bound state

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## Examples

The hierarchy of bound states in [[quantum chromodynamics]]:

* if [[quarks]] themselves were bound states (which they are not according to the [[standard model of particle physics]]) then one would call them bound states of _[[preons]]_

* [[hadrons]] are bound states of [[quarks]]

  (and of [[gluons]], while [[glueballs]] are bound states purely of [[gluons]])

  (at [[temperatures]] that these bound states decay one speaks of a _[[quark gluon plasma]]_)

  * [[mesons]] are bound states specifically of two [[quarks]]

    * [[pions]], [[rho-mesons]] and [[omega-mesons]] are bound states of an [[up quark]] with a [[down quark]]

    * [[kaons]] are bound states of a [[strange quark]] with either an [[up quark]] or a [[down quark]]

    * [[B-mesons]] are bound states of a [[bottom quark]] with either an [[up quark]] or a [[down quark]] or a [[strange quark]] or a [[charm quark]]

  * [[baryons]] are bound states specifically of three [[quarks]], or rather of one [[quark]] and a [[diquark]]

    * [[nuclei]] are bound states of three among the [[up quarks]] and the [[down quarks]]

      * [[protons]] are bound states of 2 [[up quarks]] and 1 [[down quark]]

      * [[neutrons]] are bound states of 1 [[up quarks]] and 2 [[down quark]]

* [[atomic nuclei]] are bound states of of [[nuclei]] ([[protons]] and [[neutrons]])

* [[atoms]] are [[bound states]] of [[atomic nuclei]] and [[electrons]]

  (at temperature high enough that these bound states decay, one speaks of a _[[plasma]]_)

  * in particular [[hydrogen atoms]] are bound states of a [[proton]] and an [[electron]]

* [[molecules]] are bound states of [[atoms]]

Here the last two steps go beyond [[QCD]] as they also involve [[electrons]] from [[quantum electrodynamics]].

There are also bound states purely in [[QED]]:

* [[positronium]] is a bound state of an [[electron]] and a [[positron]]

Hypothetically:

* a [[composite Higgs boson]] would be a bound state of a strongly interacting sector beyond the current [[standard model of particle physics]]

\linebreak

[[!include brane bound states -- table]]

## Related concepts

* [[binding energy]]

* [[quantum state]]

## References

On bound states in [[QED]] and [[QCD]]:

* [[Paul Hoyer]], _Bound states -- from QED to QCD_, Lectures at "Mini-school on theoretical methods in particle physics" at the Higgs Centre for Theoretical Physics, University of Edinburgh on 30 September to 4 October 2013 ([arXiv:1402.5005](https://arxiv.org/abs/1402.5005))

* Changyong Liu, _On bound states in Quantum Field Theory_ ([arXiv:1811.06457](https://arxiv.org/abs/1811.06457))

* [[Paul Hoyer]], _Journey to the Bound States_ ([arXiv:2101.06721](https://arxiv.org/abs/2101.06721))

See also

* Wikipedia, _[Bound state](http://en.wikipedia.org/wiki/Bound_state)_

[[!redirects bound states]]


boundary

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## Idea

Loosely speaking, the **boundary** of a subset $S$ of topological space $X$ consists of those points in $X$ that are neither 'fully in' $S$ nor are 'fully not in' $S$.

## Definition

### Of a subset of a topological space

For $S \subset X$ a [[subset]] of a [[topological space]] $X$, the **boundary** or **frontier** $\partial S$ of $S$ is its [[closure]] $\bar S$ minus its [[interior]] $S^\circ$:

$$
  \partial S = \bar S \backslash S^\circ
$$

Letting $\neg$ denote set-theoretic complementation, $\partial S = \neg (S^\circ \cup (\neg S)^\circ)$. It is a closed set. If we consider $\partial$ restricted to closed sets as an operation on closed sets, then it becomes a special case of the boundary operator on a [[co-Heyting algebra]]; see there for further properties.

### Of a manifold

In a [[manifold with boundary]] of [[dimension]] $n$ the boundary is the collection of points which do not have a [[neighborhood]] diffeomorphic to an [[open n-ball]], but do have a neighborhood homeomorphic to a half-ball, that is, an open ball intersected with closed half-space.

## Properties

### General

One reason behind the notation $\partial$ may be this (cf. [[co-Heyting boundary]]):

+-- {: .num_prop #Leibniz}
###### Proposition
Let $X, Y$ be topological spaces. Then for closed subsets $A \subseteq X$ and $B \subseteq Y$, the [[Leibniz rule]] $\partial (A \times B) = (\partial A \times B) \cup (A \times \partial B)$ holds.
=--

Notice the conclusion *must* fail if $A$, $B$ are *not* closed, since in this case $(\partial A \times B) \cup (A \times \partial B)$ is not closed (it doesn't include $\partial A \times \partial B$).

+-- {: .proof}
###### Proof
The interior operation preserves intersections, so $(A \times B)^\circ = ((A \times Y) \cap (X \times B))^\circ = (A^\circ \times Y) \cap (X \times B^\circ)$. Its complement is $(\neg A^\circ \times Y) \cup (X \times \neg B^\circ)$, whose intersection with $\widebar{A \times B} = A \times B$ is $(\partial A \times B) \cup (A \times \partial B)$.
=--

+-- {: .num_prop}
###### Proposition
If $A, B$ are [[connected space|connected]] open subsets of $X$ and $A \cap B$ is [[inhabited set|inhabited]], then $\partial A = \partial B$ implies $A = B$.
=--

+-- {: .proof}
###### Proof
Since $B$ is open, we have

$$B \subseteq \neg \partial B = \neg \partial A = A^\circ \cup (\neg A)^\circ,$$

where the right side is a disjoint union of open sets. $B$ is connected, so $B \subseteq A^\circ$ or $B \subseteq (\neg A)^\circ \subseteq \neg A$. The latter cannot occur since $A \cap B$ is inhabited. So $B \subseteq A^\circ \subseteq A$; by symmetry $A \subseteq B$.
=--

### Collar neighbourhood theorem

For [[topological manifolds]] and [[smooth manifolds]] [[manifold with boundary|with boudnary]], see: _[[collar neighbourhood theorem]]_.

## Some ramifications

The [Leibniz rule](#Leibniz) shows that the boundary operator is better behaved when restricted to the lattice of [[closed subspace|closed subsets]]. Since this lattice forms a [[co-Heyting algebra]], one is led to study algebraic operators axiomatizing properties of $\partial$ ([Zarycki 1927](#Zar27)) on these, the so called [[co-Heyting boundary]] operators.

Since the [[lattice of subtoposes]] of a given [[topos]] carries a co-Heyting algebra structure, it becomes possible to define (co-Heyting) boundaries of subtoposes and thereby even boundaries of the [[geometric theory|geometric theories]] that the subtoposes correspond to! Intuitively, such a boundary $\partial T$ of a theory $T$ consists of those geometric sequents that neither 'fully follow' from $T$ nor 'fully contradict' $T$.

The interior $Int(\mathcal{E}_j)$ of a subtopos $\mathcal{E}_j$ of a [[Grothendieck topos]] is defined in an exercise of [[SGA|SGA4]] as the largest [[open subtopos]] contained in $\mathcal{E}_j$. The boundary $\partial\mathcal{E}_j$ is then defined as the subtopos complementary to the (open) join of the [[dense subtopos|exterior subtopos]] $Ext(\mathcal{E}_j)$ and $Int(\mathcal{E}_j)$ in the [[lattice of subtoposes]].

The co-Heyting algebra perspective and the accompanying _mereo-logic of theories_ was proposed by [[William Lawvere]]. See the references at [[co-Heyting boundary]] for further pointers!

## Squaring things with algebraic topology

[[!include chains and cochains - table]]

## Related entries

* [[interior]]

* [[co-Heyting boundary]]

* [[mereology]]

* [[boundary of a simplex]]

* [[corner]]

## References

* [[M. Artin]], [[A. Grothendieck]], [[J. L. Verdier]], _Théorie des Topos et Cohomologie Etale des Schémas ([[SGA4]])_, LNM **269** Springer Heidelberg 1972. (exposé IV, exercise 9.4.8, pp.461-462) {#SGA4}

* {#Zar27} M. Zarycki, _Quelque notions fondamentales de l'Analysis Situs au point de vue de l'Algèbre de la Logique_ , Fund. Math. **IX** (1927) pp.3-15. ([pdf](http://matwbn.icm.edu.pl/ksiazki/fm/fm9/fm912.pdf))

[[!redirects boundary]]
[[!redirects boundaries]]

[[!redirects frontier]]
[[!redirects frontiers]]

[[!redirects topological boundary]]
[[!redirects topological boundaries]]


boundary (type theory) > history

< [[cubical type theory]]


boundary condition

## Related concepts

[[!include field theory with boundaries and defects - table]]

## References

* Wikipedia, _[Boundary condition](http://en.wikipedia.org/wiki/Boundary_condition)_

[[!redirects boundary conditions]]


boundary conformal field theory

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## Idea

The study of [[conformal field theory]] on [[manifolds with boundary]], hence conformal [[boundary field theory]], is known as _boundary conformal field theory_ (often abbreviated as BCFT).

For the special case of [[2d CFT]] understood as the [[quantum field theory]] on the [[worldsheet]] of [[strings]], boundary conformal field theory is the key ingredient of [[perturbative string theory]] with prescribed [[boundary conditions]] for the [[open strings]]. From the [[target spacetime]]-perspective these [[boundary conditions]] are interpreted as the [[D-branes]] that the [[open strings]] may "end on".

In this way 2d BCFT can serve to give an [[algebra|algebraic]] definition of [[D-branes]], independent of or complementary to their incarnation in the [[target space|target]]-[[spacetime]] [[geometry]]. In particular [[fractional D-branes]] may be described this way as boundary conditions for the [[2d CFTs]] describing [[open strings]] propagating in [[orbifolds]].

A key tool of BCFT is the _boundary state_ formalism, where one interprets [[boundary conditions]] for the [[open string]] as [[coherent states]] of the corresponding [[closed string]] [[2d CFT]]. The physical interpretation of this algebraic fact is that the [[D-brane]], being [[mass|massive]] (see also at _[[black brane]]_) emits and absorbs [[gravitons]], which, in [[string theory]], are excitations of the [[closed string]].

It has been argued that, therefore, for [[sigma-model]] [[2d CFT]]s the classification of [[equivalence classes]] of BCFT boundary states under [[renormalization group flow]] should reproduce the expected classification of [[D-branes]] by the ([[equivariant K-theory|equivariant]], [[twisted K-theory]]) [[topological K-theory]] of the [[target spacetime]] ([Moore 03, section 3](#Moore03)).

While this gives, in principle, a precise algebraic definition of [[D-brane charge]] from the [[worldsheet]]-perspective, in practice it is hard to follow the [[renormalization group flow]] of the boundary states.
The proposal has been checked in some special cases (e.g. [Braun & Schaefer-Nameki 05](#BraunSchaeferNameki05)). Other authors reported some discrepancies ([Quiroz-Stefanski 01](#QuirozStefanski01))

Specifically for [[fractional D-branes]] at $G$-[[orbifold]] [[singularities]] the relevant boundary states are [[sums]] of boundary states for each "$g$-twisted sector" of the [[closed string]] [[2d CFT]], consisting of those string configurations that are closed in the [[covering space]] up to the [[action]] of the element $g \in G$:

> graphics grabbed from [BCR 00](#BCR00)

Here the [[coefficient]] of the $g$-twisted sector in the total boundary state is proprotional to the value $\chi_V(g)$ of the [[character]] $\chi_V$ of some $G$-[[representation]] $V \in R_{\mathbb{C}}(G) = KU_G(\ast)$ (e.g. [Recknagel-Schomerus 13 (4.102)](#RecknagelSchomerus13)). This makes it plausible that the [[RG-flow]]-[[equivalence classes]] of the boundary states for [[fractional D-branes]] do coincide with the [[equivariant K-theory]] $KU_G(\ast) = R_{\mathbb{C}}(G)$ of the [[orbifold]] singularity (the [[representation ring]] of $G$).

