localization at geometric homotopies

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###Context###
#### Locality and descent
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[[!include descent and locality - contents]]
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#### Homotopy theory
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[[!include homotopy - contents]]
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#Contents#
* table of contents
{:toc}

## Definition

In general, given a [[category]] $\mathcal{C}$ and a [[class]] $W$ of [[morphisms]], one may ask for the _[[localization]]_ $\mathcal{C}[W^{-1}]$, or more specifically for the [[reflective subcategory]] of $W$-[[local objects]] (the [[reflective localization]]). Similarly for variants in [[higher category]], such as _[[localization of an (∞,1)-category]]_.

If $\mathcal{C}$ has [[finite products]], then for a given [[object]] $\mathbb{A} \in \mathcal{C}$, one may take $W \coloneqq W_{\mathbb{A}}$ to be the class of morphisms of the form

$$
  X \times (\mathbb{A} \overset{\exists!}{\to} \ast)
  \;\;\colon\;\;
  X \times \mathbb{A}
    \overset{p_1}{\longrightarrow}
  X
  \,,
$$

where $X$ is any [[object]], and where $\ast$ is the [[terminal object]], and where $(-) \times (-) \colon \mathcal{C} \times \mathcal{C} \to \mathcal{C}$ denotes the [[Cartesian product]] [[functor]].

The [[reflective localization]] at such a class of morphisms $W_{\mathbb{A}}$ is often referred to as _homotopy localization at the object $\mathbb{A}$_ or similar.

The idea is that if $\mathbb{A}$ is, or is regarded as, an [[interval object]], then "geometric" [[left homotopies]] between morphisms $X \to Y$ are, or would be, given by morphisms out of $X \times \mathbb{A}$, and hence forcing the projections $X \times \mathbb{A} \to X$ to be equivalences means forcing all morphisms to be _[[homotopy invariance|homotopy invariant]]_ with respect to $\mathbb{A}$.

Typically this is considered in the case that $\mathcal{C}$ is a [[locally presentable category]] with a [[small set]] of [[generators and relations|generating objects]] $G_i$ such that it becomes sufficient to enforce the localization only on the resulting [[small set]] of [[morphisms]] of the form $G_i \times (\mathbb{A} \to \ast)$.

## Examples

* The localization of the [[(∞,1)-category of (∞,1)-sheaves]] on the [[Nisnevich site]] at the [[affine line]] $\mathbb{A}^1$ is known as  _[[A1-homotopy theory]]_.

* The localization of [[smooth ∞-groupoids]] at the [[real line]] $\mathbb{R}^1$ is equivalently ([[geometrically discrete ∞-groupoids|geometrically discrete]]) [[∞-groupoids]].

  After realizing [[smooth ∞-groupoids]] as the [[(∞,1)-sheaves]] over [[CartSp]], $Smooth\infty Grpd \simeq Sh_{\infty}(CartSp)$, this follows from the following Prop. \ref{HomotopyLocalizationOverSiteOfAns}.

+-- {: .num_prop #HomotopyLocalizationOverSiteOfAns}
###### Proposition
**([[homotopy localization]] at $\mathbb{A}^1$ over the [[site]] of $\mathbb{A}^n$s)**

Let $\mathcal{C}$ be any [[site]] ([this Def.](geometry+of+physics+--+categories+and+toposes#Coverage)), and write $[\mathcal{C}^{op}, sSet_{Qu}]_{proj, loc}$ for its local projective [[model category of simplicial presheaves]] ([this Prop.](geometry+of+physics+--+categories+and+toposes#TopologicalLocalization)).

Assume that $\mathcal{C}$ contains an [[object]] $\mathbb{A} \in \mathcal{C}$, such that every other object is a [[finite product]] $\mathbb{A}^n \coloneqq \underset{n \; \text{factors}}{\underbrace{\mathbb{A} \times \cdots \times \mathbb{A}}}$, for some $n \in \mathbb{N}$. (In other words, assume that $\mathcal{C}$ is also the [[syntactic category]] of [[Lawvere theory]].)

Consider the  $\mathbb{A}^1$-[[homotopy localization]] ([this Def.](geometry+of+physics+--+categories+and+toposes#HomotopyLocalizationOfCombinatorialModelCategories)) of the [[(∞,1)-sheaf (∞,1)-topos]] over $\mathcal{C}$ ([this Prop.](geometry+of+physics+--+categories+and+toposes#TopologicalLocalization))

$$
  Sh_\infty(\mathcal{C})_{\mathbb{A}}
    \underoverset
      {\underset{\phantom{AA}\iota\phantom{AA}}{\hookrightarrow}}
      {\overset{L_{\mathbb{A}}}{\longleftarrow}}
      {\bot}
  Sh_\infty(\mathcal{C})
  \;\;
  \in
  Ho(CombModCat)
$$

hence the [[left Bousfield localization]] of [[model categories]]

$$
  [\mathcal{C}^{op}, sSet_{Qu}]_{proj,loc,\mathbb{A}}
    \underoverset
      {\underset{\phantom{AA}id\phantom{AA}}{\longrightarrow}}
      {\overset{id}{\longleftarrow}}
      {\phantom{{}_{Qu}}\bot_{Qu}}
  [\mathcal{C}^{op}, sSet_{Qu}]_{proj,loc}
  \;\;
  \in
  CombModCat
$$

at the set of morphisms

$$
  S
    \;\coloneqq\;
  \big\{
    \mathbb{A}^n \times \mathbb{A}
      \overset{p_1}{\longrightarrow}
    \mathbb{A}^n
  \big\}
$$

(according to [this Prop.](geometry+of+physics+--+categories+and+toposes#ExistenceOfLeftBousfieldLocalization)).

Then this is [[equivalence of (∞,1)-categories|equivalent]] ([this Def.](geometry+of+physics+--+categories+and+toposes#HoCombModCat)) to [[∞Grpd]] ([this Example](geometry+of+physics+--+categories+and+toposes#InfinityGroupoid)),

$$
  \infty Grpd
  \;\simeq\;
  Sh_\infty(\mathcal{C})_{\mathbb{A}}
    \underoverset
      {\underset{\phantom{AA}\iota\phantom{AA}}{\hookrightarrow}}
      {\overset{L_{\mathbb{A}}}{\longleftarrow}}
      {\bot}
  Sh_\infty(\mathcal{C})
  \;\;
  \in
  Ho(CombModCat)
$$

in that the ([[constant functor]] $\dashv$ [[limit]])-[[adjunction]] ([this Def.](geometry+of+physics+--+categories+and+toposes#Limits))

\[
  \label{QuillenEquivalenceInABousfLocalization}
  [\mathcal{C}^{op}, sSet_{Qu}]_{inj, loc, \mathbb{A}}
    \underoverset
      {\underset{ \underset{\longleftarrow}{\lim} }{\longrightarrow}}
      {\overset{ \phantom{AA}const\phantom{AA} }{\longleftarrow}}
      {\bot}
  sSet_{Qu}
  \;\;\;\;
  \in
  CombModCat
\]

is a [[Quillen equivalence]] ([this Def.](geometry+of+physics+--+homotopy+types#QuillenEquivalence)).

=--

+-- {: .proof}
###### Proof

First to see that (eq:QuillenEquivalenceInABousfLocalization) is a [[Quillen adjunction]]: Since we have a [[simplicial Quillen adjunction]] before localization

$$
  [\mathcal{C}^{op}, sSet_{Qu}]_{inj}
    \underoverset
      {\underset{ \underset{\longleftarrow}{\lim} }{\longrightarrow}}
      {\overset{ \phantom{AA}const\phantom{AA} }{\longleftarrow}}
      {\bot}
  sSet_{Qu}
$$

(by [this Example](geometry+of+physics+--+categories+and+toposes#HomotopyLimitOfSimplicialSets)) and since both [[model categories]] here are [[left proper model category|left proper]] [[simplicial model categories]] (by [this Prop.](geometry+of+physics+--+categories+and+toposes#SimplicialPresheavesIsProperCombinatorialSimplicial) and [this Prop.](geometry+of+physics+--+categories+and+toposes#ExistenceOfLeftBousfieldLocalization)), and since [[left Bousfield localization]] does not change the class of [[cofibrations]] ([this Def.](geometry+of+physics+--+categories+and+toposes#BousfieldLocalizationOfModelCategories)) it is sufficient to show that $\underset{\longleftarrow}{\lim}$ preserves [[fibrant objects]] (by [this Prop.](geometry+of+physics+--+categories+and+toposes#RecognitionOfSimplicialQuillenAdjunction)).

But by assumption $\mathcal{C}$ has a [[terminal object]] $\ast = \mathbb{A}^0$, which is hence the [[initial object]] of $\mathcal{C}^{op}$, so that the [[limit]] operation is given just by evaluation on that object:

$$
  \underset{\longleftarrow}{\lim}
  \mathbf{X}
  \;=\;
  \mathbf{X}(\mathbb{A}^0)
  \,.
$$

Hence it is sufficient to see that an injectively fibrant simplicial presheaf $\mathbf{X}$ is objectwise a [[Kan complex]]. This is indeed the case, by [this Prop.](geometry+of+physics+--+categories+and+toposes#ModelCategoriesOfSimplicialPresheaves).

To check that (eq:QuillenEquivalenceInABousfLocalization) is actually a [[Quillen equivalence]], we check that the [[derived adjunction unit]] and [[derived adjunction counit]] are [[weak equivalences]]:

For $X \in sSet$ any simplicial set (necessarily cofibrant), the [[derived adjunction unit]] is

$$
  X
    \overset{id_X}{\longrightarrow}
  const(X)(\mathbb{A}^0)
    \overset{ const(j_X)(\mathbb{A}^0) }{\longrightarrow}
  const(P X)(\mathbb{A}^0)
$$

where $X \overset{j_X}{\longrightarrow} P X$ is a [[fibrant replacement]] ([this Def.](geometry+of+physics+--+categories+and+toposes#FibrantCofibrantReplacementFunctorToHomotopyCategory)). But $const(-)(\mathbb{A}^0)$ is clearly the [[identity functor]] and the plain adjunction unit is the [[identity morphism]], so that this composite is just $j_X$ itself, which is indeed a weak equivalence.

For the other case, let $\mathbf{X} \in [\mathcal{C}^{op}, sSet_{Qu}]_{inj, loc, \mathbb{A}^1}$ be fibrant. This means (by [this Prop.](geometry+of+physics+--+categories+and+toposes#ExistenceOfLeftBousfieldLocalization)) that $\mathbf{X}$ is fibrant in the injective [[model structure on simplicial presheaves]] as well as in the local model structure, and is a derived-$\mathbb{A}^1$-[[local object]] ([this Def.](geometry+of+physics+--+categories+and+toposes#DerivedLocalObjects)), in that the [[derived hom-functor]] out of any $\mathbb{A}^n \times \mathbb{A}^1 \overset{p_1}{\longrightarrow} \mathbb{A}^n$  into $\mathbf{X}$ is a [[weak homotopy equivalence]]:

$$
  \mathbb{R}Hom( p_1 )
  \;\colon\;
  \mathbb{R}Hom( \mathbb{A}^n , \mathbf{X})
  \overset{\in W}{\longrightarrow}
  \mathbb{R}Hom( \mathbb{A}^n \times \mathbb{A}^1 , \mathbf{X})
$$

But since $\mathbf{X}$ is fibrant, this derived hom is equivalent to the ordinary [[hom-functor]] ([this Lemma](geometry+of+physics+--+categories+and+toposes#HomsOutOfCofibrantIntoFibrantComputeHomotopyCategory)), and hence with the [[Yoneda lemma]] ([this Prop.](geometry+of+physics+--+categories+and+toposes#YonedaLemma)) we have that

$$
  \mathbf{X}(p_1)
  \;\colon\;
  \mathbf{X}(\mathbb{A}^n)
  \overset{\in W}{\longrightarrow}
  \mathbf{X}(\mathbb{A}^{n+1})
$$

is a weak equivalence, for all $n \in \mathbb{N}$. By [[induction]] on $n$ this means that in fact

$$
  \mathbf{X}(\mathbb{A}^0)
   \overset{\in W}{\longrightarrow}
  \mathbf{X}(\mathbb{A}^n)
$$

is a weak equivalence for all $n \in \mathbb{N}$. But these are just the components of the [[adjunction counit]]

$$
  const (\mathbf{X}(\mathbb{A}^0))
   \underoverset{\in W}{\epsilon}{\longrightarrow}
  \mathbf{X}
$$

which is hence also a weak equivalence. Hence for the [[derived adjunction counit]]

$$
  const (Q \mathbf{X})(\mathbb{A}^0)
    \overset{const(p_{\mathbf{X}}(\mathbb{A}^0))}{\longrightarrow}
  const (\mathbf{X}(\mathbb{A}^0))
   \underoverset{\in W}{\epsilon}{\longrightarrow}
  \mathbf{X}
$$

to be a weak equivalence, it is now sufficient to see that the value of a [[cofibrant replacement]] $p_{\mathbf{X}}$ on $\mathbb{A}^0$ is a weak equivalence. But by definition of the weak equivalences of simplicial presheaves these are objectwise weak equivalences.

=--

## References

* {#MorelVoevodsky99} [[Fabien Morel]], [[Vladimir Voevodsky]], Def. 3.1 in _$\mathbb{A}^1$-homotopy theory of schemes_, Publications Mathématiques de l'IHÉS, Volume 90 (1999), p. 45-143  ([Numdam:PMIHES_1999__90__45_0](http://www.numdam.org/item/?id=PMIHES_1999__90__45_0) [K-Theory:0305](http://www.math.uiuc.edu/K-theory/0305/) )

For more references see also at _[[motivic homotopy theory]]_.

