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 $$g$$.