Elaborating on works by Abouzaid and Mescher, we prove that for a Morse function on a smooth compact manifold, its Morse cochain complex can be endowed with an ΩBAs-algebra structure by counting moduli spaces of perturbed Morse gradient trees. This rich structure descends to its already known A∞-algebra structure. We then introduce the notion of ΩBAs-morphism between two ΩBAs-algebras and prove that given two Morse functions, one can construct an ΩBAs-morphism between their associated ΩBAs-algebras by counting moduli spaces of two-colored perturbed Morse gradient trees. This morphism induces a standard A∞-morphism between the induced A∞-algebras. We work with integer coefficients, and provide to this extent a detailed account on the sign conventions for A∞ (resp. ΩBAs)-algebras and A∞ (resp. ΩBAs)-morphisms, using polytopes (resp. moduli spaces) which explicitly realize the dg-operadic objects encoding them. Our proofs also involve building at the level of polytopes an explicit functor from the category of ΩBAs-algebras to the category of A∞-algebras, drawing from a result by Markl and Shnider. This paper is adressed to people acquainted with either symplectic topology or algebraic operads, and written in a way to be hopefully understood by both communities. It comes in particular with a detailed survey on operads, A∞-algebras and A∞-morphisms, the associahedra and the multiplihedra, as well as some details on the usual techniques used in symplectic topology to define algebraic structures on geometrical (co)chain complexes. It moreover lays the basis for a second article in which we solve the problem of finding a satisfactory homotopic notion of higher morphisms between A∞-algebras and between ΩBAs-algebras, and show how this higher algebra of A∞ and ΩBAs-algebras naturally arises in the context of Morse theory.The associahedron K 4 and the multiplihedron J 3 ...

This paper introduces the notion of n-morphisms between two A∞-algebras, such that 0-morphisms correspond to standard A∞-morphisms and 1-morphisms correspond to A∞homotopies between A∞-morphisms. The set of higher morphisms between two A∞-algebras then defines a simplicial set which has the property of being an algebraic ∞-category. The operadic structure of n − A∞-morphisms is also encoded by new families of polytopes, which we call the n-multiplihedra and which generalize the standard multiplihedra. These are constructed from the standard simplices and multiplihedra by lifting the Alexander-Whitney map to the level of simplices. Rich combinatorics arise in this context, as conveniently described in terms of overlapping partitions. Shifting from the A∞ to the ΩBAs framework, we define the analogous notion of n-morphisms between ΩBAs-algebras, which are again encoded by the n-multiplihedra, endowed with a thinner cell decomposition by stable gauged ribbon tree type. We then realize this higher algebra of A∞ and ΩBAs-algebras in Morse theory. Given two Morse functions f and g, we construct n − ΩBAsmorphisms between their respective Morse cochain complexes endowed with their ΩBAs-algebra structures, by counting perturbed Morse gradient trees associated to an admissible simplex of perturbation data. We moreover show that any inner horn of higher morphisms arising from a count of perturbed Morse gradient trees can always be filled, not only algebraically but also geometrically.

A. We define a cellular approximation for the diagonal of the Forcey-Loday realizations of the multiplihedra, and endow them with a compatible topological cellular operadic bimodule structure over the Loday realizations of the associahedra. This provides us with a model for topological and algebraic A ∞ -morphisms, as well as a universal and explicit formula for their tensor product. We study the monoidal properties of this newly defined tensor product and conclude by outlining several applications, notably in algebraic and symplectic topology.

We define a cellular approximation for the diagonal of the Forcey-Loday realizations of the multiplihedra, and endow them with a compatible topological cellular operadic bimodule structure over the Loday realizations of the associahedra. This provides us with a model for topological and algebraic A∞-morphisms, as well as a universal and explicit formula for their tensor product. We study the monoidal properties of this newly defined tensor product and conclude by outlining several applications, notably in algebraic and symplectic topology. Résumé (La diagonale des multiplièdres et le produit tensoriel de morphismes A-infini)On définit une approximation cellulaire de la diagonale des réalisations de Forcey-Loday des multiplièdres, et on les munit d'une structure de bimodule opéradique topologique et cellulaire compatible sur les réalisations de Loday des associaèdres. On obtient ainsi un modèle algébrique et topologique pour les morphismes A-infini, de même qu'une formule universelle explicite pour leur produit tensoriel. On étudie la monoïdalité de ce nouveau produit tensoriel et on conclut en esquissant plusieurs applications en topologie algébrique et en topologie symplectique.