Fundamental groupoid

Summary

The fundamental groupoid $\Pi_1(X)$ of a topological space $X$ is a groupoid. Its objects are the points of $X$, and its morphisms are paths up to path homotopy. It records the fundamental group at every base point at once, together with the paths that compare them. It needs no choice of base point, which makes some results, such as versions of the Seifert–van Kampen theorem, easier to state and prove.