Functoriality of bornological groupoid convolution

TL;DR

This paper establishes a functorial relationship between Lie groupoid convolution algebras and differentiable stacks, demonstrating their algebraic properties and applications to noncommutative geometry.

Contribution

It introduces a monoidal functor from differentiable stacks to bornological algebras, analyzing their unitality, module properties, and applications to noncommutative tori.

Findings

Convolution algebras possess one-sided approximate units.

Convolution modules are smooth and quasi-unital.

Applications include bornological noncommutative tori.

Abstract

We show that the complete bornological convolution algebras of Lie groupoids and convolution bimodules of groupoid bibundles define a monoidal functor from the 2-category of differentiable stacks to the Morita 2-category of complete bornological algebras. The convolution algebras are generally non-unital, but are shown to possess one-sided approximate units such that the multiplication operators Mackey converge in the functional bornology of endomorphisms. This implies that the convolution algebras are self-induced and the convolution modules are smooth in the sense of R. Meyer. We also show that Lie groupoid actions that are submersive, proper, and transitive have projective convolution modules. This implies that all convolution algebras are quasi-unital. We provide a long list of examples and applications, such as to bornological noncommutative tori, which are Hopf monoids in the Morita category.