Abstract: Given a manifold possibly with punctures, we define an A_infinity category whose objects are k-tuples of Morse functions. By counting the intersection between gradient trajectories and the diagonal of the symmetric product space, we get a q-deformation. As an application, we show that the endomorphism of k disjoint cotangent fibers over a disk viewed as in the Morse category is the Hecke algebra associated to the symmetric group.
