Strong epimorphism

A strong epimorphism has the left lifting property with respect to every monomorphism. Thus every commutative square with the strong epimorphism on the left and a monomorphism on the right has a diagonal filler.

A map $e:X\to Y$ has this property if every commutative square with $e$ as its left side and a monomorphism $m:A\to B$ as its right side has a diagonal filler $t:Y\to A$. The filler is unique because $m$ is monic. This definition does not itself stipulate that $e$ is an epimorphism; binary products imply that conclusion by testing against the categorical diagonal.

In a category with binary products in a category, a map with the left lifting property against monomorphisms is an epimorphism. If $ue=ve$, lift the square with right side $\Delta$ and bottom side $⟨u,v⟩$. The two projections of its filler give $u=v$. This argument does not require equalizers.

If $gf$ has the left lifting property against monomorphisms, so does $g$. Precompose a lifting square for $g$ with $f$ and lift the composite. To check the remaining triangle, cancel the monic side of the square, rather than cancelling $f$. In particular, a monic right factor of a lifting-only strong morphism is an isomorphism.

If $e:X\to Y$ is a monomorphism and has the left lifting property against monomorphisms, apply its property to the square with both vertical arrows $e$ and both horizontal arrows identities. The filler is a two-sided inverse, so $e$ is an isomorphism.

Suppose a category has pullbacks and coequalizers and every pullback of a regular epimorphism is epic. Coequalizing the kernel pair of any morphism produces a factorization as a regular epimorphism followed by a monomorphism. Consequently every strong epimorphism is regular.

A functor between small categories is a strong epimorphism precisely when its image generates its codomain as a subcategory. It is a regular epimorphism precisely when it is surjective on objects and full. Hence a functor can be strong without being regular when its image arrows generate additional composites that have no single preimage.

A strong quotient of an object $A$ is the codomain of a strong epimorphism out of $A$. A class of objects is closed under strong quotients when every such codomain remains in the class up to isomorphism.