# Monomorphisms and epimorphisms

The ideas of injections and surjections in the context of sets and functions is paralleled in the context of categories by the concepts of monomorphisms and epimorphisms.

### Monomorphism

Let \( \cat{C} \) be a category and \( A \) and \( B \) be objects of \( \cat{C} \). A morphism \( f \in \cathom{C}(A, B) \) is said to be a monomophism iff:

For all objects \( Z \) of \( \cat{C} \) and all morphisms \( \alpha', \, \alpha'' \in \) [ what set? ]

An epimorphism is defined similarly, but with the composition order reversed.

### Epimorphism

Let \( \cat{C} \) be a category and \( A \) and \( B \) be objects of \( \cat{C} \). A morphism \( g \in \cathom{C}(A, B) \) is said to be a monomophism iff:

For all objects \( Z \) of \( \cat{C} \) and all morphisms \( \beta', \, \beta'' \in \) [ what set? ]

### Essense

If \( f \) is a monomorphism and \( f \, \alpha \) is \( f \) composed with some unknown morphism \( \alpha \), then knowing \( f \) and \( f \, \alpha \) is enough to information to recover \( \alpha \) exactly.

In other words, there is no redundancy prodived by \( f \) that allows two morphisms \( \alpha' \) and \( \alpha'' \) to compose with \( f \) such that \( f \, \alpha' \) and \( f \, \alpha'' \) produce the same morphism.

Yet another wording: no morphism can 'hide' behind \( f \).

Similarly, an epimorphism \( g \) does not afford any ambiguity to morphisms that compose after \( g \).