Bousfield-Friedlander model structure

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###Context###
#### Stable Homotopy theory
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[[!include stable homotopy theory - contents]]
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#### Model category theory
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[[!include model category theory - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

The _Bousfield-Friedlander model structure_ ([Bousfield-Friedlander 78, section 2](#BousfieldFriedlander78)) is a [[model structure for spectra]], specifically it is a standard [[model structure on sequential spectra]] in [[simplicial sets]]. An immediate variant works for [[sequential spectra]] in [[topological spaces]], see at _[[model structure on topological sequential spectra]]_.

As such, the Bousfield-Friedlander model structure [[presentable (infinity,1)-category|presents]] the [[stable (infinity,1)-category of spectra]] of [[stable homotopy theory]], hence, in particular, its [[homotopy category]] is the classical [[stable homotopy category]].

## Background on sequential spectra

### Sequential pre-spectra

Write $S^1 \coloneqq \Delta[1]/\partial\Delta[1]$ for the minimal [[simplicial set|simplicial]] [[circle]]. Write

$$
  \wedge
  \;\colon\;
  sSet^{\ast/}
  \times
  sSet^{\ast/}
  \longrightarrow
  sSet^{\ast/}
$$

for the [[smash product]] of [[pointed simplicial sets]].

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

A **[[sequential spectrum|sequential]] [[prespectrum]] in [[simplicial sets]]**, or just **[[sequential spectrum]]** for short (or even just **[[spectrum]]), is

* an $\mathbb{N}$-[[graded object|graded]] [[pointed simplicial set]] $X_\bullet$

* equipped with morphisms $\sigma_n \colon S^1 \wedge X_n \to X_{n+1}$ for all $n \in \mathbb{N}$.

 A [[homomorphism]] $f \colon X \to Y$ of spectra is a sequence $f_\bullet \colon X_\bullet \to Y_\bullet$ of homomorphisms of pointed simplicial sets, such that all [[diagrams]] of the form

$$
  \array{
    S^1 \wedge X_n &\stackrel{S^1 \wedge f_n}{\longrightarrow}& S^1 \wedge Y_n
    \\
    \downarrow^{\mathrlap{\sigma_n^X}} && \downarrow^{\mathrlap{\sigma_n^Y}}
    \\
    X_{n+1} &\stackrel{f_{n+1}}{\longrightarrow}& Y_{n+1}
  }
$$

[[commuting diagram|commute]].

Write $SeqSpec(sSet)$ for this [[category]] of sequential spectra.

=--

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

For $X \in SeqSpec(sSet)$ and $K \in $ [[sSet]], hence $K_+ \in sSet^{\ast/}$ then $X \wedge K_+$ is the sequential spectrum degreewise given by the [[smash product]] of pointed objects

$$
  (X \wedge K_+)_n \coloneqq (X_n \wedge K_+)
$$

and with structure maps given by

$$
  S^1 \wedge (X_n \wedge K_+) \simeq (S^1 \wedge X_n) \wedge K_+ \stackrel{\sigma_n \wedge K_+}{\longrightarrow} X_{n+1}\wedge K_+
  \,.
$$

=--

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

The category $SeqSpec$ of def. \ref{SequentialSpectra} becomes a [[simplicially enriched category]] (in fact an $sSet^{\ast/}$-[[enriched category]]) with [[hom objects]] $[X,Y]\in sSet$ given by

$$
  [X,Y]_n \coloneqq Hom_{SeqSpec(sSet)}(X\wedge \Delta[n]_+,Y)
  \,.
$$

=--

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

The [[stable homotopy groups]] of a [[sequential spectrum]] $X$, def. \ref{SequentialSpectra}, is the $\mathbb{Z}$-[[graded abelian groups]] given by the [[colimit]] of [[homotopy groups]] of [[geometric realizations]] of the component spaces

$$
  \pi_\bullet(X)
  \coloneqq
  \underset{\longrightarrow}{\lim}_k
  \pi_{\bullet+k}({\vert X_n \vert})
  \,.
$$

This constitutes a [[functor]]

$$
  \pi_\bullet \;\colon\; SeqSpec(sSet) \longrightarrow Ab^{\mathbb{Z}}
  \,.
$$

