In topology, a (parametrised, oriented) loop in a space is a map (a morphism in an appropriate category of spaces) to from the circle . A loop at is a loop such that for any (hence every) integer . An unparametrised loop is an equivalence class of loops, such that and are equivalent if there is an increasing automorphism of such that . An unoriented loop is an equivalence class of loops such that is equivalent to . A Moore loop has domain for some natural number (or possibly any real number) . All of these variations can be combined, of course. (A Moore loop at has instead of . Also, a Moore loop for is simply a point, so possibly there is a better way to define this to avoid making this exception. Finally, there is not much difference between unparametrised loops and unparametrised Moore loops, since we may interpret as a reparametrisation .)
In graph theory, a loop is an edge whose endpoints are the same vertex. Actually, this is a special case of the above, if we interpret as the graph with vertex and edge; in this way, the other variations become meaningful. In this context, a Moore loop is called a cycle. (However, as the only directed graph automorphism of is the identity, parametrisation is trivial for directed graphs and equivalent to orientation for undirected graphs.)
Every loop may be interpreted as a path. Sometimes a loop, say at , is defined to be a path from to . However, this is correct only in certain contexts. In graph theory, it's incorrect, but only because of terminological conventions; the idea is sound. In continuous spaces, it is also correct. However, in smooth spaces, it is not correct, since the derivatives at the endpoints should also agree; the same holds in many other more structured contexts.
Given two Moore loops and at , the concatenation of and is a Moore loop or at . If the domain of is and the domain of is , then the domain of is , and
Often we are more interested in a quotient monoid of the Moore loop monoid. If we use unparametrised loops (in which case we may use loops with domain if we wish), then we get the unparametrised loop monoid?. If is a smooth space, then we may additionally identify loops related through a thin homotopy to get the loop group. Finally, if is a continuous space and we identify loops related through any (basepoint-preserving) homotopy, then we get the fundamental group of .