Symmetric weak multicategories

TL;DR

This paper introduces symmetric weak multicategories, extending weak multicategories by incorporating a weak symmetric group action, enriching the structure with natural isomorphisms and their coherence conditions.

Contribution

It formalizes the concept of symmetric weak multicategories, providing a framework that generalizes multicategories with symmetric group actions up to cocycles.

Findings

Defines symmetric weak multicategories with natural isomorphisms

Establishes coherence conditions for the symmetric group action

Provides foundational structure for further categorical research

Abstract

A multicategory is what remains of a monoidal category when monoidal product is not available. A weak multicategory means that hom-sets are in fact categories, and in place of usual equations, there are natural isomorphisms, which have to satisfy their own equations. A symmetric weak multicategory implies a weak multicategory with a weak (up to a cocycle) action of symmetric groups.