nLab monomorphism

This entry is about the basic notion of “monic” in relation to morphisms in category theory. For the notion of “monic” in relation to polynomials in commutative algebra, see at monic polynomial.

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Idea

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

The formally dual concept is that of epimorphism, which similarly generalizes (or strengthens) the concept of surjective function.

Common jargon includes “is a mono” or “is monic” for “is a monomorphism”, and “is an epi” or “is epic” for “is an epimorphism”, and “is an iso” for “is an isomorphism”.

Definition

A morphism $f \colon X \to Y$ in some category is called a monomorphism (sometimes abbrieviated to mono, or described as being monic), if for every object $Z$ and every pair of parallel morphisms $g_1,g_2 \colon Z \to X$ then

$$\left( f \circ g_1 \,=\, f \circ g_2 \right) \;\Rightarrow \; \left( g_1 \,=\, g_2 \right) \,.$$

Stated more abstractly, this says that $f$ is a monomorphism if for every $Z$ the hom-functor $Hom(Z,-)$ takes it to an injective function between hom-sets

$$Hom(Z,X) \overset{\phantom{AA} f_\ast \phantom{AA}}{\hookrightarrow} Hom(Z,Y) \,.$$

Since injective functions are precisely the monomorphisms in Set (example below) this may be stated as saying that $f$ is a monomorphism if $Hom(Z,f)$ is a monomorphism for all objects $Z$.

Finally, $f$ being a monomorphism in a category $\mathcal{C}$ means equivalently that it is an epimorphism in the opposite category $\mathcal{C}^{op}$.

Examples

General

Example

(monomorphisms in preorders)

In a preorder, all arrows are mono because they satisfy the required condition vacuously (any pair of parallel arrows is equal in a preorder).

Example

(monomorphisms in $Set$)

The monomorphisms in the category Set of sets and functions between them are precisely the injective functions.

Example

(monomorphisms in $Cat$)

The monomorphisms in the category Cat of categories and functors between them are precisely the embeddings of categories, i.e. the injective-on-objects faithful functors.

Example

Every isomorphism is both a monomorphism and an epimorphism.

But beware that the converse fails:

Examples of monos that are epi but not iso

The following lists some examples of morphisms that are both monomorphisms and epimorphisms, but not necessarily isomorphisms.

Example

In the category of Hausdorff topological spaces, the inclusion $A \hookrightarrow X$ of a dense subspace is an epimorphism.

See this Prop. for proof.

Example

In unital Rings, the canonical inclusion $\mathbb{Z} \overset{i}{\hookrightarrow} \mathbb{Q}$ of the integers into the rational numbers is an epimorphism.

See this Prop. for proof.

Properties

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

Proposition

The following are equivalent:

• $f : x \to y$ is a monomorphism in $C$;

• $f$ is an epimorphism in the opposite category $C^{op}$;

• postcomposition with $f$ is a monomorphism in Set: that is, for all $c \in C$, $f \circ -: Hom(c,x) \to Hom(c,y)$ is an injection;

• the commuting diagram

  $$\array{ x &\stackrel{Id}{\to}& x \\ {}^{Id}\downarrow && \downarrow^{f} \\ x &\underset{f}{\to}& y }$$

  is a pullback diagram.

Proposition

If $f \colon x \to y$ and $g \colon y \to z$ are monomorphisms, so is their composite $g f$. If $g f$ is an monomorphism, so is $f$.

Proposition

Every equalizer

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

is a monomorphism.

The converse of the above proposition fails, and a mononomorphism that is the equalizer of some pair of morphisms is called a regular monomorphism.

Proposition

Monomorphisms are preserved by pullback.

(In an adhesive category they are also preserved by pushout.)

Proposition

Monomorphisms are preserved by any right adjoint functor, or more generally any functors that preserves pullbacks.

Proposition

Monomorphisms are reflected by faithful functors.

We have seen some ways in which monomorphisms get along with limits. Here is another:

Proposition

Any morphism from a terminal object is a monomorphism. The product of monomorphisms is a monomorphism.

Monomorphisms do not get along quite as well with colimits. For example, the unique morphism from the initial object is not always an monomorphism, and the canonical morphisms from the summands into a coproduct, e.g. $i_1 : x_1 \to x_1 + x_2$, are not always monomorphisms (though these results do hold in $Set$). However, the unique morphism from the initial object is a monomorphism when the initial object is strict, and the canonical morphisms into a coproduct are monomorphisms when the coproduct is a disjoint coproduct.

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 monos 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 monos are the inclusions of normal subgroups (hence the name). In any Ab-enriched category, all regular monos are normal, but not all monos need be regular.

In a Boolean topos, such as Set (in classical mathematics), any monomorphism with inhabited domain is split. Of course, no mono 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

• a category in which all morphisms are monomorphisms is called a left cancellative category

• jointly monic morphisms

References

Textbook accounts:

• Saunders MacLane, §I.5 of: Categories for the Working Mathematician, Graduate Texts in Mathematics 5 Springer (1971, second ed. 1997) [doi:10.1007/978-1-4757-4721-8]

• Francis Borceux, Section 1.7 in: Handbook of Categorical Algebra Vol. 1: Basic Category Theory [doi:10.1017/CBO9780511525858]