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Localization of categories
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.

Examples
Module theory
In the theory of modules over a commutative ring R, when R has Krull dimension ≥ 2, it can be useful to treat modules M and N as pseudo-isomorphic if M/N has support of codimension at least two.
This idea is much used in Iwasawa theory.

Derived categories
The derived category of an abelian category is much used in homological algebra. It is the localization of the category of chain complexes (up to homotopy) with respect to the quasi-isomorphisms.

Related concepts
The localization of a topological space, introduced by Dennis Sullivan, produces another topological space whose homology is a localization of the homology of the original space.
A much more general concept from homotopical algebra, including as special cases both the localization of spaces and of categories, is the Bousfield localization of a model category.
Bousfield localization forces certain maps to become weak equivalences, which is in general weaker than forcing them to become isomorphisms.
(引用終り)

要するに
1)"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."
2)"Model categories、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."
3)"Related concepts A much more general concept from homotopical algebra, including as special cases both the localization of spaces and of categories, is the Bousfield localization of a model category.
Bousfield localization forces certain maps to become weak equivalences, which is in general weaker than forcing them to become isomorphisms"
が、肝らしい ；ｐ)