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Lie's Third Theorem, asserting that each finite-dimensional Lie algebra isthe Lie algebra of a Lie group, fails in infinite dimensions. The modernaccount on this phenomenon is the integration problem for central extensions ofinfinite-dimensional Lie algebras, which in turn is phrased in terms of anintegration procedure for Lie algebra cocycles.  This paper remedies the obstructions for integrating cocycles and centralextensions from Lie algebras to Lie groups by generalising the integratingobjects. 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 locallyexponential Lie algebra (e.g., a Banach-Lie algebra) integrates to a Lie2-group in the sense that there is a natural Lie functor from certain Lie2-groups to Lie algebras, sending the integrating Lie 2-group to an isomorphicLie algebra.

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