For any topological space X let Pa(X) denote the category with objects |Pa(X)|=X and morphisms A from x to y in Pa(X)(x,y) given by continuous maps to X with domain of the form [0,r] where r is a non-negative real number, including possibly r=0. Call r the duration of path A. The identity morphism at x is the map from [0,0] to X with value x, and composition of A from x to y with duration r and B from y to z with duration s is the naturally defined map A;B from [0,r+s] to X.
The homsets Pa(X)(x,y) are "stratified" into disjoint sets of paths with the same duration. Construct a new category strictly containing Pa(X) as follows. Let G be the graph with vertices |G|=X and for objects x, y in G let the set of edges G(x,y) be the disjoint union of the underlying sets of the commutative free monoids generated by paths of the same duration. Let || denote the "parallelism" operation so that if A and B have the same duration then A||B = B||A denotes the freely generated parallelism of A and B. Define the duration of A||B to be the same duration as A and B.
Form the quotient Par(X) of the free category generated by G by the congruence which both restores the composition of Pa(X), and forces the exchange law (A||B);(C||D)=(A;C)||(B;D).
My question is whether this construction is already identified and named, perhaps as part of a more general discussion, including a universal property and an adjoint pair of functors, etc.