On the image of the almost strict Morse n-category under almost strict n-functors

TL;DR

This paper explores the structure of almost strict Morse n-categories and their images under specific functors, extending previous work on flow categories and introducing new categorical frameworks based on vector spaces and integer tuples.

Contribution

It defines two new almost strict n-categories and constructs functors from the Morse n-category to these new categories, expanding the categorical understanding of Morse theory.

Findings

Construction of categories $\\mathcal V$ and $\mathcal W$ based on vector spaces and integers.

Definition of functors from Morse n-category to these new categories.

Extension of the flow category framework to almost strict n-categories.

Abstract

In an earlier work, we constructed the almost strict Morse $n$-category $\mathcal X$ which extends Cohen $\&$ Jones $\&$ Segal's flow category. In this article, we define two other almost strict $n$-categories $\mathcal V$ and $\mathcal W$ where $\mathcal V$ is based on homomorphisms between real vector spaces and $\mathcal W$ consists of tuples of positive integers. The Morse index and the dimension of the Morse moduli spaces give rise to almost strict $n$-category functors $\mathcal F : \mathcal X \to \mathcal V$ and $\mathcal G : \mathcal X \to \mathcal W$.