Higher algebra of $A_\infty$ and $ΩB As$-algebras in Morse theory II

Abstract: This paper introduces the notion of $n$-morphisms between two $A_\infty$-algebras, such that 0-morphisms correspond to standard $A_\infty$-morphisms and 1-morphisms correspond to $A_\infty$-homotopies between $A_\infty$-morphisms. The set of higher morphisms between two $A_\infty$-algebras then defines a simplicial set which has the property of being an algebraic $\infty$-category. The operadic structure of $n-A_\infty$-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 $A_\infty$ to the $\Omega B As$ framework, we define the analogous notion of $n$-morphisms between $\Omega B As$-algebras, which are again encoded by the $n$-multiplihedra, endowed with a refined cell decomposition by stable gauged ribbon tree type. We then realize this higher algebra of $A_\infty$ and $\Omega B As$-algebras in Morse theory. Given two Morse functions $f$ and $g$, we construct $n-\Omega B As$-morphisms between their respective Morse cochain complexes endowed with their $\Omega B As$-algebra structures, by counting perturbed Morse gradient trees associated to an admissible simplex of perturbation data. We moreover show that the simplicial set consisting of higher morphisms defined by a count of perturbed Morse gradient trees is a contractible Kan complex.