Sheaf representation of monoidal categories

TL;DR

This paper establishes a sheaf-theoretic representation for small monoidal categories with certain joins, generalizing several classical representation theorems and enabling the transfer of properties to stalks.

Contribution

It introduces a sheaf representation framework for monoidal categories, unifying and extending classical results like the Stone and Takahashi representations.

Findings

Monoidal categories with universal finite joins of central idempotents are equivalent to sheaves of local monoidal categories.

Small stiff monoidal categories can embed into categories of global sections of such sheaves.

Many properties of monoidal categories are preserved in the sheaf stalks.

Abstract

Every small monoidal category with universal finite joins of central idempotents is monoidally equivalent to the category of global sections of a sheaf of local monoidal categories on a topological space. Every small stiff monoidal category monoidally embeds into such a category of global sections. An infinitary version of these theorems also holds in the spatial case. These representation results are functorial and subsume the Lambek-Moerdijk-Awodey sheaf representation for toposes, the Stone representation of Boolean algebras, and the Takahashi representation of Hilbert modules as continuous fields of Hilbert spaces. Many properties of a monoidal category carry over to the stalks of its sheaf, including having a trace, having exponential objects, having dual objects, having limits of some shape, and the central idempotents forming a Boolean algebra.