Higher categories of bordisms with geometric structures

Abstract

We introduce a system of axioms that uniquely defines an (infinity,d)-category of bordisms equipped with geometric data. The underlying manifolds of these bordisms may be smooth, complex, super, or formal smooth manifolds, as well as any class of manifolds satisfying conditions specified in this paper. We develop a general notion of a field on a manifold, encompassing structures such as Riemannian metrics, principal bundles with connection, conformal structures, and traditional tangential structures. Using this framework, we construct a symmetric monoidal (infinity,d)-category of bordisms with prescribed underlying manifolds and fields, and prove that it satisfies our axioms.