Is there a standard name for the following flavor of weak factorization system? I am interested in the data of two classes of maps $\mathfrak L$ and $\mathfrak R$ in a category $\mathcal C$ such that:
- $\mathfrak L$ is exactly the class of element having the left lifting property against all the elements of $\mathfrak R$
- $\mathfrak R$ is closed under pullback
- each morphism of $\mathcal C$ can be factored as an element of $\mathfrak L$ postcomposed with a element of $\mathfrak R$
So the main difference with actual weak factorization systems is that $\mathfrak R$ is not necessarily closed under retracts. I guess it is a notion that is fairly common (or at least the dual version, when dealing with cofibrantly generated model categories), but I have not been able to find a name for this in the litterature.