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Monomorphisms and epimorphisms

Monomorphisms and epimorphisms generalise injective and surjective functions in set theory, respectively.[cite:@enwiki:1253148463]

Definition

A morphism $f : x \to y$ is

a monomorphism if for any two morphisms $g_{1}, g_{2} : w \to x$ , $g_{1} f = g_{2} f$ implies $g_{1} = g_{2}$ . That is to say that a monomorphism is a right-cancellativeIn diagrammatic order! morphism.
an epimorphism if for any two morphisms $g_{1}, g_{2} : y \to z$ , $f g_{1} = f g_{2}$ implies $g_{1} = g_{2}$ . That is to say that an epimorphism is a left-cancellative morphism.

In any category in which objects have "underlying sets," then any underlying {in,sur}jection underlying a morphism induces a {mono,epi}morphism at the categorical level.[cite:@riehl2017category] Note that the converse does not necessarily hold, as shown in CTiC Exercise 1.6.v.

Continuing the parallels between bijections, surjections, and injections, every isomorphism is both monic and epic. Perhaps surprisingly, the converse does not hold.

Nomenclature

Adjectival forms: monomorphisms are monic and epimorphisms are epic.
Shorthand: a monomorphism may be referred to as a mono, and an epimorphism an epi.

Notation

A monomorphism from $x$ to $y$ is denoted $x ↣ y$ .
An epimorphism from $x$ to $y$ is denoted $x ↠ y$ .

Examples

Monomorphisms and epimorphisms in 𝐒𝐞𝐭

The monomorphisms of $𝐒 𝐞 𝐭$ are exactly the injective functions, and the epimorphisms are exactly the surjections:

Suppose a $𝐒 𝐞 𝐭$ -mono, $f : X \to Y$ , and two constants $x, x^{'} : {*} \to X$ , where ${*}$ denotes the singleton set. The definition of 'monomorphism' specialises such that $x f = x f^{'}$ implies $x = x^{'}$ — an injection.[cite:@riehl2017category]

Composition

Monos and epis compose; that is,

if $f : x ↣ y$ and $g : y ↣ z$ are monic, so is $f g$ .
if $f g$ is monic, then $f$ is monic.

Dually,

If $f : x ↠ y$ and $g : y ↠ z$ are epic, so is $f g$ .
If $f g$ is epic, then $g$ is epic.

The proof of this property is left to CTiC Exercise 1.2.iii.

Sections and retractions

Suppose the following diagram in a category $𝐂$ :

The arrow $s$ is called a section or right inverse of $r$ , while $r$ is called a retraction or left inverse to $s$ . Together, $s$ and $r$ express the object $x$ as a retract of $y$ .[cite:@riehl2017category]

Left-invertible morphisms are necessarily monic, but monomorphisms do not necessarily have inverses^[proof?]. To emphasise this, we give $s$ yet another label; $s$ can be called a split monomorphism.[cite:@enwiki:1253148463@riehl2017category] As you've probably inferred, $r$ is called a split epimorphism.

While functors do not generally preserve monos or epis, functors do preserve split monos and split epis.[cite:@riehl2017category]

Remarks

The axiom of choice asserts that every epimorphism in the category of sets is a split epimorphism.[cite:@riehl2017category]

References

[cite:@nlab:split_monomorphism]
[cite:@riehl2017category]