nLab atlas

Theorems

Manifolds and cobordisms

manifolds and cobordisms

Contents

Idea

An atlas is a compatible collection of coordinate charts.

Definition

In full generality, for $𝒢$ a pregeometry and $X\in {\mathrm{Sh}}_{\left(\infty ,1\right)}\left(𝒢\right)$ an object in the (∞,1)-sheaf (∞,1)-topos, an atlas for $X$ is a collection of suitable morphisms (open maps) $\left\{{U}_{i}\to X\right\}$ with ${U}_{i}\in 𝒢↪{\mathrm{Sh}}_{\left(\infty ,1\right)}\left(𝒢\right)$, such that the morphism out of the coproduct

$\coprod _{i}{U}_{i}\to X$\coprod_i U_i \to X

is an effective epimorphism.

Examples

For geometric stacks

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Revised on August 18, 2013 14:24:42 by Urs Schreiber (24.131.18.91)