=--

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

A morphism $f \colon X \longrightarrow Y$ of [[sequential spectra]], def. \ref{SequentialSpectra}, is called a [[stable weak homotopy equivalence]], if its image under the [[stable homotopy group]]-functor of def. \ref{StableHomotopyGroups} is an [[isomorphism]]

$$
  \pi_\bullet(f) \;\colon\; \pi_\bullet(X) \longrightarrow \pi_\bullet(Y)
  \,.
$$

=--

### Omega-spectra

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

A _[[Omega-spectrum]]_ is a sequential spectrum $X$, def. \ref{SequentialSpectra}, such that after [[geometric realization]]/[[Kan fibrant replacement]] the ([[smash product]] $\dahsv$ [[pointed mapping space]])-[[adjuncts]]

$$
  {\vert X_n\vert} \stackrel{}{\longrightarrow} {\vert X^{n+1}\vert}^{{\vert S^1\vert}}
$$

of the structure maps ${\vert \sigma_n\vert}$ are [[weak homotopy equivalences]].

=--

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

If a [[sequential spectrum]] $X$ is an [[Omega-spectrum]], def. \ref{OmegaSpectrum},
then its colimiting [[stable homotopy groups]], def. \ref{StableHomotopyGroups}, are attained as the actual homotopy groups of its components:

$$
  \pi_k(X) \simeq
  \simeq
  \left\{
    \array{
      \pi_k {\vert X_0 \vert} & if\; k \geq 0
      \\
      \pi_0 {\vert X_k \vert} & if \; k \lt 0
    }
  \right.
  \,.
$$

=--

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

The canonical _$\Omega$-[[spectrification]]_ $Q X$ of a [[sequential spectrum]] $X$ of [[simplicial sets]], def. \ref{SequentialSpectra}, is the operation of forming degreewise the [[colimit]] of higher [[loop space objects]] $\Omega(-)\coloneqq (-)^{S^1}$

$$
  (Q X)_n
  \coloneqq
  \underset{\longrightarrow}{\lim}_{k }
  Sing \Omega^k {\vert X_{n+k}\vert }
  \,,
$$

where $Sing$ denotes the [[singular simplicial complex]] functor.

This constitutes an [[endofunctor]]

$$
  Q \;\colon\;  SeqSpec(sSet) \longrightarrow SeqSpec(sSet)
  \,.
$$

Write

$$
 \eta \;\colon\; id \longrightarrow Q
$$

for the [[natural transformation]] given in degree $n$ by the $({\vert-\vert}\dashv Sing)$-[[adjunction unit]] followed the 0-th component map of the colimiting [[cocone]]:

$$
  (\eta_X)_n
    \;\colon\;
  X_n
    \longrightarrow
  Sing{\vert X_n\vert}
    \stackrel{\iota_0}{\longrightarrow}
  \underset{\longrightarrow}{\lim}_{k }
  Sing \Omega^k {\vert X_{n+k}\vert }
  \,.
$$

=--

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

The [[spectrification]] of def. \ref{Spectrification} satisfies

1. $Q X$ is an [[Omega-spectrum]], def. \ref{OmegaSpectrum};

1. $\eta_X \colon X \longrightarrow Q X$ is a [[stable weak homotopy equivalence]], def. \ref{StableWeakEquivalenceOfSequentialsSetSpectra};

1. if for a homomorphims of sequential spectra $f \colon X \longrightarrow Y$ each $f_n$ is a [[weak homotopy equivalence]], then also each $(Q X)_n$ is a weak homotopy equivalence;

1. $(Q\eta_X)$ is degreewise a weak homotopy equivalence.

=--

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

A homomorphism of [[sequential spectra]], def. \ref{SequentialSpectra}, is a [[stable weak homotopy equivalence]], def. \ref{StableWeakEquivalenceOfSequentialsSetSpectra}, precisely if its [[spectrification]] $Q f$ , def. \ref{PropertiesOfStandardSpectrification}, is degreewise a [[weak homotopy equivalence]].

=--

## The strict model structure on sequential spectra
 {#TheStrictModelStructure}

The [[model category]] structure on [[sequential spectra]] which [[presentable (infinity,1)-category|presents]] [[stable homotopy theory]] is the "stable model structure" discussed [below](#TheStableModelStructure). Its fibrant-cofibrant objects are (in particular) [[Omega-spectra]], hence are the proper [[spectrum objects]] among the pre-spectrum objects.

