Computads and string diagrams for $n$-sesquicategories

TL;DR

This paper develops a formal framework using computads and string diagrams to represent compositions in $n$-sesquicategories, advancing the understanding of semistrict higher categories.

Contribution

It provides an explicit description of computads for the monad defining $n$-sesquicategories and shows that their category is a presheaf category, enabling string diagram notation.

Findings

Computads form a presheaf category for the monad $T_n^{D^s}$.

String diagrams can represent arbitrary composites in $n$-sesquicategories.

This work advances the theory of string diagrams for semistrict $n$-categories.

Abstract

An $n$-sesquicategory is an $n$-globular set with strictly associative and unital composition and whiskering operations, which are however not required to satisfy the Godement interchange laws which hold in $n$-categories. In arXiv:2202.09293 we showed how these can be defined as algebras over a monad $T_n^{D^s}$ whose operations are simple string diagrams. In this paper, we give an explicit description of computads for the monad $T_n^{D^s}$ and we prove that the category of computads for this monad is a presheaf category. We use this to describe a string diagram notation for representing arbitrary composites in $n$-sesquicategories. This is a step towards a theory of string diagrams for semistrict $n$-categories.