In general, localization is a process of adding formal inverses to an algebraic structure. The localization of a category at a collection of its morphisms is – if it exists – the result of universally making all morphisms in into isomorphisms.
A classic example is the localization of a commutative ring: we can ‘localize the ring away from the prime ’ and obtain the ring , or localize it away from all primes and obtain its field of fractions: the field of rational numbers.
The terminology is odd for historical and geometric reasons: localizing at a prime means inverting things not divisible by that prime, while inverting the prime itself is called localizing away from that prime. The reason for this, as well as for the term ‘localization’, becomes more apparent when we consider examples of a more vividly geometric flavor.
For example, the ring consists of polynomial functions on the real line. If we pick a point and localize by putting in an inverse to the element , the resulting ring consists of rational functions defined everywhere on the real line except possibly at the point . This is called localization away from , or localization away from the ideal generated by .
If on the other hand we put in an inverse to every element of that is not in the ideal , we obtain the ring of rational functions defined somewhere on the real line at least at the point : namely, those without a factor of in the denominator. This is called localizing at , or localizing at the ideal .
Notice that what is literally ‘localized’ when localizing the ring is not the ring itself, but its spectrum: the spectrum becomes smaller. The spectrum of is the whole real line. When we localize away from , the resulting ring has spectrum . When we localize at , the resulting ring has spectrum .
A ring is a very special case of a category, namely a one-object Ab-enriched category. This article mainly treats the more general case of localizing an arbitrary category. The localization of a category at a class of morphisms is the universal solution to making the morphisms in into isomorphisms; it is variously written , or . In some contexts, it also could be called the homotopy category of with respect to .
A localization of by (or “at ”) is
and a functor ;
if exists, it is unique up to equivalence.
size issues: If is large, then the existence of may depend on foundations, and it will not necessarily be locally small even if is. The tools of homotopy theory, and in particular model categories, can be used to address this question (see also at homotopy category of a model category).
In such a case
this adjoint exhibits as a reflective subcategory of .
Given a symmetric closed monoidal category , a -enriched category with underlying ordinary category and a subcategory of containing the identities of , H. Wolff defines the corresponding theory of localizations. See localization of an enriched category.
There is a general construction of , if it exists, which is however hard to use. When the system has special properties, most notably when admits a calculus of fractions or a factorization system, then there are more direct formulas for the hom-sets of .
If is a category and is a set of arrows, we construct the localization of . Let be the set in corresponding to (it isn’t necessarily a category).
Let be the following directed graph:
The arrows in are written as for .
Let be the free category on . The identity arrows are given by the empty path beginning and ending at a given object. We introduce a relation on the arrows of and quotient by the equivalence relation generated by it to get .
The equivalence relation is generated by
(Continue to show the quotient by gives a category, that it is locally small, and that if is small, the quotient is small.)
David Roberts: This could probably be described as the fundamental category of 2-dimensional simplicial complex with the directed space structure coming from the 1-skeleton, which will be the path category above. In that case, we could/should probably leave out the paths of zero length.
If the class admits a calculus of fractions, then there is a simpler description of in terms of spans instead of zig-zags. The idea is that any morphism in is built from a morphism in and a morphism in :
For more on this, see the entry calculus of fractions.
Localization is especially well developed in abelian setup where several competing formalisms and input data are used. See localization of an abelian category.
The notion of localization of a category has analogs in higher category theory.
For (∞,1)-categories and the special case of reflective embeddings this is discussed in
The classical reference to localization for categories is the book by Gabriel and Zisman:
A more recent account of localization with a calculus of fractions is section 7 of
An excellent account emphasizing the interplay of the different notions (reflective subcategory, calculus of fractions, closure operator) can be found in ch. V of
The pioneering work on abelian categories, with a large part on the localization in abelian categories is
A terminological discussion prompted by question in which sense “localization” is a descriptive term or not is archived ion Forum here.
See also vols.2,3 for examples of the theory in action in abelian categories, sheaf theory etc. ↩