Higher-dimensional normalisation strategies for acyclicity

TL;DR

This paper develops higher-dimensional normalization strategies for acyclic polygraphs, providing a new framework for understanding categorical models, their acyclicity, and finiteness conditions in higher category theory.

Contribution

It introduces acyclic polygraphs and higher-dimensional normalization strategies, establishing their role in characterizing acyclicity and extending finiteness conditions to higher categories.

Findings

Acyclic polygraphs serve as complete categorical models.

Existence of a normalization strategy characterizes acyclicity.

New homotopical and homological finiteness conditions are proposed.

Abstract

We introduce acyclic polygraphs, a notion of complete categorical cellular model for (small) categories, containing generators, relations and higher-dimensional globular syzygies. We give a rewriting method to construct explicit acyclic polygraphs from convergent presentations. For that, we introduce higher-dimensional normalisation strategies, defined as homotopically coherent ways to relate each cell of a polygraph to its normal form, then we prove that acyclicity is equivalent to the existence of a normalisation strategy. Using acyclic polygraphs, we define a higher-dimensional homotopical finiteness condition for higher categories which extends Squier's finite derivation type for monoids. We relate this homotopical property to a new homological finiteness condition that we introduce here.