Multicategories from Symmetric Monoidal Categories

TL;DR

This paper demonstrates that symmetric monoidal categories have essentially unique underlying multicategories, establishing a canonical functor and a strictification process via an adjunction.

Contribution

It proves the uniqueness of underlying multicategories for symmetric monoidal categories and constructs a strictification functor through an adjunction.

Findings

Uniqueness of underlying multicategories up to isomorphism

Existence of a canonical forgetful functor

Construction of a strictification via a weak left adjoint

Abstract

This paper considers the possible underlying multicategories for a symmetric monoidal category, and shows that, up to canonical and coherent isomorphism, there really is only one. As a result, there is a well-defined forgetful functor from symmetric monoidal categories to multicategories, as long as all morphisms of symmetric monoidal categories are at least lax symmetric monoidal. The paper also shows that this forgetful functor has a weak left adjoint, and that the monad of the adjunction gives a strictification construction.