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A groupoid is a small category in which each morphism has aninverse. A topological groupoid is a groupoid in which both setsof objects and morphisms have topologies such that all maps ofgroupoid structure are continuous. The notion of monodromygroupoid of a topological groupoid  generalizes those offundamental groupoid and universal cover.  It was  earlier provedthat  the monodromy groupoid of a locally sectionable topologicalgroupoid has the structure of a topological groupoid satisfyingsome properties. In this paper a similar problem is studied forcompatible locally trivial topological groupoids

Extendibility, monodromy, and local triviality for topological groupoids