> But maybe this has not yet been actually proven?

## Results

There are some mistakes in the literature. A clean account is in [Fuchs-Schweigert-Walcher 00](#FuchsSchweigertWalcher00), [Fuchs-Kaste-Lerche-Lutken-Schweigert 00](#FuchsKasteLercheLutkenSchweigert00)

## Computer scan of Gepner-model compactifications

Discussion of [[string phenomenology]] of [[intersecting D-brane models]] [[KK-compactification|KK-compactified]] with non-geometric [[fibers]] such that the would-be string [[sigma-models]] with these [[target spaces]] are in fact [[Gepner models]] (in the sense of _[Spectral Standard Model and String Compactifications](https://www.physicsforums.com/insights/spectral-standard-model-string-compactifications/)_) is in ([Dijkstra-Huiszoon-Schellekens 04a](#DijkstraHuiszoonSchellekens04a), [Dijkstra-Huiszoon-Schellekens 04b](#DijkstraHuiszoonSchellekens04b)):

>  A plot of [[standard model of particle physics|standard model]]-like [[coupling constants]] in a computer scan of [[Gepner model]]-[[KK-compactification]] of [[intersecting D-brane models]] according to [Dijkstra-Huiszoon-Schellekens 04b](#DijkstraHuiszoonSchellekens04b).

>  The blue dot indicates the couplings in $SU(5)$-[[GUT]] theory. The faint lines are NOT drawn by hand, but reflect increased density of Gepner models as seen by the computer scan.

## Related concepts

* [[boundary field theory]]

* [[fractional brane]], [[permutation brane]]

[[!include field theory with boundaries and defects - table]]

## References

### General

The boundary state formalism in BCFT is due to

* C. G. Callan, C. Lovelace, C. R. Nappi and S. A. Yost, _Adding holes and crosscaps to the superstring_, Nucl. Phys. B 293 (1987) 83 (doi:10.1016/0550-3213(87)90065-4)

* C. G.  Callan, C. Lovelace, C. R. Nappi and S. A. Yost, _Loop corrections to superstring equations of motion_, Nucl. Phys. B308 (1988) 221 ([pdf](https://www.researchgate.net/profile/Scott_Yost/publication/222468211_Loop_Corrections_to_Superstring_Equations_of_Motion/links/5a54d5eda6fdccd72f5b4583/Loop-Corrections-to-Superstring-Equations-of-Motion.pdf))

* N. Ishibashi, _Boundary and Crosscap States in Conformal Field Theories_,  Mod. Phys. Lett. A4(1989) 251 ([pdf](https://inis.iaea.org/collection/NCLCollectionStore/_Public/20/072/20072844.pdf#page=125))

* N. Ishibashi and T. Onogi, _Open string model building_,  Nucl. Phys. B318(1989) 239 (doi:10.1016/0550-3213(89)90054-0)

* [Dixon-Friedan-Martinec-Shenker 87](#DixonFriedanMartinecShenker87)

See also

* {#RecknagelSchomerus99} [[Andreas Recknagel]], [[Volker Schomerus]], _Boundary Deformation Theory and Moduli Spaces of D-Branes_, Nucl.Phys. B545 (1999) 233-282 ([arXiv:hep-th/9811237](https://arxiv.org/abs/hep-th/9811237))

Textbook accounts

* [[Ralph Blumenhagen]], [[Erik Plauschinn]], Chapter 6 of: *Introduction to Conformal Field Theory -- With Applications to String Theory*, Lecture Notes in Physics **779**, Springer (2009) [[doi:10.1007/978-3-642-00450-6](https://doi.org/10.1007/978-3-642-00450-6)]

* {#RecknagelSchomerus13} [[Andreas Recknagel]], [[Volker Schomerus]], _Boundary Conformal Field Theory and the Worldsheet Approach to D-branes_, Cambridge 2013 ([spire:1308990](http://inspirehep.net/record/1308990))

### Relation to K-theory

The suggestion that [[renormalization group flow]]-[[equivalence classes]] of boundary states in 2d BCFT should realize the expected classification of [[D-brane charge]] in ([[equivariant K-theory|equivariant]]) [[topological K-theory]] apparntly goes back to

* {#Moore03} [[Gregory Moore]], section 3 of _K-Theory from a physical perspective_ ([arXiv:hep-th/0304018](https://arxiv.org/abs/hep-th/0304018))

This has been checked in examples in

* {#BraunSchaeferNameki05} [[Volker Braun]], [[Sakura Schafer-Nameki]], _D-Brane Charges in Gepner Models_, J. Math. Phys. 47 (2006) 092304 ([arXiv:hep-th/0511100](https://arxiv.org/abs/hep-th/0511100))

* {#KrizZayasQuiroz08} [[Igor Kriz]], Leopoldo A. Pando Zayas, Norma Quiroz, _Comments on D-branes on Orbifolds and K-theory_, Int.J.Mod.Phys.A23:933-974, 2008 ([arXiv:hep-th/0703122](https://arxiv.org/abs/hep-th/0703122))

A mismatch was claimed in

* [Quiroz-Stefanski 01](#QuirozStefanski01)

### For rational CFT, WZW models and Gepner models

For [[rational CFT]], specifically [[Gepner models]] and [[WZW models]]

* [[Ilka Brunner]], [[Michael Douglas]], Albion Lawrence, Christian Romelsberger, _D-branes on the Quintic_, JHEP 0008 (2000) 015 ([arXiv:hep-th/9906200](https://arxiv.org/abs/hep-th/9906200))

* {#FuchsSchweigertWalcher00} [[Jürgen Fuchs]], [[Christoph Schweigert]], J. Walcher, _Projections in string theory and boundary states for Gepner models_, Nucl.Phys. B588 (2000) 110-148 ([arXiv:hep-th/0003298](https://arxiv.org/abs/hep-th/0003298))

  (with emphasis on [[GSO projections]])

* {#FuchsKasteLercheLutkenSchweigert00} [[Jürgen Fuchs]], P. Kaste, [[Wolfgang Lerche]], C. Lutken, [[Christoph Schweigert]], J. Walcher, _Boundary Fixed Points, Enhanced Gauge Symmetry and Singular Bundles on K3_, Nucl.Phys.B598:57-72, 2001 ([arXiv:hep-th/0007145](https://arxiv.org/abs/hep-th/0007145))

  (with emphasis on [[gauge enhancement]] on coincident D-branes)

* Sergei E. Parkhomenko, _Free Field Construction of D-Branes in Rational Models of CFT and Gepner Models_ ([arXiv:0802.3445](https://arxiv.org/abs/0802.3445))

* [[Giovanni Felder]], [[Jürg Fröhlich]], [[Jürgen Fuchs]], [[Christoph Schweigert]], _The geometry of WZW branes_ ([arXiv:hep-th/9909030](https://arxiv.org/abs/hep-th/9909030))

* [Braun & Schaefer-Nameki 05](#BraunSchaeferNameki05)

### On orbifolds

Discussion specifically concerning [[fractional D-branes]] on [[orbifolds]] includes

* {#DixonFriedanMartinecShenker87} [[Lance Dixon]], [[Daniel Friedan]], [[Emil Martinec]], [[Stephen Shenker]], _The conformal field theory of orbifolds_, Nucl. Phys. B 282 (1987) 13. (doi:10.1016/0550-3213(87)90676-6)

* {#DiaconescuGomis99} Diaconescu, [[Jaume Gomis]], _Fractional Branes and Boundary States in Orbifold Theories_, JHEP 0010 (2000) 001 ([arXiv:hep-th/9906242](https://arxiv.org/abs/hep-th/9906242))

* {#BCR00} M. Billo', B. Craps, F. Roose, _Orbifold boundary states from Cardy's condition_, JHEP 0101:038, 2001 ([arXiv:hep-th/0011060](https://arxiv.org/abs/hep-th/0011060))

* {#QuirozStefanski01} N. Quiroz, [[Bogdan Stefanski]] Jr, _Dirichlet Branes on Orientifolds_, Phys.Rev. D66 (2002) 026002 ([arXiv:hep-th/0110041](https://arxiv.org/abs/hep-th/0110041))

* {#DijkstraHuiszoonSchellekens04a} T.P.T. Dijkstra, L. R. Huiszoon, [[Bert Schellekens]], _Chiral Supersymmetric Standard Model Spectra from Orientifolds of Gepner Models_, Phys.Lett. B609 (2005) 408-417 ([arXiv:hep-th/0403196](https://arxiv.org/abs/hep-th/0403196))

* {#DijkstraHuiszoonSchellekens04b} T.P.T. Dijkstra, L. R. Huiszoon, [[Bert Schellekens]], _Supersymmetric Standard Model Spectra from RCFT orientifolds_, Nucl.Phys.B710:3-57,2005 ([arXiv:hep-th/0411129](https://arxiv.org/abs/hep-th/0411129))

* {#MMR06} Jaydeep Majumder, Subir Mukhopadhyay, Koushik Ray, _Fractional Branes in Non-compact Type IIA Orientifolds_, JHEP0611:008, 2006 ([arXiv:hep-th/0602135](https://arxiv.org/abs/hep-th/0602135))

[[!redirects boundary conformal field theories]]

[[!redirects boundary CFT]]
[[!redirects boundary CFTs]]

[[!redirects BCFT]]
[[!redirects BCFTs]]

[[!redirects boundary state]]
[[!redirects boundary states]]

[[!redirects twisted sector]]
[[!redirects twisted sectors]]


boundary field theory

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## Idea

In [[FQFT]]-language,
as a ([[prequantum field theory|pre]]-)[[quantum field theory|quantum]] [[field theory]] in [[dimension]] $n$ is given by a functor on an [[(infinity,n)-category of cobordisms]], so a field theory with boundaries/[[branes]] is given by such a  functor on an $(\infty,n)$-category of cobordisms with marked [[boundaries]]. The assignments to the boundary data itself defines a kind of field theory of one dimension lower, which is the corresponding boundary field theory.

If the [[field theory]] is thought of as a [[sigma-model]] that describes the [[quantum mechanics]] of a [[particle]] or [[string]] or [[membrane]], etc., then the boundary condition is also called a _[[brane]]_ for that [[sigma-model]].

## Definition

See at

* _[Cobordism theorem -- For cobordisms with singuarities (boundaries/branes and defects/domain walls)](cobordism+hypothesis#ForCobordismsWithSingularities)_

* _[[QFT with defects]]_.

## Examples

* [[boundary conformal field theory]]

## Related concepts

* [[holographic principle]]

  [[AdS-CFT correspondence]]

[[!include field theory with boundaries and defects - table]]

## References

* Wikipedia, _[Boundary conformal field theory](http://en.wikipedia.org/wiki/Boundary_conformal_field_theory)_

For more see the references at _[[defect field theory]]_.

