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
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 π1(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 MxG.