nLab
strong monomorphism

Contents

Definition

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

Remarks

  • If CC 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 CC 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 YY of a diffeological space XX is again a diffeological space. If smooth, the inclusion ι:YX\iota:Y \rightarrow X is always a monomorphism, but it is a strong monomorphism if and only if YY has “enough” plots, that is if φ:UY\varphi: U\rightarrow Y is a plot if and only if the composite ιφ:UX\iota\varphi: U\rightarrow X is a plot.

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