  We study Morse theory on noncompact manifolds equipped with exhaustions by
compact pieces, defining the Morse homology of a pair which consists of the
manifold and related geometric/homotopy data. We construct a collection of
Morse data parametrized by cubes of arbitrary dimensions. From this collection,
we obtain a family of linear maps subject to some coherency conditions, which
can be packaged into a homotopy coherent diagram. We introduce a chain complex
which is a colimit for the diagram and show that it computes the Morse
homology.
