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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.

Higher category theory studies structures with morphisms between objects, morphisms between those morphisms, and further layers of comparison. In a two-dimensional example, categories are objects, functors are one-morphisms and natural transformations are two-morphisms. A two-morphism can be composed along a shared functor or alongside another natural transformation; the compatibility of these compositions is part of the structure.^[ref]

Homotopy theory gives the motivating picture: points are connected by paths, paths are compared by homotopies, and homotopies themselves can be compared by further homotopies. Replacing every path by only its homotopy class produces the fundamental groupoid and forgets these higher comparisons. Weak higher categories retain coherent comparison data when associativity holds only up to equivalence. One must specify a model and whether higher morphisms are required to be invertible; “higher category” does not impose one universal convention.