**Motivation:** In the classical construction of the derived category of an abelian category, one (roughly) starts with an abelian category $\mathcal{A}$, then considers the *quotient* category $\mathcal{K}$ of $\mathcal{A}$-chain complexes modulo homotopy and observes this is triangulated; finally, the derived category $\mathcal{D}$ is the localization of the above triangulated category with respect to some morphisms defined in terms of the triangulated structure.

I have always thought as the first process of modding out by homotopy equivalences as being a way to make homotopy-equivalent maps look the same or *inverting things which are homotopy equivalent to invertible arrows*. Then realized that this is also the aim of localising: force some arrows to become isomorphisms.

Here the **question:** is there a conceptual frame in which we can see both localisation and modding out by a relation as special cases? Or, are the two concepts better related that my vague interpretation that they both make some arrows invertible? I have seen someone in nLab saying that they are *entirely different constructions* but that left me a bit disappointed since I still believe a bit in the feeling that at the end of both days I find myself with a new category having the same objects as the old one, and some new isomorphisms. More generally: I realise that if I had a category in my hands and were willing to see some arrows become isomorphisms, I would not know which of the two approaches would be natural in my case, whence the feeling I still have some fog in my mind.

I should add that a colleague of mine observed that he would expect the first construction to be reminiscent of taking closed immersions, and the second to open immersions; which I fully agree to, but would again like to know how to think about these parallels when dealing with a category.

entirelyunrelated seems a bit unfair to me.) $\endgroup$$Cat$or something? $\endgroup$