{
  "abstract": "This paper studies (co)products and additive structures under Gabriel--Zisman localization $Q:\\mathcal A\\longrightarrow\\mathcal A[S^{-1}]$. 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 $(\\mathcal A_i,S_i)_{i\\in I}$, necessary and sufficient conditions are given for the canonical functor $\\big(\\prod_{i\\in I}\\mathcal A_i\\big)\\big[\\big(\\prod_{i\\in I}S_i\\big)^{-1}\\big]\\longrightarrow\\prod_{i\\in I}\\mathcal A_i[S_i^{-1}]$ 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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.18617",
  "author_total": 1,
  "authors": [
    {
      "name": "Chencheng Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2qh86aw49aweh.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.18617",
  "fulltext_url": null,
  "id": "2qh86aw49aweh",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.18617",
  "language": null,
  "licence": "CC0-1.0",
  "licence_label": "CC0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/publicdomain/zero/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CT"
    ],
    [
      "openalex",
      "2610|Homotopy and Cohomology in Algebraic Topology"
    ]
  ],
  "openalex": "W7213514647",
  "pdf_url": "https://arxiv.org/pdf/2609.18617",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2qh86aw49aweh.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:04Z",
      "native_id": "2609.18617",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.18617"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Gabriel--Zisman Localizations, Products, Coproducts, and Product Categories",
  "title_text": "Gabriel--Zisman Localizations, Products, Coproducts, and Product Categories",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:04Z",
  "url": "https://research.draconitic.com/paper/2qh86aw49aweh",
  "venue": "arXiv",
  "versions": []
}