Comments in the context of [[multisymplectic geometry]] are in

* [[José Zapata]], _Multisymplectic effective General Boundary Field Theory_ (2013) [[pdf](http://relativity.phys.lsu.edu/ilqgs/zapata090313.pdf), [[Zapata-MultisymplecticBoundary.pdf:file]]]

Axiomatization and examples of boundary field theories in [[local prequantum field theory]] is in

* _[[schreiber:Local prequantum field theory]]_

* {#FiorenzaValentino14} [[Domenico Fiorenza]], [[Alessandro Valentino]], _Boundary Conditions for Topological Quantum Field Theories, Anomalies and Projective Modular Functors_ ([arXiv:1409.5723](http://arxiv.org/abs/1409.5723))

Discussion of the [[BV-formalism|BV-]][[phase space]] and [[Poisson n-algebras|Poisson $\infty$-algebras]] of  [[BF theory]] and [[Einstein gravity]] on [[manifold with corners|corners]]:

* [[Alberto Cattaneo]], *Poisson Structures from Corners of Field Theories*, [talk at](CQTS#CattaneoNov2023) [[CQTS]] (22 Nov 2023) [slides: [[Cattaneo-PoissonStrucFromCorners.pdf:file]], video: [YT](https://youtu.be/_bGH09FmAxk)]

[[!redirects boundary field theories]]


boundary of a simplex

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##Idea##

In as far as the simplicial $n$-[[simplex]] $\Delta^n$ (a [[simplicial set]]) is a combinatorial model for the $n$-[[ball]], its _[[boundary]]_ $\partial \Delta^n$ is a combinatorial model for the $(n-1)$-[[sphere]].

##Definition##

The **boundary** $\partial \Delta^n$ of the simplicial $n$-[[simplex]] $\Delta^n$ is the [[simplicial set]] _generated_ from the simplicial set $\Delta^n$ minus its unique non-degenerate cell in dimension $n$.

This may equivalently be described to be degreewise the [[coequalizer]]

$$
  \coprod_{0\le i\lt j\le n}\Delta[n-2]\rightrightarrows\coprod_{0\le i\le n}\Delta[n-1]\to \partial \Delta[n]
$$

defined by the (induced [[coproduct]] maps of the) [[simplicial identities]] $d_i d_j=d_{j-1} d_i$.

Regarding $\Delta^n$ as the [[presheaf]] on the [[simplex category]] that is [[representable functor|represented]] by $[n] \in Obj(\Delta)$, then this means that $\partial \Delta^n$ is the simplicial set generated from $\Delta$ minus the [[identity]] morphism $Id_{[n]}$.

There is a canonical [[monomorphism]]

$$
  i_n : \partial \Delta^n \hookrightarrow \Delta^n
  \,,
$$

the _boundary inclusion_ .

The [[geometric realization]] of this is the inclusion of the $(n-1)$-[[sphere]] as the [[boundary]] of the $n$-[[disk]].

Simplicial boundary inclusions are one part of the [[cofibrantly generated model category|cofibrant generation]] of the classical [[model structure on simplicial sets]].

##Examples##

For low $n$ the boundaries of $n$-simplices look as follows (see also the illustrations at [[oriental]])

* $\partial \Delta^0 = \emptyset$;

* $\partial \Delta^1 = \partial\{0 \to 1\} = \{0, 1\} = \Delta^0 \sqcup \Delta^0$;

* $\partial \Delta^2 =
   \partial\left\{
      \array{
        && 1
        \\
        & \nearrow &\Downarrow& \searrow
        \\
        0 &&\to&& 2
      }
     \right\}
    =
   \left\{
      \array{
        && 1
        \\
        & \nearrow && \searrow
        \\
        0 &&\to&& 2
      }
     \right\}
  $

## Related concepts

* [[horn]], [[spine]]

[[!redirects boundaries of simplicies]]


boundary separation

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## Idea

Boundary separation is a modular reconstruction of the [[uniqueness of identity proofs]] in [[cubical type theory]]. It is a rule which implies UIP as a theorem.

## Definition

Recall that in [[cubical type theory]], there is an interval primitive $I$ with endpoints $0:I$ and $1:I$, as well as face formulas $\phi:F$ with rules which make $\phi$ behave like a formula in first-order logic ranging over the interval $I$.

Boundary separation is the following rule:

$$\frac{\Gamma \vdash A \;  \mathrm{type} \quad \Gamma \vdash r:I \quad \Gamma, \partial(r) \vdash a \equiv b:A}{\Gamma \vdash a \equiv b:A}$$

where $I$ is the interval primitive in [[cubical type theory]] and $\partial(r)$ is the boundary face formula for dimension variables:

$$\delta(r) \coloneqq r = 0 \vee r = 1$$

(The interval primitive $I$ has more points than $0$ and $1$, so it is not the case that the sequent $r:I \vdash r = 0 \vee r = 1 \;\mathrm{true}$ holds.)

There is also a typal version of boundary separation which refers to [[cubical path types]] rather than definitional equality, given by the following rule:

$$\frac{\Gamma \vdash A \;  \mathrm{type} \quad \Gamma \vdash a:A \quad \Gamma \vdash b:A \quad \Gamma \vdash r:I \quad \Gamma, \partial(r) \vdash p:a =_A b}{\Gamma \vdash p:a =_A b}$$

## Proof of UIP from boundary separation

We denote path types by $a =_A b$ and dependent path types by $a =_{i.A} b$.

Consider the following context:

$$\Gamma \coloneqq (\Delta, A \; \mathrm{type}, a:A, b:A, p:a =_{A} b, q: a =_A b)$$

...

## See also

* [[uniqueness of identity proofs]]

* [[axiom K (type theory)|axiom K]]

## References

* [[Jonathan Sterling]], [[Carlo Angiuli]], [[Daniel Gratzer]], _Cubical syntax for reflection-free extensional equality_. In Herman Geuvers, editor, _4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019)_, volume 131 of _Leibniz International Proceedings in Informatics (LIPIcs)_, pages 31:1-31:25. ([arXiv:1904.08562](https://arxiv.org/abs/1904.08562), [doi:10.4230/LIPIcs.FCSD.2019.31](https://doi.org/10.4230/LIPIcs.FCSD.2019.31))

* [[Jonathan Sterling]], [[Carlo Angiuli]], [[Daniel Gratzer]], _A Cubical Language for Bishop Sets_, Logical Methods in Computer Science, 18 (1), 2022. ([arXiv:2003.01491](https://arxiv.org/abs/2003.01491)).


bounded chain complex

+-- {: .rightHandSide}
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###Context###
#### Homological algebra
+--{: .hide}
[[!include homological algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

A [[chain complex]] $A_\bullet$ is called

* **bounded above** if there exists $n \in \mathbb{Z}$ such that $A_k = 0$ for all $k \gt n$;

* **bounded below** if there exists $n \in \mathbb{Z}$ such that $A_k = 0$ for all $k \lt n$;

* **bounded** if it is both bounded above and bounded below.

## Examples

* A [[connective chain complex]] is bounded from below.

## Related concepts

* [[category of chain complexes]]

  * [[chain complex]], [[connective chain complex]]

  * [[chain map]], [[quasi-isomorphism]]

  * [[chain homotopy]]

  * [[model structure on chain complexes]]

  * [[elliptic chain complex]]

* [[cochain complex]]

* [[filtered chain complex]]

* [[perfect chain complex]]

[[!redirects bounded chain complexes]]

[[!redirects bounded complex]]
[[!redirects bounded complexes]]

[[!redirects bounded-below chain complex]]
[[!redirects bounded-below chain complexes]]

[[!redirects bounded-below complex]]
[[!redirects bounded-below complexes]]

[[!redirects bounded-above complex]]
[[!redirects bounded-above complexes]]


bounded function

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### Context
#### Analysis
+-- {: .hide}
[[!include analysis - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A [[function]] to a [[metric space]] is _bounded_ if its [[image]] is a [[bounded subset]].

More generally there is a concept of bounded function between [[bornological sets]].

## Related concepts

* [[extreme value theorem]]

* [[rapidly decreasing function]]

* [[continuous function]], [[differentiable function]], [[smooth function]]

* [[integrable function]], [[square-integrable function]]

* [[bounded function]]

* [[compactly supported function]]

* [[measurable function]]

[[!redirects bounded functions]]

[[!redirects bounded map]]
[[!redirects bounded maps]]


bounded geometric morphism

+-- {: .rightHandSide}
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###Context###
#### Topos Theory
+--{: .hide}
[[!include topos theory - contents]]
=--
=--
=--

#Contents#
* table of contens
{:toc}

## Idea

Given a [[geometric morphism]] $f \colon \mathcal{E} \longrightarrow \mathcal{S}$, we may regard $\mathcal{E}$ as a *[[topos]] [[base topos|over]] $\mathcal{S}$* via $f$.  The geometric morphism $f$ being *bounded* is the "over $\mathcal{S}$" version of $\mathcal{E}$ being a [[Grothendieck topos]].

## Definition

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

A [[geometric morphism]] $f = (f^*\dashv f_*) \colon \mathcal{E} \longrightarrow \mathcal{S}$ between [[toposes]] is called **bounded**, if any of the following three equivalent condition holds.

* There exists an [[object]]  $B \in \mathcal{E}$ -- called a **bound** of $f$ -- such that every $A \in \mathcal{E}$
is a [[subquotient]] of an object of the form $(f^* I) \times B$ for some $I  \in \mathcal{S}$: this means that there exists a diagram

  $$
    \array{
      S &\xrightarrow{\; epi \;}& A
      \\
      {}^{\mathllap{mono}}
      \big\downarrow
      \\
      (f^* I) \times B
    }
    \,.
  $$

* {#TheGluingFibration} The [[gluing fibration]] $(\mathcal{E}/f^*)\to \mathcal{S}$ has a [fibered separating family](separating+family#fibered).

* There exists a $B\in\mathcal{E}$ such that for every $A\in\mathcal{E}$ the
composite

  $$
    f^*f_*(\tilde A^B)
      \times
    B
      \to
    \tilde A^B\times B\to \tilde A
   $$

   is an [[epimorphism]], where $\tilde A$ is the [[partial map classifier]] of $A$.

=--

+-- {: .proof}
###### Proof of the equivalence.

Lemma B3.1.6 in the [[Elephant]]

=--

If we regard $\mathcal{E}$ as a topos over $\mathcal{S}$ via $f$, then when $f$ is bounded we call $\mathcal{E}$ a **bounded $\mathcal{S}$-topos**.

## Properties

### As relative Grothendieck toposes

If $f \coloneqq \Gamma\colon \mathcal{E}\to Set$ is the [[global section]] geometric morphism of a topos (such a geometric morphism being unique if it exists), then it is bounded if and only if $\mathcal{E}$ is a [[Grothendieck topos]]. As such we can also call Grothendieck toposes "bounded [[Set]]-toposes".

More generally, bounded toposes over $\mathcal{S}$ are precisely the toposes of $\mathcal{S}$-valued [[internal sheaves]] on [[internal sites]] in $\mathcal{S}$ ([Johnstone, Section B3.3](#Johnstone)).

If $f \colon \mathcal{E}\to\mathcal{S}$ is bounded and $\mathcal{S}$ is a [[Grothendieck topos]], then $\mathcal{E}$ is a Grothendieck topos as well. This is a consequence of prop. \ref{StabilityUnderComposition}.

### Stability under composition

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

Bounded geometric morphisms are stable under [[composition]].