But for technical purposes it is useful to also be able to speak of a model structure on pre-spectra, which sees their homotopy theory as sequences of simplicial sets equipped with suspension maps, but not their stable structure. This is called the "strict model structure" for sequential spectra. It's main point is that the stable model structure of interest arises fromit via [[Bousfield localization of model categories|left Bousfield localization]].

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

Say that a homomorphism $f_\bullet \colon X_\bullet \to Y_\bullet$ in the category $SeqSpec(sSet)$, def. \ref{SequentialSpectra} is

* a **strict weak equivalence** if each component $f_n \colon X_n \to Y_n$ is a weak equivalence in the [[classical model structure on simplicial sets]] (hence a [[weak homotopy equivalence]] of [[geometric realizations]]);

* a **strict fibration** if each component $f_n \colon X_n \to Y_n$ is a fibration in the [[classical model structure on simplicial sets]] (hence a [[Kan fibration]]);

* a **strict cofibration** if the simplicial maps $f_0\colon X_0 \to Y_0$ as well as all [[pushout products]] of $f_n$ with the structure maps of $X$

  $$
    X_{n+1}\underset{S^1 \wedge X_n}{\coprod} S^1 \wedge Y_n
    \longrightarrow
    Y_{n+1}
  $$

  are cofibrations of simplicial sets in the  [[classical model structure on simplicial sets]] (i.e.: [[monomorphisms]] of simplicial sets);

=--

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

The classes of morphisms in def. \ref{ClassesOfMorphismsOfTheStrictModelStructureOnSequentialSpectra} give the structure of a [[model category]] $SeqSpec(sSet)_{strict}$, called the **strict model structure** on sequential spectra.

Moreover, this is

* a [[proper model category]];

* a [[simplicial model category]] with respect to the simplicial enrichment of prop. \ref{SimplicialEnrichment}.

=--

([Bousfield-Friedlander 78, prop. 2.2](#BousfieldFriedlander78)).

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

The representation of [sequential spectra as diagram spectra](sequential%20spectrum#AsDiagramSpectra) says that the category of sequential spectra is [[equivalence of categories|equivalently]] an [[enriched functor category]]

$$
  SeqSpec(sSet)
  \simeq
  [StdSpheres, sSet^{\ast/}]
$$

([this proposition](sequential+spectrum#SequentialSpectraAsDiagramSpectra)). Accordingly, this carries the [[projective model structure on enriched functors]], and unwinding the definitions, this gives the statement for the fibrations and the weak equivalences.

It only remains to check that the cofibrations are as claimed. To that end, consider a [[commuting square]] of sequential spectra

$$
  \array{
    X_\bullet &\stackrel{h_\bullet}{\longrightarrow}& A_\bullet
    \\
    \downarrow^{\mathrlap{f_\bullet}} && \downarrow
    \\
    Y_\bullet &\longrightarrow& B_\bullet
  }
  \,.
$$

By definition, this is equivalently a $\mathbb{N}$-collection of commuting diagrams of simplicial sets of the form

$$
  \array{
    X_n &\stackrel{h_n}{\longrightarrow}& A_n
    \\
    \downarrow^{\mathrlap{f_n}} && \downarrow
    \\
    Y_n &\longrightarrow& B_n
  }
$$

such that all structure maps are respected.

$$
  \array{
    X_n &\stackrel{\sigma_n^X}{\longrightarrow}& X_{n+1}
    \\
    \downarrow^{\mathrlap{f_n}} && \downarrow^{\mathrlap{f_{n+1}}}
    \\
    Y_n &\stackrel{\sigma_n^Y}{\longrightarrow}& Y_{n+1}
    \\
    & \searrow && \searrow
    \\
    && B_n &\stackrel{\sigma_n^B}{\longrightarrow}& B_{n+1}
  }
  \;\;\;
  \Rightarrow
  \;\;\;
  \array{
    X_n &\stackrel{\sigma_n^X}{\longrightarrow}& X_{n+1}
    \\
    & \searrow^{\mathrlap{h_n}} && \searrow^{\mathrlap{h_{n+1}}}
    \\
    && A_n &\stackrel{\sigma_n^A}{\longrightarrow}& A_{n+1}
    \\
    && \downarrow && \downarrow
    \\
    && B_n &\stackrel{\sigma_n^B}{\longrightarrow}& B_{n+1}
  }
  \,.
$$

Hence a [[lifting]] in the original diagram is a lifting in each degree $n$, such that the lifting in degree $n+1$ makes these diagrams of structure maps commute.

