A Weak $\infty$-Functor in Morse Theory

TL;DR

This paper constructs two related $ abla$-categories in Morse theory and develops a weak $ abla$-functor between them, revealing higher algebraic structures through topological quantum field theory perspectives.

Contribution

It introduces a novel weak $ abla$-category in Morse theory and constructs a weak $ abla$-functor connecting it to a strict category of Morse chain complexes.

Findings

Construction of two $ abla$-categories in Morse theory.

Development of a weak $ abla$-functor between these categories.

Revelation of higher algebraic structures via topological quantum field theory.

Abstract

In the spirit of Morse homology initiated by Witten and Floer, we construct two $\infty$-categories $\mathcal{A}$ and $\mathcal{B}$. The weak one $\mathcal{A}$ comes out of the Morse-Samle pairs and their higher homotopies, and the strict one $\mathcal{B}$ concerns the chain complexes of the Morse functions. Based on the boundary structures of the compactified moduli space of gradient flow lines of Morse functions with parameters, we build up a weak $\infty$-functor $\mathcal{F}: \mathcal{A}\rightarrow \mathcal{B}$. Higher algebraic structures behind Morse homology are revealed with the perspective of defects in topological quantum field theory.