Monodromy is the transformation of local mathematical data obtained by continuing it around a closed path. For a multivalued function, analytic continuation of a branch along a loop can return a different branch at the starting point. The resulting transformations form the monodromy group.
For example, for a positive integer 
, continuing 
once counterclockwise around the origin multiplies
by the root
of unity 
and cyclically permutes its 
branches. After 
turns, the original branch returns
(Puiseux 1850, pp. 365-368). Continuing 
once instead adds 
, and repeated turns give infinitely many distinct branches.
For a covering map, lifting loops gives a group action of the fundamental group on the fiber. This is the topological form of monodromy. More generally, monodromy describes how local data are transported around loops, including local morphisms in category theory.