Since components are parameterized over $\mathbb{N}$, this condition has solutions by [[induction]]. First of all there must be an ordinary lifting in degree 0. Then assume a lifting $l_n$ in degree $n$ has been found

$$
  \array{
    X_n &\stackrel{h_n}{\longrightarrow}& A_n
    \\
    \downarrow^{\mathrlap{f_n}} &\nearrow_{\mathrlap{l_n}}& \downarrow
    \\
    Y_n &\longrightarrow& B_n
  }
$$

the lifting $l_{n+1}$ in the next degree has to also make the following diagram commute

$$
  \array{
    X_n &\stackrel{\sigma_n^X}{\longrightarrow}& X_{n+1}
    \\
    \downarrow^{\mathrlap{f_n}} && \downarrow^{\mathrlap{h_{n+1}}}
    & \searrow
    \\
    Y_n &\stackrel{\sigma_n^Y}{\longrightarrow}& Y_{n+1}
    &&
    \\
    & \searrow^{\mathrlap{l_n}} && \searrow^{\mathrlap{l_{n+1}}} & \downarrow
    \\
    && A_n &\stackrel{\sigma_n^A}{\longrightarrow}& A_{n+1}
  }
  \,.
$$

This is a [[cocone]] under the the commuting square for the structure maps, and therefore the outer diagram is equivalently a morphism out of the [[domain]] of the [[pushout product]] $f_n \Box \sigma_n^X$, while the compatible lift $l_{n+1}$ is equivalently a lift against this pushout product:

$$
  \array{
    Y_n \underset{X_n}{\sqcup} X_{n+1}
    &\stackrel{(\sigma_n^A l_n,h_{n+1})}{\longrightarrow}&
    A_{n+1}
    \\
    \downarrow &{}^{\mathllap{l_{n+1}}}\nearrow& \downarrow
    \\
    Y_{n+1} &\stackrel{}{\longrightarrow}& B_{n+1}
  }
  \,.
$$

=--

## The stable model structure on sequential spectra
 {#TheStableModelStructure}

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

Say that a homomorphism $f_\bullet \colon X_\bullet \to Y_\bullet$ in the category $SeqSpec(sSet)$, def. \ref{SequentialSpectra} is

* a **stable weak equivalence** if it is a [[stable weak homotopy equivalence]], def. \ref{StableWeakEquivalenceOfSequentialsSetSpectra};

* a **stable cofibration** if the simplicial maps $f_0\colon X_0 \to Y_0$ as well as all [[pushout products]] of $f_n$ with the structure maps of $X$

  $$
    X_{n+1}\underset{S^1 \wedge X_n}{\coprod} S^1 \wedge Y_n
    \longrightarrow
    Y_{n+1}
  $$

  are cofibrations of simplicial sets in the [[classical model structure on simplicial sets]] (i.e.: [[monomorphisms]] of simplicial sets);

* a **stable fibration** if it is degreewise a fibration of simplicial sets, hence degreewise a [[Kan fibration]], and if in addition the naturality squares of the [[spectrification]], def. \ref{Spectrification},

  $$
    \array{
      X_n &\stackrel{}{\longrightarrow}& (Q X)_n
      \\
      \downarrow^{\mathrlap{f_n}} && \downarrow^{\mathrlap{Q f_n}}
      \\
      Y_n &\stackrel{}{\longrightarrow}& (Q Y)_n
    }
  $$

  are [[homotopy pullback]] squares (with respect to the [[classical model structure on simplicial sets]]).

=--

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

The classes of morphisms in def. \ref{ClassesOfMorphismsOfTheStableModelStructureOnSequentialSpectra} give the structure of a [[model category]] $SeqSpec(sSet)_{stable}$, called the **stable model structure** on sequential spectra.

Moreover, this is

* a [[proper model category]];

* a [[simplicial model category]] with respect to the simplicial enrichment of prop. \ref{SimplicialEnrichment}.

