Abbadini, M;
Reggio, L;
(2023)
Barr-exact categories and soft sheaf representations.
Journal of Pure and Applied Algebra
, 227
(12)
, Article 107413. 10.1016/j.jpaa.2023.107413.
Preview |
PDF
AbbadiniReggio_2022_final.pdf - Accepted Version Download (536kB) | Preview |
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 with frames. These results are used to construct 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).
Type: | Article |
---|---|
Title: | Barr-exact categories and soft sheaf representations |
Open access status: | An open access version is available from UCL Discovery |
DOI: | 10.1016/j.jpaa.2023.107413 |
Publisher version: | https://doi.org/10.1016/j.jpaa.2023.107413 |
Language: | English |
Additional information: | This work is licensed under a Creative Commons Attribution 4.0 International License. The images or other third-party material in this article are included in the Creative Commons license, unless indicated otherwise in the credit line; if the material is not included under the Creative Commons license, users will need to obtain permission from the license holder to reproduce the material. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/ |
UCL classification: | UCL UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Engineering Science UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Engineering Science > Dept of Computer Science |
URI: | https://discovery.ucl.ac.uk/id/eprint/10171231 |
Archive Staff Only
![]() |
View Item |