On fibrations of Lie groupoids

As groupoids generalize groups, motivated by group extensions we consider a kind of fibrations of Lie groupoids, called locally topological product Lie groupoid fibrations with fiber A, i.e., 1→A→G→K→1where A,G and K are Lie groupoids. Similar to the theory of group extensions, we show that the existence of locally topological product Lie groupoid fibrations with fiber A over K is obstructed by a groupoid cohomology of HΛ̄3(K,ZA), and these locally topological product Lie groupoid fibrations are classified by HΛ̄2(K,ZA) once exists. Here ZA is the center of A. This generalizes the theory of group extensions, of gerbes over manifolds/groupoids and etc.