=--

([Bousfield-Friedlander 78, theorem 2.3](#BousfieldFriedlander78)).

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

By corollary \ref{StableWeakHomotopyEquivalencesofSeqsSetSpectraIsDegreewsieWeakHomotopyEquivalencesOfSpectrification}, the stable model structure $SeqSpectra(sSet)_{stable}$ is, if indeed it exists, the [[left Bousfield localization]] of the strict model structure of prop. \ref{StrictModelStructureOnSequentialPrespectraIsModelCategory} at the morphisms that become weak equivalences under the [[spectrification]] functor $Q \colon SeqSpectra(sSet) \longrightarrow SeqSpectra(sSet)$, def. \ref{Spectrification}. By prop. \ref{PropertiesOfStandardSpectrification}  $Q$ satisfies the conditions of the [[Bousfield-Friedlander theorem]], and this implies the claim.

=--

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

A spectrum $X \in SeqSpec(sSet)_{stable}$ is

* fibrant precisely if it is an [[Omega-spectrum]], def. \ref{OmegaSpectrum}, and each $X_n$ is a [[Kan complex]];

* cofibrant precisely if all the structure maps $S^1 \wedge X_n \to X_{n+1}$ are cofibrations of simplicial sets, i.e. [[monomorphisms]].

=--

## Properties

### Fibrations and cofibrations
 {#FibrantAndCofibrantObjects}

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

A [[sequential spectrum]] $X\in SeqSpec(sSet)_{stable}$ is cofibrant precisely if all its structure morphisms $S^1 \wedge X_n \to X_{n+1}$ are [[monomorphisms]].

=--

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

A morphism $\ast \to X$ is a cofibration according to def. \ref{ClassesOfMorphismsOfTheStrictModelStructureOnSequentialSpectra} (in either the strict or stable model structure, they have the same cofibrations) if

1. $X_0$ is cofibrant; this is no condition in [[sSet]];

1. $$
     \ast_{n+1}\underset{S^1 \wedge \ast_n}{\coprod} S^1 \wedge X_n
     \longrightarrow
     X_{n+1}
   $$

   is a cofibration. But in this case the [[pushout]] reduces to just its second summand, and so this is now equivalent to

   $$
     S^1 \wedge X_n \longrightarrow X_{n+1}
   $$

   being cofibrations;  hence inclusions.

=--

### Relation to sequential spectra in $Top$ and to combinatorial spectra

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

There is a [[zig-zag]] of [[Quillen equivalences]] relating the Bousfield-Friedlander model structure $SeqSpec(sSet)_{stable}$, def. \ref{ClassesOfMorphismsOfTheStableModelStructureOnSequentialSpectra}, prop. \ref{StableModelStructureOnSequentialSpectraIsModelCategory} with standard model structures on [[sequential spectra]] in [[topological spaces]] (the [[model structure on topological sequential spectra]]) and with Kan's [[combinatorial spectra]].

=--

([Bousfield-Friedlander 78, section 2.5](#BousfieldFriedlander78)).

### Relation to symmetric spectra

There is a [[Quillen equivalence]] to the [[model structure on symmetric spectra]] ([Hovey-Shipley-Smith 00, section 4.3](#HoveyShipleySmith00), [Mandell-May-Schwede-Shipley 01, theorem 0.1](#MandellMaySchwedeShipley01)).

### Relation to excisive functors
 {#RelationToExcisiveFunctors}

There is a [[Quillen equivalence]] between the Bousfield-Friedlander model structure and a [[model structure for excisive functors]] ([Lydakis 98](#Lydakis98)).

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

Write

* [[sSet]] for the [[category]] of [[simplicial sets]];

* $sSet^{\ast/}$ for the category of [[pointed object|pointed]] simplicial sets;

* $sSet_{fin}^{\ast/}\simeq s(FinSet)^{\ast/} \hookrightarrow sSet^{\ast/}$ for the [[full subcategory]] of [[pointed object|pointed]] [[simplicial object|simplicial]] [[finite sets]].

Write

$$
  sSet^{\ast/}
  \stackrel{\overset{(-)_+}{\longleftarrow}}{\underset{u}{\longrightarrow}}
  sSet
$$

for the [[free-forgetful adjunction]], where the [[left adjoint]] functor $(-)_+$ freely adjoins a base point.

