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

This paper studies (co)products and additive structures under Gabriel–Zisman localization $Q:\mathcal{A}⟶\mathcal{A}[S^{-1}]$. 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 $\left({\prod }_{i\in I}{\mathcal{A}}_i\right)\left[\right({\prod }_{i\in I}{S}_i{)}^{-1}]⟶{\prod }_{i\in I}{\mathcal{A}}_i[S_i^{-1}]$ 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.