In the computation of some representation-theoretic numerical invariants of domestic string algebras, a finite combinatorial gadget introduced by Schr{o}er--the emph{bridge quiver} whose vertices are (representatives of cyclic permutations of) bands
and whose arrows are certain band-free strings--has been used extensively. There is a natural but ill-behaved partial binary operation, $circ$, on the larger set of emph{weak bridges} such that bridges are precisely the $circ$-irreducibles. With the goal of computing hammocks up to isomorphism in a later work we equip an even larger set of emph{weak arch bridges} with another partial binary operation, $circ_H$, to obtain a finite category. Each weak arch bridge admits a unique $circ_H$-factorization into emph{arch bridges}, i.e., the $circ_H$-irreducibles.