Write

$$
  \wedge \colon sSet^{\ast/} \times sSet^{\ast/} \longrightarrow sSet^{\ast/}
$$

for the [[smash product]] of [[pointed object|pointed]] [[simplicial sets]], similarly for its restriction to $sSet_{fin}^{\ast}$:

$$
  X \wedge Y
    \coloneqq
  cofib\left(
    \;
    \left(\,
      (u(X),\ast) \sqcup (\ast, u(Y))
    \,\right)
      \longrightarrow
     u(X) \times u(Y)
    \;
  \right)
  \,.
$$

This gives $sSet^{\ast/}$ and $sSet^{\ast/}_{fin}$ the structure of a [[closed monoidal category]] and we write

$$
  [-,-]_\ast \;\colon\; (sSet^{\ast/})^{op} \times sSet^{\ast/} \longrightarrow sSet^{\ast/}
$$

for the corresponding [[internal hom]], the pointed [[function complex]] functor.

=--

We regard all the categories in def. \ref{SimplicialSetsPointedAndFinite} canonically as [[simplicially enriched categories]], and in fact regard $sSet^{\ast/}$ and $sSet^{\ast/}_{fin}$ as $sSet^{\ast/}$-[[enriched categories]].

The category that supports a [[model structure for excisive functors]] is the  $sSet^{\ast/}$-[[enriched functor category]]

$$
  [sSet^{\ast/}_{fin}, sSet^{\ast/}]
  \,.
$$

([Lydakis 98, example 3.8, def. 4.4](#Lydakis98))

In order to compare this to to [[sequential spectra]] consider also the following variant.

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

Write $S^1_{std} \coloneqq \Delta[1]/\partial\Delta[1]\in sSet^{\ast/}$ for the standard minimal pointed simplicial [[1-sphere]].

Write

$$
  \iota \;\colon\; StdSpheres \longrightarrow sSet^{\ast/}_{fin}
$$

for the non-full $sSet^{\ast/}$-[[enriched category|enriched]] [[subcategory]] of pointed [[simplicial object|simplicial]] [[finite sets]], def. \ref{SimplicialSetsPointedAndFinite} whose

* [[objects]] are the [[smash product]] powers $S^n_{std} \coloneqq (S^1_{std})^{\wedge^n}$ (the standard minimal simplicial [[n-spheres]]);

* [[hom-objects]] are

  $$
    [S^{n}_{std}, S^{n+k}_{std}]_{StdSpheres}
    \coloneqq
    \left\{
      \array{
        \ast & for & k \lt 0
        \\
        im(S^{k}_{std} \stackrel{}{\to} [S^n_{std}, S^{n+k}_{std}]_{sSet^{\ast/}_{fin}})
        & otherwise
      }
    \right.
  $$

=--

([Lydakis 98, def. 4.2](#Lydakis98))

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

There is an $sSet^{\ast/}$-[[enriched functor]]

$$
  (-)^seq
  \;\colon\;
  [StdSpheres,sSet^{\ast/}]
    \longrightarrow
  SeqSpec(sSet)
$$

(from the category of $sSet^{\ast/}$-[[enriched presheaves|enriched copresheaves]] on the categories of standard simplicial spheres of def. \ref{CategoriesOfStandardSpheres} to the category of sequential spectra in [[sSet]], def. \ref{Spectra}) given on objects by sending $X \in [StdSpheres,sSet^{\ast/}]$ to the sequential spectrum $X^{seq}$ with components

$$
  X^{seq}_n \coloneqq X(S^n_{std})
$$

and with structure maps

$$
  \frac{S^1_{std} \wedge X^{seq}_n \stackrel{\sigma_n}{\longrightarrow} X^{seq}_n}{S^1_{std} \longrightarrow [X^{seq}_n, X^{seq}_{n+1}]}
$$

given by

$$
  S^1_{std}
    \stackrel{\widetilde{id}}{\longrightarrow}
  [S^n_{std}, S^{n+1}_{std}]
    \stackrel{X_{S^n_{std}, S^{n+1}_{std}}}{\longrightarrow}
  [X^{seq}_n, X^{seq}_{n+1}]
  \,.
$$

This is an $sSet^{\ast/}$ [[enriched category theory|enriched]] [[equivalence of categories]].

