It can be viewed as an extension of the more widely-known fundamental group; as such, it captures information about the homotopy type of a topological space.
In certain situations (such as descent theorems for fundamental groups à la Van Kampen) it is much more elegant, even indispensable for understanding something, to work with fundamental groupoids with respect to a suitable packet of base points, [,,,]Let X be a topological space.
The fundamental groupoid Π(X), or Π1(X), assigns to each ordered pair of points (p, q) in X the collection of equivalence classes of continuous paths from p to q.
In categorical terms, the assertion is that the objects p and q are in the same groupoid component if and only if the set of morphisms from p to q is nonempty.
[6] Given a topological space X, a local system is a functor from the fundamental groupoid of X to a category.