Strong monomorphism

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.

Take four distinct objects $a,b,x,d$ and six nonidentity arrows $m:a\to b$, $h:x\to b$, $r,s:b\rightrightarrows d$, $k:a\to d$, $l:x\to d$, with $rm=sm=k$ and $rh=sh=l$. These and identities are all morphisms. The only nonidentity epimorphisms have target $d$, which admits no arrow to $b$, so every lifting square against $m$ either has an identity left side or does not exist. Thus $m$ is a strong monomorphism. It is not a regular monomorphism: $h$ equalizes $r,s$ but cannot factor through $m$, and an identical pair has $1_b$ as equalizer. Every strong nonregular monomorphism in any category occurs as the image of $m$ under a faithful functor from this category. Indeed, such a monomorphism is not epic, so choose distinct $r,s$ agreeing on it; failure of its equalizer property then supplies $h$.

In a balanced category with pullbacks in a category, pull a monomorphism back across the bottom map of a lifting square whose left map is an epimorphism. The resulting monic projection is epic because its composite with the lifted top map is epic. Balancedness makes that projection invertible, producing the unique lift. Conversely, if all monomorphisms are strong monomorphisms, applying the lifting property of a bimorphism to its own square gives its inverse, so the category is balanced. Pullback stability of epimorphisms is not needed.

A morphism $m:A\to B$ is strict when every $g:C\to B$ satisfying $hm=km\Rightarrow hg=kg$ for all parallel arrows $h,k$ out of $B$ factors uniquely through $m$. Every regular monomorphism is strict by its equalizer property. A strict monomorphism is strong: in a lifting square with an epimorphism, cancellation of the epimorphism makes the bottom arrow admissible, and its unique factorization gives the diagonal.

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

This is the equalizer-defined subclass of monomorphisms.

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.