nLab category of fractions




In general, localization is a process of freely adjoining inverses to an algebraic structure. In category theory, the localization 𝒞[W 1]\mathcal{C}[W^{-1}] of a category 𝒞\mathcal{C} at a class WW of its morphisms is – if it exists – the result of universally making all morphisms in WW into isomorphisms.

Some authors call any localization of a category a category of fractions (Gabriel & Zisman, §1.1; Borceux 1994 I, Def. 5.2.1), while other authors reserve this name for the case that WW admits a “calculus of fractions” (e.g. Fritz 2011). Conversely, some authors use “localization” to refer exclusively to the latter case (e.g. Verdier 1996, Def. 2.2.5; Stacks Project).

Yet other authors take “localization” to mean specifically “reflective localizations”, by default (e.g. Lurie 2009).


A classic example is the localization of a commutative ring: we can ‘localize the ring \mathbb{Z} away from the prime 22’ and obtain the ring [12]\mathbb{Z}[\frac{1}{2}], or localize it away from all primes and obtain its field of fractions: the field \mathbb{Q} 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 [x]\mathbb{R}[x] consists of polynomial functions on the real line. If we pick a point aa \in \mathbb{R} and localize [x]\mathbb{R}[x] by putting in an inverse to the element (xa)(x-a), the resulting ring consists of rational functions defined everywhere on the real line except possibly at the point aa. This is called localization away from aa, or localization away from the ideal II generated by (xa)(x-a).

If on the other hand we put in an inverse to every element of [x]\mathbb{R}[x] that is not in the ideal II, we obtain the ring of rational functions defined somewhere on the real line at least at the point aa: namely, those without a factor of (xa)(x-a) in the denominator. This is called localizing at aa, or localizing at the ideal II.

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 [x]\mathbb{R}[x] is the whole real line. When we localize away from aa, the resulting ring has spectrum {a}\mathbb{R} - \{a\}. When we localize at aa, the resulting ring has spectrum {a}\{a\}.

The case of localizing \mathbb{Z} can also be interpreted geometrically in a similar way, using scheme theory and arithmetic geometry.

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 CC at a class of morphisms WW is the universal solution to making the morphisms in WW into isomorphisms; it is variously written C[W 1]C[W^{-1}], W 1CW^{-1}C or L WCL_W C. In some contexts, it also could be called the homotopy category of CC with respect to WW.


Let CC be a category and WMor(C)W \subset Mor(C) a collection of morphisms.


A localization of CC by WW (or “at WW”) is

  • a (generally large, see below) category C[W 1]C[W^{-1}];

  • and a functor Q:CC[W 1]Q : C \to C[W^{-1}];

  • such that

    • for all wWw \in W, Q(w)Q(w) is an isomorphism;

    • for any category AA and any functor F:CAF : C \to A such that F(w)F(w) is an isomorphism for all wWw \in W, there exists a functor F W:C[W 1]AF_W : C[W^{-1}] \to A and a natural isomorphism FF WQF \simeq F_W \circ Q;

    • the map between functor categories

      ()Q:Funct(C[W 1],A)Funct(C,A)(-)\circ Q : Funct(C[W^{-1}], A) \to Funct(C,A)

      is full and faithful for every category AA.


  • if C[W 1]C[W^{-1}] exists, it is unique up to equivalence.

  • In 2-categorical language, C[W 1]C[W^{-1}] is the coinverter of the canonical natural transformation sts\to t, where s,t:WCs,t:W\to C are the “source” and “target” functors and WW is considered as a full subcategory of the arrow category C 2C ^{\mathbf{2}}.


Contrary to what is sometimes asserted, the localization C[W 1]C[W^{-1}] may generally be constructed when CC is large, even only in ZF where “large” means definable by class formulas. See Remark below. However, the localization might not be locally small, even if CC is. The tools of homotopy theory, and in particular model categories, can be used to address this issue (see also at homotopy category of a model category).

Reflective localization

A special class of localizations are reflective localizations, those where the functor CL WCC \to L_W C has a full and faithful right adjoint L WCCL_W C \hookrightarrow C.

In such a case

L WCQC L_W C \stackrel{\overset{Q}{\leftarrow}}{\hookrightarrow} C

this adjoint exhibits L WCL_W C as a reflective subcategory of CC.

One shows that L WCL_W C is – up to equivalence of categories – the full subcategory on the WW-local objects, and this property precisely characterizes such reflective localizations.

More on this is at reflective localization, reflective subcategory, reflective sub-(∞,1)-category, and reflective factorization system.

Localizations of enriched categories

Given a symmetric closed monoidal category VV, a VV-enriched category AA with underlying ordinary category A 0A_0 and a subcategory Σ\Sigma of A 0A_0 containing the identities of A 0A_0, H. Wolff defines the corresponding theory of localizations. See localization of an enriched category.


There is a general construction of C[W 1]C[W^{-1}], if it exists, which is however hard to use. When the system WW has special properties, most notably when WW admits a calculus of fractions or a factorization system, then there are more direct formulas for the hom-sets of C[W 1]C[W^{-1}].

