Barr-Exact Categories and Soft Sheaf Representations
Marco Abbadini, Luca Reggio

TL;DR
This paper extends sheaf representation theory from algebraic varieties to Barr-exact categories, using K-sheaves on lattices and frames to include non-algebraic examples and recover known point-free sheaf representations.
Contribution
It generalizes sheaf representation theory to Barr-exact categories, introducing K-sheaves on lattices and frames for a broader class of structures.
Findings
Constructed sheaf representations for the dual of compact ordered spaces.
Reproduced Banaschewski and Vermeulen's sheaf representation of Gelfand rings.
Extended sheaf representation framework to non-algebraic categories.
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 this 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…
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Taxonomy
TopicsAlgebraic structures and combinatorial models · Rings, Modules, and Algebras
