莫尔斯理论中$A_\infty$和$ΩB As$ -代数的高等代数II
Higher algebra of $A_\infty$ and $ΩB As$-algebras in Morse theory II
摘要：
引入了两个$A_\infty$ -代数之间的$A_\infty$-态射的概念，使得0-态射对应于标准$A_\infty$ -态射，1-态射对应于$A_\infty$ -态射之间的$A_\infty$ -同伦.两个$A_\infty$ -代数之间的高阶态射的集合则定义了一个具有代数$\infty$ -范畴性质的单纯集。$n-A_\infty$ -态射的运算结构也由新的多面体族编码，我们称之为$n$ -多重面体，它推广了标准的多重面体。这些都是从标准的单纯形和multiplihedra通过提升Alexander-Whitney映射到单纯形的水平。丰富的组合数学在这种情况下出现，如方便地描述的重叠分区。从$A_\infty$转移到$ΩB As$的框架，我们定义了类似的概念之间的$n$ -态射的$ΩB As$ -代数，这是再次编码的$n$ -多面体，赋予了一个细化的细胞分解的稳定规范带状树类型。然后我们在莫尔斯理论中实现了$A_\infty$和$ΩB As$ -代数的这个高等代数。给定两个Morse函数$f$和$g$，通过计算与扰动数据的容许单形相关联的扰动Morse梯度树，构造了它们各自的Morse余链复形之间的$n-ΩB As$ -态射，并赋予它们的$ΩB As$ -代数结构.此外，我们表明，由一个计数的扰动Morse梯度树定义的高级态射组成的单纯集是一个可收缩的Kan复形。
英文摘要：
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 $ΩB As$ 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 $A_\infty$ 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.
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