Folded Morse flow trees

We present an approach to Morse theory on symmetric products of surfaces using the notion of folded ribbon trees. We introduce an $A_{\mathrm{\infty }}$-category with objects defined as $\kappa$-tuples of Morse functions, where the differential of the tuple has no self-intersection. We show that when the graph of the differential of the $\kappa$-tuple of Morse functions on $T^{\ast }{\mathbb{R}}^2$ is the wrapped $\kappa$ disjoint cotangent fibers, its endormorphism is the Hecke algebra associated to the symmetric group ${\mathfrak{S}}_{\kappa }$.