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Math and science::Algebra::Aluffi

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? ]

[\[ ? \implies ? \].]

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? ]

[\[ ? \implies ? \]]

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 \( g \).