We classify the indecomposable injective E(n)⁎E(n)-comodules, where E(n) is the Johnson-Wilson homology theory. They are suspensions of the $J_{n,r}=E\left(n\right)⁎\left(M_{r}E\left(r\right)\right)$, where 0 ≤ r ≤ n, with the endomorphism ring of $J_{n,r}$ being $\widehat{E\left(r\right)}*\widehat{E\left(r\right)}$, where $\widehat{E\left(r\right)}$ denotes the completion of E(r).

The purpose of this paper is to show how central extensions of (possibly infinite-dimensional) Lie algebras integrate to central extensions of étale Lie 2-groups in the sense of [Get09, Hen08]. In finite dimensions, central extensions of Lie algebras integrate to central extensions of Lie groups, a fact which is due to the vanishing of ${\pi }_2$ for each finite-dimensional Lie group. This fact was used by Cartan (in a slightly other guise) to construct the simply connected Lie group associated to each finite-dimensional...