= Fundamental groupoid
{title2=$\Pi_1(X)$}
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.<Algebraic Topology by Allen Hatcher>[section 1.1, Paths and Homotopy, printed pages 25–28]{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.
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