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 be a category and and be objects of
. A morphism is said to be a
monomophism iff:
For all objects of and all morphisms [ what set? ]
[.]
An epimorphism is defined similarly, but with the composition order reversed.
Epimorphism
Let be a category and and be objects of
. A morphism is said to be a
monomophism iff:
For all objects of and all morphisms [ what set? ]
[]
Essense
If is a monomorphism and is composed with some unknown morphism , then knowing
and is enough to information to recover exactly.
In other words, there is no redundancy prodived by that allows two
morphisms and to compose with such that
and produce the same morphism.
Yet another wording: no morphism can 'hide' behind .
Similarly, an epimorphism does not afford any ambiguity to morphisms
that compose after .