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arXiv · 2609.18617

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 $Q:\mathcal A\longrightarrow\mathcal A[S^{-1}]$. 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.

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Chencheng Zhang. 2026-09-16. Gabriel--Zisman Localizations, Products, Coproducts, and Product Categories. https://arxiv.org/abs/2609.18617

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