nLab
sheaves on a simplicial topological space

under construction

Contents

Definition

For X :Δ opX_\bullet : \Delta^{op} \to Top a simplicial object topological space, write Sh(X n)Sh(X_n) for the category of sheaves on (the category of open subsets) X nX_n.

The category Sh(X )Sh(X_\bullet) of sheaves on the simplicial space is defined to be the category whose

  • objects are

    • collections (S nSh(X n)) n(S_n \in Sh(X_n))_n

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

      S(α):X(α) *S nS m S(\alpha) : X(\alpha)^* S_n \to S_m
    • such that

      • S(Id [n])=Id S nS(Id_{[n]}) = Id_{S_n};

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

        X(β) *X(α) *S n X(β) *S(α) X(β) *S m S(β) X(βα) *S m S(βα) S k \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 nT n)(S_n \to T_n) of morphisms of sheaves, compatible with all structure maps.

Examples

For CC a topological category and NC:Δ opTopN C : \Delta^{op} \to Top its nerve, Sh(N C)Sh(N_\bullet C) is the classifying topos for CC-torsors. see classifying topos of a localic groupoid.

Revised on December 13, 2010 14:04:37 by Anonymous Coward (141.20.212.217)