I'll get straight to the point: the simplex category $\Delta$ can be given the structure of a site by defining $S$ to be a sieve on an object $[n]$ if all morphisms selected by $S$ are injective on set level and if the union of their images as functions of sets equals $[n]$.
Then sheaves $F$ on $\Delta$ are simplicial sets. The locality condition of a sheaf on $\Delta$ corresponds to faces of $F$ being uniquely defined by the faces that they are attached to. I.e if we have faces $f,f' \in F[n]$ with $d_i(f)=d_i(f')$ for all $i$ then $f = f'$.
I don't really know what to make of this if I want to prove that all Kan complexes are sheaves and all sheaves are Kan complexes though or if the injective condition on the morphisms selected by the sieves is even needed
Thanks for any input!