nLab
sheaves on a simplicial topological space

under construction

Definition

For $X_{•}:{\Delta }^{\mathrm{op}}\to$ Top a simplicial object topological space, write $\mathrm{Sh}(X_n)$ for the category of sheaves on (the category of open subsets) $X_n$.

The category $\mathrm{Sh}(X_{•})$ of sheaves on the simplicial space is defined to be the category whose

• objects are

  • collections $(S_n\in \mathrm{Sh}(X_n){)}_n$

  • equipped for each $\alpha :[n]\to [m]$ with morphisms

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

  • such that

    • $S({\mathrm{Id}}_{[n]})={\mathrm{Id}}_{S_n}$;

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

      $$\begin{array}{ccc}X(\beta {)}^{*}X(\alpha {)}^{*}{S}_n& \stackrel{X(\beta {)}^{*}S(\alpha )}{\to }& X(\beta {)}^{*}{S}_m\\ \simeq \downarrow & & {\downarrow }^{S(\beta )}\\ X(\beta \alpha {)}^{*}{S}_m& \underset{S(\beta \alpha )}{\to }& S_k\end{array}$$\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_m &\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 $NC:{\Delta }^{\mathrm{op}}\to \mathrm{Top}$ its nerve, $\mathrm{Sh}(N_{•}C)$ is the classifying topos for $C$-torsors. see classifying topos of a localic groupoid.