General construction

Given CC a category and WMor(C)W \,\subset\, Mor(C) is a set of morphisms (not necessarily forming a subcategory), we construct the localization of CC. Let W opW^{op} denote the corresponding set in the opposite category C opC^{op}.

Let GG be the following directed graph:

The arrows in W op(x,y)W^{op}(x,y) are written as f¯\overline{f} for fW(y,x)f\in W(y,x).

Let 𝒫G\mathcal{P}G be the free category on GG. The identity morphisms are given by the empty path beginning and ending at a given object. We introduce a relation on the morphisms of 𝒫G\mathcal{P}G and quotient by the equivalence relation \sim generated by it to get C[W 1]C[W^{-1}].

The equivalence relation \sim is that generated by

  • for all objects xx of CC,

    (x;id x;x)(x;;x)(x;id_x;x) \sim (x;\emptyset;x)
  • for all f:xyf:x\to y and g:yzg:y\to z in CC,

    (x;f,g;z)(x;gf;z)(x;f,g;z)\sim (x;g\circ f;z)
  • for all f:xyf:x\to y in WW,

    (x;f,f¯;x)(x;id x;x)(x;f,\overline{f};x)\sim (x;id_x;x)


    (y;f¯,f;y)(y;id y;y)(y;\overline{f},f;y)\sim (y;id_y;y)

With a little care, this general construction can be enacted even for large categories CC, where “large” means given by a class formula in ZF. First, to define the free category 𝒫G\mathcal{P} G on a large graph GG, notice that an arrow of 𝒫G\mathcal{P} G is a certain partial function or functional relation from \mathbb{N} to Arr(G)Arr(G) satisfying some conditions given by ZF formulas; each such relation is a “set” by the replacement axiom, and the collection of such relations forms a definable class of sets. Second, in passing to the quotient, we remark that while the \sim-equivalence classes of morphisms in 𝒫G\mathcal{P} G might be large (hence themselves could not be elements of a class), we can always use Scott's trick and consider instead those representatives in each equivalence class that have minimal ZF rank (using crucially the axiom of foundation); this collection of representatives is a set. Thus the collection of morphisms in the quotient may be identified with a definable class of sets.

The rub is that the localization might not be locally small, even if CC is.

Construction when there is a calculus of fractions

If the class WW admits a calculus of fractions, then there is a simpler description of C[W 1]C[W^{-1}] in terms of spans instead of zig-zags. The idea is that any morphism f:xzf: x \to z in C[W 1]C[W^{-1}] is built from a morphism f 2:yzf_2 : y \to z in CC and a morphism f 1:yxf_1 : y \to x in WW:

xf 1yf 2z x \stackrel{f_1}{\longleftarrow} y \stackrel{f_2}{\longrightarrow} z

For more on this, see the entry calculus of fractions.

Dorette Pronk has extended this idea to construct a bicategories of fractions where a class of 1-arrows is sent to equivalences.

In abelian categories

Localization is especially well developed in abelian setup where several competing formalisms and input data are used. See localization of an abelian category.


  • The localization of the product category of two categories with weak equivalences is (if it exists) the product of their localizations

    (𝒞×𝒟)[(W 𝒞×W 𝒟) 1](𝒞[W 𝒞 1])×(𝒟[W 𝒟 1]). (\mathcal{C} \times \mathcal{D})[ (W_{\mathcal{C}} \times W_{\mathcal{D}})^{-1} ] \;\; \simeq \;\; ( \mathcal{C}[W_{\mathcal{C}}^{-1}] ) \times ( \mathcal{D}[W_{\mathcal{D}}^{-1}] ) \,.

    This is because localization is a reflector into the exponential ideal of minimal categories with weak equivalences.


In higher category theory

The notion of localization of a category has analogs in higher category theory.

For 2-categories:

For (∞,1)-categories and the special case of reflective embeddings this is discussed in

Every locally presentable (∞,1)-category is presented by a combinatorial model category. Accordingly, there is a model for the localization of (,1)(\infty,1)-categories in terms of these models. This is called

  • left Bousfield localization of model categories?

See also localization of a simplicial model category?.


Authors who say “category of fractions” for the result of inverting any class of morphisms:

Authors who say “localization” for the result of inverting any class of morphisms:

and reserve “category of fractions” for the case that WW admits a calculus of fractions:

See also:

Authors who say “localization” for inverting with respect to a calculus of fractions:

With focus on localization of abelian categories and of triangulated categories

Review of localization in homotopy theory:

For triangulated categories:

A terminological discussion prompted by question in which sense “localization” is a descriptive term or not is archived ion nnForum here.

A formal implementation of Gabriel-Zisman localization in ZFC, which in turn is implemented in the proof assistant Coq is in

A HoTT-Coq-formalization of left-exact reflective sub-(∞,1)-categories (localization of an (∞,1)-category) in homotopy type theory is in

An original account of localization of commutative rings and of p-local homotopy theory is

  • Dennis Sullivan, Localization, Periodicity and Galois Symmetry (The 1970 MIT notes) edited by Andrew Ranicki, K-Monographs in Mathematics, Dordrecht: Springer (pdf)

Last revised on December 27, 2023 at 13:46:48. See the history of this page for a list of all contributions to it.