nLab
atlas

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Context

Higher geometry

higher geometry / derived geometry

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Manifolds and cobordisms

manifolds and cobordisms

Definitions

Genera and invariants

Theorem

Contents

Idea

An atlas is a compatible collection of coordinate charts.

Definition

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

∐ iU i→X\coprod_i U_i \to X

is an effective epimorphism.

Examples

For manifolds

For geometric stacks

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Revised on April 3, 2013 14:57:29 by Urs Schreiber (82.169.65.155)
Edit | Back in time (2 revisions) | See changes | History | Views: Print | TeX | Source | Linked from: category algebra, (infinity,n)-category of cobordisms, cobordism, supermanifold, topological stack, manifold, Carlos Simpson, cobordism hypothesis, sphere, HQFT, transversal maps, symplectic manifold, cobordism category, cobordism cohomology theory, Riemannian manifold, (2,1)-dimensional Euclidean field theory, complex supermanifold, pseudo-Riemannian manifold, Hopf algebroid, Riemann surface, Banach manifold, foliation, Thom spectrum, manifold with boundary, bibundle, tubular neighbourhood, Pontrjagin-Thom collapse map, decalage, Thom isomorphism, normal bundle, local addition, smooth manifold, 2d TQFT, Fenchel-Nielsen coordinates, genus, cobordism ring, manifolds and cobordisms - contents, mapping class group, smooth structure, differentiable manifold, surface, vielbein, B-bordism, torus, circle, symmetric space, coordinate system, synthetic differential infinity-groupoid, atlas, diffeomorphism group, cohesive (infinity,1)-topos -- infinitesimal cohesion, complex manifold, embedding of smooth manifolds, concrete smooth infinity-groupoid, infinite-dimensional manifold, submanifold, concrete structure, contents of contents, hole argument, branched manifold, 2-framing, String bordism, Berkovich space, Hopf fibration, G2 manifold, generalized Calabi-Yau manifold, convenient manifold, ILH manifold, smooth groupoid, twisted smooth cohomology in string theory, Stein manifold, spin bordism, framed manifold, signature genus, Kervaire invariant, Hilbert manifold, geometrization conjecture, elliptization conjecture, low-dimensional topology, geometry of physics, generalized G2-manifold, projective manifold, prequantum field theory, super Riemann surface, closed manifold, Lipschitz map, non-Hausdorff manifold, hyperbolic manifold, extended geometric quantization of 2d Chern-Simons theory, higher Atiyah groupoid, higher prequantum geometry