**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](https://faculty.math.illinois.edu/K-theory/0563/spre.pdf) and [Lurie](http://people.math.harvard.edu/%7Elurie/papers/highertopoi.pdf), 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!



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


