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.Comment: 34 pages, essentially revised, to appear in Adv. Mat