These ideas were first made explicit in complex analysis. In the process of analytic continuation, a function that is an analytic function *F*(*z*) in some open subset *E* of the *punctured disk* *D* given

- 0 < |
*z*| < 1

*F*(*z*) = log*z*

- Re(
*z*) > 0

- |
*z*| = 0.5

*F*(*z*)+2π*i*.

In the case of a covering map, we look at it as a special case of a fibration, and use the *homotopy lifting property* to 'follow' paths on the base space *X* (we assume it path-connected for simplicity) as they are lifted up into the cover *C*. If we follow round a loop based at *x* in *X*, which we lift to start at *c* above *x*, we'll end at some *c** again above *x*; it is quite possible that *c* ≠ *c**, and to code this one considers the action of the fundamental group π_{1}(*X*,*x*) as a permutation group on the set of all *c*, as **monodromy group** in this context.

In differential geometry, an analogous role is played by parallel transport. In a principal bundle *B* over a smooth manifold *M*, a connection allows 'horizontal' movement from fibers above *m* in *M* to adjacent ones. The effect when applied to loops based at *m* is to define a **holonomy** group of translations of the fiber at *m*; if the structure group of *B* is *G*, it is a subgroup of *G* that measures the deviation of *B* from the product bundle *M*x*G*.