Higher algebra of $A_{\infty }$ and $\Omega BAs$-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 $\Omega BAs$ framework, we define the analogous notion of $n$-morphisms between $\Omega BAs$-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 $\Omega BAs$-algebras in Morse theory. Given two Morse functions $f$ and $g$, we construct $n-\Omega BAs$-morphisms between their respective Morse cochain complexes endowed with their $\Omega BAs$-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.

79 pages - Correction of the construction of the n-multiplihedra - Addition of section 2.3. and 2.4.4 discussing the links between the HOM-simplicial set of higher morphisms with the simplicial set defined by applying Faonte's $A_{\infty }$-nerve functor to the $A_{\infty }$-category of $A_{\infty }$-functors with pre-natural transformations between them. arXiv admin note: text overlap with arXiv:2102.06654

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