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

Martin Markl

TL;DR

This work extends operadic category theory by introducing thick operadic categories with arities in arbitrary finite sets and establishing when their operads and algebras are equivalent to the standard thin framework. It develops cloven and unital structures, and constructs extension and restriction functors between thin and thick theories, proving a natural equivalence of operad and algebra categories under these extensions. The framework is applied to graphs and modular operads, yielding non-skeletal models and explicit constructions (including determinant-based and odd variants) that connect to Kodù’s modular operad formalism. The results broaden applicability to non-ordered combinatorial structures and clarify how to transfer between skeletal and non-skeletal settings while preserving the operadic algebraic data.

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.

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

TL;DR

This work extends operadic category theory by introducing thick operadic categories with arities in arbitrary finite sets and establishing when their operads and algebras are equivalent to the standard thin framework. It develops cloven and unital structures, and constructs extension and restriction functors between thin and thick theories, proving a natural equivalence of operad and algebra categories under these extensions. The framework is applied to graphs and modular operads, yielding non-skeletal models and explicit constructions (including determinant-based and odd variants) that connect to Kodù’s modular operad formalism. The results broaden applicability to non-ordered combinatorial structures and clarify how to transfer between skeletal and non-skeletal settings while preserving the operadic algebraic data.

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.

Paper Structure

This paper contains 7 sections, 18 theorems, 35 equations, 1 figure.

Table of Contents

  1. Thick operadic categories
  2. Cloven operadic categories
  3. Cloven unitality
  4. Operads
  5. Algebras
  6. The equivalences
  7. Graphs, modular operads, &c

Key Result

Proposition 15

Let ${\boldsymbol {\mathbf O}}$ be a thick cloven operadic category. If ${\boldsymbol {\mathbf O}}$ is left unital, the fiber functor ${\it Fib}_{\{x\}} : {\boldsymbol {\mathbf O}}/{\boldsymbol U\!}^{\ \{x\}}_c \to {\boldsymbol {\mathbf O}}$ is the domain functor for each $c \in \pi_0({\boldsymbol

Figures (1)

  • Figure 1: The local terminal object $\star_X$ of ${\mathbf {GR}}$ with $X = \{a,b,c,d\}$.

Theorems & Definitions (52)

  • Definition 1
  • Definition 2
  • Example 3
  • Definition 4
  • Remark 5
  • Definition 6
  • Example 8
  • Definition 9
  • Example 10
  • Definition 14
  • ...and 42 more