Anick Resolution for Lawvere Theories from Algebraic Discrete Morse Theory

TL;DR

This paper applies algebraic discrete Morse theory to compute homology of Lawvere theories, extending known resolutions to higher dimensions and deriving bounds on axiomatizations.

Contribution

It introduces a novel application of Morse theory to Lawvere theories, extending resolutions to higher dimensions and providing new homological bounds.

Findings

Derived homological inequalities bounding the number of axioms

Reinterpreted known resolutions via collapsing of bar resolutions

Extended resolutions to higher dimensions

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.