=--

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

Assume that $f : \mathcal{F} \to \mathcal{S}$ is bounded by $B\in\mathcal{F}$, and  $g:\mathcal{G}\to\mathcal{F}$ is bounded by $C\in\mathcal{G}$. Let $A\in\mathcal{G}$. Then there exist $J\in \mathcal{F}$ and $I\in\mathcal{S}$, and  subquotient spans
$g^*J\times C\leftarrow\bullet\rightarrow A$
and
$f^*I\times B\leftarrow\bullet\rightarrow J$.
By applying $g^*(-)\times C$ to the second subquotient and forming a pullback, we get the diagram
$$
\begin{matrix}
\bullet & \to & \bullet & \to A \\
\downarrow && \downarrow \\
\bullet & \rightarrow & g*J\times C \\
\downarrow \\
g^*f^* I\times g^* B \times C
\end{matrix}
$$
where the vertical arrows are monos and the horizontal ones are epis (using the fact that epis are stable under $g^*$, products, and pullbacks), from which we can see that $f g$ is bounded by $g^*B\times C$.

=--

Almost all geometric morphisms in practice are bounded, so that often when people work in the 2-category [[Topos]] of toposes and geometric morphisms, they mean that the geometric morphisms are bounded. See [[unbounded topos]] for the few examples of unbounded geometric morphisms.

## Related concepts

* [[base topos]]
* [[internal site]]
* [[internal sheaf]]
* [[unbounded topos]]

## References

definition B3.1.7 in

* {#Johnstone} [[Peter Johnstone]], _[[Sketches of an Elephant]]_

[[!redirects bounded geometric morphisms]]

[[!redirects bounded topos]]
[[!redirects bounded toposes]]
[[!redirects bounded topoi]]


bounded linear logic

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Foundations
+-- {: .hide}
[[!include foundations - contents]]
=--
#### $(0,1)$-Category theory
+-- {: .hide}
[[!include (0,1)-category theory - contents]]
=--
#### Monoidal categories
+-- {: .hide}
[[!include monoidal categories - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A kind of [[linear logic]] in which the [[exponential modality]] is indexed by, say, the [[natural numbers]] to indicate the availability of the number of copies of a premise.

## Related concepts

* [[graded modality]]

## References

* [[Jean-Yves Girard]], [[Andre Scedrov]], [[Philip J. Scott]], _Bounded linear logic: a modular approach to polynomial-time computability_, Theoretical Computer Science Volume 97, Issue 1, 20 April 1992, Pages 1-66 (doi:10.1016/0304-3975(92)90386-T)

* [[Ugo Dal Lago]], [[Martin Hofmann]], _Bounded Linear Logic, Revisited_, Logical Methods in Computer Science, Volume 6, Issue 4 (December 18, 2010) lmcs:1064 ([arXiv:0904.2675](https://arxiv.org/abs/0904.2675))

* _Bounded Linear Logic Workshop_, Fontainebleau, 2-4 December 2013, ([webpage](https://www.cs.bham.ac.uk/~drg/bll.html))

[[!redirects bounded linear logics]]


bounded operator

+-- {: .rightHandSide}
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### Context
#### Operator algebra
+-- {: .hide}
[[!include AQFT and operator algebra contents]]
=--
#### Functional analysis
+-- {: .hide}
[[!include functional analysis - contents]]
=--
#### AQFT
+-- {: .hide}
[[!include AQFT and operator algebra contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

In a (Hausdorff) [[topological vector space]] one can consider [[bounded sets]]: a set $S$ is bounded if it is absorbed by any open neighborhood $U$ of zero (i.e. a dilated multiple $\lambda U = \{\lambda x | x\in U\}$ contains $S$). This specializes to the usual definition of a bounded set in a [[normed vector space]]: a set is bounded if it is contained in a ball of some finite radius $r \gt 0$.

A [[linear operator]] $A: V_1\to V_2$ between topological vector spaces is **bounded** if it sends each bounded set in $V_1$ to a bounded set in $V_2$. For [[normed vector space|normed spaces]], this is equivalent to saying that it sends the unit ball to a bounded set. Between finite dimensional normed spaces, every linear operator is bounded. A linear operator between any two normed linear spaces is bounded iff it is [[continuous map|continuous]].

There is also a rich theory for [[unbounded operators]] on [[Hilbert spaces]].

The [[quotient]] of the bounded operators by the [[compact operators]] is called the _[[Calkin algebra]]_.

## Properties

Every bounded operator on a [[Hilbert space]] has a [[polar decomposition]].

The _[[Gelfand-Naimark theorem]]_ says that every [[C*-algebra]] is [[isomorphism|isomorphic]] to a $C^\ast$-algebra of [[bounded linear operators]] on a [[Hilbert space]].

## Examples

Important classes of bounded operators are the [[compact operators]], [[trace-class operator | trace-class operators]] and [[Hilbert-Schmidt operator | Hilbert-Schmidt operators]].

## Related concepts

* [[linear operator]]

* **bounded linear operator** / [[unbounded linear operator]]

* [[operator algebra]]

## References

* wikipedia: [bounded operator](http://en.wikipedia.org/wiki/Bounded_operator)

category: analysis
[[!redirects bounded operator]]
[[!redirects bounded operators]]

[[!redirects bounded linear operator]]
[[!redirects bounded linear operators]]
[[!redirects bounded linear function]]
[[!redirects bounded linear functions]]
[[!redirects bounded linear map]]
[[!redirects bounded linear maps]]

[[!redirects algebra of bounded operators]]
[[!redirects algebras of bounded operators]]

[[!redirects continuous operator]]
[[!redirects continuous operators]]


bounded set

+-- {: .rightHandSide}
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### Context
#### Analysis
+-- {: .hide}
[[!include analysis - contents]]
=--
=--
=--

# Bounded sets
* table of contents
{: toc}

## Idea

The term 'bounded' has several meaning in different branches of mathematics.  For a general axiomatic approach to boundedness, see [[bornological set]].  Here we list definitions in various fields.

## In metric spaces

+-- {: .num_defn #metric}
###### Definition
Let $E$ be a [[metric space]].  A [[subset]] $B \subseteq E$ is **bounded** if there is some [[real number]] $r$ such that $d(x,y) \lt r$ for all $x, y \in B$.
=--

This generalises immediately to pseudometric spaces, quasimetric spaces, extended metric spaces, and most generally to [[Lawvere metric spaces]].

We can also generalise to [[gauge spaces]]:
+-- {: .num_defn #metric}
###### Definition
Let $E$ be a [[gauge space]].  A [[subset]] $B \subseteq E$ is **bounded** if there is some [[real number]] $r$ such that $d(x,y) \lt r$ for all $x, y \in B$ and all gauging distances $d$.
=--

This generalises immediately to quasigauge spaces.

The family of all bounded sets of a quasigauge space (and hence of the more particular kinds of spaces above) defines a [[bornology]] on its [[underlying set]].

## In topological vector spaces

+-- {: .num_defn #bset}
###### Definition
Let $E$ be a [[LCTVS]].  A subset $B \subseteq E$ is **bounded** if whenever $U \subseteq E$ is a [[neighbourhood]] of $0$ then there is some [[real number]] $r$ such that $B \subseteq r U$.
=--

The family of all bounded sets of a LCTVS defines a [[bornology]] on its [[underlying set]].

## Related concepts

* [[totally bounded metric space]]

* [[bounded function]]

[[!redirects bounded set]]
[[!redirects bounded sets]]
[[!redirects bounded subset]]
[[!redirects bounded subsets]]
[[!redirects bounded subspace]]
[[!redirects bounded subspaces]]


Bourbaki

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### Context
#### Mathematics
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[[!include mathematicscontents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Overview

The __Association of Collaborators of Nicolas Bourbaki__ (*l\'Association des collaborateurs de Nicolas Bourbaki*) is a collective of mathematicians based in France, who in the 20th century set a standard for a structural and abstract approach to exposition of pure mathematics.  They published a series "Elements of mathematics" of 10 multi-volume monographs under the pseudonym 'Nicolas Bourbaki', consolidating known results.  They also held ---and still hold--- a largely independent seminar on contemporary research.

The members of the group at any one time are required to be under 50 years of age, which is considered the only fixed rule of the group[^fine]; the membership is taken under acceptance of an invitation by existing members.

A bit of history can be found at:

* [en.wikipedia](http://en.wikipedia.org/wiki/Nicolas_Bourbaki),

* [fr.wikipedia](http://fr.wikipedia.org/wiki/Nicolas_Bourbaki),

* [britannica](http://www.britannica.com/EBchecked/topic/75700/Nicolas-Bourbaki)

and references therein.

According to [[Jean Dieudonné]], the Bourbaki series of expositions is dedicated to "dead", i.e. stable mathematics which is not likely to change soon and which has wider importance, rather than being of a specialized character. Most successful was the exposition of the chapters on [[Lie theory]], especially concerning [[Lie algebras]], whose style was later followed by much of the literature on the subject. See more in [[Dieudonné](#Dieudonne)].

Bourbaki has been blamed for following too formal an approach. Indeed, the  books are void of much motivation and application, apart from a few of introductions and few chapters on history.

Apart from the Bourbaki volumes, there is also a lively _Bourbaki seminar_ which takes place in Paris on certain Saturdays, is open to public and has non-Bourbaki members as invited speakers who present and discuss topics,chosen in advance, of recent development in mathematics. The expositions are subsequently published.

## Selected writings
 {#SelectedWritings}

On [[general topology]], [[uniform structures]], [[topological groups]] and the [[real numbers]] with some [[real analysis]]:

* {#Bourbaki71} [[Nicolas Bourbaki]], *General topology*, Elements of Mathematics III, Springer (1971, 1990, 1995) [[doi:10.1007/978-3-642-61701-0](https://doi.org/10.1007/978-3-642-61701-0)]

On [[Lie groups]] and [[Lie algebras]]:

* [[Nicolas Bourbaki]], _Lie groups and Lie algebras -- Chapters 1-3_, Springer (1975, 1989) [[ISBN:9783540642428](https://link.springer.com/book/9783540642428)]

On [[topological vector spaces]]:

* [[Nicolas Bourbaki]], *Topological Vector Spaces*, Chapters 1–5, Masson (1981), Springer (2003) [[doi:10.1007/978-3-642-61715-7](https://doi.org/10.1007/978-3-642-61715-7)]

On [[algebraic topology]] ([[groupoids]], [[fundamental groupoids]]):

* [[Nicolas Bourbaki]], _Topologie Algébrique_, Chapitres 1 à 4, Springer, (1998, 2016)  [[doi:10.1007/978-3-662-49361-8](https://doi.org/10.1007/978-3-662-49361-8), ISBN 978-3-662-49361-8]

(...)

## Bourbaki's notion of _structure_ and the relation to category theory
 {#StructureAndCategoryTheory}

A central point of view of Bourbaki is the emphasis of mathematics as the study of *[[structure]]*, as in the approach developed by Göttingen mathematicians in the 1920s and 1930s, such as [[Emmy Noether]] and [[Emil Artin]], and written up by van der Waerden in _Moderne Algebra_ (see [Corry](#Corry)). This approach was very influential in the mainstream mathematics of the second half of the 20th century.

However, Bourbaki did not embrace [[category theory]], which may be thought of as being the essence of that structural approach (cf. [[structuralism]]), though some of the [[universal properties]] treated in category theory in fact first appeared in early editions of Bourbaki. Instead, Bourbaki proposed its own formalization of the notion of "structure" (in the book *Theory of Sets*), which however neither caught on nor does it seem to have been taken very seriously by the group itself. As documented by [[Corry](#Corry)], the Elements pays only the slightest lip service to the Bourbaki formal notion of structure in its vast development (in the books after *Theory of Sets* when they are considered in their intended logical order).

