Simple string diagrams and n-sesquicategories

Abstract

We define a monad $T_n^{{\mathrm{D}}^{\mathrm{s}}}$ 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 $T_n^{{\mathrm{D}}^{\mathrm{s}}}$ , 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.