nLab microflexible sheaf

Contents

Context

Manifolds and cobordisms

(,1)(\infty,1)-Topos Theory

(∞,1)-topos theory

Background

Definitions

Characterization

Morphisms

Extra stuff, structure and property

Models

Constructions

structures in a cohesive (∞,1)-topos

Contents

Definition

Denote by Emb nEmb_n the site of nn-dimensional smooth manifolds and open embeddings.

An (∞,1)-sheaf F:Emb n opTopF\colon Emb_n^op\to Top of topological spaces is microflexible if for any closed inclusion KKK\to K' of compact spaces, the induced map F(K)F(K)F(K')\to F(K) is a Serre microfibration.

An (∞,1)-sheaf F:Emb n opTopF\colon Emb_n^op\to Top of topological spaces is flexible if for any closed inclusion KKK\to K' of compact spaces, the induced map F(K)F(K)F(K')\to F(K) is a Serre fibration.

Gromov’s theorem

Given an open manifold MM, the inclusion of microflexible sheaves into flexible sheaves on the slice site site Emb n/MEmb_n/M is an equivalence of (∞,1)-categories.

Literature

The original reference is

  • M. L. Gromov?, Stable mappings of foliations into manifolds, Mathematics of the USSR-Izvestiya 3:4 (1969), 671-694. doi.

The canonical reference is Section 2.2.1 of

See also

Last revised on May 9, 2022 at 13:00:33. See the history of this page for a list of all contributions to it.