[[!redirects homotopy localization]]
[[!redirects homotopy localizations]]

[[!redirects homotopy localization at an object]]
[[!redirects homotopy localizations at an object]]

[[!redirects localization at an object]]
[[!redirects localizations at an object]]

[[!redirects localization at objects]]
[[!redirects localizations at objects]]


Localization in Homotopy Type Theory

* [[J. Daniel Christensen]], [[Morgan Opie]], [[Egbert Rijke]], [[Luis Scoccola]]:

  **Localization in Homotopy Type Theory**

  Higher Structures **4** 1 (2020) 1-32

  [arXiv:1807.04155](https://arxiv.org/abs/1807.04155)

on [[localization]] in [[homotopy type theory]].

> **Abstract.** We study [[localization]] [[p-localization|at a prime]] in [[homotopy type theory]], using [[endomorphism|self maps]] of the [[circle]]. Our main result is that for a [[pointed]], [[simply connected]] [[type]] $X$, the natural map $X \to X_{(p)}$ induces algebraic localizations on all [[homotopy groups]]. In order to prove this, we further develop the theory of [[reflective subuniverses]]. In particular, we show that for any reflective subuniverse $L$, the subuniverse of $L$-separated types is again a reflective subuniverse, which we call $L_0$. Furthermore, we prove results establishing that $L_0$ is almost [[left exact]]. We next focus on localization with respect to a map, giving results on preservation of [[coproducts]] and [[connectivity]]. We also study how such localizations interact with other reflective subuniverses and [[orthogonal factorization systems]]. As key steps towards proving the main theorem, we show that localization at a prime commutes with taking loop spaces for a pointed, simply connected type, and explicitly describe the localization of an [[Eilenberg-Mac Lane space]] $K(G, n)$ with $G$ abelian. We also include a partial converse to the main theorem.

## See also ##

* [[Modalities in homotopy type theory]]

* [[homotopy type theory - references]]

category: reference


localization of a 2-category

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#### 2-Category theory
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[[!include 2-category theory - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

The generalization of the concept of _[[localization of categories]]_ from [[categories]] to [[2-categories]].

## Definition

+-- {: .num_defn #2Localization}
###### Definition
**(localization of a 2-category)**

Let $\mathcal{C}$ be a [[2-category]] and let $W \subset 1Mor(\mathcal{C})$ be a sub-[[class]] of its [[1-morphisms]].

Then [[generalized the|the]] the _2-localization_ x is, if it exists,

1. a [[2-category]] $\mathcal{C}[W^{-1}]$;

1. a [[2-functor]] $\mathcal{C} \overset{\gamma}{\longrightarrow} \mathcal{C}[W^{-1}]$

such that

1. $\gamma$ sends all elements of $W$ to an [[equivalence in a 2-category|equivalence in the 2-category]] $\mathcal{C}[W^{-1}]$;

1. $\gamma$ is [[universal property|universal with this property]]: precomposition with $\gamma$ induces for every [[2-category]] $\mathcal{D}$ an [[equivalence of 2-categories]]

   $$
     (-) \circ \gamma
     \;\colon\;
      2Func(\mathcal{C}[W^{-1}], \mathcal{D})
      \overset{\simeq}{\longrightarrow}
      2Func(\mathcal{C},\mathcal{D})_{W}
      \subset
      2Func(\mathcal{C}, \mathcal{D})
   $$

   from the [[2-functor 2-category]] from $\mathcal{C}[W^{-1}]$ to $\mathcal{D}$ to the [[full sub-2-category]] of the [[2-functor 2-category]] from $\mathcal{C}$ to $\mathcal{D}$ on those [[2-functors]] that send element of $W$ to [[equivalences in a 2-category|equivalences]].

=--

## Examples

### Homotopy 2-category of combinatorial model categories
  {#Homotopy2CategoryOfCombinatorialModelCategories}

+-- {: .num_example #2CategoryOfModelCategories}
###### Definition
**(the [[2-category]] of [[combinatorial model category|combinatorial]] [[model categories]])**

Write

1. $ModCat$ for the [[2-category]] whose [[objects]] are [[model categories]], [[1-morphisms]] are [[left adjoint]] [[functors]] of [[Quillen adjunctions]] and [[2-morphisms]] are [[natural transformations]].

1. $CombModCat \subset ModCat$ for the [[full sub-2-category]] on the [[combinatorial model categories]].

=--

+-- {: .num_remark #LocalPresentationOfCombinatorialModelCategories}
###### Remark
**(local presentation of combinatorial model categories)**

By [[Dugger's theorem]], we may choose for every $\mathcal{C} \in CombModCat$ a [[simplicial set]] $\mathcal{S}$ and a [[Quillen equivalence]]

$$
  \mathcal{C}^p
  \;\coloneqq\;
  [\mathcal{S}^{op}, sSet]_{proj,loc}
  \overset{\simeq_{Qu}}{\longrightarrow}
  \mathcal{C}
$$

from the local projective [[model structure on sSet-enriched presheaves]] over $\mathcal{S}$.

=--

+-- {: .num_theorem #Homotopy2CategoryOf2CatOfCombinatorialModelCategories}
###### Theorem
**(the [[homotopy 2-category]] of [[combinatorial model categories]])**

The [[2-localization]] (Def. \ref{2Localization})

$$
  CombModCat\big[QuillenEquivalences^{-1}\big]
$$

of the [[2-category]] of [[combinatorial model categories]] (Def. \ref{2CategoryOfModelCategories}) at the [[Quillen equivalences]] exists. Up to [[equivalence of 2-categories]], it has the same [[objects]] as $CombModCat$ and for any $\mathcal{C}, \mathcal{D} \in CombModCat$ its [[hom-category]] is the [[localization of categories]]

$$
  CombModCat\big[QuillenEquivalences^{-1}\big](\mathcal{C}, \mathcal{D})
  \;\simeq\;
  ModCat( \mathcal{C}^p, \mathcal{D}^p )\big[\{QuillenHomotopies\}^{-1}\big]
$$

of the category of [[left Quillen functors]] and [[natural transformations]] between local presentations $\mathcal{C}^p$ and $\mathcal{D}^p$ (Remark \ref{LocalPresentationOfCombinatorialModelCategories}) at those [[natural transformation]] that on [[cofibrant objects]] have components that are [[weak equivalences]] ("Quillen homotopies").

=--

This is the statement of [Renaudin 06, theorem 2.3.2](#Renaudin06).

For $\mathcal{C}$ a [[2-category]] write

1. $\mathcal{C}_0$ for the 1-category obtained by discarding all [[2-morphisms]];

1. $\pi_0^{iso}(\mathcal{C})$ for the [[1-category]] obtained by identifying isomorphic [[2-morphisms]].

+-- {: .num_prop}
###### Proposition

The composite 1-functor

$$
  CombModCat_0
   \longrightarrow
  \pi_0^{iso}(CombModCat)
    \overset{\pi_0^{iso}(\gamma)}{\longrightarrow}
  \pi_0^{iso}( CombModCat[QuillenEquivalences^{-1}] )
$$

induced from the [[2-localization]] of Theorem \ref{Homotopy2CategoryOf2CatOfCombinatorialModelCategories} exhibits the ordinary [[localization of a category]] of the 1-category $CombModCat$ at the [[Quillen equivalences]], hence [[Ho(CombModCat)]].

=--

This is the statement of [Renaudin 06, cor. 2.3.8](#Renaudin06).

## Related concepts

* [[bicategory of fractions]]

* [[homotopy 2-category]]

* [[localization of an enriched category]]

## References

* {#Renaudin06} [[Olivier Renaudin]], Section 1.2 of: *Plongement de certaines théories homotopiques de Quillen dans les dérivateurs*, J. Pure Appl. Alg. __213__:10 (2009) 1916--1935
([arXiv:math/0603339](https://arxiv.org/abs/math/0603339), [doi:10.1016/j.jpaa.2009.02.014](https://doi.org/10.1016/j.jpaa.2009.02.014))

Via [[bicategories of fractions]]:

* [[Dorette A. Pronk]], Section 2 of: _Etendues and stacks as bicategories of fractions_, Comp. Math. **102** 3 (1996) pp.243-303. ([numdam:CM_1996__102_3_243_0](http://www.numdam.org/item/?id=CM_1996__102_3_243_0))

Another related construction

* M. E. Descotte, E. J. Dubuc, M. Szyld, _A localization of bicategories via homotopies_, Theory and Applications of Categories __35__, 2020, No. 23, 845--874, [TAC](http://tac.mta.ca/tac/volumes/35/23/35-23abs.html) [[zoranskoda:MR4112764]]

Yer another related subject

* M. E. Descotte, [[E. J. Dubuc]], M. Szyld, _Model bicategories and their homotopy bicategories_, Adv. Math. __404__B, (2022) 108455 [arXiv:1805.07749](https://arxiv.org/abs/1805.07749) [doi](https://doi.org/10.1016/j.aim.2022.108455)

[[!redirects localizations of a 2-category]]
[[!redirects localizations of 2-categories]]

[[!redirects 2-localization of a 2-category]]
[[!redirects 2-localizations of a 2-category]]
[[!redirects 2-localizations 2-categories]]

[[!redirects 2-localization]]
[[!redirects 2-localizations]]

[[!redirects pseudo-localization]]
[[!redirects pseudo-localizations]]

[[!redirects pseudolocalization]]
[[!redirects pseudolocalizations]]


localization of a category

Localization of a category is an ambiguous term that can refer to one of the two following notions:

* [[category of fractions]].  This is traditional terminology, used in the books by Gabriel–Zisman and Borceux.

* [[reflective localization]].  This meaning is especially prevalent in the literature on [[quasicategories]], following Lurie's [[Higher Topos Theory]].  However, some authors use “localization” to mean the [[quasicategory of fractions]] instead.

[[!redirects localizations of a category]]
[[!redirects localizations of categories]]

[[!redirects localization of categories]]
[[!redirects localizations of categories]]


localization of a commutative ring

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###Context###
#### Algebra
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[[!include higher algebra - contents]]
=--
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=--

#Contents#
* table of contents
{:toc}

## Idea

The _[[localization of a ring|localization]]_ of a  [[commutative ring]] $R$ at a [[set]] $S$ of its [[elements]] is a new ring $R[S^{-1}]$ in which the elements of $S$ become invertible ([[unit|units]]) and which is [[universal property|universal]] with this property.

When interpreting a ring under [[Isbell duality]] as the [[ring of functions]] on some [[space]] $X$ (its [[spectrum of a commutative ring|spectrum]]), then localizing it at $S$ corresponds to restricting to the complement of the subspace $Y \hookrightarrow X$ on which the elements in $S$ vanish.

See also _[[commutative localization]]_ and _[[localization of a ring]] (noncommutative)_.

\begin{remark}\label{Terminology}
**Localization "at" and "away from"**

The common terminology in algebra is as follows.

For $S$ a set of [[primes]], "localize at $S$" means "invert what is not divisible by $S$"; so for $p$ prime, localizing "at $p$" means considering only $p$-[[torsion subgroup|torsion]].

Adjoining inverses $[S^{-1}]$ is pronounced "localized away from $S$". Inverting a [[prime]] $p$ is localizing away from $p$, which means ignoring $p$-torsion.

See also lecture notes such as ([Gathmann](#Gathmann)) and see at _[[localization of a space]]_ for more discussion of this.

Evidently, this conflicts with more-categorial uses of "localized"; "inverting weak equivalences" is called localization, by obvious analogy, and is written as "localizing at weak equivalences". This is confusing! It's also weird: since a ring is a one-object Ab-enriched category with morphisms "multiply-by", the localization-of-the-category $R$ "at $p$" (or its Ab-enriched version, if saying that is necessary) really means the localization-of-the-ring R "away from p".

\end{remark}

## Definition

### For commutative rings

Let $R$ be a [[commutative ring]]. Let $S \hookrightarrow U(R)$ be a [[multiplicative subset]] of the underlying [[set]].

The following gives the [[universal property]] of the localization.

\begin{definition}
{#UniversalPropertyOfLocalizationOfARing}
The **localization** $L_S \colon R \to R[S^{-1}]$ is a [[homomorphism]] to another commutative ring $R[S^{-1}]$ such that

* for all elements $s \in S \hookrightarrow R$ the image $L_S(s) \in R[S^{-1}]$ is invertible (is a [[unit]]);

* for every other homomorphism $R \to \tilde R$ with this property, there is a unique [[homomorphism]] $R[S^{-1}] \to \tilde R$ such that we have a [[commuting diagram]]

   $$
     \array{
       R &\stackrel{L_S}{\to}& R[S^{-1}]
       \\
       & \searrow & \downarrow
       \\
       && \tilde R
     }
     \,.
   $$
\end{definition}

The special case of inverting an element $r$ of $R$, in which $S$ is the set $\{ r, r^{2}, r^{3}, \ldots \}$, is discussed at [[localisation of a commutative ring at an element]]. See also for example [Sullivan 70, first pages](#Sullivan70).

\begin{remark}
The [[Isbell duality|formal duals]] $Spec(R[S^{-1}]) \longrightarrow Spec(R)$ of the localization maps $R \longrightarrow R[S^{-1}]$ (under forming [[spectrum of a commutative ring|spectra]]) serve as the standard [[open immersion of schemes|open immersions]] that define the [[Zariski topology]] on [[algebraic varieties]].
\end{remark}

Explicitly:

\begin{definition}
The localization of a [[commutative ring]] $R$ at a [[multiplicative subset]] $S$ is the [[commutative ring]] whose underlying  [[set]] is the set of [[equivalence classes]] on $R \times S$ under the [[equivalence relation]]

$$
  (r_1, s_1) \sim (r_2, s_2)
  \;\;\Leftrightarrow\;\;
  \exists u \in S
  \;
  (r_1 s_2- r_2 s_1) u = 0 \;\in R
  \,.
$$

Write $r s^{-1}$ for the [[equivalence class]] of $(r,s)$. On this set, addition and multiplication is defined by

$$
  r_1 s_1^{-1} + r_2 s_2^{-1} \coloneqq (r_1 s_2 + r_2 s_1) (s_1 s_2)^{-1}
$$

$$
  (r_1 s_1^{-1})(r_2 s_2^{-1}) \coloneqq r_1 r_2 (s_1 s_2)^{-1}
  \,.
$$
\end{definition}

(e.g. [[The Stacks Project|Stacks Project, def. 10.9.1]])

\begin{remark}
The above definitions also work for non-commutative [[rings]] $R$ as well, so long as the [[multiplicative subset]] $S$ is a [[submonoid]] of the [[center]] $Z(R)$ of the multiplicative monoid of $R$.
\end{remark}

### For $E_\infty$-rings
 {#ForEInfinityRings}

(...) By the lifting property of etale morphisms for $E_\infty$-rings, see [here](étale+morphism+of+E-∞+rings#LocalizationOfRings). (...)

