Cobordism categories of manifolds with disc-presheaf structures

The study of cobordism categories of manifolds has had important consequences on the topology of manifolds, in particular regarding the cohomology
of moduli spaces of manifolds in certain stable ranges. An interesting variant are embedded cobordism categories, where we require all closed manifolds
be embedded in a background manifold W, and where cobordisms are taken in W \times I. Using ideas from embedding calculus, we propose a notion of
a cobordism category of manifolds with certain disc-presheaf structures, which generalizes the classical cobordism category, and conjecturally recovers
the homotopy type of embedded cobordism categories.