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Let $\mathcal{A}$ be a locally small category with colimits of small filtered diagrams. For the purposes of this question, an $\mathcal{A}$-presheaf on a topological space $X$ is a functor $\Omega (X) …

To avoid confusing myself, I will write $L : \mathcal{C} \to \mathcal{C} [\mathcal{W}^{-1}]$ for the localising functor and $R : \mathcal{C} [\mathcal{W}^{-1}] \to \mathcal{C}$ for its right adjoint. …

One of the fundamental results in commutative algebra is the following: Let $M$ be an $A$-module. Then $M = 0$ if and only if $M_\mathfrak{m} = 0$ for all maximal ideals $\mathfrak{m} \trianglelef …

Here is a reformulation/generalisation of G. Stefanich's counterexample, showing that sheaf-locality can fail very dramatically once we leave the realm of locally finitely presentable categories. More …