Sheaf representation of monoidal categories
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.
Cite
@article{arxiv.2106.08896,
title = {Sheaf representation of monoidal categories},
author = {Rui Soares Barbosa and Chris Heunen},
journal= {arXiv preprint arXiv:2106.08896},
year = {2023}
}
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41 pages