Operads for Symmetric Monoidal Categories

TL;DR

This paper explicitly describes the operad that characterizes symmetric monoidal categories, enabling a structured understanding of their hierarchy and paving the way for further analysis of their multicategorical structures.

Contribution

It provides an explicit operadic description of symmetric monoidal categories and situates it within a hierarchy of stricter categorical concepts.

Findings

Explicit operad for symmetric monoidal categories

Placement within a hierarchy of categorical concepts

Foundation for analyzing multicategories

Abstract

This paper gives an explicit description of the categorical operad whose algebras are precisely symmetric monoidal categories. This allows us to place the operad in a sequence of four, and therefore a sequence of four successively stricter concepts of symmetric monoidal category. A companion paper will use this operadic presentation to describe a vast array of underlying multicategories for a symmetric monoidal category.