Abstract: In this talk, I will describe a 2-functor, called "2-linearization", from spans of groupoids into 2-vector spaces (C-linear abelian categories). Using groupoids representing moduli stacks of flat connections, this gives rise, for every finite group G, to an "extended topological quantum field theory", a 2-functorial invariant for manifolds with corners. I will also discuss how to extend this to compact Lie groups, including measures on stacks, and a generalization of the category of distintegrations (a nice category of measure spaces) to stacks.