nLab
atlas

Context

Higher geometry

Manifolds and cobordisms

Contents

Idea

An atlas is a compatible collection of coordinate charts.

Definition

In full generality, for 𝒢\mathcal{G} a pregeometry and XSh (,1)(𝒢)X \in Sh_{(\infty,1)}(\mathcal{G}) an object in the (∞,1)-sheaf (∞,1)-topos, an atlas for XX is a collection of suitable morphisms (open maps) {U iX}\{U_i \to X\} with U i𝒢Sh (,1)(𝒢)U_i \in \mathcal{G} \hookrightarrow Sh_{(\infty,1)}(\mathcal{G}), such that the morphism out of the coproduct

iU iX \coprod_i U_i \to X

is an effective epimorphism.

Examples

For manifolds

For geometric stacks

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