Localization away from a suitably tame ideal may be understood as the [[dR-shape modality]] in the [[cohesion]] of [[E-infinity arithmetic geometry]]:

[[!include arithmetic cohesion -- table]]

## Examples

* [[field of fractions]]

## Related concepts

* [[completion of a ring]]

* [[localization]],

* [[localization of a monoid]]

* [[localization of an abelian group]]

* [[localization of a module]]

* [[localization of a ring]], [[localization of a category]]

* [[Cohn localization]], [[Ore localization]].

* [[Bousfield localization of spectra]]

  * [[localization of a space]] (and of a [[spectrum]])

  * [[p-localization]], [[p-completion]]

* [[quotient ring]]

## References
 {#References}

A classical set of lecture notes:

* {#Sullivan70} [[Dennis Sullivan]], _Localization, Periodicity and Galois Symmetry_ (The 1970 MIT notes) edited by [[Andrew Ranicki]], K-Monographs in Mathematics, Dordrecht: Springer ([pdf](http://www.maths.ed.ac.uk/~aar/surgery/gtop.pdf))

Concretely on localization of commutative rings:

* {#Sullivan05} [[Dennis Sullivan]], Section 1 of: _Geometric topology: localization, periodicity and Galois symmetry_, volume 8 of K- Monographs in Mathematics. Springer, Dordrecht, 2005. The 1970 MIT notes, Edited and with a preface
by [[Andrew Ranicki]] ([pdf](http://www.maths.ed.ac.uk/~aar/books/gtop.pdf))

Discussion in [[constructive mathematics]]:

* [[Henri Lombardi]], [[Claude Quitté]] chapter 2 section 2 in:  *Commutative algebra: Constructive methods (Finite projective modules)* Translated by Tania K. Roblo, Springer (2015) [[doi:10.1007/978-94-017-9944-7](https://link.springer.com/book/10.1007/978-94-017-9944-7), [pdf](http://hlombardi.free.fr/CACM.pdf)]

Further review:

* {#Gathmann} Andreas Gathmann, _Localization_ ([[GathmannLocalization.pdf:file]])

* [[Joseph Neisendorfer]] _A Quick Trip through Localization_ ([pdf](https://www.math.rochester.edu/people/faculty/jnei/localization.pdf))

Other accounts of the basics include

* {#Gathmann} Andreas Gathmann, _Localization_ ([[GathmannLocalization.pdf:file]])

* [[The Stacks Project]], 10.9. _Localization_

* Yoshifumi Tsuchimoto, Review of commutative (usual) affine schemes. _[Localization of a commutative ring](http://www.math.kochi-u.ac.jp/docky/bourdoki/NAS/nas002/node4.html)_

[[!redirects localizations of a commutative ring]]
[[!redirects localisation of a commutative ring]]
[[!redirects localization of commutative rings]]
[[!redirects Localization of commutative rings]]
[[!redirects localizations of commutative rings]]
[[!redirects Localizations of commutative rings]]


localization of a model category > history


localization of a module

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _localization of a module_ is the result of application of an [[additive functor|additive]] [[localization functor]] on a [[category of modules]] over some [[ring]] $R$.

When $R$ is a [[commutative ring]] of functions, and under the [interpretation of modules as generalized vector bundles](module#RelationToVectorBundlesInIntroduction) the _localization_ of a module corresponds to the restriction of the bundle to a subspace of its base space.

## Definition

### For modules over rings

For $R$ a (possibly noncommutative) unital ring, let $\mathcal{A} = R$[[Mod]] be the [[category]] $R$-[[modules]]. Here $R$ may be the [[structure sheaf]] of some [[ringed topos]] and accordingly the modules may be [[sheaves of modules]].

Consider a [[reflective localization]] functor

$$
  Q^* = Q^*_\Sigma \colon \mathcal{A}\to \Sigma^{-1}\mathcal{A}
$$

with [[right adjoint]] $Q_*$. The application of this functor to a [[module]] $M\in \mathcal{A}$ is some object $Q^*(M)$ in the localized category $\Sigma^{-1}\mathcal{A}$, which is up to isomorphism determined by its image $Q_* Q^*(M)$.

The __localization map__ is the component of the [[unit of an adjunction|unit of the adjunction]] (usually denoted by $i$, $j$ or $\iota$ in this setup) $\iota_M : M\to Q_* Q^*(M)$.

Depending on an author or a context, we say that a (reflective) localization functor of category of modules is __flat__ if either $Q^*$ is also left [[exact functor]], or more strongly that the composed endofunctor $Q_* Q^*$ is exact. For example, [[Gabriel localization]] is flat in the first, weak sense, and left or right Ore localization is flat in the second, stronger, sense.

### For chain complexes

... Greenlees-May duality...

### For spectra ($\mathbb{S}$-modules)
 {#ForSpectra}

+-- {: .num_prop}
###### Claim

Suppose that $L \colon Spectra \to Spectra$ is a [[smashing localization]] given by [[smash product]] with some [[spectrum]] $T$. Write $F$ for the [[homotopy fiber]]

$$
  F \longrightarrow \mathbb{S} \longrightarrow T
  \,.
$$

Then there is a [[fracture diagram]] of operations

$$
  \array{
    T \wedge (-) && \longleftarrow  &&  [T,-]
    \\
    & \nwarrow && \swarrow
    \\
    && \mathbb{S}
    \\
    & \swarrow & & \nwarrow
    \\
    [F,-] && \longleftarrow && F \wedge (-)
  }
$$

where $[F,-]$ and $T \wedge (-) \colon Spectr \to Spectra$ are [[idempotent (∞,1)-monads]] and $[T,-]$, $[F,-]$ are idempotent $\infty$-comonad, the diagonals are [[homotopy fiber sequences]].

=--

([[Charles Rezk]], [MO comment,August 2014](http://mathoverflow.net/a/178316/381))

+-- {: .num_example}
###### Example

For $T = S \mathbb{Z}[p^{-1}]$ the [[Moore spectrum]] of the [[integers]] [[localization of a ring|localized away from]] $p$, then

$$
  F = \Sigma^{-1} S (\mathbb{Z}[p^{-1}]/\mathbb{Z})
   \to
  \mathbb{S}
   \to
  S \mathbb{Z}[p^{-1}]
$$

and hence

* $\Sigma^{-1} S (\mathbb{Z}[p^{-1}]/\mathbb{Z}) \wedge (-)$ is $p$-[[torsion approximation]];

* $[\Sigma^{-1} S (\mathbb{Z}[p^{-1}]/\mathbb{Z}),-]$ is $p$-[[completion of a module|completion]];

* $S \mathbb{Z}[p^{-1}] \wedge (-)$ is localization away from $p$ ($p$-[[rationalization]])

* $[T,-]$ is forming $p$-[[adic residual]].

$$
  \array{
    &&  localization\;away\;from\;\mathfrak{a} && \stackrel{}{\longrightarrow} && \mathfrak{a}\;adic\;residual
    \\
    & \nearrow & & \searrow & & \nearrow && \searrow
    \\
    && && X &&  &&
    \\
    & \searrow &  & \nearrow & & \searrow && \nearrow
    \\
    && formal\;completion\;at\;\mathfrak{a}\; && \longrightarrow && \mathfrak{a}\;torsion\;approximation
  }
  \,,
$$

=--

With ([Bousfield 79, prop.2.5](#Bousfield79))

### For $\infty$-modules over $E_\infty$-rings

Let $A$  be an [[E-∞ ring]] and $\mathfrak{a} \subset \pi_0 A$ a [[generators and relations|finitely generated]] ideal of its underlying [[commutative ring]].

+-- {: .num_defn #LocalInfinityModule}
###### Definition

An $A$-[[∞-module]] $N$ is an _$\mathfrak{a}$-local module_ if for every $\mathfrak{a}$-[[torsion module]] $T$ (def. \ref{TorsionInfinityModule}), the [[derived hom space]]

$$
 Hom_A(T,N) \simeq \ast
$$

is contractible.

=--

([Lurie "Completions", def. 4.1.9](#LurieCompletions)).

+-- {: .num_prop #LocalizationAwayByColimit}
###### Proposition

For $\mathfrak{a} = (a)$ generated from a single element, then the [localization of an (∞,1)-ring](localization+of+a+commutative+ring#ForEInfinityRings)-map $A \to A[a^{-1}]$ is given by the [[(∞,1)-colimit]] over the sequence of right-multiplication with $a$

$$
  A[a^{-1}]
  \simeq
  \underset{\rightarrow}{\lim}
  (
    A \stackrel{\cdot a}{\longrightarrow} A
    \stackrel{\cdot a}{\longrightarrow} A
    \stackrel{\cdot a}{\longrightarrow}  \cdots
  )
  \,.
$$

=--

([Lurie "Completions", remark 4.1.11](#LurieCompletions))

+-- {: .num_prop}
###### Proposition

The [[full sub-(∞,1)-category]]

$$
  A Mod_{\mathfrak{a}loc}
  \hookrightarrow
  A Mod
$$

of [[∞-modules]] [[localization of a module|local]] away from $\mathfrak{a}$ is [[reflective sub-(∞,1)-category|reflective]]. The reflector

$$
  \Pi_{\mathfrak{a}dR} \colon A Mod \longrightarrow A Mod_{\mathfrak{a}loc}
$$

is called _localization_.

=--

+-- {: .num_prop}
###### Proposition

There is a [[natural transformation|natural]] [[homotopy fiber sequence]]

$$
  ʃ_{\mathfrak{a}} \longrightarrow id \longrightarrow ʃ_{\mathfrak{a}dR}
$$

relating $\mathfrak{a}$-[[torsion approximation]] on the left with $\mathfrak{a}$-localization on the right.

=--

## Properties

### Eilenberg-Watts theorem

By the [[Eilenberg-Watts theorem]], if $\mathcal{A}= R$[[Mod]] then the localization of a module

$$
  Q^*(M) = Q^*(R)\otimes_R M
$$

is given by forming the [[tensor product of modules]] with the localizatin of the ring $R$, regarded as a module over itself.

If the localization is a left [[Ore localization]] or [[commutative localization]] at a set $S\subset R$ then $Q^*(R) = S^{-1} R$ is the [[localization of a ring|localization of the ring]] itself
and hence in this case the localization of the module

$$
  Q^*(M) = S^{-1} R\otimes_R M
$$

is given by [[extension of scalars]] along the localization map $R \to S^{-1}R$ of the ring itself.

In these cases there are also direct constructions of $Q^*(M)$ (not using to $Q^*(R)$) which give an isomorphic result, also denoted by $S^{-1}M$.

### Relation with torsion approximation and completion

[[!include arithmetic cohesion -- table]]

## Related concepts

* [[localization of an abelian group]]

* [[localization of a ring]], [[Ore localization]], [[Gabriel localization]], [[Cohn localization]]

* [[localization of a space]] (and of a [[spectrum]])

* [[localization of a category]] (= localization functor)

* [[locally free module]]

## Literature

Standard discussion over [[commutative rings]] is for instance in

* {#Gathmann} Andreas Gathmann, _Localization_ ([[GathmannLocalization.pdf:file]])

Discussion in the general case of [[noncommutative geometry]] is in

* [[Z. ?koda]], _Noncommutative localization in noncommutative geometry_,  London Math. Society Lecture Note Series 330 ([pdf](http://www.maths.ed.ac.uk/~aar/books/nlat.pdf)), ed.  A. Ranicki; pp. 220--313, [math.QA/0403276](http://arxiv.org/abs/math.QA/0403276)

Discussion in the context of [[spectra]] originates in

* {#BousfieldKan72} [[Aldridge Bousfield]], [[Daniel Kan]], _[[Homotopy limits, completions and localizations]]_, Lecture Notes in Mathematics, Vol 304, Springer 1972

* {#Bousfield79} [[Aldridge Bousfield]], _The localization of spectra with respect to homology_ , Topology vol 18 (1979) ([pdf](http://www.uio.no/studier/emner/matnat/math/MAT9580/v12/undervisningsmateriale/bousfield-topology-1979.pdf))

* {#Sullivan05} [[Dennis Sullivan]], _Geometric topology: localization, periodicity and Galois symmetry_, volume 8 of K- Monographs in Mathematics. Springer, Dordrecht, 2005. The 1970 MIT notes, Edited and with a preface
by [[Andrew Ranicki]] ([pdf](http://www.maths.ed.ac.uk/~aar/books/gtop.pdf))

Discussion in the context of [[higher algebra]] is in

* {#LurieCompletions} [[Jacob Lurie]], section 4 of _[[Proper Morphisms, Completions, and the Grothendieck Existence Theorem]]_

[[!redirects localizations of a module]]
[[!redirects localizations of modules]]


localization of a monoid

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### Monoid theory
+-- {: .hide}
[[!include monoid theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The concept of [[localization]] of a monoid, the [[localization]] of a [[category]] with a single object. Could be generalized to localization of [[monoid objects]] in a [[cartesian monoidal category]] $C$. The localization of a ring is a localization of a monoid object in [[Ab]].

## Definition

### For commutative monoids

Let $(M,1,*)$ be a commutative [[monoid object]] in $Set$ and let $S$ be a commutative [[subobject|submonoid]] of $M$. Then the __localization__ of $M$ __away from__ $S$, $S^{-1}M$, is the set of equivalences on $M \times S$, $(m_1,s_1) \sim (m_2,s_2)$ such that there is an element $u:S$ where $m_1 \cdot s_2 \cdot u = m_2 \cdot s_1 \cdot u$.