Discussion of this point can be found in [[CTList](#CTList)]. There [[Colin McLarty]] writes:

> Bourbaki's first publication was

> Bourbaki, N. [1939]: _Théorie des ensembles, Fascicules de résultats_, Paris: Hermann, Paris.

> It is very sketchy on "structures," and uses no notion of mapping between structures except isomorphisms. Their actual theory of structures first appeared in

> Bourbaki, N. [1957]: _Théorie des ensembles_, Chapter 4, Paris: Hermann.

> That theory was a rear-guard action meant to give an alternative to category theory.  As i mentioned before, Weil was citing the categorical idea, and thinking about finding an in-house alternative to it, already in 1951.  By 1957 Grothendieck, and Cartier, and Chevalley, probably Dieudonné, and others, all saw that category theory was more agile than these structure, simpler, and more to the point, plus it had a natural "higher order" aspect in the theory of functors which was actually more useful in practice than categories alone.

> Cartier has justly said it would have been a huge job to formulate all Bourbaki's ideas in terms of categories and functors.  It would have called for a lot of ideas which were only invented in the coming years.

> It was relatively easy to give Bourbaki's theory of structures -- because it never really worked at all even for Bourbaki's purposes (as Corry documents in detail).  Naturally it is easier to give an unusable theory of structures than to work out the ways categories and functors would actually be used. [[McLarty, Category Theory List 2012](http://permalink.gmane.org/gmane.science.mathematics.categories/7298)]

## References

* [website](https://www.bourbaki.fr/) of the group

Accounts of the history of the Bourbaki project include the following:

* {#Dieudonne} [[Jean Dieudonné]], _The work of Nicolas Bourbaki_, Amer. Math. Monthly __77__ (1970) 134-145 [[MR252179](http://www.ams.org/mathscinet-getitem?mr=252179), [doi](http://dx.doi.org/10.2307/2317325)]

* Bourbaki's biography at math tutor: the [pre-war years](http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Bourbaki_1.html), the [after-war years](http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Bourbaki_2.html)

* [[Armand Borel]]: _[25 years with Bourbaki](http://www.ega-math.narod.ru/Bbaki/Bourb3.htm)_

* Amir D. Aczel, _The artist and the mathematician_: the story of Nicolas Bourbaki, the genius mathematician who never existed, Thunder's Mouth Press, New York, 2006, viii+239 pp. [MR2008h:01020](http://www.ams.org/mathscinet-getitem?mr=2339913), [SIAM review](http://www.siam.org/news/news.php?id=1078)

* The [Bourbaki archives](http://mathdoc.emath.fr/archives-bourbaki/) contain many draughts (and discussions thereof) of Bourbaki's texts

Discussion of Bourbaki's notion of _[[mathematical structure]]_ and the relation (or not) to [[category theory]]:

* Jean-Michel Kantor, _Bourbaki's Structures and Structuralism_, The Mathematical Intelligencer __33__:1 (2011), 1, [doi](http://dx.doi.org/10.1007/s00283-010-9173-4)

* [[Leo Corry]], *Mathematical Structures from Hilbert to Bourbaki: The Evolution of an Image of Mathematics*, in: *Changing Images of Mathematics in History. From the French Revolution to the new Millenium* Harwood Academic Publishers (2001) 167-186 [[ISBN:9780415868273](https://www.routledge.com/Changing-Images-in-Mathematics-From-the-French-Revolution-to-the-New-Millennium/Bottazini-Dalmedico/p/book/9780415868273), [pdf](https://www.leocorry.com/_files/ugd/e6fef7_15c7077b08ec4f249fb40eb961256ced.pdf)]

* {#Corry} [[Leo Corry]]: _Nicolas Bourbaki: Theory of structures_ [[pdf](http://www.tau.ac.il/~corry/publications/articles/pdf/bourbaki-structures.pdf)] Chapter 7  of: *Modern Algebra and the Rise of Mathematical Structures*, Springer (2004) [[doi:10.1007/978-3-0348-7917-0](https://doi.org/10.1007/978-3-0348-7917-0)]

* {#CTList} Category Theory List, Staffan Angere, _[Bourbaki & category theory](https://web.archive.org/web/20160427073919/http://comments.gmane.org/gmane.science.mathematics.categories/7283)_

* David Aubin, _The Withering Immortality of Nicolas Bourbaki: A Cultural Connector at the Confluence of Mathematics, Structuralism, and the Oulipo in France_, Science in Context 10 (1997), 297-342. ([pdf](http://www.math.jussieu.fr/~daubin/publis/1997.pdf))
 {#Aubin}

[^fine]: According to Liliane Beaulieu, a pre-eminent historian of Bourbaki, this rule was however violated many times, and she could find no written trace of the rule's being formally adopted. See [Aubin](#Aubin), footnote 3.

category: people

[[!redirects Bourbaki]]
[[!redirects Nicolas Bourbaki]]
[[!redirects N. Bourbaki]]
[[!redirects N Bourbaki]]
[[!redirects Association of Collaborators of Nicolas Bourbaki]]
[[!redirects Association of collaborators of Nicolas Bourbaki]]
[[!redirects association of collaborators of Nicolas Bourbaki]]
[[!redirects The Association of Collaborators of Nicolas Bourbaki]]
[[!redirects The association of collaborators of Nicolas Bourbaki]]
[[!redirects the association of collaborators of Nicolas Bourbaki]]
[[!redirects Association des Collaborateurs de Nicolas Bourbaki]]
[[!redirects Association des collaborateurs de Nicolas Bourbaki]]
[[!redirects association des collaborateurs de Nicolas Bourbaki]]
[[!redirects L'Association des Collaborateurs de Nicolas Bourbaki]]
[[!redirects l'Association des Collaborateurs de Nicolas Bourbaki]]
[[!redirects L'association des collaborateurs de Nicolas Bourbaki]]
[[!redirects l'Association des collaborateurs de Nicolas Bourbaki]]
[[!redirects l'association des collaborateurs de Nicolas Bourbaki]]
[[!redirects L'Association des Collaborateurs de Nicolas Bourbaki]]
[[!redirects l'Association des Collaborateurs de Nicolas Bourbaki]]
[[!redirects L'association des collaborateurs de Nicolas Bourbaki]]
[[!redirects l'Association des collaborateurs de Nicolas Bourbaki]]
[[!redirects l'association des collaborateurs de Nicolas Bourbaki]]

[[!redirects Éléments de Mathématique]]


Bousfield equivalence

+-- {: .rightHandSide}
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### Context
#### Equality and Equivalence
+-- {: .hide}
[[!include equality and equivalence - contents]]
=--
#### Stable Homotopy theory
+--{: .hide}
[[!include stable homotopy theory - contents]]
=--
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

Two [[generalized homology theories]] $E_\bullet$, $F_\bullet$, hence [[spectra]] $E$, $F$ are called _Bousfield equivalent_ if the [[homology groups]] of both always vanish simultaneously, hence if for every [[homotopy type]]/[[spectrum]] $X$ we have $E_\bullet(X) \simeq 0$ precisely if $F_\bullet(X) \simeq 0$.

## Examples

### Global arithmetic fracture square

There is a Bousfield equivalence

$$
  S (\mathbb{Q}/\mathbb{Z}) \simeq_{Bousf} \vee_p S \mathbb{F}_p
$$

between the [[Moore spectrum]] of the quotient $\mathbb{Q}/\mathbb{Z}$ and the [[coproduct]] of the Moore spectra of all [[cyclic groups]]/[[finite fields]] of [[prime]] order (e.g. [Strickland 12, MO comment](http://mathoverflow.net/a/91057/381)).

This goversn the global [[fracture theorem|arithmetic fracture theorem]] [in stable homotopy theory](fracture+theorem#InStableHomotopyTheory).

### Morava E-theory and Morava K-theory

For all $n$, the $n$th [[Morava E-theory]] $E(n)$ is [[Bousfield equivalence|Bousfield equivalence]] to $E(n-1) \times K(n)$, where the last factor is $n$th [[Morava K-theory]].

([Lurie 10, lect. 23, prop. 1](#LurieLect23))

## References

The concept of Bousfield classes is due to (see at [[Bousfield localization of spectra]])

* {#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))

and was named such in

* {#Ravenel84} [[Douglas Ravenel]], _Localization with respect to certain periodic homology theories_, American Journal of Mathematics, Vol. 106, No. 2, (Apr., 1984), pp. 351-414 ([pdf](https://www.math.rochester.edu/people/faculty/doug/mypapers/loc.pdf))

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

* {#LurieLect23} [[Jacob Lurie]], _[[Chromatic Homotopy Theory]]_, Lecture series 2010; Lecture 23 _The Bousfield Classes of $E(n)$ and $K(n)$_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture23.pdf))

[[!redirects Bousfield equivalences]]


Bousfield lattice

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#### Homotopy theory
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[[!include homotopy - contents]]
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#Contents#
* table of contents
{:toc}

## Definition

Given a [[well-generated triangular category|well-generated]] [[monoidal category|tensor]] [[triangulated category]] $(T,\otimes, 1)$, let the _Bousfield class_ of an object $X$, denoted $\langle X\rangle$, be the class $\{ Y\in obj(T): X\otimes Y=0\}$.  It was proven by Ohkawa that if $T$ is the [[stable homotopy category]], $(\mathcal{S},\wedge,S)$, then the collection of all Bousfield classes is a [[set]] of [[cardinality]] at most $\beth_2$.  It was proven more generally by Iyengar and Krause that such a collection is always a set and not a [[proper class]] when $T$ is well-generated. This set has a partial ordering on it and the structure of a complete lattice. This lattice is called the _Bousfield lattice_ of $T$, denoted $B_T$.

Note that, perhaps by some abuse of notation, $B_T\subseteq\mathbf{Loc}(T)$, the collection of [[localizing subcategories]], since every Bousfield class is a localizing subcategory. However, the question of whether or not every localizing subcategory is a Bousfield class is still open in general.

## The Distributive Lattice of the Bousfield Lattice

Within $B_T$, there is a [[distributive lattice]] $DL_T$ that is precisely all _Bousfield idempotent_ objects of $T$.  That is, $DL_T$ is precisely the objects $X$ such that $\langle X\otimes X\rangle=\langle X\rangle$.  Because $B_T$ is an affine [[quantale]], it follows that $DL_T$ is a [[frame]].  In particular, $DL_T$ is a [[distributive lattice]], so by the work of Stone, it corresponds to a coherent space.  For more on this, see [[Stone duality]].

##References

* S. B. Iyengar and H. Krause, _The Bousfield Lattice of a Triangulated Category and Stratification_ ([arXiv:1105.1799](http://arxiv.org/abs/1105.1799))

[[!redirects Bousfield Lattice]]
[[!redirects bousfield lattice]]


Bousfield localization

Bousfield localization is a sophisticated version of the general idea of [[localization]].  We can localize a category by formally inverting certain morphisms: for example, when forming the [[homotopy category]] of a [[model category]], where we invert morphisms called 'weak equivalences'.  But Bousfield localization is a subtler process.  In the case of a model category, Bousfield localization allows us to *make more morphisms count as weak equivalences*.  There is a related notion of Bousfield localization for [[triangulated categories]].

## Left Bousfield localization

Left Bousfield localizations model [[reflective localizations]] of [[(∞,1)-categories]].

