PlanetMath: initial source

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initial source

(Definition)

Let $\mathbf{A}$ be a concrete category over $\mathbf{X}$ . A source $(A\overset{f_i}\to A_i)_{i\in I}$ in $\mathbf{A}$ is called initial provided that an $\mathbf{X}$ -morphism $f:\vert B \vert\to\vert A \vert$ is an $\mathbf{A}$ -morphism whenever each composite $f_i\circ f:\vert B \vert\to\vert A_i \vert$ is an $\mathbf{A}$ -morphism.

The dual notion is called a final sink.

A source in the category of topological spaces $\mathbf{Top}$ is initial if and only if has the initial topology with respect to the family .

A topological space is completely regular if and only if the source $S(X,\mathbb{R})$ , consisting of all continuous maps from to the real line, is initial (in the construct $\mathbf{Top}$ ); and is a Tychonoff space if and only if $S(X,\mathbb{R})$ is an initial mono-source.

A similar characterization holds for epireflective subcategories of $\mathbf{Top}$ .

Bibliography

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

"initial source" is owned by kompik.

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Other names:

final sink

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Cross-references: subcategories, characterization, similar, Tychonoff space, line, real, continuous maps, completely regular, initial topology, topological spaces, category, composite, source, concrete category
There are 2 references to this entry.

This is version 4 of initial source, born on 2006-06-30, modified 2007-06-17.
Object id is 8110, canonical name is InitialMonosource.
Accessed 1145 times total.

Classification:

AMS MSC:

18A05 (Category theory; homological algebra :: General theory of categories and functors :: Definitions, generalizations)

Pending Errata and Addenda

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