Quotients of monodromy groupoids

The main object of this extended abstract is to define the quotient groupoid of the monodromy groupoid for a topological group-groupoid using the categorical assignment between crossed modules and group-groupoids and state some results without in detail. The idea of monoromy principle is that a local morphism f on a topological structure G is extended not only to G itself but also to some simply connected cover of G [12, Theorem 2, Chapter 2]. A version of this result was developed in [13] for Lie groups. The notion of monodromy groupoid was introduced by J. Pradines in a series of the works [27, 28, 29, 30] to generalize a standard construction of a simply connected Lie group from a Lie algebra to a corresponding construction of a Lie groupoid from a Lie algebroid (see also [16, 17, 18, 19, 25, 33]). For more discussions about holonomy and monodromy groupoids we refer the readers to [7] and [8]. Let G be a topological groupoid. The monodromy groupoid denoted by Mon(G) is defined by Mackenzie in [17, p.67-70] as a disjoint union of the universal covers of the stars Gx’s. The topological aspect of the monodromy groupoid was developed in [20] using the holonomy technics of [1] and written in Lie groupoid case including topological groupoids in [9] and [10].