fpf.tools
Library / Mathematical Thinking DPF
Jump to passage
In this reading
Source · lines 831–843

Link to current text

Published source confirmed at last check

Source changed 2026-10-03 07:42:37 UTC · snapshot created 2026-10-03 07:43:27 UTC · last check 2026-10-03 07:50:10 UTC

MATH.1:4.3 - Define composition by concatenation

For paths p:x→y and q:y→z, write p;q for the path obtained by putting the list of q after the list of p. Here the semicolon is read in execution order: first p, then q.

Composition is defined when the full intermediate object agrees. Its source is x and its target is z. If the endpoints disagree, repair the proposed sequence, supply a connecting arrow that is actually permitted, or return the failed connection.

Concatenation gives two laws:

  • id_x;p=p=p;id_y for p:x→y, because an empty list adds no generator.
  • (p;q);r=p;(q;r) for three consecutively composable paths, because both sides contain the same three lists in the same order.

These arguments establish identity and associativity for this construction. They can be reused while the definitions stay unchanged. They leave the order of the generators intact: p;q and q;p can differ or one can be undefined. A structure with objects, arrows and these laws is a category; the path construction gives the free category on the generating graph.