For a given category C and a topological space X, the constant stack on X
with stalk C is the stack of locally constant sheaves with values in C. Its
global objects are classified by their monodromy, a functor from the Poincare
groupoid of X to C. In this paper we recall these notions from the point of
view of higher category theory and then define the 2-monodromy of a locally
constant stack with values in a 2-category as a 2-functor from the homotopy
2-groupoid into the 2-category. We show that 2-monodromy classifies locally
constant stacks on a reasonably well-behaved space X. As an application, we
show how to recover from this classification the cohomological version of a
classical theorem of Hopf, and we extend it to the non abelian case.Comment: 43 pages. This is a revised version of our preprint RIMS 1432
(11-2003
