# 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

$\Lambda^n_k\to \mathcal{E}(U)$
| $\phantom{aaaaaa}$|
V $\phantom{aaaaw}$V
$\Delta^n\to \mathcal{B}(U)$
there exists a covering family $(V_i\to U)_i$ such that in the induced square

$\Lambda^n_k\to \Pi_i\mathcal{E}(V_i)$
| $\phantom{aaaaaa}$|
V $\phantom{aaaaw}$V
$\Delta^n\to \Pi_i\mathcal{B}(V_i)$
has a lifting $\Delta^n\to \Pi_i\mathcal{E}(V_i)$

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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.