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# Homology, Homotopy and Applications

## Volume 7 (2005)

### Number 1

### Higher monodromy

Pages: 109 – 150

DOI: http://dx.doi.org/10.4310/HHA.2005.v7.n1.a7

#### Authors

#### Abstract

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 fundamental groupoid $Π_1(X)$ 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 $C$ as a 2-functor from the homotopy 2-groupoid $Π_2(X)$ to $C$. 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.

#### Keywords

monodromy representation, algebraic topology, stacks, category theory, non abelian cohomology

#### 2010 Mathematics Subject Classification

14A20, 18G50, 55Pxx