Homology, Homotopy and Applications
Volume 7 (2005)
Pages: 109 – 150
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