  We prove several combinatorial results on path algebras over discrete
structures related to directed graphs. These results are motivated by Morse
theory on a manifold with boundary and, more generally, by Floer theory on a
configuration space with boundary. Their purpose is to organize cobordism
relationships among moduli spaces in order to define new algebraic invariants.
We discuss applications to the Morse and Fukaya categories, and to work with
John Baldwin on a bordered monopole Floer theory.