* [[left Bousfield localization of model categories]]

* [[Bousfield localization of spectra]]

* [[Bousfield localization of triangulated categories]]

## Right Bousfield localization

Right Bousfield localizations model [[coreflective localizations]] of [[(∞,1)-categories]].

* [[right Bousfield localization of model categories]]

[[!redirects Bousfield localizations]]


Bousfield localization of spectra

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###Context###
#### Cohomology
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[[!include cohomology - contents]]
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#### Stable Homotopy theory
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[[!include stable homotopy theory - contents]]
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#### Higher algebra
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[[!include higher algebra - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

_Bousfield localization of spectra_ refers generally to [[localization of an (∞,1)-category|localizations]] of the [[stable (∞,1)-category of spectra]] (hence [[Bousfield localization of model categories|Bousfield localization]] of [[model categories of spectra]]) at the collection of [[morphisms]] which become [[equivalences]] under [[smash product of spectra|smash product]] with a given [[spectrum]] $E$. Since any such $E$ represents a [[generalized homology theory]], this may also be thought of  _$E$-[[homology localization]]_.

More specifically, if the [[stable (∞,1)-category of spectra]] is [[presentable (infinity,1)-category|presented]] by a ([[stable model category|stable]]) [[model category]], then the [[localization of an (∞,1)-category|∞-categorical localization]] can be presented by the operation of  _[[Bousfield localization of model categories]]_. The original article ([Bousfield 79](#Bousfield79)) essentially considered localization at the level of [[homotopy categories]].

Specifically, for $E \in Spec$ a [[spectrum]], the _Bousfield localization at $E$_ of another [[spectrum]] $X$ is the [[universal property|universal map]]

$$
  X \longrightarrow L_E X
$$

to the _$E$-[[local spectrum]]_ $L_E X$, with the property that for $Y$ any $E$-acyclic spectrum in that $Y \wedge E \simeq 0$, every morphism $Y \longrightarrow L_E X$ is null-homotopic (a [[zero morphism]] in the [[stable (∞,1)-category of spectra]]). (see for instance [Lurie 10, Example 4](#Lurie10))

For $E = M \mathbb{Z}_p$ the [[Moore spectrum]] of the [[cyclic group]] $\mathbb{Z}_p \coloneqq \mathbb{Z}/p\mathbb{Z}$ for some [[prime number]] $p$, this $E$-localization is _[[p-completion]]_. (see for instance [Lurie 10, Examples 7 and 8](#Lurie10))

## Definition

We write $Ho(Spectra)$ for the [[stable homotopy category]]. For $X,Y \in Ho(Spectra)$ two [[spectra]] we write $[X,Y] \in $ [[Ab]] for its [[hom-object|hom-]][[abelian groups]], and $[X,Y]_\bullet \coloneqq [\Sigma^\bullet X,Y]$ for the corresponding [[graded abelian group]] ([def.](Introduction+to+Stable+homotopy+theory+--+1-1#GradedAbelianGroupStructureOnHomsInTheHomotopyCategory)).

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

Let $E \in Ho(Spectra)$ be a [[spectrum]]. Say that

1. a spectrum $X$ is **$E$-acyclic** if the [[smash product of spectra|smash product]] with $E$ is [[zero object|zero]], $E \wedge X \simeq 0$;

1. a morphism $f \colon X \to Y$ of spectra is an **$E$-equivalence** if $E \wedge f \;\colon\; E \wedge X \to E \wedge Y$ is an [[isomorphism]] in $Ho(Spectra)$, hence if $E_\bullet(f)$ is an isomorphism in $E$-[[generalized homology]];

1. a spectrum $X$ is **$E$-local** if the following equivalent conditions hold

   1. for every $E$-equivalence $f$ then $[f,X]_\bullet$ is an isomorphism;

   1. every [[morphism]] $Y \longrightarrow X$ out of an $E$-acyclic spectrum $Y$ is [[zero morphism|zero]] in $Ho(Spectra)$;

=--

([Bousfield 79, §1](#Bousfield79)) see also for instance ([Lurie, Lecture 20, example 4](#Lurie10))

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

The two conditions in the last item of def. \ref{EAcyclicAndLocalSpectra} are indeed equivalent.

=--

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

Notice that $A \in Ho(Spectra)$ being $E$-acyclic means equivalently that the unique morphism $0 \longrightarrow A$ is an $E$-equivalence.

Hence one direction of the claim is trivial. For the other direction we need to show that for $[-,X]_\bullet$ to give an isomorphism on all $E$-equivalences $f$, it is sufficient that it gives an isomorphism on all $E$-equivalences of the form $0 \to A$.

Given a morphism $f \colon A \to B$, write $B \longrightarrow B/A$ for its [[homotopy cofiber]]. Then since $Ho(Spectra)$ is a [[triangulated category]] ([prop.](Introduction+to+Stable+homotopy+theory+--+1-1#StableHomotopyCategoryIsTriangulated)) the defining axioms of triangulated categories ([def.](Introduction+to+Stable+homotopy+theory+--+1-1#CategoryWithCofiberSequences), [lemma](Introduction+to+Stable+homotopy+theory+--+1-1#TrianglesMayBeShiftedToTheLeft)) give that there is a [[commuting diagram]] of the form

$$
  \array{
    0
      &\longrightarrow&
    A
      &\overset{id}{\longrightarrow}&
    A
      &\overset{}{\longrightarrow}&
    0
      &\overset{}{\longrightarrow}&
    \Sigma A
    \\
    \downarrow
      &&
    \downarrow^{\mathrlap{id}}
      &&
    \downarrow^{\mathrlap{f}}
      &&
    \downarrow
      &&
    \downarrow^{\mathrlap{id}}
    \\
    \Sigma^{-1} B/A
      &\longrightarrow&
    A
      &\underset{f}{\longrightarrow}&
    B
      &\longrightarrow&
    B/A
      &\longrightarrow&
    \Sigma A
  }
  \,,
$$

where both the top as well as the bottom are [[homotopy cofiber sequences]]. Hence applying $[-,X]_\bullet$ to this diagram in $Ho(Spectra)$ yields a diagram of [[graded abelian groups]] of the form

$$
  \array{
    0
      &\longleftarrow&
    [A,X]_\bullet
      &\longleftarrow&
    [A,X]_\bullet
      &\longleftarrow&
    0
      &\longleftarrow&
    [A,X]_{\bullet+1}
    \\
    \uparrow
      &&
    \uparrow^{\mathrlap{id}}
      &&
    \uparrow^{\mathrlap{[f,X]_\bullet}}
      &&
    \uparrow
      &&
    \uparrow^{\mathrlap{id}}
    \\
    [B/A,X]_{\bullet+1}
      &\longleftarrow&
    [A,X]_\bullet
      &\longleftarrow&
    [B,X]_\bullet
      &\longleftarrow&
    [B/A,X]_\bullet
      &\longleftarrow&
    [A,X]_{\bullet+1}
  }
  \,,
$$

where now both horizontal sequences are [[long exact sequences]] ([prop.](Introduction+to+Stable+homotopy+theory+--+1-1#LongFiberSequencesOfMapsOfSpectra)).

Hence if $[B/A,X]_\bullet \longrightarrow 0$ is an isomorphism, then all four outer vertical morphisms in this diagram are isomorphisms, and then the [[five-lemma]] implies that also $[f,X]_\bullet$ is an isomorphism.

Hence it is now sufficient to observe that with $f \colon A \to B$ an $E$-equivalence, then its homotopy cofiber $B/A$ is $E$-acyclic.

To see this, notice that by the [[tensor triangulated category|tensor triangulated]] structure on $Ho(Spectra)$ ([prop.](Introduction+to+Stable+homotopy+theory+--+1-2#TensorTriangulatedStructureOnStableHomotopyCategory)) the [[smash product of spectra|smash product]] with $E$ preserves homotopy cofiber sequences, so that there is a homotopy cofiber sequence

$$
  E \wedge A
    \overset{E \wedge f}{\longrightarrow}
  E \wedge B
    \longrightarrow
  E \wedge (B/A)
    \longrightarrow
  E \wedge \Sigma A
  \,.
$$

But if the first morphism here is an isomorphism, then the axioms of a [[triangulated category]] ([def.](Introduction+to+Stable+homotopy+theory+--+1-1#CategoryWithCofiberSequences)) imply that $E \wedge B / A \simeq 0$. In detail: by the axioms we may form the morphism of homotopy cofiber sequences

$$
  \array{
    E \wedge A
      &\overset{E \wedge f}{\longrightarrow}&
    E \wedge B
      &\longrightarrow&
    E \wedge B/A
      &\longrightarrow&
    E \wedge \Sigma A
    \\
    \downarrow^{\mathrlap{id}}
      &&
    \downarrow^{\mathrlap{(E\wedge f)^{-1}}}
      &&
    \downarrow
      &&
    \downarrow^{\mathrlap{id}}
    \\
    E \wedge A
      &\underset{id}{\longrightarrow}&
    E \wedge A
      &\longrightarrow&
    0
      &\longrightarrow&
    E \wedge \Sigma A
  }
  \,.
$$

Then since two of the three vertical morphisms on the left are isomorphisms, so is the third ([lemma](Introduction+to+Stable+homotopy+theory+--+1-1#TwoOutOfThreeForMorphismsOfDistinguishedTriangles)).

=--

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

Given $E,X \in Ho(Spectra)$, then an **$E$-localization** of $X$ is

1. an $E$-local spectrum $L_E X$

1. an $E$-equivalence $X \longrightarrow L_E X$.

according to def. \ref{EAcyclicAndLocalSpectra}.

=--

We discuss now that $E$-Localizations always exist. The key to this is the following lemma \ref{KappaCellSpectrumWitnessingELocalization}, which asserts that a spectrum being $E$-local is equivalent to it being $A$-null, for some "small" spectrum $A$:

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

For every [[spectrum]] $E$ there exists a spectrum $A$ such that any spectrum $X$ is $E$-local (def. \ref{EAcyclicAndLocalSpectra}) precisely if it is $A$-null, i.e.

$$
  X \;is\; E\text{-local}
    \;\;\;\;
  \Leftrightarrow
    \;\;\;\;
  [A,X]_\ast = 0
$$

and such that

1. $A$ is $E$-acyclic (def. \ref{EAcyclicAndLocalSpectra});

1. there exists an infinite [[cardinal number]] $\kappa$ such that $A$ is a $\kappa$-[[CW spectrum]] (hence a [[CW spectrum]] ([def.](Introduction+to+Stable+homotopy+theory+--+1-1#CWSpectrum)) with at most $\kappa$ many cells);

1. the class of $E$-acyclic spectra (def. \ref{EAcyclicAndLocalSpectra}) is the class generated by $A$ under

   1. [[wedge sum]]

   1. the relation that if in a [[homotopy cofiber sequence]] $X_1 \to X_2 \to X_3$ two of the spectra are in the class, then so is the third.

=--

([Bousfield 79, lemma 1.13 with lemma 1.14](#Bousfield79)) review includes ([Bauer 11, p.2,3](Bauer11), [VanKoughnett 13, p. 8](#VanKoughnett13))

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

For $E \in Ho(Spectra)$ any [[spectrum]], every spectrum $X$ sits in a [[homotopy cofiber sequence]] of the form

$$
  G_E(X)
    \longrightarrow
  X
    \overset{\eta_X}{\longrightarrow}
  L_E(X)
  \,,
$$

and [[natural transformation|natural]] in $X$, such that

1. $G_E(X)$ is $E$-acyclic,

1. $L_E(X)$ is $E$-local,

according to def. \ref{EAcyclicAndLocalSpectra}.

