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Higher monodromy
#
Higher monodromy

##
Pietro Polesello and Ingo Waschkies

For a given category $\catc$ and
a topological space $X$, the constant stack
on $X$ with stalk $\catc$ is the stack of locally constant sheaves with values
in $\catc$. Its global objects are classified by their monodromy, a functor
from the fundamental groupoid $\Pi_1(X)$ to $\catc$. 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
$\Catc$ as a 2-functor from the homotopy 2-groupoid $\Pi_2(X)$ to $\Catc$.
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.

Homology, Homotopy and Applications, Vol. 7(2005), No. 1, pp. 109-150
http://www.rmi.acnet.ge/hha/volumes/2005/n1a7/v7n1a7.dvi (ps, dvi.gz, ps.gz, pdf)
ftp://ftp.rmi.acnet.ge/pub/hha/volumes/2005/n1a7/v7n1a7.dvi (ps, dvi.gz, ps.gz, pdf)