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PreprintCC0arXiv

Gabriel--Zisman Localizations, Products, Coproducts, and Product Categories

  • Chencheng Zhang
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Abstract

This paper studies (co)products and additive structures under Gabriel--Zisman localization Q:π’œβŸΆπ’œ[Sβˆ’1]. Examples show that localization may destroy existing (co)products or fail to preserve (co)products that remain, even when Q is additive. For a set-indexed family (π’œi,Si)i∈ I, necessary and sufficient conditions are given for the canonical functor (∏i∈ Iπ’œi)[(∏i∈ ISi)βˆ’1]⟢∏i∈ Iπ’œi[Siβˆ’1] to be full or faithful. These criteria yield compatibility with I-indexed (co)products for localizations admitting a calculus of fractions or arising from model categories, provided the relevant (co)products exist and the designated classes of morphisms are closed under them.

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Cite

BibTeXRIS
BibTeX
@misc{zhang2026gabrielzisman,
  title = {{Gabriel--Zisman Localizations, Products, Coproducts, and Product Categories}},
  author = {Chencheng Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.18617},
  eprint = {2609.18617},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.18617},
  note = {CC0},
}
RIS
TY  - UNPB
TI  - Gabriel--Zisman Localizations, Products, Coproducts, and Product Categories
AU  - Chencheng Zhang
PY  - 2026
DA  - 2026/09/16
PB  - arXiv
DO  - 10.48550/arxiv.2609.18617
UR  - https://arxiv.org/abs/2609.18617
AB  - This paper studies (co)products and additive structures under Gabriel--Zisman localization Q:π’œβŸΆπ’œ[S^βˆ’1]. Examples show that localization may destroy existing (co)products or fail to preserve (co)products that remain, even when Q is additive. For a set-indexed family (π’œ_i,S_i)_i∈ I, necessary and sufficient conditions are given for the canonical functor (∏_i∈ Iπ’œ_i)[(∏_i∈ IS_i)^βˆ’1]⟢∏_i∈ Iπ’œ_i[S_i^βˆ’1] to be full or faithful. These criteria yield compatibility with I-indexed (co)products for localizations admitting a calculus of fractions or arising from model categories, provided the relevant (co)products exist and the designated classes of morphisms are closed under them.
ER  -