\documentstyle{article} \begin{document} This is a further point in reply to the report on Lie groupoids by Andre Joyal. Pradines' groupoid analogue of the functor {Lie algebras}$\rightarrow$ {simply connected Lie groups} has subtleties which do not appear in the group case. These are stated in his CR Note of 1966, and an outline of the constructions was explained by him to me in the years from 1981. 1)Holonomy: From a Lie algebroid there may be obtained a locally Lie groupoid, i.e. a groupoid $G$ with a Lie structure on a subset $W$ of $G$ containing the identities of $G$. However, unlike the group case, the simple conditions which arise naturally for this situation do not imply that the Lie structure extends to give a Lie groupoid structure on $G$. Instead, under suitable conditions, a new groupoid, the holonomy groupoid $Hol(G,W)$ is obtained which has a Lie structure and which maps to $G$. This groupoid is the minimal "covering" of $(G,W)$ which has a Lie structure. Complete statements and proofs for the topological case are given in Aof and Brown, "The holonomy groupoid of a locally topological groupoid" Top. Appl. 47 (1992) 97-113. (This is essentially an account of Th\'eor\`eme 1 of Pradines Note.) 2) The natural analogue of a simply connected toplogical group is a topological groupoid whose stars (the inverse images of the source map) are simply connected. This we call star simply connected. The problem is then to construct from a Lie groupoid $G$ a star simply connected Lie groupoid $\tilde{G}$ and morphism $p:G\rightarrow \tilde{G}$ which is a universal covering morphism on stars. The existence of this is a part of the statement of Th'eor\`eme 2 of the same Note. Full details of the construction and full prooofs of its properties are in a Bangor preprint 93.09, R Brown and O Mucuk, ``The monodromy groupoid of a Lie groupoid". This also discusses the Lie case of holonomy (in line with Pradines' Note). The point is that is not hard to construct a groupoid which is the universal cover of each star; the problem is to get a topology making it a Lie groupoid. The proof given follows Pradines' outline (given verbally) in using holonomy arguments. 3) For applications to foliations, one needs to recognise that a foliation on a paracompact manifold gives rise to a locally Lie groupoid. This is proved in Bangor preprint 93.10 by Brown and Mucuk. These two preprints have not yet been duplicated, but I can send a TEX file of 93.09, a postscript file of 93.10 (3MB) (this has some pictures), or hard copies to anyone interested. These results formed a part of Mucuk's thesis (Jan, 1993). The locally trivial case of the monodromy construction is dealt with in Mackenzie's book ``Lie groupoids and Lie algebroids in differential geometry" (Cambridge, 1987), by a different method. Kock and Moerdijk also have work on related ideas for local equivalence relations. \end{document} Ronnie Brown +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++