# Elementary example of right localization of functor

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?

## 1 Answer

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.