ag.algebraic geometry - Local fibration vs. stalkwise fibration

These are equivalent.

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.

(This is addressed in the introduction to the paper by Jardine, "Boolean localization in practice" (Documenta Mathematica, v.1), where he tells you what to do even if you don't have enough points!)

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