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International Press:PO Box 43502:Somerville, MA 02143:(617)623-3016, (617)623-3101, EMAIL: [email protected], INTERNET: http://www.intlpress.com, Fax: (617)623-3101
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
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