Morse theory for loop-free categories

TL;DR

This paper develops a Morse theory framework for loop-free categories, extending discrete Morse-Bott theory, and establishes homological collapsing results to derive Morse inequalities, thus partially answering a longstanding question.

Contribution

It introduces a Morse theory for loop-free categories, including homological versions of Quillen's Theorem A, cellular categories, and vector fields, expanding the theoretical toolkit for category theory.

Findings

Homological version of Quillen's Theorem A for loop-free categories

Definition of cellular categories and vector fields in this context

Homological collapsing theorem leading to Morse inequalities

Abstract

We extend discrete Morse-Bott theory to the setting of loop-free (or acyclic) categories. First of all, we state a homological version of Quillen's Theorem A in this context and introduce the notion of cellular categories. Second, we present a notion of vector field for loop-free categories. Third, we prove a homological collapsing theorem in the absence of critical objects in order to obtain the Morse inequalities. Examples are provided through the exposition. This answers partially a question by T. John: whether there is a Morse theory for loop-free (or acyclic) categories? [14].