For a given category $\catc$ and a topological space $X$, the constant stack on $X$ with stalk $\catc$ is the stack of locally constant sheaves with values in $\catc$. Its global objects are classified by their monodromy, a functor from the fundamental groupoid $\Pi_1(X)$ to $\catc$. 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 $\Catc$ as a 2-functor from the homotopy 2-groupoid $\Pi_2(X)$ to $\Catc$. 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.
Homology, Homotopy and Applications, Vol. 7(2005), No. 1, pp. 109-150
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