Let be a topological space. Its Moore path category has
Its set of all morphisms consists of pairs where is continuous, and is constant on . The source of is and the target of is . The number may be called the shape of . We may compose and to obtain where for and for . Identities are paths of shape .
The advantage of this definition as pairs is partly in giving a topology on , but also in iteration.
The reference below defines as a strict cubical -category. It also has connections, which satisfy all the laws except cancellation of and under composition. This structure seems a sensible home for -paths in for all , and has the advantage over simplicial or globular versions of “-groupoids” of easily encompassing multiple compositions.