  Consider the topologically enriched category of compact smooth manifolds
(possibly with corners), with morphisms given by codimension zero smooth
embeddings. Now formally identify any object X with its thickening X x [-1,1].
We prove that the resulting infinity-category of thickened smooth manifolds is
equivalent to the infinity-category of finite spaces over BO. (This is one
formalization of the philosophy that embedding questions become
homotopy-theoretic upon passage to higher dimensions.) The central tool is a
geometric construction of pushouts in this infinity-category, carried out with
an eye toward proving analogous results in exact symplectic geometry. Notably,
the proof never invokes smooth approximation nor any h-principle.
