Higher Monodromy

Details Higher Monodromy Higher Monodromy

2004-07-29

arxiv.org/abs/math/0407507v1

Homology Homotopy Appl. 7 (2005), no. 1, 109--150.

Mathematics

Algebraic Topology

43 pages. This is a revised version of our preprint RIMS 1432 (11-2003)

Scientific paper

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 Poincare groupoid of 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 as a 2-functor from the homotopy 2-groupoid into the 2-category. 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.

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