Cofibrations of functors

Question

Let $\cal M$ and $\cal N$ be model categories, $S,T:\cal M\to N$ functors, and $\alpha:S\to T$ a natural transformation. Say that $\alpha$ is a <blank> cofibration if for any cofibration $i:A\to B$ in $\cal M$, the "Leibniz" pushout corner map

$$ S(B) \cup_{S(A)} T(A) \longrightarrow T(B)$$

is a cofibration in $\cal N$ that is acyclic if $i$ is. Has this notion been studied? Does it have a name?

Some notes:

• If $H:\cal P\times \cal M\to \cal N$ is a Quillen two-variable adjunction (such as the tensor product of a monoidal model category, or the copower of an enriched model category), then $H(j,-) : H(C,-) \to H(D,-)$ is a <blank> cofibration for any cofibration $j:C\to D$ in $\cal P$.
• <blank> cofibrations are closed under pushout and transfinite composites in $[{\cal M, N}]$. In fact they are the maps in $[{\cal M,N}]$ having the left lifting property with respect to a certain class of "Leibniz right Kan extension" maps defined from a cofibration in $\cal M$ and a fibration in $\cal N$ one of which is acyclic. But I see no obvious reason for these classes of maps to form a weak factorization system.
• If $S$ (hence $T$) is <blank> cofibrant, then $\alpha:S\to T$ is a <blank> cofibration if and only if the induced functor $\cal M \to \cal N^{\bf 2}$ is left Quillen for the Reedy model structure on $\cal N^{\bf 2}$ where the arrow of $\bf 2$ points "up". I think the cofibrancy condition can be removed by using instead some "hybrid Reedy-ish" weak factorization systems on $\cal N^{\bf 2}$.

Answer 1

When $\mathcal{M}$ and $\mathcal{N}$ are combinatorial, this class $\mathcal{C}$ is indeed the left class of a weak factorization system on the category of all left adjoints from $\mathcal{M}$ to $\mathcal{N}$. I have a proof written down somewhere, but if I recall correctly, the basic idea is to choose generating (acyclic) cofibrations $I$ ($J$) for $\mathcal{M}$ and use these to write down $\mathcal{C}$ as the preimage under a suitable left adjoint $$\mathrm{Fun^L}(\mathcal{M}, \mathcal{N}) \to \prod_{f \in I} \mathcal{N}^{\cdot \to \cdot} \times \prod_{g \in J} \mathcal{N}^{\cdot \to \cdot}$$ where the two $\mathcal{N}^{\cdot \to \cdot}$s are equipped with (different) suitable model structures cooked up to produce the class of morphisms under consideration. Then you can apply the result of M. Makkai, J. Rosický, Cellular categories, to conclude that $\mathcal{C}$ is the left class of a weak factorization system.

The weak factorization systems of an injective model category structure on diagrams can be seen as a special case of this construction, so you are going to need to do something relatively difficult to prove this.