Categorified central extensions, \'etale Lie 2-groups and Lie's Third Theorem for locally exponential Lie algebras

TL;DR

This paper develops a framework to integrate infinite-dimensional Lie algebra cocycles into Lie 2-groups, overcoming classical obstructions and establishing universal objects that generalize Lie's Third Theorem.

Contribution

It introduces a categorified approach to integrate central extensions of infinite-dimensional Lie algebras into Lie 2-groups, providing universal solutions to the integration problem.

Findings

Obstructions to integrating cocycles are remedied by categorification.

Lie 2-groups serve as universal integrating objects for certain Lie algebras.

A Mackey-complete locally exponential Lie algebra integrates into a Lie 2-group.

Abstract

Lie's Third Theorem, asserting that each finite-dimensional Lie algebra is the Lie algebra of a Lie group, fails in infinite dimensions. The modern account on this phenomenon is the integration problem for central extensions of infinite-dimensional Lie algebras, which in turn is phrased in terms of an integration procedure for Lie algebra cocycles. This paper remedies the obstructions for integrating cocycles and central extensions from Lie algebras to Lie groups by generalising the integrating objects. Those objects obey the maximal coherence that one can expect. Moreover, we show that they are the universal ones for the integration problem. The main application of this result is that a Mackey-complete locally exponential Lie algebra (e.g., a Banach-Lie algebra) integrates to a Lie 2-group in the sense that there is a natural Lie functor from certain Lie 2-groups to Lie algebras, sending the integrating Lie 2-group to an isomorphic Lie algebra.