nLab microflexible sheaf

Definition

Denote by $Emb_n$ the site of $n$-dimensional smooth manifolds and open embeddings.

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

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

Gromov’s theorem

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

Related concepts

• h-principle

• Serre microfibration

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

• Mikhael Gromov, Partial differential relations, 1986, doi.

See also

• John Francis, The H-principle for microflexible sheaves, 2010 (pdf)

• Alexander Kupers, Section 2 of: Three applications of delooping to H-principles, Geom Dedicata 202 (2019) 103–151 (arXiv:1701.06788, doi:10.1007/s10711-018-0405-7)