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In this paper, we summarize a general method of transforming DG structuresinto higher structures on the various complexes related to the reduced barresolution of a given quiver algebra using algebraic Morse theory. As anapplication, we describe the $A_\infty$-structures of toupie algebras.Additionally, for certain special toupie algebras, we also prove that theirdouble homological duals are isomorphic to their associated graded algebras.

A note on higher structures on the complexes associated to quiver algebras with applications to toupie algebras