Abstract
It has long been known that a key ingredient for a sheaf representation of a universal algebra A consists in a distributive lattice of commuting congruences on A. The sheaf representations of universal algebras (over stably compact spaces) that arise in this manner have been recently characterised by Gehrke and van Gool (J. Pure Appl. Algebra, 2018), who identified the central role of the notion of softness.
In this paper, we extend the scope of the theory by replacing varieties of algebras with Barr-exact categories, thus encompassing a number of ``non-algebraic'' examples. Our approach is based on the notion of K-sheaf: intuitively, whereas sheaves are defined on open subsets, K-sheaves are defined on compact ones. Throughout, we consider sheaves on complete lattices rather than spaces; this allows us to obtain point-free versions of sheaf representations whereby spaces are replaced with frames.
These results are used to obtain sheaf representations for the dual of the category of compact ordered spaces, and to recover Banaschewski and Vermeulen's point-free sheaf representation of commutative Gelfand rings (Quaest. Math., 2011).
In this paper, we extend the scope of the theory by replacing varieties of algebras with Barr-exact categories, thus encompassing a number of ``non-algebraic'' examples. Our approach is based on the notion of K-sheaf: intuitively, whereas sheaves are defined on open subsets, K-sheaves are defined on compact ones. Throughout, we consider sheaves on complete lattices rather than spaces; this allows us to obtain point-free versions of sheaf representations whereby spaces are replaced with frames.
These results are used to obtain sheaf representations for the dual of the category of compact ordered spaces, and to recover Banaschewski and Vermeulen's point-free sheaf representation of commutative Gelfand rings (Quaest. Math., 2011).
Original language | English |
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Article number | 107413 |
Number of pages | 41 |
Journal | Journal of Pure and Applied Algebra |
Volume | 227 |
Issue number | 12 |
Early online date | 8 May 2023 |
DOIs | |
Publication status | Published - Dec 2023 |
Keywords
- Soft sheaves
- Sheaf representations
- Regular categories
- Barr-exact categories
- K-sheaves
- Sheaves over complete lattices