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arXiv · 2106.08896

Sheaf representation of monoidal categories

Rui Soares Barbosa·Chris Heunen
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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.

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Rui Soares Barbosa, Chris Heunen. 2021-06-16. Sheaf representation of monoidal categories. https://doi.org/10.1016/j.aim.2023.108900

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