Cloven operadic categories: An approach to operadic categories with cardinalities in finite unordered sets

TL;DR

This paper introduces a new class of operadic categories with finite set cardinalities, providing a framework that unifies and extends existing operad theories, especially for graphs and modular operads.

Contribution

It develops a novel approach to operadic categories with finite set cardinalities, connecting them to established operad theories and applications in graph and modular operads.

Findings

Operadic categories with finite set cardinalities are shown to be equivalent to standard operad frameworks.

The approach naturally applies to categories of graphs and modular operads.

Conditions for equivalence of operad theories are established.

Abstract

We introduce and study operadic categories with cardinalities in finite sets and establish conditions under which their associated theories of operads and algebras are equivalent to the standard framework introduced in 2015 by Batanin and Markl. Our approach is particularly natural in applications to the operadic category of graphs and the related category of modular operads and their clones.