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”.
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
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
Since injective functions are precisely the monomorphisms in Set (example 1 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 $\mathcal{C}^{op}$.
(monomorphisms in $Set$)
The monomorphisms in the category Set of sets and functions between them are precisely the injective functions.
Every isomorphism is both a monomorphism and an epimorphism.
But beware that the converse fails:
A morphism that is both a monomorphism and an epimorphism need not be an isomorphism.
For instance in the categories Ring or CRing, then the defining inclusion $\mathbb{Z} \hookrightarrow \mathbb{Q}$ of the ring of integers into that of rational numbers is both a monomorphism and an epimorphism, but clearly not an isomorphism.
We list the following properties without their (easy) proof. The proofs can be found spelled out (dually) at epimorphism.
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 $g$.
Every equalizer
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.
Monomorphisms are preserved by pullback.
Monomorphisms are preserved by right adjoint functors.
Monomorphisms are reflected by faithful functors.
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:
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).
isomorphism classes of monomorphism define subobjects.
a category in which all morphisms are monomorphisms is called a left cancellative category