[Paper Review] Associative algebras and broken lines

Jacob Lurie, Hiro TANAKA|arXiv (Cornell University)|May 24, 2018

Topological and Geometric Data Analysis1 references3 citations

TL;DR

This paper introduces the moduli stack of broken lines, $Δ\mathsf{Broken}$, as a geometric framework for studying Morse theory via associative algebra structures. It establishes that factorizable sheaves on $Δ\mathsf{Broken}$ correspond precisely to nonunital $A_\infty$-algebras in any compactly generated $Δ\infty$-category, providing a foundational step toward an equation-free construction of the Morse complex on compact Riemannian manifolds.

ABSTRACT

Inspired by Morse theory, we introduce a topological stack Broken, which we refer to as the moduli stack of broken lines. We show that Broken can be presented as a Lie groupoid with corners and provide a combinatorial description of sheaves on Broken with values in any compactly generated infinity-category C. Moreover, we show that factorizable C-valued sheaves (with respect to a natural semigroup structure on the stack Broken) can be identified with nonunital A-infinity-algebras in C. This is a first step in a program whose goal is to present an `equation-free' construction of the Morse complex associated to a compact Riemannian manifold.

Motivation & Objective

• To develop a geometric framework for Morse theory using topological stacks and broken lines.
• To define and characterize the moduli stack of broken lines, $Δ\mathsf{Broken}$, as a Lie groupoid with corners.
• To classify sheaves on $Δ\mathsf{Broken}$ with values in a compactly generated $Δ\infty$-category $Δ{C}$.
• To establish a correspondence between factorizable $Δ{C}$-valued sheaves on $Δ\mathsf{Broken}$ and nonunital $A_\infty$-algebras in $Δ{C}$.
• To lay the groundwork for an equation-free construction of the Morse complex on compact Riemannian manifolds.

Proposed method

• Introduces the concept of a broken line as a topological space with a directed $Δ{R}$-action and finite fixed point set, generalizing the standard line $[-∞,\infty]$.
• Defines families of broken lines over a base space $S$ as topological spaces with continuous $Δ{R}$-actions and projection maps, allowing degenerations.
• Constructs the moduli stack $Δ\mathsf{Broken}$ as a topological stack classifying such families, proving it is representable by a Lie groupoid with corners.
• Provides a combinatorial description of sheaves on $Δ\mathsf{Broken}$ via the twisted arrow category and left Kan extensions.
• Uses Day convolution and lax sheaf structures to relate factorizable sheaves to $A_\infty$-algebra structures.
• Applies homotopy invariance and sheafification techniques to prove that the correspondence between factorizable sheaves and nonunital $A_\infty$-algebras is a homotopy equivalence.

Experimental results

Research questions

• RQ1 How can the moduli stack of broken lines be constructed as a geometric object with good categorical and differential-geometric properties?
• RQ2 What is the combinatorial structure of sheaves on the stack $Δ\mathsf{Broken}$ with values in a compactly generated $Δ\infty$-category $Δ{C}$?
• RQ3 How does the semigroup structure on $Δ\mathsf{Broken}$, given by concatenation of broken lines, relate to algebraic structures in $Δ{C}$?
• RQ4 Can factorizable sheaves on $Δ\mathsf{Broken}$ be classified in terms of algebraic objects such as $A_\infty$-algebras?
• RQ5 What role does homotopy invariance play in establishing the equivalence between sheaf-theoretic and algebraic structures on $Δ\mathsf{Broken}$?

Key findings

• The moduli stack $Δ\mathsf{Broken}$ is representable by a Lie groupoid with corners, providing a well-behaved geometric model for families of broken lines.
• Sheaves on $Δ\mathsf{Broken}$ with values in a compactly generated $Δ\infty$-category $Δ{C}$ admit a combinatorial description via the twisted arrow category of the category of linear orders.
• Factorizable $Δ{C}$-valued sheaves on $Δ\mathsf{Broken}$ are equivalent to nonunital $A_\infty$-algebras in $Δ{C}$, establishing a deep link between geometry and homotopical algebra.
• The correspondence between factorizable sheaves and nonunital $A_\infty$-algebras is realized as a homotopy equivalence, verified via sheafification and homotopy invariance of the presheaf $Δ{G}$.
• The proof relies on showing that the restriction map along the inclusion of linear orders into the twisted arrow category is a homotopy equivalence, using the fact that sheafification preserves values on points.
• The construction is applied to a family of broken lines over $S = \mathbf{R}_{\geq 0}$, where fibers degenerate from unbroken to broken lines, illustrating the stack's ability to capture degenerations.