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

TL;DR

This paper develops a higher algebraic framework for $A_ abla$ and $ abla B As$-algebras, introducing new polytopes called $n$-multiplihedra, and applies this to Morse theory to construct and analyze higher morphisms between Morse complexes.

Contribution

It introduces the notion of $n$-morphisms for $A_ abla$ and $ abla B As$-algebras, constructs the $n$-multiplihedra polytopes, and applies these concepts to Morse theory to define and study higher morphisms.

Findings

Higher morphisms form a simplicial set with algebraic $ abla$-category structure.

The $n$-multiplihedra generalize standard multiplihedra via polytopes derived from simplices.

The simplicial set of higher morphisms in Morse theory is a contractible Kan complex.

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.