Higher categories of bordisms with geometric structures

TL;DR

This paper defines a comprehensive axiomatic framework for an (infinity,d)-category of bordisms with various geometric structures, unifying multiple manifold types and field configurations.

Contribution

It introduces a set of axioms that uniquely characterize an (infinity,d)-category of bordisms with geometric data, accommodating diverse manifold classes and structures.

Findings

Constructed a symmetric monoidal (infinity,d)-category of bordisms with prescribed fields.

Proved the category satisfies the introduced axioms.

Unified various geometric structures within a single categorical framework.

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.