Weak Monomorphism

A morphism $f:{{X}\to_{u}{Y}}$ in a Displayed Category $\mathcal{E}\rightarrowtriangle\mathcal{B}$ is a weak monomorphism if for all $v:{{C}\to{A}}$ and $g,h:{{Z}\to_{v}{X}}$, $f\circ g=f\circ h$ implies that $g=h$.

In more plain terms, weak monomorphisms are Monic, but only for morphisms of $\mathcal{E}$ that lie over the same morphism. This property is quite useful, and can often replace an assumption that $f$ is cartesian.

Naming

The name "weak mono" was chosen as it is similar to the property of $f$ being a Weak Cartesian Morphism, though "hypomonic" could also suffice. However, the argument could be made that this is a bad name.

Properties

• Every Cartesian Morphism is weakly monic, and thus every invertible morphism is weakly monic.

• If $f:{{Y}\to_{u}{Z}}$ and $g:{{X}\to_{v}{Y}}$ are weakly monic, then $f\circ g$ is weakly monic.

  Let $h,k:{{A}\to_{w}{X}}$ such that $f\circ g\circ h=f\circ g\circ k$. Both $g\circ h$ and $g\circ k$ are displayed over the same map, so $g\circ h=g\circ k$, at which point we get $h=k$.

• If $f\circ g$ is weakly monic, then so is $g$.

  Let $h,k:{{}_{w}(X,Y)}$ be a pair morphisms with $g\circ h=g\circ k$. $f\circ g$ is weakly monic, so it suffices to show that $f\circ g\circ h=f\circ g\circ k$, which follows from our assumption.

• Postcomposition with a weak mono is Injective; this suggests that weak monos are the "right" notion of monomorphism in a displayed category, not Total Monomorphisms.

References

• This property is mentioned without name in Categorical Logic and Type Theory, Exercise 1.1.1 (ii).