The smash product of monoidal theories

TL;DR

This paper generalizes the tensor product of monoidal theories using a topological approach, introducing a smash product of pointed directed spaces that captures higher-dimensional coherence in algebraic structures.

Contribution

It provides a topological interpretation and generalization of the tensor product of props as a smash product of pointed directed spaces, linking combinatorial and categorical structures.

Findings

Topological interpretation of tensor product as smash product

Systematic construction of higher-dimensional coherence cells

Web of adjunctions connecting various categorical structures

Abstract

The tensor product of props was defined by Hackney and Robertson as an extension of the Boardman-Vogt product of operads to more general monoidal theories. Theories that factor as tensor products include the theory of commutative monoids and the theory of bialgebras. We give a topological interpretation (and vast generalisation) of this construction as a low-dimensional projection of a "smash product of pointed directed spaces". Here directed spaces are embodied by combinatorial structures called diagrammatic sets, while Gray products replace cartesian products. The correspondence is mediated by a web of adjunctions relating diagrammatic sets, pros, probs, props, and Gray-categories. The smash product applies to presentations of higher-dimensional theories and systematically produces higher-dimensional coherence cells.