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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 C be a category and A and B be objects of C. A morphism fHomC(A,B) is said to be a monomophism iff:

For all objects Z of C and all morphisms α,α [ what set? ]

[??.]

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

Epimorphism

Let C be a category and A and B be objects of C. A morphism gHomC(A,B) is said to be a monomophism iff:

For all objects Z of C and all morphisms β,β [ what set? ]

[??]

Essense

If f is a monomorphism and fα is f composed with some unknown morphism α, then knowing f and fα is enough to information to recover α exactly.

In other words, there is no redundancy prodived by f that allows two morphisms α and α to compose with f such that fα and fα 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.