Inclusion of $1$-presheaves into $\infty$-presheaves preserves pushouts?

Question

Let $\mathcal{R}$ be a $1$-category. Assume that one has a pushout of representable $1$-presheaves $\mathrm{y} A \cup_{\mathrm{y} B} \mathrm{y} C$ in $\mathsf{PSh}(\mathcal{R})$. Under which conditions does then the inclusion $\mathsf{PSh}(\mathcal{R}) \to \mathcal{P}(\mathcal{R})$ into $\infty$-preheaves preserve this pushout?

The reason why I am asking this, is the claim in Notation 8.3 of Barwick‘s and Schommer-Pries‘ unicity paper which says that a specific class (called $S_{0}$) of specific morphisms in $\mathcal{P}(\mathcal{R})$ induced by some pushouts in $\mathcal{R}$ is the essential image under the above functor of a similarly defined class in $\mathsf{PSh}(\mathcal{R})$. For that to make sense, I think that the above must hold for the pushouts appearing there. Note also that in the situation, we know that the pushout $A \cup_{B} C$ exists in $\mathcal{R}$.

Answer 1

The inclusion from discrete presheaves to spaces doesn't preserve all pushouts, even when $\mathcal R$ is a point.

But it does preserves some pushouts. The usual condition is to check that at least one of $yB \to yA$ or $yB \to yC$ is a levelwise monomorphism of sets. Then because the Kan-Quillen model structure is left proper with monomorphisms for cofibrations, it follows that the pushout is a levelwise pushout of spaces, and hence a pushout of presheaves of spaces.

Note that for $yB \to yA$ (say) to be a levelwise monomorphism of sets is equally for $B \to A$ to be a monomorphism in $\mathcal R$.

I haven't checked carefully, but I suspect that this criterion suffices to imply that all the pushouts considered by Barwick and Schommer-Pries are preserved by the inclusion from discrete presheaves to presheaves of spaces.