higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
function algebras on ∞-stacks?
derived smooth geometry
A semi-algebraic set is much like a variety over the real numbers, a subset of a Cartesian space $\mathbb{R}^n$ being the joint vanishing locus in of a finite set of polynomials intersected with inequalities.
A semi-algebraic manifold is much like a corresponding scheme, a space that is locally like a semi-algebraic set.
The Fulton-MacPherson compactifications $FM_n\left( \mathbb{R}^d\right)$ of ordered configuration spaces of points are semi-algebraic manifolds (Lambrechts-Volic 14, Prop. 5.2).
Moreover, the canonical forgetful functions $FM_{n + k}\left( \mathbb{R}^d\right) \longrightarrow FM_{n}\left( \mathbb{R}^d\right)$ are semi-algebraic fiber bundles (Lambrechts-Volic 14, Theorem 5.8).
A theory of differential forms on semi-algebraic manifolds was sketched in Appendix 8 of
Details are provided, the de Rham complex of semi-algebraic differential forms on a semi-algebraic manifold is constructed in and their real homotopy theory is studied in
Discussion of the example of Fulton-MacPherson compactifications of configuration spaces of points includes
See also