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Anick Resolution for Lawvere Theories from Algebraic Discrete Morse Theory

Mirai Ikebuchi

Abstract

Inspired by Brown's collapsing method (or discrete Morse theory) to obtain a free resolution of $\bbZ$ over the monoid ring $\bbZ M$, we apply algebraic discrete Morse theory to compute the homology groups of Lawvere theories, which is defined as Tor of a certain module. We reinterpret known partial free resolutions arising from complete term rewriting systems in terms of collapsing of the normalized bar resolution. This perspective yields homological inequalities that bound the number of equational axioms in presentations and recovers classical results, such as lower bounds for group axiomatizations. Our main contribution is to extend these resolutions to higher dimensions.

Anick Resolution for Lawvere Theories from Algebraic Discrete Morse Theory

Abstract

Inspired by Brown's collapsing method (or discrete Morse theory) to obtain a free resolution of over the monoid ring , we apply algebraic discrete Morse theory to compute the homology groups of Lawvere theories, which is defined as Tor of a certain module. We reinterpret known partial free resolutions arising from complete term rewriting systems in terms of collapsing of the normalized bar resolution. This perspective yields homological inequalities that bound the number of equational axioms in presentations and recovers classical results, such as lower bounds for group axiomatizations. Our main contribution is to extend these resolutions to higher dimensions.

Paper Structure

This paper contains 15 sections, 34 theorems, 58 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Lawvere theories
  3. Ringoids, modules, and algebraic discrete Morse theory
  4. Preliminaries on ringoids and modules
  5. Algebraic discrete Morse theory for ringoids
  6. Bar resolution for Lawvere theories
  7. Unnormalized bar resolution
  8. Normalized bar resolution
  9. Anick resolution
  10. Abstract rewriting systems
  11. Review of Anick resolutions
  12. Terms
  13. Observations
  14. Morse matching
  15. Morse inequalities

Key Result

Lemma 2.9

For sorts $X_1,\dots,X_m\in S$, let $H = \{\vec{\omega} \in \mathrm{Mor}(\mathrm{Syn}\ab(\Sigma)) \mid \text{$\vec{\omega}$ is efficient and its codomain is } X_1\times\dots\times X_m\}$. We define an equivalence relation $\sim_\Pi$ on $H$ as follows: $\vec{\omega} \sim_\Pi \vec{\omega}'$ iff there Moreover, for any $[\vec{\omega}]_{\sim_\Pi} \in H_{\sim_\Pi}$ with $\phi([\vec{\omega}]_{\sim_\Pi}

Figures (1)

  • Figure 1:

Theorems & Definitions (86)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Example 2.4: Abelian groups
  • Example 2.5: Left modules over a monoid
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Lemma 2.9
  • proof
  • ...and 76 more