morphism

monomorphism

A morphism f with hook colon A right arrow B is called a monomorphism if, for every object C and all morphisms g comma h colon C right arrow A, the following holds.

f with hook composed with g equals f with hook composed with h right arrow g equals h

epimorphism

A morphism f with hook colon A right arrow B is called an epimorphism if, for every object C and all morphisms g comma h colon B right arrow C, the following holds.

g composed with f with hook equals h composed with f with hook right arrow g equals h

bimorphism

A morphism f with hook colon A right arrow B is called a bimorphism if it is both a monomorphism and an epimorphism.

section

A morphism f with hook colon A right arrow B is called a section if there exists a morphism g colon B right arrow A such that g composed with f with hook equals id sub A.

retraction

A morphism f with hook colon A right arrow B is called a retraction if there exists a morphism g colon B right arrow A such that f with hook composed with g equals id sub B.

isomorphism

A morphism f with hook colon A right arrow B is called an isomorphism if there exists a morphism g colon B right arrow A such that g composed with f with hook equals id sub A and f with hook composed with g equals id sub B.