nLab
monomorphism

Idea

The notion of monomorphism is the generalization of the notion of injective map of sets from the category Set to arbitrary categories.

Definition

A monomorphism in a category $C$ is a morphism $f:X\to Y$ such that, equivalently,

• $f$ is an epimorphism in the opposite category $C^{\mathrm{op}}$.

• for any object $A$ the induced morphism $f_{*}:C(A,X)\to C(A,Y)$ is injective.

The last condition here states the usual arrow-theoretic way to say monomorphism:

The morphism $f:X\to Y$ is mono precisely if for all $g,h:A\to X$ such that $f_{*}(h):A\stackrel{h}{\to }X\stackrel{f}{\to }Y$ equals $f_{*}(g):A\stackrel{g}{\to }X\stackrel{f}{\to }Y$ we have $g=h$.

Properties

We list the following properties without their (easy) proof. The proofs can be found spelled out (dually) at epimorphism.

Proposition

Every equalizer

$$t\to x\overrightarrow{\to }y$$t \to x \stackrel{\to}{\to} y

is a monomorphism.

Proposition

Monomorphisms are preserved by pullback.

Proposition

Monomorphisms are preserved by right adjoint functors.

Variations

At epimorphism there is a long list of variations on the concept of epimorphism. Each of these, of course, has a dual notion for monomorphism, but the ones most commonly used are:

• split monomorphism = morphism which has a retraction
• normal monomorphism = a kernel of some morphism (in a category with zero morphisms)
• regular monomorphism = an equalizer of some parallel pair
• strong monomorphism = a monomorphism right orthogonal to epimorphisms
• monomorphism

Frequently, regular and strong monics coincide. For instance, this is the case in any quasitopos, and also in Top, where they are the subspace inclusions (the plain monomorphisms are the injective functions).

Sometimes, all monomorphisms are regular—this seems to happen a bit more frequently than for epimorphisms. For instance, this is the case in any pretopos (including any topos, such as Set), but also in any abelian category, and also in the category Grp.

In Ab and in any abelian category, all monomorphisms are normal. But this is not so in Grp, where (despite the fact that all monomorphisms are regular), the normal monics are the inclusions of normal subgroups (hence the name). In any Ab-enriched category, all regular monics are normal, but not all monics need be regular.

In a Boolean topos, such as Set (in classical mathematics), any monomorphism with inhabited domain is split. Of course, no monic with empty domain and inhabited codomain can be split (in contrast to the dual case, where it can happen that all epimorphisms split – the axiom of choice).

Related concepts

• isomorphism classes of monomorphism define subobjects.

• monomorphism in an (∞,1)-category, n-monomorphism

• image, coimage