Simple string diagrams and n-sesquicategories

TL;DR

This paper introduces a monad based on simple string diagrams to define and characterize n-sesquicategories, providing a new algebraic framework for their compositional structure and an inductive description.

Contribution

It defines a monad encoding n-dimensional string diagrams and characterizes n-sesquicategories as its algebras, offering a new algebraic and inductive perspective.

Findings

Defined a monad $T_n^{D^s}$ with string diagram operations

Described n-sesquicategories as algebras over this monad

Provided an inductive characterization of n-sesquicategories

Abstract

We define a monad $T_n^{\operatorname{D^s}}$ whose operations are encoded by simple string diagrams and we define $n$-sesquicategories as algebras over this monad. This monad encodes the compositional structure of $n$-dimensional string diagrams. We give a generators and relations description of $T_n^{\operatorname{D^s}}$ , which allows us to describe $n$-sesquicategories as $n$-globular sets equipped with associative and unital composition and whiskering operations. One can also see them as strict $n$-categories without interchange laws. Finally we give an inductive characterization of $n$-sesquicategories.