On fibrations of Lie groupoids

Abstract

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 $\mathsf{A}$, i.e., \[ 1\rightarrow {\sf A} \rightarrow {\sf G} \rightarrow {\sf K}\rightarrow 1 \] where $\mathsf{A},\mathsf{G}$ and $\mathsf{K}$ are Lie groupoids. Similar to the theory of group extensions, we show that the existence of locally topological product Lie groupoid fibrations with fiber $\mathsf{A}$ over $\mathsf{K}$ is obstructed by a groupoid cohomology of $H_{\bar{\Lambda }}^3(\mathsf{K},Z_{\mathsf{A}})$, and these locally topological product Lie groupoid fibrations are classified by $H_{\bar{\Lambda }}^2(\mathsf{K},Z_{\mathsf{A}})$ once exists. Here $Z_{\mathsf{A}}$ is the center of $\mathsf{A}$. This generalizes the theory of group extensions, of gerbes over manifolds/groupoids and etc.

28 pages; fonts modified; title changed; comments are welcome; final version; to appear in J. Geom. Phys