We prove that the algebra of singular cochains on a smooth manifold, equipped with the cup product, is equivalent to the A1
structure on the Lagrangian Floer cochain group associated to the zero section in the cotangent bundle. More generally, given
embeddings with isomorphic normal bundles of a closed manifold B into manifolds Q1 and Q2, we construct a differential graded
category from the singular cochains of these spaces, and prove that it is equivalent to the A1 category obtained by considering
exact Lagrangian embeddings of Q1 and Q2 which intersect cleanly along B.