2021/02/17 by Thibaut Mazuir, Mazuir, Thibaut · 1 citation
Computer Science · Mathematics · #Topological and Geometric Data Analysis #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2102.08996
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 ΩB As framework, we define the analogous notion of n-morphisms between ΩB 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_∞ and ΩB As-algebras in Morse theory. Given two Morse functions f and g, we construct n-ΩB As-morphisms between their respective Morse cochain complexes endowed with their ΩB 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.