Title: Floer homology with boundary.

Abstract: We extend the TQFT structure of monopole Floer homology to cobordisms with 
multiple ends, equipped with (degenerating) families of metrics.  The story is 
complicated by the fact that the configuration space has boundary, consisting of 
reducible monopoles.  We show how to package the cobordism relations among the resulting 
moduli spaces into algebraic structure, using a notion of path DGA on a directed 
hypergraph.  Our approach is motivated by, and applies to, the finite-dimensional model: 
Morse homology on a manifold with boundary.  We discuss some applications.