Localizing Restriction Categories
Abstract
The localizing/separating factorization, which is the analogue, for restriction categories, to the surjective/embedding factorization for topoi, is developed at both the mere restriction category and join restriction category level. The factorization system at both the ordinary and join level is approached using the notion of a density, which is a stable and universal suborder of the restriction order. The main technical results of the paper construct the join localization and establish its basic properties. The paper gives several examples of localizations. In particular, examples of localization on commutative rigs are provided which give a way to construct categories of schemes. The Zariski spectrum and the Pierce spectrum are, in particular, developed.
Related
- Override and Update in Restriction Categories
- Localizing a Dirac operator via J-holomorphic curves
- Localizing quantum information
- Localizing the Gardam unit: the support geometry of units in F_2[P] and its non-unique-product relatives
- Localizing Global Discrepancies: Marginal Contributions and Contextual Anomaly Detection
BibTeX
@misc{cockett2026localizing,
title = {{Localizing Restriction Categories}},
author = {Robin Cockett},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.30553},
eprint = {2610.30553},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.30553},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Localizing Restriction Categories
AU - Robin Cockett
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.30553
UR - https://arxiv.org/abs/2610.30553
AB - The localizing/separating factorization, which is the analogue, for restriction categories, to the surjective/embedding factorization for topoi, is developed at both the mere restriction category and join restriction category level. The factorization system at both the ordinary and join level is approached using the notion of a density, which is a stable and universal suborder of the restriction order. The main technical results of the paper construct the join localization and establish its basic properties. The paper gives several examples of localizations. In particular, examples of localization on commutative rigs are provided which give a way to construct categories of schemes. The Zariski spectrum and the Pierce spectrum are, in particular, developed.
ER -