Write $m s^{-1}$ for the [[equivalence class]] of $(m,s)$. On this set, the product is defined by

$$
  (m_1 s_1^{-1}) \cdot (m_2 s_2^{-1}) \coloneqq (m_1 \cdot m_2) (s_1 \cdot s_2)^{-1}
  \,.
$$

### For non-commutative monoids

Analogue of [[noncommutative localization]] for noncommutative monoid objects in $Set$ instead of $Ab$.

...

## Group completion

The [[group completion]] of a commutative monoid $M$ is the localization of $M$ away from $M$.

## Related concepts

* [[localization]]

* [[localization of a ring]]


localization of a ring

> This entry is meant to be about the general notion of localization of a possible noncommutative ring (a special case of *[[Cohn localization]]*). For the more restrictive but traditional notion for commutative rings see at *[[localization of a commutative ring]]*.

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Given a (possibly noncommutative) unital [[ring]] $R$ one may ask to universally force some [[subset]] $S$ of its elements to become multiplicatively [[invertible element|invertible]] in that there is a "localized" ring $S^{-1}R$ equipped with a [[universal construction|universal]] [[localization]] [[homomorphism]] $R\to S^{-1}R$ under which all elements in $S$ are mapped to [[units]].

This is a special case of and generalizes to the notion of *[[Cohn localization]]* where one may also force certain [[matrices]] with [[coefficients]] in the ring to become invertible.

Often one inverts elements in a left or right [[Ore subset]] $S\subset R$ in which case the localized ring is expressed by fractions as naively expected, in which case one speaks of *[[Ore localization]]*.

This [[Ore subset|Ore condition]] is automatic for [[commutative rings]] which leads to the notion of *[[localization of a commutative ring]]*.

Another special case is known as *[[Martindale localization]]*.

## Definition

Let $S$ be any [[subset]] of a [[ring]] $R$.

Say that a ring [[homomorphism]] $f \colon R \to A$ is _$S$-inverting_ if the [[image]] of $S$ under $f$ is contained in the [[units]] $A^\times \subset A$, i.e. if for every $s \in S$ there is a $t \in A$ so that $t \cdot f(s) = 1 = f(s) \cdot t$ in $A$.

Then the _localization of $R$ with respect to $S$_ is a ring homomorphism $h \colon R \to R_S$ which is [[initial object|initial]] with respect to such $S$-inverting ring homomorphisms.

By this defining [[universal property]] the localization is unique up to [[isomorphism]], when it exists.  Its existence is a special case of [[universal localization]]/[[Cohn localization]], a general abstract construction.

If $S$ is an [[Ore set]], then the localization of $R$ with respect to $S$ has an explicit description in terms of fractions, see at *[[Ore localization]]*.

If $S$ is a [[submonoid]] of the [[center]] $Z(R)$ of the multiplicative monoid of $R$, then the localization of $R$ at $S$ follows the same definition as that of [[localization of a commutative ring]].

## Examples

* The localization of a ring at a multiplicative submonoid $S$ which contains $0$ is the [[trivial ring]].

## References

See the references at *[[Cohn localization]]*, going back to

* [[Paul M. Cohn]], _Free rings and their relations_, Academic Press (1971) [[pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/cohnfree.pdf)]

* [[Paul M. Cohn]], _Inversive localization in noetherian rings_, Communications on Pure and Applied Mathematics __26__ 5-6 (1973 ) 679-691 [[doi:10.1002/cpa.3160260510](http://dx.doi.org/10.1002/cpa.3160260510)]

with more discussion in:

* [[V. Retakh]], R. Wilson, _Advanced course on quasideterminants and universal localization_ (2007) [[[RetakhWilson-Quasideterminants.pdf:file]]]

*  [[Andrew Ranicki]] (ed.),  _Noncommutative localization in algebra and topology_, (Proceedings of Conference at ICMS, Edinburgh, 29-30 April 2002) London Math. Soc. Lecture Notes Series **330** Cambridge University Press (2006) [[pdf](http://www.maths.ed.ac.uk/~aar/books/nlat.pdf)]

and see also at *[[noncommutative localization]]*.

For references on the [[localization of commutative rings]] see [there](localization+of+a+commutative+ring#References).

See also:

* [[Zoran Škoda]], _Noncommutative localization in noncommutative geometry_,  London Math. Society Lecture Note Series 330 ([pdf](http://www.maths.ed.ac.uk/~aar/books/nlat.pdf)), ed.  A. Ranicki; pp. 220--313, [math.QA/0403276](http://arxiv.org/abs/math.QA/0403276).

[[!redirects localizations of a ring]]

[[!redirects localization of rings]]
[[!redirects localizations of rings]]

[[!redirects localisation of a ring]]
[[!redirects localisations of a ring]]
[[!redirects localisation of rings]]
[[!redirects localisations of rings]]

[[!redirects left localization of a ring]]
[[!redirects left localizations of a ring]]
[[!redirects left localization of rings]]
[[!redirects left localizations of rings]]

[[!redirects left localisation of a ring]]
[[!redirects left localisations of a ring]]
[[!redirects left localisation of rings]]
[[!redirects left localisations of rings]]

[[!redirects right localization of a ring]]
[[!redirects right localizations of a ring]]
[[!redirects right localization of rings]]
[[!redirects right localizations of rings]]

[[!redirects right localisation of a ring]]
[[!redirects right localisations of a ring]]
[[!redirects right localisation of rings]]
[[!redirects right localisations of rings]]


localization of a simplicial model category > history

see *[[localization of a model category]]* or *[[Bousfield localization of model categories]]*


localization of a space

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Homotopy theory
+-- {: .hide}
[[!include homotopy - contents]]
=--
=--
=--

# Localization of a space
* table of contents
{: toc}

## Idea

The **localization of a space** (really: [[homotopy type]], [[∞-groupoid]]) or [[spectrum]] with respect to some [[prime numbers]] is a [[homotopy theory|homotopical]] analogue of the notion of [[localization of a commutative ring]] (or rather of [[localization of a module]]), specifically the ring $\mathbb{Z}$ of [[integers]], with respect to some [[prime numbers]].  In good cases, at least, this localization acts on the [[homotopy groups|homotopy]] and/or [[homology groups]] as algebraic localization.  (For spaces, there are multiple inequivalent notions of localization, although all agree on [[nilpotent spaces]].)

Localization in this sense is closely related to [[Bousfield localization]].  The localization of spectra is a [[Bousfield localization of spectra]], while one of the constructions of localization of spaces is a Bousfield localization of model categories.  The present notion of localization should not be confused with the [[completion of a space]], which is a different sort of Bousfield localization.

## Preliminaries

### Sets of primes

For all of this page,

* let $T$ be a [[set]] of [[prime numbers]],

* write $\neg T$ for the set of primes not in $T$.

+-- {: .num_defn}
###### Definition

Write $\mathbb{Z}_T$ for the [[commutative ring|ring]] of [[integers]] [[localization of a commutative ring|localized]] by inverting all primes in $\neg T$, i.e. the subring of $\mathbb{Q}$ whose denominators are products of primes in $\neg T$.

=--

+-- {: .num_example}
###### Example

The most important cases are:

* $T = \{p\}$ for a prime $p$.  In this case, $T$-localization will be **localization at $p$** or **$p$-localization**.

* $T = \neg \{p\}$, the set of all primes except $p$.  In this case, $T$-localization will be **localization away from $p$**.

* $T = \emptyset$.  In this case, $T$-localization will be **[[rationalization]]**.

In these cases:

*  $\mathbb{Z}_\emptyset = \mathbb{Q}$ is the [[rational numbers]];

* $\mathbb{Z}_{\neg\{p\}} = \mathbb{Z}[\frac{1}{p}]$;

* $\mathbb{Z}_{\{p\}} = \mathbb{Z}_{(p)}$ is the integers localized at the [[prime ideal]] $(p)$.

=--

+-- {: .num_remark}
###### Remark

The analogous theory of the [[completion of a space]] involves the [[cyclic groups]] $\mathbb{Z}/p\mathbb{Z}$ -- written $\mathbb{F}_p$ when regarded as a [[finite field]] -- and/or the [[p-adic integers]] $\mathbb{Z}_p$ instead of $\mathbb{Z}_T$ (e.g. [Lurie "Proper morphisms", section 4](#LurieProper)).

For the relation of that to completion see remark \ref{RelationToCompletion} below.

Hence beware the subtle but crucial difference in what a subscript means, depending on which symbol is being subscribed and whether there are parenthesis or not:

* $\mathbb{Z}_{(p)}$: inverting all primes except $p$;

* $\mathbb{F}_p$: quotient by $p$;

* $\mathbb{Z}_p$: [[formal completion]] at $p$.

=--

### Local groups

+-- {: .num_defn #TLocalGroup}
###### Definition

A [[group]] $G$ is said to be **$T$-local** if the $p^{th}$ power map $G\to G$ is a bijection for all $p\in \neg T$.

=--

+-- {: .num_prop #CharacterizationOfAbelianLocalGroups}
###### Proposition

If $G$ is [[abelian group|abelian]], then this map is a group [[homomorphism]] and is  generally written additively as multiplication by $p$.  In this case the following are equivalent:

* $G$ is $T$-local;

* $G$ admits a structure of $\mathbb{Z}_T$-[[module]] (necessarily unique);

* The [[tensor product]] $G\otimes \mathbb{Z}/p\mathbb{Z}$ with the [[cyclic group]] of order $p$ is equal to zero for all $p\in\neg T$.

=--

+-- {: .num_prop}
###### Remark

The second characterization in prop. \ref{CharacterizationOfAbelianLocalGroups} implies that $T$-local abelian groups are [[reflective subcategory|reflective]] in [[Ab]]: the reflection is the [[extension of scalars]] functor $(\mathbb{Z}_T \otimes -)$.  In fact, $T$-local nonabelian groups are also reflective in [[Grp]], but the construction is less pretty.

=--

## Definitions

### Localization of spectra

+-- {: .num_defn #TLocalSpectrum}
###### Definition

A [[spectrum]] $X$ is called **$T$-local** (or $\mathbb{Z}_T$-local, if there is potential for confusion) if its [[homotopy groups]] are $T$-local abelian groups, def. \ref{TLocalGroup}.

The **$T$-localization** of a spectrum is its [[reflective sub-(infinity,1)-category|reflection]] into $T$-local spectra.

=--

+-- {: .num_remark}
###### Remark

The $T$-localization, def. \ref{TLocalSpectrum}, may be constructed as the [[Bousfield localization of spectra]] with respect to the [[Moore spectrum]] $S(\mathbb{Z}_T)$ (e.g. [Bauer 11, Example 1.7](#Bauer11)).

It can also be constructed as a [[Bousfield localization of model categories]] where we invert the maps that induce an isomorphism on [[generalized homology]] with [[coefficients]] in $H \mathbb{Z}_T$; these are called **$\mathbb{Z}_T$-homology isomorphisms**. See at _[[homology localization]]_.

=--

### Localization of nilpotent spaces

The presence of the nonabelian group $\pi_1$ makes the theory of localization of unstable spaces more subtle than that of spectra.  If spaces are [[simply connected]], of course, then this is not a problem.  More generally, it suffices to consider [[simple spaces]]: those where $\pi_1$ is abelian and acts trivially on the higher homotopy groups.  Even more generally, it suffices to consider [[nilpotent spaces]], whose definition is more complicated; the reader is encouraged to think about simple or even simply connected spaces.

For a nilpotent space $Z$, the following conditions are equivalent.  When they hold, we say that $Z$ is **$T$-local**.  (See [May-Ponto, Theorem 6.1.1](#MayPonto).)

1. Whenever $X\to Y$ induces an isomorphism on homology with $\mathbb{Z}_T$-coefficients, the induced map $[Y,Z] \to [X,Z]$ is a bijection.

2. Each [[homotopy group]] $\pi_n(Z)$ is a $T$-local group.

3. Each [[homology group]] $H_n(Z,\mathbb{Z})$ is a $T$-local group.

For a general nilpotent space $X$, the following properties of a map $\phi:X\to Y$, with $Y$ nilpotent and $T$-local, are equivalent.  Such a map exists and is unique up to homotopy, and we call it the **$T$-localization** of $X$ at $T$.  (See [May-Ponto, Theorem 6.1.2](#MayPonto).)

1. The induced map $[Y,Z] \to [X,Z]$ is a bijection for all $T$-local nilpotent spaces $Z$.

2. $\phi$ induces an isomorphism on homology with $\mathbb{Z}_T$-coefficients.

3. The induced map $\pi_n(X) \to \pi_n(Y)$ is an algebraic $T$-localization for all $n$.

4. The induced map $H_n(X,\mathbb{Z}) \to H_n(Y,\mathbb{Z})$ is an algebraic $T$-localization for all $n$.

### Localization of non-nilpotent spaces

For non-nilpotent spaces, there are multiple inequivalent definitions of localization (see [May-Ponto, Remark 19.3.11](#MayPonto)).  For instance:

* One can do a [[Bousfield localization of model categories]] with respect to the class of $\mathbb{Z}_T$-homology isomorphisms.  In this case, the "$T$-local spaces" are those $Z$ such that $[Y,Z] \to [X,Z]$ is an isomorphism for all $\mathbb{Z}_T$-homology isomorphisms $X\to Y$ (the first characterization of $T$-local nilpotent spaces above).

* One can construct a [[totalization]] of the [[cosimplicial object]] induced by the "free simplicial $\mathbb{Z}_T$-module" monad.  This construction is due to [Bousfield-Kan](#BousfieldKan72).

* One can do a [[Bousfield localization of model categories]] with respect to the maps $p:S^1\to S^1$ for all $p\in \neg T$.  This construction is due to [Casacuberta-Peschke](#CasacubertaPeschke).  In this case, the "$T$-local spaces" are those such that $\pi_1(X)$ is a $T$-local group and acts "$T$-[[p-local module|locally]]" on each $\pi_n(X)$.

