nLab sheaves on a simplicial topological space

under construction

Definition

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

    $$S(\alpha) : X(\alpha)^* S_n \to S_m$$

  • such that

    • $S(Id_{[n]}) = Id_{S_n}$;

    • for every $\alpha : [n] \to [m]$ and $\beta : [m] \to [k]$ the diagram

      $$\array{ X(\beta)^* X(\alpha)^* S_n &\stackrel{X(\beta)^* S(\alpha)}{\to}& X(\beta)^* S_m \\ {\mathllap{\simeq}}\downarrow && \downarrow^{\mathrlap{S(\beta)}} \\ X(\beta \alpha)^* S_n &\underset{S(\beta \alpha)}{\to}& S_k }$$

      commutes;

• morphisms are collections $(S_n \to T_n)$ of morphisms of sheaves, compatible with all structure maps.

Examples

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.