Functorial aspects of the reconstruction of Lie groupoids from their bisections

TL;DR

This paper explores the functorial relationship between Lie groupoids and their bisections, establishing an adjunction and near-equivalence of categories, thus deepening understanding of their reconstruction.

Contribution

It demonstrates the functorial nature of reconstructing Lie groupoids from bisections and develops an adjunction and near-equivalence between their categories.

Findings

The reconstruction process is functorial for morphisms fixing the base.

An adjunction between Lie groupoids and Lie groups acting on the base is established.

The adjunction is promoted to an almost equivalence of categories.

Abstract

To a Lie groupoid over a compact base, the associated group of bisection is an (infinite-dimensional) Lie group. Moreover, under certain circumstances one can reconstruct the Lie groupoid from its Lie group of bisections. In the present article we consider functorial aspects of these construction principles. The first observation is that this procedure is functorial (for morphisms fixing the base). Moreover, it gives rise to an adjunction between the category of Lie groupoids over a fixed base and the category of Lie groups acting on the base. In the last section we then show how to promote this adjunction to almost an equivalence of categories.