One can of course state the other two characterizations of "$T$-local space" from the nilpotent case even in the non-nilpotent case, as is done by [Sullivan](#Sullivan70), but these characterizations are no longer equivalent to the first one, and it is not clear whether corresponding "localizations" exist.

## Properties

### Relation to formal completion
 {#RelationToCompletion}

+-- {: .num_remark #RelationToCompletion}
###### Remark
**(relation to completion)**

A $\neg\{p\}$-local spectrum is also called **$\mathbb{Z}/p\mathbb{Z}$-acyclic**.  According to the general theory of [[Bousfield localization of spectra]], they are "dual" to the "$\mathbb{Z}/p\mathbb{Z}$-local spectra", in the sense that $X$ is $\mathbb{Z}/p\mathbb{Z}$-local if every map $Y \to X$ out of a $\mathbb{Z}/p\mathbb{Z}$-acyclic $Y$ is [[null homotopy|null homotopic]].

$\mathbb{Z}/p\mathbb{Z}$-local spectra are also known as *$p$-complete* spectra, and are the [[Bousfield localization of spectra]] at the [[Moore spectrum]] $S \mathbb{Z}/p\mathbb{Z}$ (e.g. [Bauer 11, Example 1.7](#Bauer11)).

This may be regarded as a consequence of the [[mod p Whitehead theorem]].

=--

See ([May-Ponto, example 19.2.3](#MayPonto), [Lurie, example 8](#Lurie),[Lurie "Proper morphisms", section 4](#LurieProper)).

In summary:

[[!include localization and completion -- table]]

+-- {: .num_remark}
###### Remark

In terms of [[arithmetic geometry]] this may be understood as follows:

1. $\mathbb{F}_p = \mathbb{Z}/(p \mathbb{Z})$ is the [[ring of functions]] exactly on the point $(p)\in $ [[Spec(Z)]]

1. $\mathbb{Z}_p$ is the functions on the [[formal neighbourhood]] of $(p)$.

1. $\mathbb{Z}_{(p)}$ is the ring of functions defined on the open complement of the point $(p)$, hence on an [[open neighbourhood]].

So the localization at the Moore spectrum of one of these rings localizes to the formal neighbourhood of their support. For $\mathbb{F}_p$ the support is the closed point, and so the localization there is infinitesimally bigger than that, as given by the $p$-adic completion. On the other hand the support of $\mathbb{Z}_{(p)}$ is open and hence contains its own formal neighbourhood, so localizing here just gives the plain localization.

=--

### Basic properties

* [[fracture theorem]]

### In chromatic homotopy theory

Many of the basic constructions and theorems in [[chromatic homotopy theory]] apply to [[finite spectrum|finite]] $p$-local spectra, such as

* [[telescopic localization]]

* [[periodicity theorem]]

* [[chromatic convergence theorem]]

## Examples

* [[Brown-Peterson spectrum]]

## Related concepts

* [[p-completion]]

* [[finite homotopy type]], [[finite spectrum]]

* [[nilpotent homotopy type]]

* [[rational homotopy type]]

## References

* {#Sullivan70} [[Dennis Sullivan]], _Localization, Periodicity and Galois Symmetry_ (The 1970 MIT notes) edited by [[Andrew Ranicki]], K-Monographs in Mathematics, Dordrecht: Springe ([pdf](http://www.maths.ed.ac.uk/~aar/surgery/gtop.pdf))

* {#BousfieldKan72} [[Aldridge Bousfield]], [[Daniel Kan]], _[[Homotopy limits, completions and localizations]]_, Lecture Notes in Mathematics, Vol 304, Springer 1972

* {#MayPonto} [[Peter May]], [[Kate Ponto]], _More concise algebraic topology: Localization, completion, and model categories_ ([pdf](http://www.maths.ed.ac.uk/~aar/papers/mayponto.pdf))

* {#CasacubertaPeschke} [[Carles Casacuberta]], [[Georg Peschke]], *Localizing with respect to self maps of the circle*, Trans. Amer. Math. Soc. **339** (1993) 117-140  [[doi:10.1090/S0002-9947-1993-1123451-X](https://doi.org/10.1090/S0002-9947-1993-1123451-X), [pdf](http://www.ub.edu/topologia/casacuberta/articles/cpes.pdf)]

* {#Lurie} [[Jacob Lurie]], _[[Chromatic Homotopy Theory]]_, Lecture series 2010, Lecture 20 _Bousfield localization_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture20.pdf))

* {#Bauer11} [[Tilman Bauer]], _Bousfield localization and the Hasse square_ ([pdf](http://math.mit.edu/conferences/talbot/2007/tmfproc/Chapter09/bauer.pdf))

* {#LurieProper} [[Jacob Lurie]], section 4 of _[[Proper Morphisms, Completions, and the Grothendieck Existence Theorem]]_

* [[Emily Riehl]], §14.4 in: *[[Categorical Homotopy Theory]]*, Cambridge University Press (2014) [[doi:10.1017/CBO9781107261457](https://doi.org/10.1017/CBO9781107261457), [pdf](http://www.math.jhu.edu/~eriehl/cathtpy.pdf)]

See also

* Wikipedia, _[Localization of a topological space](http://en.wikipedia.org/wiki/Localization_of_a_topological_space)_

and see at

* [[Bousfield localization of spectra]] for more.

Formalization in [[homotopy type theory]]:

* [[J. Daniel Christensen]], [[Morgan Opie]], [[Egbert Rijke]], [[Luis Scoccola]], _Localization in Homotopy Type Theory_, Higher Structures, 4(1) (2020), 1-32 ([arXiv:1807.04155](https://arxiv.org/abs/1807.04155))

[[!redirects p-localization]]
[[!redirects p-localizations]]
[[!redirects p-localisation]]
[[!redirects p-localisations]]

[[!redirects p-local]]
[[!redirects p-locally]]

[[!redirects p-local spectrum]]
[[!redirects p-local spectra]]

[[!redirects p-local space]]
[[!redirects p-local spaces]]

[[!redirects p-local homotopy type]]
[[!redirects p-local homotopy types]]

[[!redirects localization of spaces]]
[[!redirects localizations of spaces]]


localization of a type

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### Induction
+-- {: .hide}
[[!include induction - contents]]
=--
=--
=--

\tableofcontents

## Idea

On [[localization]] in [[homotopy type theory]].

## Definition

### Localization at a function

Consider a [[function]] $f:S \to T$. We say that a [[type]] $X$ is **$f$-local** if the function

$$\lambda g . g (f) : (T \to X) \to (S \to X)$$

is an [[equivalence of types]].

The localization of a type $X$ at $f$, $L_f(X)$, is the [[higher inductive type]] generated by

* the function $\mathrm{tolocal}:X \to L_f(X)$
* the function $\mathrm{localextend}: (S \to L_f(X)) \to (T \to L_f(X))$
* the dependent function
$$\mathrm{localextension}:\prod_{h:S \to L_f(X)} \prod_{s:S} \mathrm{localextend}(h)(f(s)) = h(s)$$
* the dependent function
$$\mathrm{localunextension}:\prod_{g:T \to L_f(X)} \prod_{t:T} \mathrm{localextend}(g \circ f)(t) = g(t)$$
* the dependent function
$$\mathrm{localtriangle}:\prod_{g:T \to L_f(X)} \prod_{s:S} \mathrm{localunextension}(g)(f(s)) = \mathrm{localextension}(g \circ f)(s)$$

### Localization at a type

The localization $L_S(X)$ of a type $X$ at a type $S$ is the localization $L_{u_S^\mathbb{1}}$ of $X$ at the unique function $u_S^\mathbb{1}:S \to \mathbb{1}$ from $S$ to the [[unit type]] $\mathbb{1}$, or equivalently, is the type $L_S(X)$ for which the canonical function $\mathrm{const}:L_S(X) \to (S \to L_S(X))$ is an [[equivalence of types]].

### Localization at a family of functions

Consider a [[family]] $f:\prod_{(i:I)} S(i) \to T(i)$ of functions. We say that a [[type]] $X$ is **$F$-local** if the function

$$\lambda g . g (f(i)) : (S(i) \to X) \to (T(i) \to X)$$

is an [[equivalence of types]] for all (i : I).

The following [[higher inductive type]] can be shown to be a [[reflection]] of all types into the local types, constructing the [[localization]] of the category of types at the given family of functions.

    Inductive localize X :=
    | to_local : X -> localize X
    | local_extend : forall (i:I) (h : S i -> localize X),
        T i -> localize X
    | local_extension : forall (i:I) (h : S i -> localize X) (s : S i),
        local_extend i h (f i s) == h s
    | local_unextension : forall (i:I) (g : T i -> localize X) (t : T i),
        local_extend i (g o f i) t == g t
    | local_triangle : forall (i:I) (g : T i -> localize X) (s : S i),
        local_unextension i g (f i s) == local_extension i (g o f i) s.

The first constructor gives a function from `X` to `localize X`, while the other four specify exactly that `localize X` is local (by giving adjoint equivalence data to the function that we want to become an equivalence).  See [this blog post](http://homotopytypetheory.org/2011/12/06/inductive-localization/) for details.  This construction is also already interesting in extensional type theory.

### Localization at a family of types

The localization $L_F(X)$ of a type $X$ at a family of types $F:\prod_{i:I} S(i)$ is the family of localizations $L_F(-) \coloneqq \prod_{i:I} L_{u_{S(i)}^\mathbb{1}}(-)$ of $X$ at the family of unique functions $u_{S(i)}^\mathbb{1}:S(i) \to \mathbb{1}$ from $S(i)$ to the [[unit type]] $\mathbb{1}$, or equivalently, is the family of types $L_{S(i)}(X)$ for which each canonical function $\mathrm{const}:L_{S(i)}(X) \to (S(i) \to L_{S(i)}(X))$ is an [[equivalence of types]].

## Examples

* The [[bracket type]] is the localization at the unique function $\mathbb{2} \to \mathbb{1}$ from the [[two-valued type]] to the [[unit type]].

* More generally, the [[n-truncation modality]] is the localization at the unique function $S^{n + 1} \to \mathbb{1}$ from the $(n + 1)$-dimensional [[sphere type]] to the [[unit type]].

* The [[shape modality]] which expresses [[real cohesion]] is the localization at the unique function $\mathbb{R} \to \mathbb{1}$ from the [[Dedekind real numbers]] to the [[unit type]].

* The $n$-[[shape modality]] is the localization at the two unique functions $\mathbb{R} \to \mathbb{1}$ and $S^{n + 1} \to \mathbb{1}$.

* Let $\mathcal{R}$ be an [[Archimedean ordered Artinian local ring]] whose Archimedean subfield $\mathbb{R} \subseteq \mathcal{R}$ is the [[Dedekind real numbers]], and let $\mathcal{D}$ be the type of all [[infinitesimals]] in $\mathcal{R}$. The [[infinitesimal shape modality]] is the localization at the unique function $\mathcal{D} \to \mathbb{1}$.

## See also

* [[higher inductive type]]

* [[localization]]

* [[shape modality]], [[axiom of cohesion]]

* [[bracket type]], [[set truncation]], [[n-truncation modality]]

* [[localization at an object]]

## References

* [[Univalent Foundations Project]], *[[HoTT book|Homotopy Type Theory – Univalent Foundations of Mathematics]]* (2013)

* [[Egbert Rijke]], [[Michael Shulman]], [[Bas Spitters]], *Modalities in homotopy type theory* ([arXiv:1706.07526](https://arxiv.org/abs/1706.07526))

* [[Daniel Christensen]], [[Morgan Opie]], [[Egbert Rijke]], [[Luis Scoccola]], *Localization in Homotopy Type Theory* ([arXiv:1807.04155](https://arxiv.org/abs/1807.04155))

* [[David Jaz Myers]], Orbifolds as microlinear types in synthetic differential cohesive homotopy type theory ([arXiv:2205.15887](https://arxiv.org/abs/2205.15887))

[[!redirects localization of a type at a function]]
[[!redirects localization of a type at a type]]
[[!redirects localization of a type at a family of functions]]


localization of abelian groups

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
#### Homological algebra
+--{: .hide}
[[!include homological algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The general concept of [[localization]] applied to the ([[derived category|derived]]) [[category of abelian groups]] yield the concept of _localization of abelian groups_.

The two main examples are

1. classical localization at/aways from primes;

1. completion at a prime

at [[prime numbers]] $p$. Here "classical $p$-localization" is localization at the morphism $0 \to \mathbb{Z}/p\mathbb{Z}$, while $p$-completion is localization at the morphism $0 \to \mathbb{Z}[p^{-1}]$.

## Definition

+-- {: .num_remark #ExtAb}
###### Remark

Recall that [[Ext]]-groups $\Ext^\bullet(A,B)$ between [[abelian groups]] $A, B \in $ [[Ab]] are concentrated in degrees 0 and 1 ([prop.](free+resolution#AbelianGroupHasFreeResolutionOfLength2)). Since

$$
  Ext^0(A,B) \simeq Hom(A,B)
$$

is the plain [[hom-functor]], this means that there is only one possibly non-vanishing Ext-group $Ext^1$, therefore often abbreviated to just "$Ext$":

$$
  Ext(A,B)
   \coloneqq
  Ext^1(A,B)
  \,.
$$

=--

+-- {: .num_defn #AbelianGroupLocal}
###### Definition

Let $K$ be an [[abelian group]].

Then an [[abelian group]] $A$ is called **$K$-local** if all the [[Ext]]-groups from $K$ to $A$ vanish:

$$
  Ext^\bullet(K,A) \simeq 0
$$

hence equivalently (remark \ref{ExtAb}) if

$$
  Hom(K,A) \simeq 0
  \;\;\;\;\;
    and
  \;\;\;\;\;
  Ext(K,A) \simeq 0
  \,.
$$

A [[homomorphism]] of abelian groups $f \colon B \longrightarrow C$ is called **$K$-local** if for all $K$-local groups $A$ the function

$$
  Hom(f,A) \;\colon\; Hom(B,A) \longrightarrow Hom(A,A)
$$

is a [[bijection]].

