In topology, a (parametrised, oriented) path in a space is a map (a morphism in an appropriate category of spaces) to from the unit interval . A path from to is a path such that and . An unparametrised path is an equivalence class of paths, such that and are equivalent if there is an increasing automorphism of such that . An unoriented path is an equivalence class of paths such that is equivalent to . A Moore path has domain for some natural number (or, more commonly, any non-negative real number) . All of these variations can be combined, of course. (For unoriented paths, one usually says ‘between and ’ instead of ‘from to ’. Also, a Moore path from to has instead of . Finally, there is not much difference between unparametrised paths and unparametrised Moore paths, since we may interpret as a reparametrisation .)
In graph theory, a path is a list of edges, each of which ends where the next begins. Actually, this is a special case of the above, if we use Moore paths and interpret as the linear graph with vertices and edges; in this way, the other variations become meaningful. (However, as the only directed graph automorphism of is the identity, parametrisation is trivial for directed graphs and equivalent to orientation for undirected graphs. Note that a non-Moore path is simply an edge, one of the fundamental ingredients of a graph.)
Given a Moore path from to and a Moore path from to , the concatenation of and is a Moore path or from to . If the domain of is and the domain of is , then the domain of is , and
Often we are more interested in a quotient category of the Moore path category. If we use unparametrised paths (in which case we may use paths with domain if we wish), then we get the unparametrised path category. If is a smooth space, then we may additionally identify paths related through a thin homotopy to get the path groupoid. Finally, if is a continuous space and we identify paths related through any (endpoint-preserving) homotopy, then we get the fundamental groupoid of .
In graph theory, the Moore path category is known as the free category on the graph.