Abstract

String diagrams are a powerful computational tool, most commonly used in the context of monoidal categories and bicategories. I will talk about extending this to higher dimensions. The natural setting for n-dimensional string diagrams should be some form of semistrict n-category, where composition operations, corresponding to stacking of diagrams, are strictly associative and unital, but the interchange laws hold only up to coherent equivalence. The idea is to define a semistrict n-category as something which admits composites for labeled string diagrams. The first step is to develop a theory of n-sesquicategories. These encode only the compositional structure of string diagrams, without interchange laws. I will explain how to define these as algebras over a monad whose operations are simple string diagrams and how a theory of normal forms for the terms of the associated computads leads to a proof that the category of computads is a presheaf category. The second step, which is still work in progress, is to add operations implementing weak versions of the interchange laws, obtaining the desired notion of semistrict n-category. In dimension 3, this recovers the notion of Gray 3-category.