A unified framework for generalized multicategories

TL;DR

This paper introduces a unified framework for generalized multicategories using monads on double categories, unifying various existing notions and simplifying the theoretical landscape.

Contribution

It proposes a novel approach employing monads on double categories to unify and clarify the diverse definitions of generalized multicategories.

Findings

Unifies various notions of generalized multicategories.

Simplifies the theoretical framework for multicategories.

Provides a comprehensive approach applicable to multiple contexts.

Abstract

Notions of generalized multicategory have been defined in numerous contexts throughout the literature, and include such diverse examples as symmetric multicategories, globular operads, Lawvere theories, and topological spaces. In each case, generalized multicategories are defined as the "lax algebras" or "Kleisli monoids" relative to a "monad" on a bicategory. However, the meanings of these words differ from author to author, as do the specific bicategories considered. We propose a unified framework: by working with monads on double categories and related structures (rather than bicategories), one can define generalized multicategories in a way that unifies all previous examples, while at the same time simplifying and clarifying much of the theory.