Checking that (hyper) sheafification is fibrant in local projective model structure on simplicial presheaves

Question

Context and Notation

Let $X$ be a manifold and $\mathcal{C} = Op(X)$ be the category of open subsets of $X$ with inclusions. I then consider the projective model structure on the (simplicial model) category, $sPre$, of simplicial presheaves, $\mathcal{C}^{op} \to sSet$, and its simplicial mapping space (i.e. homotopy function complex), $\underline{sPre}(-,-)$.

To (left) localize this projective model structure I can choose a collection of morphisms between cofibrant objects and in this case I will choose some subfunctors, $\mathcal{U} \hookrightarrow r_{\bullet} V $, where $V \in obj(\mathcal{C})$ and $r_{\bullet}$ is the (simplicial) Yoneda embedding, $\mathcal{C} \xrightarrow{r_{\bullet}} sPre$. Let $S= \{ \mathcal{U} \hookrightarrow r_{\bullet} V \}$ be such a collection of maps between cofibrant objects in $sPre$, where $V$ and $\mathcal{U}$ are varying.

A simplicial presheaf $G \in obj(sPre)$ is then called $S$-local if each map $\mathcal{U} \hookrightarrow r_{\bullet} V$ induces a weak equivalence of simplicial sets, $$G(V) = \underline{sPre}(r_{\bullet} V, G)\to \underline{sPre}(\mathcal{U}, G). $$

Now, having been mostly learning from papers like DuHoIs and Lurie, it seems that there should be a "one-step-sheafification" construction. Perhaps I am oversimplifying the setup needed, but if I were to just not necessarily care whether I recovered the usual local projective model structure, I want to know if the following sheafification construction is still $S$-local in the current context.

My Question Given the setup above, let $F$ be a simplicial presheaf and define, $$F^{\dagger}(V) = hocolim_{\mathcal{U} \hookrightarrow r_{\bullet} V} \underline{sPre}(\mathcal{U}, F)$$

Now I feel as though it should be due to some abstract nonsense that $F^{\dagger}$ is $S$-local but I don't see it. In other words, it should be true that for each $\mathcal{Y} \hookrightarrow r_{\bullet}Z$ in $S$ we have a weak equivalence,

$$hocolim_{\mathcal{U} \hookrightarrow r_{\bullet} Z} \underline{sPre}(\mathcal{U}, F)= F^{\dagger}(Z) = \underline{sPre}(r_{\bullet} Z, F^{\dagger})\to \underline{sPre}(\mathcal{Y}, F^{\dagger}). $$

Why is this a weak equivalence or what am I missing / doing wrong? I know that I could have stated my question more generally but in the event my context hints at why something is true in my special setting, that would be great.

Thank you!

Answer 1

There are two ways to make this construction work.

The first way is to iterate the step $F↦F^†$ transfinitely many times. The reason that a single iteration of $F↦F^†$ is not sufficient is that while $F^†$ does add the missing data that prevents $F$ from satisfying the lifting property for elements of $S$, the newly added parts of $F^†$ may prevent it from satisfying the lifting property itself.

By iterating $F↦F^†$ sufficiently many times, we circumvent this problem by ensuring that any map from the domain or codomain of an element of $S$ factors through some intermediate stage of the transfinite composition. Thus, it suffices to iterate $F↦F^†$ for $α$ steps, where $α$ is a regular cardinal such that all domains and codomains of $S$ are $α$-small.

The other way is applicable when $S$ forms a Grothendieck topology. In this case, the $S$-localization functor can be computed using Verdier's hypercovering theorem, with the same homotopy colimit except that $U→r$ now has to run over all hypercovers generated by $S$. This construction does not require iteration, it does everything in one step.