# Category Theory Construction

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.

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By parallellism of A and B you mean a formal pairing of two arrows? With the semantic interpretation of computing both in parallell? –  Mikael Vejdemo-Johansson Mar 26 '10 at 21:38
Please pick a better title. At best, your title should include a condensed version of your question. Remember that titles are about as long as tweets: for example, this entire comment fits in a title. –  Theo Johnson-Freyd Mar 26 '10 at 23:16
I have never encountered the "parallelism" operation. I assume you simply mean the addition operation in your free commutative monoids? –  Theo Johnson-Freyd Mar 26 '10 at 23:17
Incidentally, why are you thinking about this particular category? Wouldn't something like the category enriched in $\mathbb R_{\geq 0}$-graded commutative monoids that is freely generated by your category ${\rm Pa}(X)$ (which is over the one-object category $\mathbb R_{\geq 0}$) be just as useful? –  Theo Johnson-Freyd Mar 26 '10 at 23:26
Think of X as the state space of a classical (non-statistical) thermodynamical system. Then Pa(X) is the category of all paths in state space. Two paths of the same duration between the same two states could represent a thermal process on one hand, and a mechanical process on the other. For example, expanding the gas behind a piston in a cylinder in contact with a heat source. Yes, I am writing the "addition" operation as || to suggest the parallel performance of two processes. A pretentious title would be something like "Basic construction in Categorical Macrodynamics". –  Ellis D Cooper Mar 27 '10 at 21:48