=--

([Bousfield 79, theorem 1.1](#Bousfield79)) see also for instance ([Lurie, Lecture 20, example 4](#Lurie10))

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

Consider the $\kappa$-[[CW-spectrum]] spectrum $A$ whose existence is asserted by lemma \ref{KappaCellSpectrumWitnessingELocalization}. Let

$$
  I_A
  \coloneqq
  \{A \to Cone(A)\}
$$

denote the set containing as its single element the canonical morphism (of [[sequential spectra]]) from $A$ into the standard [[cone]] of $A$, i.e.  the cofiber

$$
  Cone(A)
    \coloneqq
  cofib( A \to A \wedge I_+ )
    \simeq
  A \wedge I
$$

of the inclusion of $A$ into its standard [[cylinder spectrum]] ([def.](Introduction+to+Stable+homotopy+theory+--+1-1#StandardCylinderSpectrumSequential)).

Since the standard cylinder spectrum on a CW-spectrum is a [[good cylinder object]] ([prop.](Introduction+to+Stable+homotopy+theory+--+1-1#CylinderSpectrumOverCWSpectrumIsGood)) this means ([lemma](Introduction+to+Stable+homotopy+theory+--+P#HomsOutOfCofibrantIntoFibrantComputeHomotopyCategory)) that for $X$ any fibrant sequential spectrum, and for $A \longrightarrow X$ any morphism, then an extension along the cone inclusion

$$
  \array{
    A &\longrightarrow& X
    \\
    \downarrow & \nearrow
    \\
    Cone(A)
  }
$$

equivalently exhibits a null-homotopy of the top morphism. Hence the $(A \to Cone(A))$-[[injective objects]] in $Ho(Spectra)$ are precisely those spectra $X$ for which $[A,X]_\bullet \simeq 0$.

Moreover, due to the degreewise nature of the smash tensoring $Cone(A) = A \wedge I$ ([def](Introduction+to+Stable+homotopy+theory+--+1-1#TensoringAndPoweringOfSequentialSpectra)), the inclusion morphism $A \to Cone(A)$ is degreewise the inclusion of a [[CW-complex]] into its standard cone, which is a [[relative cell complex]] inclusion ([prop.](Introduction+to+Stable+homotopy+theory+--+P#TopologicalCylinderOnCWComplexIsGoodCylinderObject)).

By [this lemma](Introduction+to+Stable+homotopy+theory+--+1-1#kappaCellSpectrumIsKappaSmall) the $\kappa$-[[cell spectrum]] $A$ is _$\kappa$-small object_ ([def.](Introduction+to+Stable+homotopy+theory+--+P#ClassOfMorphismsWithSmallDomains)) with respect to morphisms of spectra which are degreewise [[relative cell complex]] inclusion  [[small object argument]] .

Hence the [[small object argument]] applies ([prop.](Introduction+to+Stable+homotopy+theory+--+P#SmallObjectArgument)) and gives for every $X$ a factorization of the terminal morphism $X \to \ast$ as an $I_A$-[[relative cell complex]] ([def.](Introduction+to+Stable+homotopy+theory+--+P#TopologicalCCellComplex)) followed by an $I_A$-[[injective morphism]] ([def.](Introduction+to+Stable+homotopy+theory+--+P#RightLiftingProperty))

$$
  X
    \overset{I_A Cell}{\longrightarrow}
  L_E X
    \overset{I_A Inj}{\longrightarrow}
  \ast
  \,.
$$

By the above, this means that $[A, L_E X] = 0$, hence by lemma \ref{KappaCellSpectrumWitnessingELocalization} that $L_E X$ is $E$-local.

It remains to see that the [[homotopy fiber]] of $X \to L_E X$ is $E$-acyclic: By the [[tensor triangulated category|tensor triangulated]] structure on $Ho(Spectra)$ ([prop.](Introduction+to+Stable+homotopy+theory+--+1-2#TensorTriangulatedStructureOnStableHomotopyCategory)) it is sufficient to show that the [[homotopy cofiber]] is $E$-acyclic (since it differs from the homotopy fiber only by suspension). By the [[pasting law]], the homotopy cofiber of a [[transfinite composition]] is the transfinite composition of a sequence of homotopy pushouts. By lemma \ref{KappaCellSpectrumWitnessingELocalization} and applying the pasting law again, all these homotopy pushouts produce $E$-acyclic objects. Hence we conclude by observing that the the transfinite composition of the morphisms between these $E$-acyclic objects is $E$-acyclic. Since by construction all these morphisms are relative cell complex inclusions, this follows again with the compactness of the $n$-spheres ([lemma](Introduction+to+Stable+homotopy+theory+--+P#CompactSubsetsAreSmallInCellComplexes)).

=--

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

The morphism $X \to L_E (X)$ in prop. \ref{ELocalizationCofiber} exhibits an $E$-localization of $X$ according to def. \ref{ELocalizationOfSpectra}

=--

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

It only remains to show that $X \to L_E X$ is an $E$-equivalence. By the  [[tensor triangulated category|tensor triangulated]] structure on $Ho(Spectra)$ ([prop.](Introduction+to+Stable+homotopy+theory+--+1-2#TensorTriangulatedStructureOnStableHomotopyCategory)) the [[smash product of spectra|smash product]] with $E$ preserves homotopy cofiber sequences, so that

$$
  E \wedge G_E X
   \longrightarrow
  E \wedge X
    \overset{E \wedge \eta_X}{\longrightarrow}
  E \wedge L_E X
    \longrightarrow
  E \wedge \Sigma G_E X
$$

is also a homotopy cofiber sequence. But now $E \wedge G_E X \simeq 0$ by prop. \ref{ELocalizationCofiber}, and so the axioms ([def.](Introduction+to+Stable+homotopy+theory+--+1-1#CategoryWithCofiberSequences)) of the [[triangulated category|triangulated structure]] on $Ho(Spectra)$ ([prop.](Introduction+to+Stable+homotopy+theory+--+1-1#StableHomotopyCategoryIsTriangulated)) imply that $E \wedge \eta$ is an isomorphism.

=--

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

Hence where $L_E$ is traditionally called "$E$-localization", $G_E$ might be called "$E$-acyclification", though that terminology is not used commonly.

=--

## Properties

### Localization at Moore spectra of abelian groups

A basic special case of $E$-localization of spectra is given for $E = S A$ the [[Moore spectrum]] of an [[abelian group]] $A$ ([Bousfield 79, section 2](#Bousfield79)). For $A = \mathbb{Z}_{(p)}$ this is [[p-localization]] and for $A = \mathbb{F}_p$ this is [[p-completion]], see examples \ref{pLocalization} and \ref{pCompletion} below for more.

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

For $A_1$ and $A_2$ two [[abelian groups]] then the following are equivalent

1. the Bousfield localizations at their [[Moore spectra]] are equivalent

   $$
     L_{S A_1} \simeq L_{S A_2}
     \,;
   $$

1. $A_1$ and $A_2$ have the same _type of acyclicity_, meaning that

   1. every [[prime number]] $p$ is invertible in $A_1$ precisely if it is in $A_2$;

   1. $A_1$ is a [[torsion group]] precisely if $A_2$ is.

=--

([Bousfield 79, prop. 2.3](#Bousfield79)) recalled e.g. in ([VanKoughnett 13, prop. 4.2](#VanKoughnett13)).

This means that given an abelian group $A$ then

* either $A$ is not torsion, then

  $$
    L_{S A} \simeq L_{S \mathbb{Z}[I^{-1}]}
    \,,
  $$

  where $I$ is the set of primes invertible in $A$ and $\mathbb{Z}[I^{-1}] \hookrightarrow \mathbb{Q}$ is the [[localization of a ring|localization]] at these primes of the [[integers]];

* or $A$ is torsion, then

  $$
   L_{S A }\simeq L_{S(\oplus_q \mathbb{F}_q )  }
   \,,
  $$

  where the [[direct sum]] is over all [[cyclic groups]] of order $q$, for $q$ a prime not invertible in $A$.

### Relation to nilpotent completion

Let $E$ be an [[E-∞ ring]] and let $X$ be any [[spectrum]]

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

There is a canonical map

$$
  L_E X \stackrel{}{\longrightarrow} \underset{\leftarrow}{\lim}_n (E^{\wedge^{n+1}_S}\wedge_S X)
$$

from the $E$-[[Bousfield localization of spectra]] of $X$ into the [[totalization]] of the canonical [[cosimplicial object|cosimplicial]] [[spectrum]] (see at _[[nilpotent completion]]_).

=--

We now consider conditions for this morphism to be an [[equivalence]].

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

For $R$ a [[ring]], its _core_ $c R$ is the [[equalizer]] in

$$
  c R
    \longrightarrow
  R
    \stackrel{\longrightarrow}{\longrightarrow}
  R \otimes R
  \,.
$$

=--

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

Let $E$ be a [[connective spectrum|connective]] [[E-∞ ring]] such that the core of $\pi_0(E)$, def. \ref{CoreOfARing}, is either of

* the [[localization of a ring|localization]] of the [[integers]] at a set $J$ of [[primes]], $c \pi_0(E) \simeq \mathbb{Z}[J^{-1}]$;

* $\mathbb{Z}_n$ for $n \geq 2$.

Then the map in remark \ref{CanonicalMapFromELocalizationToTotalization} is an equivalence

$$
  L_E X \stackrel{\simeq}{\longrightarrow}
  \underset{\leftarrow}{\lim}_n (E^{\wedge^{n+1}_S}\wedge_S X)
  \,.
$$

=--

([Bousfield 79](#Bousfield79)) see also for instance ([Bauer 11, p.2](#Bauer11))

For more discussion of [[E-infinity geometry|E-infinity]] (derived) [[formal completions]] via totalizations of [[Amitsur complexes]], see ([Carlsson 07](completion+of+a+module#Carlsson07)).

### Fracture theorem

The [[fracture theorem]] says how Bousfield localization at a [[coproduct]]/[[wedge sum]] of spectra is a [[homotopy pullback]] of Bousfield localization separately. See at _[[fracture theorem]]_ for more on this.

## Examples

### General

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

For $E$ an [[E-∞ ring]], every [[∞-module]] $X$ over $E$ is $E$-local, def. \ref{EAcyclicAndLocalSpectra}.

=--

(e.g. [Lurie, Lecture 20, example 5](#Lurie10))

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

For $E$ an [[E-∞ algebra]] over an [[E-∞ ring]] $S$ and for $X$ an $S$-[[∞-module]], consider the dual [[Cech nerve]] [[cosimplicial object]]

$$
  E^{\wedge_S^{\bullet+1}}\wedge_S X \;\colon\; \Delta \longrightarrow Spectra
  \,.
$$

By example \ref{EModulesAreELocal} each term is $E$-local, so that the map  to the  [[totalization]]

$$
  X \longrightarrow \underset{\leftarrow}{\lim} E^{\wedge_S^{\bullet+1}} \wedge_S X
$$

factors through the $E$-localization of $X$

$$
  X \longrightarrow
  L_E X
  \longrightarrow
  \underset{\leftarrow}{\lim} E^{\wedge_S^{\bullet+1}} \wedge_S X
  \,.
$$

Under suitable condition the second map here is indeed an [[equivalence]], in which case the [[totalization]] of the dual [[Cech nerve]] exhibits the $E$-localization. This happens for instance in the discussion of the [[Adams spectral sequence]], see the examples given there.

