发表机构
Shanghai Jiao Tong University（上海交通大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究 Gabriel--Zisman 局部化对（余）积与加法结构的影响，给出典范函子为满或忠实的充要条件，并证明在分数演算或模型范畴情形下局部化与（余）积相容。
AI 中文摘要
本文研究了在 Gabriel--Zisman 局部化 $Q:\mathcal A\longrightarrow\mathcal A[S^{-1}]$ 下的（余）积与加法结构。例子表明，即使 $Q$ 是加性的，局部化也可能破坏已有的（余）积，或无法保持仍然存在的（余）积。对于集合指标族 $(\mathcal A_i,S_i)_{i\in I}$，本文给出了典范函子 $\big(\prod_{i\in I}\mathcal A_i\big)\big[\big(\prod_{i\in I}S_i\big)^{-1}\big]\longrightarrow\prod_{i\in I}\mathcal A_i[S_i^{-1}]$ 为满函子或忠实函子的充分必要条件。这些判据使得对于具有分数演算或由模型范畴产生的局部化，在相关（余）积存在且指定的态射类在其下封闭时，局部化与 $I$-指标（余）积相容。
英文摘要
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.
