Diagrammatic calculus and generalized associativity for higher-arity tensor operations

Abstract

In this paper we investigate a ternary generalization of associativity by defining a diagrammatic calculus of hypergraphs that extends the usual notions of tensor networks, categories and relational algebras. Our key insight is to approach higher associativity as a confluence property of hypergraph rewrite systems. In doing so we rediscover the ternary structures known as heaps and are able to give a more comprehensive treatment of their emergence in the context of dagger categories and their generalizations. This approach allows us to define a notion of ternary category and heapoid, where morphisms bind three objects simultaneously, and suggests a systematic study of higher arity forms of associativity.