# Higher monodromy

## Pietro Polesello and Ingo Waschkies

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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