Relative category structure on (Set valued) presheaves

Question

Suppose $(\mathcal{C},\mathcal{W})$ is a relative category (we can assume $\mathcal{C}$ is small for the matter). Is there any work which deal with constructing a relative category structure on $\mathbf{P}(\mathcal{C})$ - the category of $\mathbf{Set}$-valued presheaves on $\mathcal{C}$ extending the structure on $\mathcal{C}$?

Answer 1

The usual constructions of Grothendieck homotopy theory (as presented by Maltsiniotis and Cisinski) can be easily extended to the setting of relative categories.

Recall that given a small category $A$, we can turn the category of presheaves of sets on $A$ into a relative category by creating weak equivalences using the right adjoint functor $ι$ from presheaves of sets on $A$ to small categories that send a presheaf $F$ of sets on $A$ to the category of elements of $F$, i.e., the comma category $A/F$.

This construction continues to make sense if $A$ is a relative category. In this case, the comma category $A/F$ admits a relative category structure created by the forgetful functor $A/F→A$.

Now we can turn presheaves of sets on $A$ into a relative category by declaring that a morphism of presheaves $F→G$ is a weak equivalence if the morphism of relative categories $A/F→A/G$ is a Dwyer–Kan equivalence of relative categories.

To the best of my knowledge, there are no attempts in the literature to study the resulting relative category of presheaves, although it would be interesting to know whether there are, for example, any interesting analogues of weak test categories, i.e., relative categories $A$ for which the left adjoint $ι^*$ of $ι$ preserves and reflects weak equivalences, and the adjunction $ι^*⊣ι$ is a Dwyer–Kan equivalence of relative categories.