality $\mathcal S\circ r=\mathcal S\circ s$ follows .

A sink $\mathcal S=((f_i)_{i\in I},A)$ is called an episink if for any pair $r,s:A\to B$ of morphisms whenever $r\circ\mathcal S=s\circ\mathcal S$ .

A monosource $\mathcal S$ is called extremal monosource, if the following holds: Whenever $\mathcal S=\overline{\mathcal S}\circ e$ for an epimorphism , then is an isomorphism.

An episink $\mathcal S$ is called extremal episink if the following holds: Whenever $\mathcal S=m\circ\overline{\mathcal S}$ pre for a monomorphism , tak is an isomorphism.

Every limit is an extremal monosource, a colimit is an extremal episink.

Bibliography

1: J. Adámek, H. Herrlich, and G. Strecker.
Abstract and Concrete Categories.
Wiley, New York, 1990.

"source" is owned by kompik.

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