Higher monodromy

Abstract

For a given category $\mathsf{C}$ and a topological space $X$, the constant stack on $X$ with stalk $\mathsf{C}$ is the stack of locally constant sheaves with values in $\mathsf{C}$. Its global objects are classified by their monodromy, a functor from the fundamental groupoid $\Pi_1(X)$ to $\mathsf{C}$. 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 $\mathsf{C}$ as a 2-functor from the homotopy 2-groupoid $\Pi_2(X)$ to $\mathsf{C}$. 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.