>>806
>See also
>Bousfield localization

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おれも いま読んでる　（＾＾

下記の”Localization of a category”が、重要キーワードだな

https://en.wikipedia.org/wiki/Localization_of_a_category
Localization of a category
In mathematics, localization of a category consists of adding to a category inverse morphisms for some collection of morphisms, constraining them to become isomorphisms. This is formally similar to the process of localization of a ring; it in general makes objects isomorphic that were not so before. In homotopy theory, for example, there are many examples of mappings that are invertible up to homotopy; and so large classes of homotopy equivalent spaces[clarification needed]. Calculus of fractions is another name for working in a localized category.

Introduction and motivation

Localization of categories

The above examples of localization of R-modules is abstracted in the following definition. In this shape, it applies in many more examples, some of which are sketched below.
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This procedure, however, in general yields a proper class of morphisms between X and Y.
Typically, the morphisms in a category are only allowed to form a set. Some authors simply ignore such set-theoretic issues.

Model categories
A rigorous construction of localization of categories, avoiding these set-theoretic issues, was one of the initial reasons for the development of the theory of model categories: a model category M is a category in which there are three classes of maps; one of these classes is the class of weak equivalences. The homotopy category Ho(M) is then the localization with respect to the weak equivalences. The axioms of a model category ensure that this localization can be defined without set-theoretical difficulties.

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