A projective-cofibrant replacement in $\mathsf{sSet}^{\mathcal{C}^{\text{op}}}$ such that $\operatorname{ev}_0Q(S) \cong \operatorname{ev}_0S$

Question

Consider the category of simplicial presheaves $\mathsf{sSet}^{\mathcal{C}^{\text{op}}}$ endowed with the projective model structure, i.e. weak equivalences and fibrations are point-wise.

I need a functorial cofibrant replacement $Q:\mathsf{sSet}^{\mathcal{C}^{\text{op}}} \to \mathsf{sSet}^{\mathcal{C}^{\text{op}}}$ with the property that $\operatorname{ev}_0Q(S) \cong \operatorname{ev}_0S$ are isomorphic in $\mathsf{Set}^{\mathcal{C}^{\text{op}}}$ for every simplicial presheaf $S$.

Actually, I would be satisfied if it worked just for all $S$ constant in the simplicial direction, so just for all $S:n \mapsto P$ for some $P \in \mathsf{Set}^{\mathcal{C}^{\text{op}}}$.

Question 1

Does such a cofibrant replacement exist? If it does, a reference too would be awesome.

Question 2

Before someone answers I add a related less general question. Does there exist a cofibrant replacement (not necessarily functorial) of the point $* \in \mathsf{sSet}^{\mathcal{C}^{\text{op}}}$ with the property that $\operatorname{ev}_0Q(*) \cong * \in \mathsf{Set}^{\mathcal{C}^{\text{op}}}$ ?

Answer 1

Before someone answers I add a related less general question. Does there exist a cofibrant replacement (not necessarily functorial) of the point ∗∈sSetCop with the property that ev0Q(∗)≅∗∈SetCop ?

No. Cofibrations in the projective model structure on simplicial presheaves on C are retracts of transfinite compositions of cobase changes of generating cofibrations, which are given by tensoring a representable presheaf on C with a generating cofibration ∂Δ^n→Δ^n of simplicial sets.

In particular, if a cofibrant replacement of the terminal simplicial presheaf has a terminal presheaf of sets in degree 0, then the terminal presheaf is a retract of a coproduct of representables.

It is easy to construct examples of C for which there no maps from the terminal presheaf to a coproduct of representables. Indeed, the latter amounts to saying the taking the limit of such a coproduct (consider as a functor from C^op) produces the empty presheaf.

This argument can be leveraged to give a complete description of projectively cofibrant simplicial presheaves, see Necessary conditions for cofibrancy in global projective model structure on simplicial presheaves.