> (**Beware** that here it would seem more natural to use $Ext^\bullet$ instead of $Hom$, but we do use $Hom$. See ([Neisendorfer 08, remark 3.2](#Neisendorfer08)).

A homomorphism of abelian groups

$$
  \eta \;\colon\; A \longrightarrow L_K A
$$

is called a **$K$-localization** if

1. $L_K A$ is $K$-local;

1. $\eta$ is a $K$-local morphism.

=--

We now discuss two classes of examples of localization of abelian groups

1. _[Classical localization at/away from primes](#ClassicalLocalizationOfAbelianGroups)_;

1. _[Formal completion at primes](#FormalCompletionOfAbelianGroups)_.

###### Classical localization at/away from primes
 {#ClassicalLocalizationOfAbelianGroups}

For $n \in \mathbb{N}$, write $\mathbb{Z}/n\mathbb{Z}$ for the [[cyclic group]] of [[order of a group|order]] $n$.

+-- {: .num_lemma #OutOfCyclicGroupExt1}
###### Lemma

For $n \in \mathbb{N}$ and $A \in Ab$ any [[abelian group]], then

1. the [[hom-object|hom-group]] out of $\mathbb{Z}/n\mathbb{Z}$ into $A$ is the $n$-[[torsion subgroup]] of $A$

   $$
     Hom(\mathbb{Z}/n\mathbb{Z}, A)
       \simeq
     \{
       a \in A \;\vert\; p \cdot a = 0
     \}
   $$

1. the first [[Ext]]-group out of $\mathbb{Z}/n\mathbb{Z}$ into $A$ is

   $$
     Ext^1(\mathbb{Z}/n\mathbb{Z},A)
       \simeq
     A/n A
     \,.
   $$

=--

+-- {: .proof}
###### Proof

Regarding the first item: Since $\mathbb{Z}/p\mathbb{Z}$ is generated by its element 1 a morphism $\mathbb{Z}/p\mathbb{Z} \to A$ is fixed by the image $a$ of this element, and the only relation on 1 in $\mathbb{Z}/p\mathbb{Z}$ is that $p \cdot 1 = 0$.

Regarding the second item:

Consider the canonical [[free resolution]]

$$
  0 \to \mathbb{Z} \overset{p \cdot (-)}{\longrightarrow} \mathbb{Z} \longrightarrow \mathbb{Z}/p\mathbb{Z} \to 0
  \,.
$$

By the general discusson of [[derived functors in homological algebra]] this exhibits the [[Ext]]-group in degree 1 as part of the following [[short exact sequence]]

$$
  0
  \to
  Hom(\mathbb{Z},A)
  \overset{Hom(n\cdot(-),A)}{\longrightarrow}
  Hom(\mathbb{Z}, A)
  \longrightarrow
  Ext^1(\mathbb{Z}/n\mathbb{Z},A)
  \to 0
  \,,
$$

where the morphism on the left is equivalently $A \overset{n \cdot (-)}{\to} A$.

=--

+-- {: .num_example}
###### Example

An [[abelian group]] $A$ is $\mathbb{Z}/p\mathbb{Z}$-local precisely if the operation

$$
  p \cdot (-)
  \;\colon\;
  A \longrightarrow A
$$

of multiplication by $p$ is an [[isomorphism]], hence if "$p$ is invertible in $A$".

=--

+-- {: .proof}
###### Proof

By the first item of lemma \ref{OutOfCyclicGroupExt1} we have

$$
  Hom(\mathbb{Z}/p\mathbb{Z}, A)
    \simeq
  \{
    a \in A \;\vert\; p \cdot a = 0
  \}
$$

By the second item of lemma \ref{OutOfCyclicGroupExt1} we have

$$
  Ext^1(\mathbb{Z}/p\mathbb{Z},A)
    \simeq
  A/p A
  \,.
$$

Hence by def. \ref{AbelianGroupLocal} $A$ is $\mathbb{Z}/p\mathbb{Z}$-local precisely if

$$
  \{
    a \in A \;\vert\; p \cdot a = 0
  \}
  \simeq
  0
$$

and if

$$
  A / p A \simeq 0
  \,.
$$

Both these conditions are equivalent to multiplication by $p$ being invertible.

=--

+-- {: .num_defn #InvertingPrimes}
###### Definition

For $J \subset \mathbb{N}$ a set of [[prime numbers]], consider the [[direct sum]] $\underset{p \in J}{\oplus} \mathbb{Z}/p\mathbb{Z}$ of [[cyclic groups]] of [[order of a group|order]] $p$.

The operation of $\underset{p \in J}{\oplus} \mathbb{Z}/p\mathbb{Z}$-localization of abelian groups according to def. \ref{AbelianGroupLocal} is called **inverting the primes** in $J$.

Specifically

1. for $J = \{p\}$ a single prime then $\mathbb{Z}/p\mathbb{Z}$-localization is called **localization away from $p$**;

1. for $J$ the set of all primes except $p$ then $\underset{p \in J}{\oplus} \mathbb{Z}/p\mathbb{Z}$-localization is called **localization at $p$;

1. for $J$ the set of all primes, then $\underset{p \in J}{\oplus} \mathbb{Z}/p\mathbb{Z}$-localizaton is called **[[rationalization]]**..

=--

+-- {: .num_defn #LocalizationOfIntegersAtSetOfPrimes}
###### Definition

For $J \subset Primes \subset \mathbb{N}$ a [[set]] of [[prime numbers]], then

$$
  \mathbb{Z}[J^{-1}]
    \hookrightarrow
  \mathbb{Q}
$$

denotes the [[subgroup]] of the [[rational numbers]] on those elements which have an expression as a fraction of natural numbers with denominator a product of elements in $J$.

This is the abelian group underlying the [[localization of a commutative ring]] of the ring of integers at the set $J$ of primes.

If $J = Primes - \{p\}$ is the set of all primes _except_ $p$ one also writes

$$
  \mathbb{Z}_{(p)}
    \coloneqq
  \mathbb{Z}[Primes - \{p\}]
  \,.
$$

Notice the parenthesis, to distinguish from the notation $\mathbb{Z}_{p}$ for the [[p-adic integers]] (def. \ref{pAdicIntegers} below).

=--

+-- {: .num_remark #ClassicalLocalizationSeenFromSpecZ}
###### Remark

The terminology in def. \ref{InvertingPrimes} is motivated by the following perspective of [[arithmetic geometry]]:

Generally for $R$ a [[commutative ring]], then an $R$-[[module]] is equivalently a [[quasicoherent sheaf]] on the [[spectrum of a commutative ring|spectrum of the ring]] $Spec(R)$.

In the present case $R = \mathbb{Z}$ is the [[integers]] and [[abelian groups]] are identified with $\mathbb{Z}$-modules. Hence we may think of an abelian group $A$ equivalently as a [[quasicoherent sheaf]] on [[Spec(Z)]].

The "[[ring of functions]]" on [[Spec(Z)]] is the integers, and a point in $Spec(\mathbb{Z})$ is labeled by a [[prime number]] $p$, thought of as generating the ideal of functions on [[Spec(Z)]] which vanish at that point. The [[residue field]] at that point is $\mathbb{F}_p = \mathbb{Z}/p\mathbb{Z}$.

Inverting a prime means [[forcing]] $p$ to become invertible, which, by this characterization, it is precisely _away_ from that point which it labels. The localization of an abelian group at $\mathbb{Z}/p\mathbb{Z}$ hence corresponds to the restriction of the corresponding quasicoherent sheaf over $Spec(\mathbb{Z})$ to the complement of the point labeled by $p$.

Similarly localization _at_ $p$ is localization away from all points except $p$.

See also at _[[function field analogy]]_ for more on this.

=--

+-- {: .num_prop}
###### Proposition

For $J \subset \mathbb{N}$ a set of [[prime numbers]], a homomorphism of abelian groups $f \;\colon\; A \longrightarrow B $ is local (def. \ref{AbelianGroupLocal}) with respect to  $\underset{p \in J}{\oplus} \mathbb{Z}/p\mathbb{Z}$ (def. \ref{InvertingPrimes}) if under [[tensor product of abelian groups]] with $\mathbb{Z}[J^{-1}]$ (def. \ref{LocalizationOfIntegersAtSetOfPrimes}) it becomes an [[isomorphism]]

$$
  f \otimes \mathbb{Z}[J^{-1}]
    \;\colon\;
  A \otimes \mathbb{Z}[J^{-1}]
    \overset{\simeq}{\longrightarrow}
  B \otimes \mathbb{Z}[J^{-1}]
  \,.
$$

Moreover, for $A$ any abelian group then its $\underset{p \in J}{\oplus} \mathbb{Z}/p\mathbb{Z}$-localization exists and is given by the canonical projection morphism

$$
  A
    \longrightarrow
  A \otimes \mathbb{Z}[J^{-1}]
  \,.
$$

=--

(e.g. [Neisendorfer 08, theorem 4.2](#Neisendorfer08))

###### Formal completion at primes
 {#FormalCompletionOfAbelianGroups}

We have seen above in remark \ref{ClassicalLocalizationSeenFromSpecZ} that classical localization of abelian groups at a prime number is geometrically interpreted as restricting a [[quasicoherent sheaf]] over [[Spec(Z)]] to a single point, the point that is labeled by that prime number.

Alternatively one may restrict to the "infinitesimal neighbourhood" of such a point. Technically this is called the _[[formal neighbourhood]]_, because its ring of functions is, by definition, the ring of [[formal power series]] (regarded as an [[adic ring]] or [[pro-ring]]). The corresponding operation on abelian groups is accordingly called _[[formal completion]]_ or _[[adic completion]]_ or just _completion_, for short, of abelian groups.

It turns out that if the abelian group is [[finitely generated object|finitely generated]] then the operation of [[p-completion]] coincides with an operation of _localization_ in the sense of def. \ref{AbelianGroupLocal}, namely localization at the [[p-primary group|p-primary component]] $\mathbb{Z}(p^\infty)$ of the group $\mathbb{Q}/\mathbb{Z}$ (def. \ref{ZpInfinity} below). On the one hand this equivalence is useful for deducing some key properties of [[p-completion]], this we discuss below. On the other hand this situation is a shadow of the relation between [[localization of spectra]] and [[nilpotent completion of spectra]], which is important for characterizing the convergence properties of [[Adams spectral sequences]].

+-- {: .num_defn #AdicCompletionOfAbelingGroups}
###### Definition

For $p$ a [[prime number]], then the **[[p-adic completion]]** of an [[abelian group]] $A$ is the abelian group given by the [[limit]]

$$
  A^\wedge_p
   \coloneqq
  \underset{\longleftarrow}{\lim}
  \left(
    \cdots
      \longrightarrow
    A / p^3 A
      \longrightarrow
    A / p^2 A
      \longrightarrow
    A/p A
  \right)
  \,,
$$

where the morphisms are the evident [[quotient]] morphisms. With these understood we often write

$$
  A^\wedge_p \coloneqq \underset{\longleftarrow}{\lim}_n A/p^n A
$$

for short. Notice that here the indexing starts at $n = 1$.

=--

+-- {: .num_example #pAdicIntegers}
###### Example

The [[p-adic completion]] (def. \ref{AdicCompletionOfAbelingGroups}) of the [[integers]] $\mathbb{Z}$ is called the **[[p-adic integers]]**, often written

$$
  \mathbb{Z}_p
    \coloneqq
  \mathbb{Z}^\wedge_p
    \coloneqq
  \underset{\longleftarrow}{\lim}_n \mathbb{Z}/p^n \mathbb{Z}
  \,,
$$

which is the original example that gives the general concept its name.

With respect to the canonical [[ring]]-structure on the integers, of course $p \mathbb{Z}$ is a prime ideal.

Compare this to the ring $\mathcal{O}_{\mathbb{C}}$ of [[holomorphic functions]] on the [[complex plane]]. For $x \in \mathbb{C}$ any point, it contains the prime ideal generated by $(z-x)$ (for $z$ the canonical [[coordinate]] function on $\mathbb{z}$). The [[formal power series ring]] $\mathbb{C}[ [(z.x)] ]$ is the [[adic completion]] of $\mathcal{O}_{\mathbb{C}}$ at this ideal. It has the interpretation of functions defined on a [[formal neighbourhood]] of $X$ in $\mathbb{C}$.

Analogously, the [[p-adic integers]] $\mathbb{Z}_p$ may be thought of as the functions defined on a [[formal neighbourhood]] of the point labeled by $p$ in [[Spec(Z)]].

=--

+-- {: .num_lemma #pAdicIntegersAspExtensionofFpByThemselves}
###### Lemma

There is a [[short exact sequence]]

$$
  0
    \to
  \mathbb{Z}_p
    \overset{p \cdot (-)}{\longrightarrow}
  \mathbb{Z}_p
    \longrightarrow
  \mathbb{Z}/p\mathbb{Z}
    \to
  0
  \,.
$$

=--

+-- {: .proof}
###### Proof

Consider the following [[commuting diagram]]

$$
  \array{
     \vdots && \vdots && \vdots
     \\
     \downarrow
       &&
     \downarrow
       &&
     \downarrow
     \\
     \mathbb{Z}/p^3\mathbb{Z}
       &\overset{p\cdot (-)}{\longrightarrow}&
     \mathbb{Z}/p^4 \mathbb{Z}
       &\longrightarrow&
     \mathbb{Z}/p\mathbb{Z}
     \\
     \downarrow
       &&
     \downarrow
       &&
     \downarrow
     \\
     \mathbb{Z}/p^2\mathbb{Z}
       &\overset{p\cdot (-)}{\longrightarrow}&
     \mathbb{Z}/p^3 \mathbb{Z}
       &\longrightarrow&
     \mathbb{Z}/p\mathbb{Z}
     \\
     \downarrow
       &&
     \downarrow
       &&
     \downarrow
     \\
     \mathbb{Z}/p\mathbb{Z}
       &\overset{p\cdot (-)}{\longrightarrow}&
     \mathbb{Z}/p^2 \mathbb{Z}
       &\longrightarrow&
     \mathbb{Z}/p\mathbb{Z}
     \\
     \downarrow
       &&
     \downarrow
       &&
     \downarrow
     \\
     0
       &\longrightarrow&
     \mathbb{Z}/p\mathbb{Z}
       &\longrightarrow&
     \mathbb{Z}/p\mathbb{Z}
  }
  \,.
$$

Each horizontal sequence is exact. Taking the [[limit]] over the vertical sequences yields the sequence in question. Since limits commute over limits, the result follows.

