nLab
strong monomorphism

Definition

A strong monomorphism in a category $C$ is a monomorphism which is right orthogonal to any epimorphism. The dual notion is, of course, strong epimorphism.

Remarks

• If $C$ has coequalizers, then any morphism which is right orthogonal to epimorphisms must automatically be a monomorphism.

• Every regular monomorphism is strong.

• 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 $\iota :Y\to X$ is always a monomorphism, but it is a strong monomorphism if and only if $Y$ has “enough” plots, that is if $\phi :U\to Y$ is a plot if and only if the composite $\iota \phi :U\to X$ is a plot.