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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 $ A$ is an object and $ f_i:A\to 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 $ 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 $ r=s$.

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 $ r=s$ 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 $ e$, then $ e$ 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 $ m$, tak $ m$ 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.



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See Also: universal mapping property

Also defines:  source, monosource, extremal monosource, sink, extremal episink

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Cross-references: colimit, limit, monomorphism, isomorphism, epimorphism, equality, codomain, domain, class, indexed by, morphisms, mean, object, category
There are 48 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 4001 times total.

Classification:
AMS MSC18A05 (Category theory; homological algebra :: General theory of categories and functors :: Definitions, generalizations)

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