  We provide a complete solution to the problem of extending a local Lie
groupoid to a global Lie groupoid. First, we show that the classical Mal'cev's
theorem, which characterizes local Lie groups that can be extended to global
Lie groups, also holds in the groupoid setting. Next, we describe a
construction that can be used to obtain any local Lie groupoid with integrable
algebroid. Last, our main result establishes a precise relationship between the
integrability of a Lie algebroid and the failure in associativity of a local
integration. We give a simplicial interpretation of this result showing that
the monodromy groups of a Lie algebroid manifest themselves combinatorially in
a local integration, as a lack of associativity.
