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

The dual notion is called a final sink.

A source $(A,f_i)_I$ in the category of topological spaces $\Top$ is initial if and only if $A$ has the initial topology with respect to the family $(f_i)_I$ .

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

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

Bibliography

1

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