a covering space
A covering space is a topological way to make precise the idea of 'several sheets sitting over a base'. You have a base space X (think of the punctured plane) and a bigger space Y with a map p from Y down to X (the covering map). The defining property is that every point of X has a small neighbourhood whose preimage in Y is a disjoint stack of copies, each mapped down homeomorphically — like pancakes stacked over a griddle, each pancake a faithful copy of the patch below. Locally Y looks like several disjoint copies of X, even though globally Y can be connected and quite different in shape.
The link to our subject is direct: the Riemann surface of a multivalued function, with its sheets and projection back down to the z-plane, is essentially a covering space (a branched one, where the branch points are special). Continuation along a path corresponds to a fundamental property of coverings called path lifting: given a path downstairs in X and a starting point upstairs in Y, there is a unique way to lift the path to a path in Y starting at that point. Lifting a LOOP downstairs may not give a loop upstairs — you can come back over the same base point but on a different sheet — and that mismatch is exactly the monodromy, the branch change you pick up going around. So the whole path-dependence story is the geometry of lifting paths through a covering map.
Coverings turn analytic continuation into clean topology, and the controlling object is the fundamental group of the base, the group of loops up to deformation. There is a precise dictionary (the classification of covering spaces): connected coverings of X correspond to subgroups of its fundamental group, and the monodromy of loops gives a representation of that group permuting the sheets. For the punctured plane the fundamental group is the integers (counting how many times a loop winds around the puncture), which is why the logarithm's monodromy is exactly addition of 2 pi i per winding. Among all coverings one is the biggest and simplest of all — the universal cover.
The map p(w) = e^w from the complex plane down to the punctured plane (it misses only 0) is a covering. Over each nonzero z the preimages are all the values of log z, spaced 2 pi i apart vertically — infinitely many sheets. Lifting a loop that winds once around 0 raises the starting w by 2 pi i: the loop downstairs is not a loop upstairs, which is the logarithm's monodromy seen as path lifting.
Exp maps the plane onto the punctured plane as a covering; lifting a winding loop shifts you by 2 pi i — log's monodromy.
A covering map is a local homeomorphism but generally NOT global — the whole point is that loops downstairs can fail to be loops upstairs. Strictly, a Riemann surface with branch points is a branched cover; it is an honest covering only away from the branch points.