# nLab sheaves on a simplicial topological space

under construction

# Contents

## Definition

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

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

• objects are

• collections $\left({S}_{n}\in \mathrm{Sh}\left({X}_{n}\right){\right)}_{n}$

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

$S\left(\alpha \right):X\left(\alpha {\right)}^{*}{S}_{n}\to {S}_{m}$S(\alpha) : X(\alpha)^* S_n \to S_m
• such that

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

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

$\begin{array}{ccc}X\left(\beta {\right)}^{*}X\left(\alpha {\right)}^{*}{S}_{n}& \stackrel{X\left(\beta {\right)}^{*}S\left(\alpha \right)}{\to }& X\left(\beta {\right)}^{*}{S}_{m}\\ \simeq ↓& & {↓}^{S\left(\beta \right)}\\ X\left(\beta \alpha {\right)}^{*}{S}_{m}& \underset{S\left(\beta \alpha \right)}{\to }& {S}_{k}\end{array}$\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 }
• morphisms are collections $\left({S}_{n}\to {T}_{n}\right)$ 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, $\mathrm{Sh}\left({N}_{•}C\right)$ is the classifying topos for $C$-torsors. see classifying topos of a localic groupoid.

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