Acyclicity conditions on pasting diagrams

TL;DR

This paper investigates various acyclicity conditions on higher-categorical pasting diagrams within regular directed complexes, identifying the weakest conditions for free generation and analyzing their stability under key operations.

Contribution

It introduces an apparently weakest acyclicity condition ensuring free generation of the associated $$-category and studies the stability of these conditions under fundamental operations.

Findings

Identified the weakest acyclicity condition for free generation.

Established conditions under which the $$-category aligns with Steiner's augmented directed chain complex.

Analyzed stability of acyclicity conditions under pasting, suspensions, Gray products, joins, and duals.

Abstract

We study various acyclicity conditions on higher-categorical pasting diagrams in the combinatorial framework of regular directed complexes. We present an apparently weakest acyclicity condition under which the $\omega$-category presented by a diagram shape is freely generated in the sense of polygraphs. We then consider stronger conditions under which this $\omega$-category is equivalent to one obtained from an augmented directed chain complex in the sense of Steiner, or consists only of subsets of cells in the diagram. Finally, we study the stability of these conditions under the operations of pasting, suspensions, Gray products, joins and duals.