manifolds and cobordisms
cobordism theory, Introduction
(2,1)-quasitopos?
structures in a cohesive (∞,1)-topos
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.
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.
The canonical reference is Section 2.2.1 of
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)