The Monodromy Groupoid of a Lie Groupoid

R esum e: Nous d emontrons que, sous des circomstances g en erales, l'union disjoint des couvertes universales des etoiles d'un groupo de de Lie admet le structure d'un groupo de de Lie auquel que le projection a une propri et e de monodromie sur les extensions des morphismes emousse. Ca compl etes une conte d etailles des r esultats annonc es par J. Pradines. Introduction The notion of monodromy groupoid which we describe here arose from the grand scheme of J. Pradines in the early 1960s to generalise the standard construction of a simply connected Lie group from a Lie algebra to a corresponding construction of a Lie groupoid from a Lie algebroid, a notion rst deened by Pradines. These results were published as 20, 21, 22, 23]. The recent survey by Mackenzie 17] puts these results in context. The construction by Pradines involved several steps. One was the passage from the innnitesimal Lie algebroid to a locally Lie groupoid. This we will not deal with here. Next was the passage from the locally Lie groupoid to a Lie groupoid. In the case of groups, this is a simple, though not entirely trivial, step, and is part of classical theory. However, in the groupoid case, instead of the locally 1 Lie structure extending directly, there is a groupoid lying over the original one and which is minimal with respect to the property that the Lie structure globalises to it. This groupoid may be called the holonomy groupoid of the locally Lie groupoid. This new result is the main content of Th eor eme 1 of the rst note 20]. Its construction is given in detail (but in the topological case) in 2] (see section 5 below). Finally, there is a need to obtain a maximal Lie groupoid analogous to the universal covering group in the group case, and in some sense locally isomorphic to any globalisation of the locally Lie structure. This groupoid may be called the monodromy groupoid, or star universal covering groupoid. Any globalisation of the locally Lie structure originally given is sandwiched between the holonomy and monodromy groupoids by star universal covering morphisms. A feature of universal covering groups is the classical Monodromy Principle. This is an important tool for extending local morphisms on simply connected topological groups, and is formulated for example in Chevalley 8], p.46, so as to be useful also for constructing maps …