=--

We now consider a concept of $p$-completion that is in general different from def. \ref{AdicCompletionOfAbelingGroups}, but turns out to coincide with it in [[finitely generated object|finitely generated]] abelian groups.

+-- {: .num_defn  #pInvertedInZ}
###### Definition

For $p$ a [[prime number]], write

$$
  \mathbb{Z}[1/p]
   \coloneqq
  \underset{\longrightarrow}{\lim}
  \left(
    \mathbb{Z}
      \overset{p \cdot (-)}{\longrightarrow}
    \mathbb{Z}
      \overset{p \cdot (-)}{\longrightarrow}
    \mathbb{Z}
      \overset{}{\longrightarrow}
    \cdots
  \right)
$$

for the [[colimit]] (in [[Ab]]) over iterative applications of multiplication by $p$ on the [[integers]].

This is the [[abelian group]] generated by formal expressions $\frac{1}{p^k}$ for $k \in \mathbb{N}$, subject to the relations

$$
  (p \cdot n) \frac{1}{p^{k+1}}
  =
  n \frac{1}{p^k}
  \,.
$$

Equivalently it is the abelian group underlying the [[localization of a ring|ring localization]] $\mathbb{Z}[1/p]$.

=--

+-- {: .num_defn #AbelianGrouppComplete}
###### Definition

For $p$ a [[prime number]], then localization of abelian groups (def. \ref{AbelianGroupLocal}) at $\mathbb{Z}[1/p]$ (def. \ref{pInvertedInZ}) is called **$p$-completion of abelian groups**.

=--

+-- {: .num_lemma #CharacterizingpCompleteAbelianGroupsBypSequence}
###### Lemma

An [[abelian group]] $A$ is $p$-complete according to def. \ref{AbelianGrouppComplete} precisely if both the [[limit]] as well as the [[lim^1]] of the sequence

$$
  \cdots \overset{}{\longrightarrow}
  A \overset{p}{\longrightarrow}
  A \overset{p}{\longrightarrow}
  A \overset{p}{\longrightarrow} A
$$

vanishes:

$$
  \underset{\longleftarrow}{\lim}
  \left(
      \cdots \overset{}{\longrightarrow}
  A \overset{p}{\longrightarrow}
  A \overset{p}{\longrightarrow}
  A \overset{p}{\longrightarrow} A
  \right)
   \simeq
  0
$$

and

$$
  \underset{\longleftarrow}{\lim}^1
  \left(
      \cdots \overset{}{\longrightarrow}
  A \overset{p}{\longrightarrow}
  A \overset{p}{\longrightarrow}
  A \overset{p}{\longrightarrow} A
  \right)
   \simeq
  0
  \,.
$$

=--

+-- {: .proof}
###### Proof

By def. \ref{AbelianGroupLocal} the group $A$ is $\mathbb{Z}[1/p]$-local precisely if

$$
  Hom(\mathbb{Z}[1/p], A) \simeq 0
  \;\;\;\;\;\;\;
  and
  \;\;\;\;\;\;\;
  Ext^1(\mathbb{Z}[1/p], A) \simeq 0
  \,.
$$

Now use the colimit definition $\mathbb{Z}[1/p] = \underset{\longrightarrow}{\lim}_n \mathbb{Z}$ (def. \ref{pInvertedInZ}) and the fact that the [[hom-functor]] sends colimits in the first argument to limits to deduce that

$$
  \begin{aligned}
    Hom(\mathbb{Z}[1/p], A)
    & =
    Hom( \underset{\longrightarrow}{\lim}_n \mathbb{Z}, A  )
    \\
    & \simeq
    \underset{\longleftarrow}{\lim}_n Hom(\mathbb{Z},A)
    \\
    & \simeq
    \underset{\longleftarrow}{\lim}_n A
  \end{aligned}
  \,.
$$

This yields the first statement. For the second, use that for every [[cotower]] over abelian groups $B_\bullet$ there is a [[short exact sequence]] of the form

$$
  0
    \to
  \underset{\longleftarrow}{\lim}^1_n Hom(B_n, A)
    \longrightarrow
  Ext^1( \underset{\longrightarrow}{\lim}_n B_n, A )
    \longrightarrow
  \underset{\longleftarrow}{\lim}_n Ext^1( B_n, A)
    \to
  0
$$

(by [this lemma](Introduction+to+Stable+homotopy+theory+--+S#lim1AndExt1)).

In the present case all $B_n \simeq \mathbb{Z}$, which is a [[free abelian group]], hence a [[projective object]], so that all the [[Ext]]-groups out of it vannish. Hence by exactness there is an isomorphism

$$
  Ext^1( \underset{\longrightarrow}{\lim}_n \mathbb{Z}, A )
   \simeq
  \underset{\longleftarrow}{\lim}^1_n Hom(\mathbb{Z}, A)
   \simeq
  \underset{\longleftarrow}{\lim}^1_n A
  \,.
$$

This gives the second statement.

=--

+-- {: .num_example #pPrimaryGroupsArePComplete}
###### Example

For $p$ a [[prime number]], the [[p-primary group|p-primary]] [[cyclic groups]] of the form $\mathbb{Z}/p^n \mathbb{Z}$ are $p$-complete (def. \ref{AbelianGrouppComplete}).

=--

+-- {: .proof}
###### Proof

By lemma \ref{CharacterizingpCompleteAbelianGroupsBypSequence} we need to check that

$$
  \underset{\longleftarrow}{\lim}
  \left(
    \cdots
      \overset{p}{\longrightarrow}
    \mathbb{Z}/p^n \mathbb{Z}
      \overset{p}{\longrightarrow}
    \mathbb{Z}/p^n \mathbb{Z}
      \overset{p}{\longrightarrow}
    \mathbb{Z}/p^n \mathbb{Z}
  \right)
  \simeq
  0
$$

and

$$
  \underset{\longleftarrow}{\lim}^1
  \left(
    \cdots
      \overset{p}{\longrightarrow}
    \mathbb{Z}/p^n \mathbb{Z}
      \overset{p}{\longrightarrow}
    \mathbb{Z}/p^n \mathbb{Z}
      \overset{p}{\longrightarrow}
    \mathbb{Z}/p^n \mathbb{Z}
  \right)
  \simeq
  0
  \,.
$$

For the first statement observe that $n$ consecutive stages of the tower compose to the [[zero morphism]]. First of all this directly implies that the limit vanishes, secondly it means that the [[tower]] satisfies the [[Mittag-Leffler condition]] ([def.](Introduction+to+Stable+homotopy+theory+--+S#MittagLefflerCondition)) and this implies that the $\lim^1$ also vanishes ([prop.](Introduction+to+Stable+homotopy+theory+--+S#Lim1VanihesUnderMittagLeffler)).

=--

+-- {: .num_defn #ZpInfinity}
###### Definition

For $p$ a [[prime number]], write

$$
  \mathbb{Z}(p^\infty)
    \coloneqq
  \mathbb{Z}[1/p]/\mathbb{Z}
$$

(the [[p-primary group|p-primary]] part of $\mathbb{Q}/\mathbb{Z}$), where $\mathbb{Z}[1/p] = \underset{\longrightarrow}{\lim}(\mathbb{Z}\overset{p}{\to} \mathbb{Z} \overset{p}{\to} \mathbb{Z} \to \cdots )$ from def. \ref{pInvertedInZ}.

Since [[colimits]] commute over each other, this is equivalently

$$
  \mathbb{Z}(p^\infty)
    \simeq
  \underset{\longrightarrow}{\lim}
  ( 0 \hookrightarrow \mathbb{Z}/p\mathbb{Z} \hookrightarrow \mathbb{Z}/p^2 \mathbb{Z} \hookrightarrow \cdots )
  \,.
$$

=--

+-- {: .num_theorem #pCompletionOfAbelianGroupsByHomsOutOfZpinfinity}
###### Theorem

For $p$ a [[prime number]], the $\mathbb{Z}[1/p]$-localization

$$
  A \longrightarrow L_{\mathbb{Z}[1/p]} A
$$

of an abelian group $A$ (def. \ref{pInvertedInZ}, def. \ref{AbelianGroupLocal}), hence the $p$-completion of $A$ according to def. \ref{AbelianGrouppComplete}, is given by the morphism

$$
  A \longrightarrow Ext^1( \mathbb{Z}(p^\infty), A )
$$

into the first [[Ext]]-group into $A$ out of $\mathbb{Z}(p^\infty)$ (def. \ref{ZpInfinity}), which appears as the first [[connecting homomorphism]] $\delta$ in the [[long exact sequence]] of [[Ext]]-groups

$$
  0
    \to
  Hom(\mathbb{Z}(p^\infty),A)
    \longrightarrow
  Hom(\mathbb{Z}[1/p],A)
    \longrightarrow
  Hom(\mathbb{Z},A)
    \overset{\delta)}{\longrightarrow}
  Ext^1(\mathbb{Z}(p^\infty), A)
    \to
  \cdots
  \,.
$$

that is induced (via [this prop.](derived+functor+in+homological+algebra#LongExactSequenceOfRightDerivedFunctorsFromShortExactSequence)) from the defining [[short exact sequence]]

$$
  0
    \to
  \mathbb{Z}
    \longrightarrow
  \mathbb{Z}[1/p]
    \longrightarrow
  \mathbb{Z}(p^\infty)
    \to
  0
$$

(def. \ref{ZpInfinity}).

=--

e.g. ([Neisendorfer 08, p. 16](#Neisendorfer08))

+-- {: .num_prop}
###### Proposition

If $A$ is a [[finitely generated module|finitely generated]] [[abelian group]], then its $p$-completion (def. \ref{AbelianGrouppComplete}, for any [[prime number]] $p$) is equivalently its [[p-adic completion]] (def. \ref{AdicCompletionOfAbelingGroups})

$$
  \mathbb{Z}[1/p] A
   \simeq
  A^\wedge_p
  \,.
$$

=--

+-- {: .proof}
###### Proof

By theorem \ref{pCompletionOfAbelianGroupsByHomsOutOfZpinfinity} the $p$-completion is $Ext^1(\mathbb{Z}(p^\infty),A)$. By def. \ref{ZpInfinity} there is a [[colimit]]

$$
  \mathbb{Z}(p^\infty)
  =
  \underset{\longrightarrow}{\lim}
  \left(
    \mathbb{Z}/p\mathbb{Z}
      \to
    \mathbb{Z}/p^2 \mathbb{Z}
      \to
    \mathbb{Z}/p^3 \mathbb{Z}
      \to
    \cdots
  \right)
  \,.
$$

Together this implies, by [this lemma](Introduction+to+Stable+homotopy+theory+--+S#lim1AndExt1), that there is a [[short exact sequence]] of the form

$$
  0
    \to
  \underset{\longleftarrow}{\lim}^1 Hom(\mathbb{Z}/p^n \mathbb{Z},A)
    \longrightarrow
  X^\wedge_p
    \longrightarrow
  \underset{\longleftarrow}{\lim}_n Ext^1(\mathbb{Z}/p^n \mathbb{Z}, A)
    \to
  0
  \,.
$$

By lemma \ref{OutOfCyclicGroupExt1} the [[lim^1]] on the left is over the $p^n$-[[torsion subgroups]] of $A$, as $n$ ranges. By the assumption that $A$ is finitely generated, there is a [[maximum]] $n$ such that all torsion elements in $A$ are annihilated by $p^n$. This means that the [[Mittag-Leffler condition]] ([def.](Introduction+to+Stable+homotopy+theory+--+S#MittagLefflerCondition)) is satisfied by the [[tower]] of $p$-torsion subgroups, and hence the [[lim^1]]-term vanishes ([prop.](Introduction+to+Stable+homotopy+theory+--+S#Lim1VanihesUnderMittagLeffler)).

Therefore by exactness of the above sequence there is an [[isomorphism]]

$$
  \begin{aligned}
    L_{\mathbb{Z}[1/p]}X
      & \simeq
    \underset{\longleftarrow}{\lim}_n Ext^1(\mathbb{Z}/p^n \mathbb{Z}, A)
    \\
    & \simeq
    \underset{\longleftarrow}{\lim}_n A/p^n A
  \end{aligned}
  \,,
$$

where the second isomorphism is by lemma \ref{OutOfCyclicGroupExt1}.

=--

+-- {: .num_prop #pDivisibleGroupsHaveVanishingpCompletion}
###### Proposition

If $A$ is a $p$-[[divisible group]] in that $A \overset{p \cdot (-)}{\longrightarrow} A$ is an [[isomorphism]], then its $p$-completion (def. \ref{AbelianGrouppComplete}) vanishes.

