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I am learning about a general framework for derived functors from Hotta et al., D-modules, Perverse Sheaves, and Representation Theory, Appendix B.

$\newcommand{\CC}{\mathcal C} \newcommand{\DD}{\mathcal D}$ Let $\CC$ be a category and $S$ be a multiplicative set of morphisms satisfying the Ore condition, so that we may form the localization $\CC_S$, with the associated functor $Q: \CC \to \CC_S$. The key definition is:

A right localization of $F: \CC \to \DD$ is a functor $F_S: \CC_S \to \DD$ representing $$ Fun(\CC_S, \DD) \to Set , \quad G \mapsto Hom_{(\CC,\DD)}(F, G \circ Q).$$

Concretely, this entails the existence of $\tau: F \to F_S \circ Q$ such that the composition $$ Hom(F_S, G) \to Hom(F_s \circ Q, G \circ Q) \to^{\tau^\ast} Hom(F, G \circ Q)$$ is the identity.

I understand this is used to define right derived functors (when we are localizing with respect to quasi-isomorphisms in the homotopy category) but are there more elementary examples of right (or left) localizations of functors? For instance, if $\CC$ is a one-element category enriched over $\mathrm{Ab}$ (i.e. a ring) and $\CC \to \DD$ is an additive functor, does this notion of localization coincide with classical localization of (noncommutative) rings?

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Yes, there are many elementary examples, and even in the book you cite, at the start of section B.5, they say "This problem can be easily solved if # = + or − and F is an exact functor" (i.e. if working in the bounded derived category). The theory goes back to a book by Gabriel and Zisman from the 1960s spelling out the language of a Calculus of Fractions, and Quillen's work on model categories from shortly afterwards. This theory is a categorical version of classical localization in ring theory, and both rely on an Ore condition.

Classical localization of a ring at a multiplicative subset (or, more precisely, a derived version of localization) is a special case of the calculus of fractions approach. Bill Dwyer nicely spells this out here. It also appears in Lurie's Higher Algebra (7.2.4.20), and in this paper.

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