under construction
For $X_\bullet : \Delta^{op} \to$ Top a simplicial object topological space, write $Sh(X_n)$ for the category of sheaves on (the category of open subsets) $X_n$.
The category $Sh(X_\bullet)$ of sheaves on the simplicial space is defined to be the category whose
objects are
collections $(S_n \in Sh(X_n))_n$
equipped for each $\alpha : [n] \to [m]$ with morphisms
such that
$S(Id_{[n]}) = Id_{S_n}$;
for every $\alpha : [n] \to [m]$ and $\beta : [m] \to [k]$ the diagram
morphisms are collections $(S_n \to T_n)$ of morphisms of sheaves, compatible with all structure maps.
For $C$ a topological category and $N C : \Delta^{op} \to Top$ its nerve, $Sh(N_\bullet C)$ is the classifying topos for $C$-torsors. see classifying topos of a localic groupoid.