This is formally similar to the process of localization of a ring; it in general makes objects isomorphic that were not so before.
In many situations, it is meaningful to replace C by another category C' in which certain morphisms are forced to be isomorphisms.
is the localization of R with respect to the (multiplicatively closed) subset S consisting of all powers of r,
These relations turn the map going in the "wrong" direction into an inverse of f. This "calculus of fractions" can be seen as a generalization of the construction of rational numbers as equivalence classes of pairs of integers.
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.
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.
Some authors also define a localization of a category C to be an idempotent and coaugmented functor.
Serre introduced the idea of working in homotopy theory modulo some class C of abelian groups.
This meant that groups A and B were treated as isomorphic, if for example A/B lay in C. 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.
It is the localization of the category of chain complexes (up to homotopy) with respect to the quasi-isomorphisms.
This quotient category can be constructed as a localization of A by the class of morphisms whose kernel and cokernel are both in B.
An isogeny from an abelian variety A to another one B is a surjective morphism with finite kernel.
To call this a direct sum decomposition, we should work in the category of abelian varieties up to isogeny.