Every strong monomorphism is extremal; the converse is true if $C$ has pushouts.

Examples

A nice example of strong monomorphisms in a category are the subspace inclusions in the category of diffeological spaces. In this setting, any subset $Y$ of a diffeological space $X$ is again a diffeological space. If smooth, the inclusion $\iota:Y \rightarrow X$ is always a monomorphism, but it is a strong monomorphism if and only if $Y$ has “enough” plots, that is if $\varphi: U\rightarrow Y$ is a plot if and only if the composite $\iota\varphi: U\rightarrow X$ is a plot.

Revised on October 20, 2010 17:48:42
by Urs Schreiber
(131.211.232.170)