The paper under review makes an important contribution to infinite-dimensional Lie theory, by finding a category of group-like objects that allow one to integrate a locally convex Lie algebra \({\mathfrak g}\) which is locally exponential, in the sense that it has a circular convex neighbourhood \(U\) of 0 which is a local Lie group with respect to a locally defined multiplication \(\ast\) with the property \((tx)\ast(sx)=(t+s)x\) if \(x\in U\) and \(\max\{| t|,| s|,| t+s|\}\leq1\) and such that the second-order term in the Taylor expansion of \(\ast\) at \((0,0)\in U\times U\) is \(\frac{1}{2}[x,y]\). The significance of this notion in the theory of infinite-dimensional Lie groups modeled on locally convex spaces is discussed in the survey paper by K.-H. Neeb [Jpn. J. Math. (3) 1, No. 2, 291–468 (2006; Zbl 1161.22012)].
On the other hand, a 2-group is a small groupoid \({\mathcal G}\) endowed with a multiplication functor \(\otimes:{\mathcal G}\times{\mathcal G}\to{\mathcal G}\), an inversion functor \(\bar{}:{\mathcal G}\to{\mathcal G}\), a distinguished object \({\mathbb I}\), and a family of natural isomorphisms \(\alpha_{g,h,k}:(g\otimes h)\otimes k\to g\otimes(h\otimes k)\), where \(g,h,k\) are objects of \({\mathcal G}\), subject to the assumptions: \({\mathbb I}=\bar{\mathbb I}\), \(g\otimes{\mathbb I}=g={\mathbb I}\otimes g\), \(g\otimes\bar g=\bar g\otimes g={\mathbb I}\) on objects and morphisms; the pentagon identities \(\alpha_{g,h,k\otimes l}\circ\alpha_{g\otimes h,k,l} =(\text{id}_g\otimes\alpha_{h,k,l}) \circ\alpha_{g,h\otimes k,l} \circ(\alpha_{g,h,k}\otimes\text{id}_l)\); the requirement that \(\alpha_{g,\bar g,g}=\text{id}_g\), \(\alpha_{\bar g,g,\bar g}=\text{id}_{\bar g}\), and \(\alpha_{g,h,k}\) is an identity whenever one of \(g,h,k\) is \({\mathbb I}\).
A Lie 2-group is a pair \(({\mathcal G},{\mathcal U})\) consisting of a 2-group \({\mathcal G}\) and a full subcategory \({\mathcal U}\) containing \({\mathbb I}\), satisfying the following conditions: (1) \({\mathcal U}\) is endowed with the structure of a smooth 2-space, also called infinite-dimensional Lie groupoid. That is, the sets of objects and morphisms of \({\mathcal U}\) are smooth manifolds modeled on locally convex spaces, the source and target maps are smooth surjective submersions in a strong sense, and the other structure maps are smooth. (2) There exists a full subcategory \({\mathcal V}\) of \({\mathcal U}\) whose set of objects is open in the set of objects of \({\mathcal U}\), such that \({\mathbb I}\in{\mathcal V}\), \(\bar{\mathcal V}={\mathcal V}\), \({\mathcal V}\otimes{\mathcal V}\subseteq{\mathcal U}\), the functors \(\otimes|_{{\mathcal V}\times{\mathcal V}}:{\mathcal V}\times{\mathcal V}\to{\mathcal U}\) and \(\bar{}|_{\mathcal V}:{\mathcal V}\to{\mathcal V}\) are smooth, and moreover \({\mathcal V}\) generates \({\mathcal G}\).
One says that \(({\mathcal G},{\mathcal U})\) is an étale Lie 2-group if all of the structure maps of \({\mathcal U}\) are local diffeomorphisms. Under this assumption, one notes that the multiplication functor defines the structure of a local infinite-dimensional Lie group on the set of objects of \({\mathcal G}\), hence it gives rise to a locally convex Lie algebra, to be thought of as the Lie algebra of \(({\mathcal G},{\mathcal U})\). This is actually a functor from the suitably defined categories of étale Lie 2-groups to the category of Lie algebras (Remark 5.13).
With this terminology at hand, we can now state one of the main results of the paper under review, which is an infinite-dimensional version of S. Lie’s third theorem: If \({\mathfrak g}\) is a Mackey-complete locally exponential Lie algebra, then there exists an étale Lie 2-group whose Lie algebra is isomorphic to \({\mathfrak g}\) (Theorem 6.5). The paper concludes with a short section collecting brief presentations of perspectives on further development of the theory. Besides this, many other questions naturally arise. To mention one, it would be interesting to establish the version in this infinite-dimensional setting of the uniqueness of the connected, simply connected Lie group associated with a finite-dimensional Lie algebra.