 Fundamental groupoid

The fundamental groupoid ${\Pi }_1(X)$ of a topological space $X$ has the points of $X$ as objects and endpoint-preserving homotopy classes of paths as morphisms. Composition is concatenation, and reversing a path gives an inverse. Concatenation need not be strictly associative as a parametrized path, but becomes associative on these homotopy classes. This makes ${\Pi }_1(X)$ a groupoid.

The automorphism group of a point $x$ is the fundamental group ${\pi }_1(X,x)$. Unlike a single based fundamental group, the fundamental groupoid simultaneously records all path components and transport between basepoints. The standard path-homotopy constructions supply the composition and inverse laws.^[ref]

For the circle, fix endpoints $x$ and $y$ and a reference path between them. Every other homotopy class differs by an integer number of turns; choosing the reference identifies this morphism set with $\mathbb{Z}$, but the identification is not canonical. For a two-point discrete space, there are two objects and only identity morphisms. Their individual automorphism groups are both trivial, yet the groupoid still records two disconnected components. A continuous map $X\to Y$ induces a functor ${\Pi }_1(X)\to {\Pi }_1(Y)$ by composing paths with that map.

In algebraic topology, the fundamental groupoid is a generalization of the fundamental group. While the fundamental group is associated with a single point in a space and considers loops based at that point, the fundamental groupoid captures the idea of paths and homotopies between points in a topological space. ### Definition 1. **Topological Space**: Given a topological space \( X \), we consider all its points.