(see also e.g. [Bauer 11, p. 2](#Bauer11))

=--

### Rationalization

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

Bousfield localization at the [[Moore spectrum]]/[[Eilenberg-MacLane spectrum]] $S \mathbb{Q}\simeq H\mathbb{Q}$ of the [[rational numbers]] is [[rationalization]] to [[rational homotopy theory]].

The corresponding $\mathbb{Q}$-acyclification (remark \ref{Acyclification}) is _[[torsion approximation]]_.

=--

e.g. ([Bauer 11, example 1.7](#Bauer11))

### $p$-Localization

For $p$ a [[prime number]] write $\mathbb{Z}_{(p)}$ for the  [[localization of a ring|localization]] of the [[integers]] _at_ $(p)$,  for the ring of integers [[localization of a ring|localized at]] $p$, hence with all primes _except_ $p$ inverted; equivalently the subring of the [[rational numbers]] with denominator not divisible by $p$.

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

The Bousfield localization at the [[Moore spectrum]] $S \mathbb{Z}_{(p)}$ is [[p-localization]].

=--

([Bousfield 79](#Bousfield79)), [Bauer 11, example 1.7](#Bauer11)). See at _[[localization of a space]]_ for details on this.

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

$p$-localization is a [[smashing localization]]:

$$
  L_{S \mathbb{Z}_{(p)}} X
  \simeq
  S \mathbb{Z}_{(p)} \wedge X
  \,.
$$

=--

([Bousfield 79, prop. 2.4](#Bousfield79)) recalled e.g. as ([van Koughnett 13, prop. 4.3](#VanKoughnett13)).

### $p$-Completion

For $p \in \mathbb{N}$ a [[prime number]],
write

$$
  \mathbb{F}_p = \mathbb{Z}/(p)
$$

for the [[cyclic group]]/[[finite field]] of [[order of a group|order]] $p$.

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

Write

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

for the [[localization of a ring|localization]] of the [[integers]] away from $p$ followed by the quotient by $\mathbb{Z}$.

=--

e.g. ([Bousfield 79, p. 6](#Bousfield79))

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

The [[short exact sequence]] of [[abelian groups]]

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

induces the [[homotopy fiber sequence]] (in [[spectra]]) of [[Moore spectra]]

$$
  \Omega S(\mathbb{Z}/p^\infty) \longrightarrow S\mathbb{Z} \longrightarrow S(\mathbb{Z}[p^{-1}])
 \,.
$$

As in [Bousfield 79, p. 6](#Bousfield79) one also writes

$$
  S^{-1} (\mathbb{Z}/p^\infty)
  \coloneqq
  \Omega S(\mathbb{Z}/p^\infty)
  \,.
$$

=--

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

The localization of spectra at the [[Moore spectrum]] $S\mathbb{F}_p$ is given by the [[mapping spectrum]] out of $S^{-1} \mathbb{Z}/p^\infty$:

$$
  L_{S \mathbb{F}_p} X
  \simeq
  [\Omega S \mathbb{Z}/p^\infty, X]
  \,.
$$

=--

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

Fact: $\mathbb{F}_p$-localizaton is [[p-completion]], e.g. [Lurie "Proper Morphisms...", section 4](#LurieProper).

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

Let

$$
  E \coloneqq H \mathbb{Z}/p\mathbb{Z}
$$

be the corresponding [[Eilenberg-MacLane spectrum]]. Then a spectrum which corresponds to a [[chain complex]] under the [stable Dold-Kan corespondence](module+spectrum#StableDoldKanCorrespondence) is $E$-local, def. \ref{AcyclicAndLocal}, if that chain complex has [[chain homology]] groups being $\mathbb{Z}[p^{-1}]$-modules.

The $E$-localization of a spectrum in this case is  _[[p-completion]]_.

=--

(e.g. [Lurie, Lecture 20, example 8](#Lurie10))

More generally

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

Bousfield localization at the [[Moore spectrum]] $S \mathbb{F}_p$ is [[p-completion]] to [[p-adic homotopy theory]].

=--

E.g. ([Bauer 11, example 1.7](#Bauer11)). See at _[[localization of a space]]_ for more on this.

### Telescopic localization

* [[telescopic localization]]

### Chromatic localization

* [[chromatic localization]]

## Related concepts

* [[smashing localization]]

* [[chromatic filtration]], [[chromatic layer]]

* [[KU-local stable homotopy theory]]

* [[K(n)-local stable homotopy theory]]

## References

Original articles are

* {#Bousfield79} [[Aldridge Bousfield]], _The localization of spectra with respect to homology_, Topology Volume 18 Issue 4 (1979) (<a href="https://doi.org/10.1016/0040-9383(79)90018-1">doi:10.1016/0040-9383(79)90018-1</a>, [pdf](http://www.uio.no/studier/emner/matnat/math/MAT9580/v12/undervisningsmateriale/bousfield-topology-1979.pdf))

* {#Ravenel84} [[Douglas Ravenel]], _Localization with respect to certain periodic homology theories_, American Journal of Mathematics, Vol. 106, No. 2, (Apr., 1984), pp. 351-414 ([jstor:2374308](https://www.jstor.org/stable/2374308), [pdf](https://www.math.rochester.edu/people/faculty/doug/mypapers/loc.pdf))

Discussion in terms of [[Bousfield localization of model categories]] [[model category of spectra|of spectra]] appears in

* {#EKMM97} [[Anthony Elmendorf]], [[Igor Kriz]], [[Michael Mandell]], [[Peter May]], chapter VIII of _[[Rings, modules and algebras in stable homotopy theory]]_ 1997 ([pdf](http://www.math.uchicago.edu/~may/BOOKS/EKMM.pdf))

* {#SchwedeShipley02} [[Stefan Schwede]], [[Brooke Shipley]], lemma 4.1 in _A uniqueness theorem for stable homotopy theory_, Mathematische Zeitschrift 239 (2002), 803-828 ([arXiv:math/0012021](https://arxiv.org/abs/math/0012021))

see also

* [[Carles Casacuberta]], [[Javier Gutiérrez]], _Homotopical Localizations of Module Spectra_, Transactions of the American Mathematical Society
Vol. 357, No. 7 (Jul., 2005), pp. 2753-2770  ([jstor](http://www.jstor.org/stable/3845180))

Review:

* [[Nerses Aramian]], _Bousfield Localization_ ([pdf](http://www.math.uiuc.edu/~aramyan2/bousfield.pdf))

* {#Bauer11} [[Tilman Bauer]], _Bousfield localization and the Hasse square_ ([pdf](http://math.mit.edu/conferences/talbot/2007/tmfproc/Chapter09/bauer.pdf), [[Bauer_BousfieldLocalization.pdf:file]]), Chapter 6 in: [[Christopher Douglas]], [[John Francis]], [[André Henriques]], [[Michael Hill]] (eds.), _Topological Modular Forms_, Mathematical Surveys and Monographs Volume 201, AMS 2014  ([ISBN:978-1-4704-1884-7](https://bookstore.ams.org/surv-201))

* {#VanKoughnett13} [[Paul VanKoughnett]], _Spectra and localization_, 2013 ([[VanKoughnettLocalization.pdf:file]])

* [[Tyler Lawson]], *An introduction to Bousfield localization* ([arXiv:2002.03888](https://arxiv.org/abs/2002.03888)) in: [[Andrew J. Blumberg]], [[Teena Gerhardt]], [[Michael A. Hill]] (eds.), *[[Stable categories and structured ring spectra]]*  MSRI Book Series, Cambridge University Press

Discussion in the general context of [[higher algebra]]/[[stable homotopy theory]]:

* {#Lurie10} [[Jacob Lurie]], _[[Chromatic Homotopy Theory]]_, Lecture notes (2010) ([web](http://www.math.harvard.edu/~lurie/252x.html)),  Lecture 20 _Bousfield localization_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture20.pdf))

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

Discussion specifically of [[K(n)-local spectra]] includes

* [[Michael Hopkins]], [[Jacob Lurie]], _[[Ambidexterity in K(n)-Local Stable Homotopy Theory]]_

See also

section 2.4 of

* Holger Reeker, _On K(1)-local SU-bordism_ ([arXiv:0907.4299](http://arxiv.org/abs/0907.4299))

Discussion specifically of [[KU-local stable homotopy theory]]:

* [[Jeroen van der Meer]], Chapter 4 of: _A Stack-Theoretic Perspective on $\mathrm{KU}$-Local Stable Homotopy Theory_, 2019 ([pdf](https://www.universiteitleiden.nl/binaries/content/assets/science/mi/scripties/master/2018-2019/jvd-meer-thesisroot.pdf))

[[!redirects local spectrum]]
[[!redirects local spectra]]

[[!redirects localization of a spectrum]]
[[!redirects localization of spectra]]

[[!redirects localization of spectra]]
[[!redirects localizations of spectra]]


Bousfield localization of triangulated categories

#Contents#
* automatic table of contents goes here
{:toc}

## Definition ##

+-- {: .un_defn}
###### Definition (triangulated subcategory)

A **triangulated subcategory** $A$ of a [[triangulated category]] $B$ is a nonempty [[subcategory]] closed under suspension of objects and such that for all objects $X,Y$ in $A$ if $X\to Y\to Z\to X[1]$ is a distinguished triangle in $B$, then $Z$ is in $A$.

A triangulated subcategory $A$ is called **thick** if with any object in $A$ it contains all its direct summands in $B$.

=--

+-- {: .un_defn}
###### Definition (Verdier quotient)

For $A \subset B$ a triangulated subcategory, the [[Verdier quotient]] category $B/A$ which is a triangulated category equipped with a canonical functor $Q^*:B\to B/A$ that is also triangulated (additive and preserving distinguished triangles) and universal among all triangulated functors $B\to D$ which send objects of $A$ to objects isomorphic to $0$.

=--

The Verdier quotient $B/A$  has the property that the only objects whose images in $B/A$ are isomorphic to the [[zero object]] are the objects from $A$.

+-- {: .un_defn}
###### Definition (Bousfield localization)

Given a thick subcategory $A\subset B$, we say that the **Bousfield localization** exists if the Verdier quotient functor $Q^*$ has a right [[adjoint functor]] $Q_*$ which is then (automatically) triangulated and fully faithful.

=--

To amplify, writing $B_{loc} := B/A$ a Bousfield localization of a triangulated category $B$ is in particular an [[adjunction]]

$$
  B_{loc} \stackrel{\stackrel{}{\leftarrow}}{\hookrightarrow}
  B
  \,.
$$

Compare this to [[Bousfield localization of model categories]], noticing that most triangulated categories arise as [[homotopy category of an (infinity,1)-category|homotopy categories]] of [[stable (infinity,1)-category|stable (∞,1)-categories]], hence of [[homotopy category|homotopy categories]] of the [[model category|model categories]] presenting these.

Also note, to a tensored, triangulated category, one may associate a [[Bousfield lattice]] which has deep connections to the above topic.

## References ##

* [[Henning Krause]], _Localization theory for triangulated categories_, [arXiv/0806.1324](http://arxiv.org/abs/0806.1324)

* [[Amnon Neeman]], _Triangulated categories_, chapter 9

* Amnon Neeman, _The connection between the $K$-theory localization theorem of Thomason, Trobaugh and Yao and the smashing subcategories of Bousfield and Ravenel_ ([pdf](http://archive.numdam.org/article/ASENS_1992_4_25_5_547_0.pdf))

[[!redirects Bousfield localization of a triangulated category]]