Homology, Homotopy and Applications

Volume 7 (2005)

Number 1

Higher monodromy

Pages: 109 – 150

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


Pietro Polesello (Dipartimento di Matematica, Università di Padova, Italy)

Ingo Waschkies (Laboratoire J.A. Dieudonné, Université de Nice Sophia-Antipolis, Nice, France)


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.


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

2010 Mathematics Subject Classification

14A20, 18G50, 55Pxx

Published 1 January 2005