Sheaf representation of monoidal categories

Chris Heunen, Rui Soares Barbosa

Research output: Contribution to journalArticlepeer-review

Abstract / Description of output

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.
Original languageEnglish
Article number108900
Number of pages53
JournalAdvances in Mathematics
Issue number1 March 2023
Early online date8 Feb 2023
Publication statusPublished - 1 Mar 2023

Keywords / Materials (for Non-textual outputs)

  • Monoidal category
  • Central idempotent
  • Sheaf
  • Representation theorem


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