Monomorphism

A monomorphism $m:A\to B$ is left-cancellative: $mu=mv$ implies $u=v$. It represents a subobject of $B$.

A split monomorphism $f:A\to B$ has a left inverse $r:B\to A$ with $rf=1_A$. Every functor preserves split monomorphisms.

An absolute monomorphism is a morphism sent to a monomorphism by every functor out of its category. The absolute monomorphisms are exactly the split monomorphisms.

An object $A$ is a retract of $B$ when there are morphisms $s:A\to B$ and $r:B\to A$ with $rs=1_A$. The map $s$ is a split monomorphism and $r$ is a split epimorphism.

A strong monomorphism has the right lifting property with respect to every epimorphism. In every commutative square with an epimorphism on the left and the strong monomorphism on the right, there is a unique diagonal filler; uniqueness follows from monicity.

A regular monomorphism is a morphism that occurs as the equalizer of some parallel pair. Every regular monomorphism is a strong monomorphism.

Whenever the relevant pullback exists, the intersection of two strong subobjects of an object is strong. More generally, an arbitrary small intersection of strong subobjects is strong when it exists: lift into each subobject and use their common composite into the ambient object to obtain a map into the limit.

In the categorical usage where no model structure is specified, an anodyne morphism may mean a morphism that is both a monomorphism and an epimorphism. It need not be an isomorphism unless the category is balanced.

An object $B$ is saturated with respect to anodyne morphisms when it is injective against all of them: every map $A^{\prime }\to B$ extends across every anodyne morphism $A^{\prime }\to A$.

Suppose a complete well-powered category has enough objects saturated with respect to anodyne morphisms. Inside a saturated object containing $A$, intersect all strong subobjects containing $A$. The resulting object is saturated, the map from $A$ to it is anodyne, and its extension property makes it the reflection of $A$ into the full subcategory of saturated objects.