  We define a basic class of algebras which we call homotopy path algebras. We
find that such algebras always admit a cellular resolution and detail the
intimate relationship between these algebras, stratifications of topological
spaces, and entrance/exit paths. As examples, we prove versions of homological
mirror symmetry due to Bondal-Ruan for toric varieties and due to
Berglund-H\"ubsch-Krawitz for hypersurfaces with maximal symmetry. We also
demonstrate that a form of shellability implies Koszulity and the existence of
a minimal cellular resolution. In particular, when the algebra determined by
the image of the toric Frobenius morphism is directable, then it is Koszul and
admits a minimal cellular resolution.
