Monomorphisms and epimorphisms
The ideas of injections and surjections in the context of sets and functions are 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 monomorphism iff:
For all objects \( Z \) of \( \cat{C} \) and all morphisms \( \alpha', \, \alpha'' \in \) \( \cathom{C}(Z, A) \)
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 an epimorphism iff:
For all objects \( Z \) of \( \cat{C} \) and all morphisms \( \beta', \, \beta'' \in \) \( \cathom{C}(B, Z) \)
Essence
If \( f \) is a monomorphism and \( f \, \alpha \) is \( f \) composed with some unknown morphism \( \alpha \), then knowing \( f \) and \( f \, \alpha \) is enough information to recover \( \alpha \) exactly.
In other words, there is no redundancy provided 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 \( g \).
\( \lnot \) (monomorphic \( \land \) epimorphic \( \iff \) isomorphic)
Functions between sets are injective iff they are monomorphisms, and they are surjective iff they are epimorphisms. Consequently, a function is isomorphic iff it is monomorphic and epimorphic.
This bi-implication relationship, (monomorphic ∧ epimorphic ⇔ isomorphic), is true for the category \( \mathrm{Set} \), but it is not true for categories in general. For example, the category formed by the relation ≤ on the integers: all morphisms are both monomorphisms and epimorphisms, but only a subset of morphisms, the identities, are isomorphisms.
Injective relationship between morphisms
The following diagram tries to hint at the idea that a monomorphism \( g \) induces an injective map forming composition morphisms when \( g \) is the second morphism. An analogous situation describes an epimorphism inducing an injection when the epimorphism is applied first in a composition.
Example
\(\phi: \mathbb{Z}/3\mathbb{Z} \to S_3 \). Monomorphic without a left-inverse.
There are non-trivial monomorphisms from the cyclic group \(C_3\) to the symmetric group \(S_3\), and its set function is injective. However, one cannot go the other way.
In category \( \cat{Set}\), a function is monomorphic iff injective iff it has a left-inverse. In other categories, however, stricter requirements on what constitutes a homomorphism may either prevent there being a left inverse, or may weed out other candidate mappings allowing a non-injective map to be monomorphic. In the current example, a monomorphism with an injective set function has no inverse, as the possible inverting set functions don't qualify as a group homomorphism.