Spaces over BO are thickened manifolds

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 OE 1; 1.We prove that the resulting 1-category of thickened smooth manifolds is equivalent to the 1-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 1-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.57N65, 57R40 1. Introduction 4287 2. From smooth manifolds to spaces over BO 4293 3. Pushouts of thickened manifolds 4297 4. Proofs of the main results 4315 References 43191 IntroductionThroughout this work, we adopt the convention that a smooth manifold may have boundary and corners, and that all connected components of a manifold have equal dimension.Consider the 1-category Mfld whose objects are smooth, compact manifolds X, up to the relationand where morphisms are codimension-zero smooth embeddings -ie, smooth embeddings whose derivatives pointwise have trivial cokernel.We do not require that the maps respect boundary or corner strata in any way.See Notation 2.8 and 2.9 for a model of this 1-category.Remark 1.1 Let X D OE 1; 1 n be the cube and Y D D n the closed unit disk.Appropriate scalings define a smooth codimension-zero embedding of X inside Y , and of Y inside X.These are equivalences in Mfld ,as the composite X !Y !X is isotopic to the identity, and likewise for Y !X !Y .In particular, in Mfld , a point -which is equivalent to OE 1; 1 n by (1-1) -is equivalent to a disk of any dimension.