=--

+-- {: .proof}
###### Proof

By theorem \ref{pCompletionOfAbelianGroupsByHomsOutOfZpinfinity} the localization morphism $\delta$ sits in a [[long exact sequence]] of the form

$$
  0
    \to
  Hom(\mathbb{Z}(p^\infty),A)
    \longrightarrow
  Hom(\mathbb{Z}[1/p],A)
    \overset{\phi}{\longrightarrow}
  Hom(\mathbb{Z},A)
    \overset{\delta}{\longrightarrow}
  Ext^1(\mathbb{Z}(p^\infty), A)
    \to
  \cdots
  \,.
$$

Here by def. \ref{pInvertedInZ} and since the [[hom-functor]] takes [[colimits]] in the first argument to [[limits]], the second term is equivalently the [[limit]]

$$
  Hom(\mathbb{Z}[1/p],A)
    \simeq
  \underset{\longleftarrow}{\lim}
  \left(
     \cdots
     \to
     A \overset{p \cdot (-)}{\longrightarrow} A \overset{p \cdot (-)}{\longrightarrow} A
  \right)
  \,.
$$

But by assumption all these morphisms $p \cdot (-)$ that the limit is over are [[isomorphisms]], so that the limit collapses to its first term, which means that the morphism $\phi$ in the above sequence is an [[isomorphism]]. But by exactness of the sequence this means that $\delta = 0$.

=--

+-- {: .num_cor}
###### Corollary

Let $p$ be a [[prime number]]. If $A$ is a [[finite abelian group]], then its $p$-completion (def. \ref{AbelianGrouppComplete}) is equivalently its [[p-primary group|p-primary part]].

=--

+-- {: .proof}
###### Proof

By the [[fundamental theorem of finite abelian groups]], $A$ is a finite [[direct sum]]

$$
  A
   \simeq
  \underset{i}{\oplus}
  \mathbb{Z}/p_i^{k_i}\mathbb{Z}
$$

of [[cyclic groups]] of [[order of a group|ordr]] $p_i^{k_1}$ for $p_i$ [[prime numbers]] and $k_i \in \mathbb{N}$ ([thm.](finite+abelian+group#FiniteAbelianGroupIsDirectSumOfCyclics)).

Since finite direct sums are equivalently [[products]] ([[biproducts]]: [[Ab]] is an [[additive category]]) this means that

$$
  Ext^1( \mathbb{Z}(p^\infty), A )
  \simeq
  \underset{i}{\prod}
  Ext^1( \mathbb{Z}(p^\infty), \mathbb{Z}/p_i^{k_1}\mathbb{Z} )
  \,.
$$

By theorem \ref{pCompletionOfAbelianGroupsByHomsOutOfZpinfinity} the $i$th factor here is the $p$-completion of $\mathbb{Z}/p_i^{k_i}\mathbb{Z}$, and since $p \cdot(-)$ is an isomorphism on $\mathbb{Z}/p_i^{k_i}\mathbb{Z}$ if $p_i \neq p$ (because its [[kernel]] evidently vanishes), prop. \ref{pDivisibleGroupsHaveVanishingpCompletion} says that in this case the factor vanishes, so that only the factors with $p_i = p$ remain. On these however $Ext^1(\mathbb{Z}(p^\infty),-)$ is the identity by example \ref{pPrimaryGroupsArePComplete}.

=--

## Related concepts

* [[localization of rings]]

* [[localization of a space]]

* [[Bousfield localization of spectra]]

## References

* {#Neisendorfer08} [[Joseph Neisendorfer]] _A Quick Trip through Localization_, in _Alpine perspectives on algebraic topology_, Third Ariolla Conference 2008 ([pdf](https://www.math.rochester.edu/people/faculty/jnei/localization.pdf))

* {#MayPonto} [[Peter May]], [[Kate Ponto]], section 10.1 of _More concise algebraic topology: Localization, completion, and model categories_ ([pdf](http://www.maths.ed.ac.uk/~aar/papers/mayponto.pdf))

[[!redirects localization of an abelian group]]

[[!redirects localizations of abelian groups]]


localization of an (infinity,1)-category

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

As for [[localization]] of ordinary [[categories]], there are slightly different notions of what a localization of an [[(∞,1)-category]] is.

One definition is in terms of [[reflective (∞,1)-subcategories]]:

A _[[localization]]_ , in this sense, of an [[(∞,1)-category]] $C$ is a functor $L : C \to C_0$ to an $(\infty,1)$-subcategory $C_0 \hookrightarrow C$ such that with $c$ any object there is a morphism connecting it to its localization

$$
  c \to L(c)
$$

in a suitable way. This "suitable way" just says that
$f$ is left adjoint to the fully faithful inclusion functor.

Since localizations are entirely determined by which morphisms in $C$ are sent to equivalences in $C_0$, they can be thought of as sending $C$ to the result of "inverting" all these morphisms, a process familiar from forming the [[homotopy category]] of a [[homotopical category]].

## Definition

+-- {: .un_defn}
###### Definition

An [[(∞,1)-functor]] $L : C \to C_0$ is called a **localization** of the [[(∞,1)-category]] $C$ if it has a right [[adjoint (∞,1)-functor]] $i : C_0 \hookrightarrow C$ that is [[full and faithful (∞,1)-functor|full and faithful]].

$$
  (L \dashv i) : C_0 \stackrel{\overset{L}{\leftarrow}}{\underset{i}{\hookrightarrow}}
  C
  \,.
$$

In other words: $L$ is a localization if it is the **reflector** of a [[reflective (∞,1)-subcategory]] $C_0 \hookrightarrow C$.

=--

This is [[Higher Topos Theory|HTT, def. 5.2.7.2]].

## Examples

* Localizations of $(\infty,1)$-categories are modeled by the notion of left [[Bousfield localization of model categories]].

  One precise statement is: localizations of [[(∞,1)-category of (∞,1)-presheaves]] $C = PSh_{(\infty,1)}(K)$ are [[presentable (∞,1)-category|presented]] by the left Bousfield localizations of the global projective [[model structure on simplicial presheaves]] on the [[simplicial category]] incarnation of $K$.

* [[∞-stackification]] (or [[(∞,1)-sheafification]]) is the localization of an [[(∞,1)-category]] of [[(∞,1)-presheaves]] to the $(\infty,1)$-subcategory [[(infinity,1)-category of (infinity,1)-sheaves|of (∞,1)-sheaves]].

* [[cohomology localization]]

* [[homotopy localization]]

## References

This is the topic of  of

* [[Jacob Lurie]], §5.2.7 & §5.5.4 of: *[[Higher Topos Theory]]* (2009)

* [[Jacob Lurie]], pp. 485 of: *[[Higher Algebra]]* (2017)

With an eye towards [[modal homotopy type theory]]:

* [[Marco Vergura]], *Localization Theory in an Infinity Topos*, 2019 ([pdf](https://ir.lib.uwo.ca/cgi/viewcontent.cgi?article=8583&context=etd), [ir.lib.uwo.ca:etd/6257](https://ir.lib.uwo.ca/etd/6257))

Via a [[calculus of fractions]] for [[quasi-categories]]:

* [[Daniel Carranza]], [[Chris Kapulkin]], [[Zachery Lindsey]], *Calculus of Fractions for Quasicategories* [[arXiv:2306.02218](https://arxiv.org/abs/2306.02218)]

[[!redirects localization of an (∞,1)-category]]
[[!redirects localisation of an (infinity,1)-category]]
[[!redirects localization of an (∞,1)-category]]
[[!redirects localizations of an (infinity,1)-category]]
[[!redirects localizations of an (∞,1)-category]]
[[!redirects localisations of an (infinity,1)-category]]
[[!redirects localizations of an (∞,1)-category]]
[[!redirects localizations of (infinity,1)-categories]]
[[!redirects localizations of (∞,1)-categories]]
[[!redirects localisations of (infinity,1)-categories]]
[[!redirects localizations of (∞,1)-categories]]
[[!redirects localization of (infinity,1)-categories]]
[[!redirects localization of (∞,1)-categories]]
[[!redirects localisation of (infinity,1)-categories]]
[[!redirects localization of (∞,1)-categories]]

[[!redirects localization of a quasi-category]]
[[!redirects localization of quasi-categories]]


localization of an abelian category

Abstract localization functors among abelian categories have several descriptions. Additional descriptions exist if in addition the category is [[Grothendieck category|Grothendieck]].

A nonempty subcategory of an abelian category is __thick__ (in the sense of [[Pierre Gabriel]]; called _dense_ in Popescu) if it is closed under subobjects, quotients and extensions (in particular it is full and abelian). Some authors say Serre subcategory for a thick subcategory, though a stronger version of the notion of Serre subcategory may be appropriate (and is occasionally so defined) if the Abelian category is not the full subcategory of modules over a ring or ringoid (when the two notions agree).

Following [[Jean-Pierre Serre]], given a thick subcategory $T$, define the __quotient category__ $A/T$ whose objects are the objects of $A$ and where the morphisms in $A/T$ are defined by
$$\mathrm{Hom}_{A/T}(X,Y) := \mathrm{colim}\, \mathrm{Hom}_A(X',Y/Y'),$$
where the colimit is over all $X',Y'$ in $A$ such that $Y'$ and $X/X'$ are in $T$. There is a canonical __quotient functor__ $Q: A\to A/T$ which is the identity on objects. The quotient category $A/T$ is abelian.

* [[Pierre Gabriel]], [[Des catégories abéliennes]], Bulletin de la Société Mathématique de France, 90 (1962), p. 323-448, [numdam](http://www.numdam.org/item?id=BSMF_1962__90__323_0)
* [[Francis Borceux]], Handbook of categorical algebra
* Chapter 15 in [[Carl Faith]], _Algebra: Rings, Modules, Categories_, vol. 1,  Grundlehren der mathematischen Wissenschaften **190**, Springer (1973) [[doi:10.1007/978-3-642-80634-6](https://doi.org/10.1007/978-3-642-80634-6)]
* N. Popescu, [[Abelian categories with applications to rings and modules]], London Mathematical Society Monographs 3, Academic Press 1973, xii + 470 pp

[[!redirects localization in abelian categories]]
[[!redirects localization in an abelian category]]


localization of an enriched category

Given a symmetric closed monoidal category $V$, a $V$-enriched category $A$ with underlying ordinary category $A_0$ and a subcategory $\Sigma$ of $A_0$
containing the identities of $A_0$, H. Wolff defines the corresponding theory of localizations.

* [[Harvey Wolff]], _$V$-localizations and $V$-triples_, Dissertation, University of Illinois-Urbana, 1970.
* H. Wolff, _$V$-localizations and $V$-monads_, J. Alg. __24__, 405-438, 1973, [MR310041](http://mathscinet.ams.org/mathscinet-getitem?mr=310041), doi; _V-localizations and $V$-monads. II_, Pacific J. Math. 63 (1976), no. 2, 579–589, [MR412253](https://mathscinet.ams.org/mathscinet-getitem?mr=412253), [euclid](http://projecteuclid.org/getRecord?id=euclid.pjm/1102867417);  _$V$-localizations and $V$-Kleisli algebras_, Manuscripta Math. __16__ (1975), no. 3, 203--228, [MR382383](https://mathscinet.ams.org/mathscinet-getitem?mr=382383), [doi](https://doi.org/10.1007/BF01164425)

While Wolff in principle defines localizations more generally, most of the theory is developed for reflective localizations, i.e. when the counit of the 2-adjunction is iso of $V$-categories. For such a $V$-enriched category $C$,

* [[F. Borceux]], C. Quinteiro, _A theory of enriched sheaves_, [[Cahiers]] de Topologie et Géométrie Différentielle Catégoriques, __37__ no. 2 (1996), p. 145-162, [MR1394507](http://www.ams.org/mathscinet-getitem?mr=1394507), [numdam](http://www.numdam.org/item?id=CTGDC_1996__37_2_145_0)

consider reflective $V$-localizations which preserve finite limits of the enriched category of presheaves $[C^{op},V]$, and relate them to an enriched version of Grothendieck topology on $C$, and to a "universal closure operation" on $[C^{op},V]$. See also under [[enriched sheaf]].

[[!redirects enriched localization]]


localization of model categories

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Model category theory
+--{: .hide}
[[!include model category theory - contents]]
=--
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

In the way that [[model categories]] serve as presentations for their [[homotopy category of a model category|homotopy categories]] or more generally of their [[simplicial localization|associated]] [[(infinity,1)-categories|$\infty$-categories]] so the general notion of *localization of a model category* [[Hirschhorn (2002), Def. 3.1.1](#Hirschhorn02)] serves to invert morphisms in the [[homotopy category]] such as to present [[localization of an (infinity,1)-category|localization of]] that $\infty$-category.

To quote [Hirschhorn (2002), p. xi](#Hirschhorn02):

> 'Localizing a model category with respect to a class of maps does not mean making the maps into isomorphisms; instead, it means making the images of those maps in the homotopy category into isomorphisms. Since the image of a map in the homotopy category is an isomorphism if and only if the map is a weak equivalence, localizing a model category with respect to a class of maps means making maps into weak equivalences.'

In practice, the most prominent notion of localization of model categories is *[[Bousfield localization of model categories]]* (see there for more), which is a special case, by [Hirschhorn (2002), Thm. 3.3.19](#Hirschhorn02).

## Examples

* [[localization of a simplicial model category]]

* [[Bousfield localization of model categories]]

## References

The main account on the general notion of localization of model categories is:

* {#Hirschhorn02} [[Philip Hirschhorn]], Section 3 of: _[[Model Categories and Their Localizations]]_, AMS Math. Survey and Monographs **99** (2002) [[ISBN:978-0-8218-4917-0](https://bookstore.ams.org/surv-99-s/), [pdf toc](http://www.gbv.de/dms/goettingen/360115845.pdf), [pdf](https://people.math.rochester.edu/faculty/doug/otherpapers/pshmain.pdf), [pdf](http://www.maths.ed.ac.uk/~aar/papers/hirschhornloc.pdf)]

> [[p. 47](#Hirschhorn02):] In Section 3.1 we define left and right localizations of model categories, and explain the connection between left localizations, local objects, and local equivalences [...] in Section 3.3 we discuss (left and right) Bousfield localizations of categories, a special case of left and right localizations (see Theorem 3.3.19).

For more references dedicated to the special case of [[Bousfield localization of model categories]] see [there](Bousfield+localization+of+model+categories#References).

[[!redirects localizations of model categories]]

[[!redirects localization of a model category]]
[[!redirects localizations of a model category]]