Editing Monomorphism (section)

==Related concepts==
There are also useful concepts of ''regular monomorphism'', ''extremal monomorphism'', ''immediate monomorphism'', ''strong monomorphism'', and ''split monomorphism''. 

* A monomorphism is said to be '''regular''' if it is an [[equaliser (mathematics)|equalizer]] of some pair of parallel morphisms. 
* A monomorphism <math>\mu</math> is said to be '''extremal'''{{sfn|Borceux|1994}} if in each representation <math>\mu=\varphi\circ\varepsilon</math>, where <math>\varepsilon</math> is an epimorphism, the morphism <math>\varepsilon</math> is automatically an [[isomorphism]]. 
* A monomorphism <math>\mu</math> is said to be '''immediate''' if in each representation <math>\mu=\mu'\circ\varepsilon</math>, where <math>\mu'</math> is a monomorphism and <math>\varepsilon</math> is an epimorphism, the morphism <math>\varepsilon</math> is automatically an [[isomorphism]].
* [[File:Diagram-orthogonality-2.jpg|thumb]] A monomorphism <math>\mu:C\to D</math> is said to be '''strong'''{{sfn|Borceux|1994}}{{sfn|Tsalenko|Shulgeifer|1974}} if for any epimorphism <math>\varepsilon:A\to B</math> and any morphisms <math>\alpha:A\to C</math> and <math>\beta:B\to D</math> such that <math>\beta\circ\varepsilon=\mu\circ\alpha</math>, there exists a morphism <math>\delta:B\to C</math> such that <math>\delta\circ\varepsilon=\alpha</math> and <math>\mu\circ\delta=\beta</math>.
* A monomorphism <math>\mu</math> is said to be '''split''' if there exists a morphism <math>\varepsilon</math> such that <math>\varepsilon\circ\mu=1</math> (in this case <math>\varepsilon</math> is called a left-sided inverse for <math>\mu</math>).