The insidious bicategory of algebra bundles

TL;DR

This paper constructs a bicategory of algebra bundles over manifolds, addressing the challenge of defining composition with non-invertible bimodules and non-semisimple algebras, and explores monoidal structures and dualizability.

Contribution

It provides a complete construction of a bicategory of algebra bundles, solving the problem of composition for non-invertible bimodules and non-semisimple algebras.

Findings

Well-defined composition law for bimodule bundles established

Addressed symmetric monoidal structures in the bicategory

Analyzed dualizability of algebra bundles

Abstract

In this paper we construct a bicategory of (super) algebra bundles over a smooth manifold, where the 1-morphisms are bundles of bimodules. The main point is that naive definitions of bimodule bundles will not lead to a well-defined composition law in such a bicategory, at least not if non-invertible bimodules and non-semisimple algebras are desired. This problem has not been addressed so far in the literature. We develop a complete solution, and also address symmetric monoidal structures as well as the corresponding questions of dualizability.