# Local fibration vs. stalkwise fibration

Let $$\mathbf{C}$$ be a Grothendieck site with enough points. Let $$p:\mathcal{E}\to \mathcal{F}$$ be a map of simplicial presheaves on $$\mathbf{C}$$. Is it true that $$p$$ is a local (Kan) fibration if and only if it is a stalkwise fibration?

To fix terminology:

$$p$$ is called stalkwise fibration if for each point $$q$$ the map $$q^*(\mathcal{E})\to q^*(\mathcal{B})$$ is a fibration.

$$p$$ is called local fibration if for each $$U\in \mathbf{C}$$ and each commutative diagram

$$\require{AMScd} \begin{CD} \Lambda^n_k @>>> \mathcal{E}(U)\\ @VVV @VVV \\ \Delta^n @>>> \mathcal{B}(U) \end{CD}$$

there exists a covering family $$(V_i\to U)_i$$ such that in the induced square

$$\require{AMScd} \begin{CD} \Lambda^n_k @>>> \Pi_i\mathcal{E}(V_i)\\ @VVV @VVV \\ \Delta^n @>>> \Pi_i\mathcal{B}(V_i) \end{CD}$$

has a lifting $$\Delta^n\to \Pi_i\mathcal{E}(V_i)$$.

If $$K$$ is a simplicial set, and $$\mathcal{F}$$ is a simplicial presheaf, then there's a presheaf of sets $$\mathcal{F}^K$$, defined by $$(\mathcal{F}^K)(U) = \hom(K, \mathcal{F}(U))$$, where $$\hom$$ is maps of simplicial sets.
The important observation here is that if $$K$$ is a finite simplicial set, then formation of this gadget commutes with sheafification: $$q^*(\mathcal{F}^K)\approx (q^*\mathcal{F})^K$$. This is because $$\mathcal{F} \mapsto \mathcal{F}^K$$ is computed as a finite limit, if $$K$$ is finite.
Now consider the map of presheaves of sets $$f: \mathcal{E}^{\Delta^n} \to \mathcal{E}^{\Lambda^n_k}\times_{\mathcal{B}^{\Lambda^n_k}} \mathcal{B}^{\Delta^n}$$. Your map $$p$$ is a local fibration if the sheafification of $$f$$ is an epimorphism; the map $$p$$ is a stalkwise fibration if $$q^*(f)$$ is a surjection for each point $$q$$. If you have enough points, these mean the same thing.