Operadic categories and quasi-Gr\"obner categories

TL;DR

This paper explores the extension of quasi-Gr"obner categories to operadic categories, providing new examples, and investigates their role in encoding homotopy coherent structures, with implications for understanding operads.

Contribution

It introduces new quasi-Gr"obner categories based on operadic categories and shows set-operads form a subcategory, opening avenues for further research.

Findings

New examples of quasi-Gr"obner categories from operadic categories

Set-operads form a full subcategory of operadic categories

Open problems proposed for future exploration

Abstract

Quasi-Gr\"obner categories were introduced by Sam and Snowden to unify treatment of categories in representation stability. We give new examples of quasi-Gr\"obner categories. Most of these categories are operadic categories of Batanin and Markl which are used to encode homotopy coherent structures. This suggests that other operadic categories might also be quasi-Gr\"obner. Additionally, we show that set-operads form a full subcategory of the category of operadic categories. We state several open problems.