=--

([Lydakis 98, prop. 4.3](#Lydakis98))

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

The [[adjunction]]

$$
  (\iota_\ast \dashv \iota^\ast)
  \;\colon\;
  [sSet^{\ast/}_{fin}, sSet^{\ast/}]_{Ly}
    \stackrel{\overset{\iota_\ast}{\longleftarrow}}{\underset{\iota^\ast}{\longrightarrow}}
  [StdSpheres, sSet^{\ast/}]
    \underoverset{\simeq}{(-)^{seq}}{\longrightarrow}
  SeqSpec(sSet)_{stable}
$$

(given by restriction $\iota^\ast$ along the defining inclusion $\iota$ of def. \ref{CategoriesOfStandardSpheres} and by left [[Kan extension]] $\iota_\ast$ along $\iota$ and combined with the equivalence $(-)^{seq}$ of prop. \ref{SequentialSpectraAsSimplicialFunctorsOnStandardSpheres}) is a  [[Quillen adjunction]] and in fact a [[Quillen equivalence]] between the [[Bousfield-Friedlander model structure]] on sequential spectra and Lydakis' [[model structure for excisive functors]].

=--

([Lydakis 98, theorem 11.3](#Lydakis98)) For more details see at _[[model structure for excisive functors]]_.

## Related concepts

* [[model structure on topological sequential spectra]]

## References

The original construction is due to

* {#BousfieldFriedlander78} [[Aldridge Bousfield]], [[Eric Friedlander]], _Homotopy theory of $\Gamma$-spaces, spectra, and bisimplicial sets_, Springer Lecture Notes in Math., Vol. 658, Springer, Berlin, 1978, pp. 80-130. ([pdf](https://web.math.rochester.edu/people/faculty/doug/otherpapers/bousfield-friedlander.pdf), [[BousfieldFriedlanderSpectra.pdf:file]])

Generalization of this model structure from sequential pre-spectra in [[sSet]]$^{\ast/}$ to sequential spectra in more general [[proper model category|proper]] [[pointed category|pointed]] [[simplicial model categories]] is in

* {#Schwede97} [[Stefan Schwede]], _Spectra in model categories and applications to the algebraic cotangent complex_, Journal of Pure and Applied Algebra 120 (1997) 77-104 ([pdf](http://www.math.uni-bonn.de/people/schwede/modelspec.pdf))

Discussion of the [[Quillen equivalence]] to the [[model structure on excisive functors]] (which does have a [[symmetric smash product of spectra]]) is in

* {#Lydakis98} Lydakis, _Simplicial functors and stable homotopy theory_ Preprint, available via Hopf archive, 1998 ([pdf](http://hopf.math.purdue.edu/Lydakis/s_functors.pdf))

Discussion of the [[Quillen equivalence]] to the [[model structure on symmetric spectra]] is in

* {#HoveyShipleySmith00} [[Mark Hovey]], [[Brooke Shipley]], [[Jeff Smith]], _Symmetric spectra_, J. Amer. Math. Soc. 13 (2000), 149-208 ([arXiv:math/9801077](http://arxiv.org/abs/math/9801077))

* {#MandellMaySchwedeShipley01} [[Michael Mandell]], [[Peter May]], [[Stefan Schwede]], [[Brooke Shipley]], theorem 0.1 of _Model categories of diagram spectra_, Proceedings of the London Mathematical Society, 82 (2001), 441-512 ([pdf](http://www.math.uchicago.edu/~may/PAPERS/mmssLMSDec30.pdf))

[[!redirects Bousfield-Friedlander model category]]

[[!redirects Bousfield-Friedlander model structures]]


Bousfield-Friedlander theorem

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Model category theory
+--{: .hide}
[[!include model category theory - contents]]
=--
#### Modalities, Closure and Reflection
+-- {: .hide}
[[!include modalities - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[model category]]-theory, the _Bousfield-Friedlander theorem_ ([Bousfield-Friedlander 78, theorem A.7](#BousfieldFriedlander78), [Bousfield 01, theorem 9.3](#Bousfield01)) states that if an [[endofunctor]] $Q \colon \mathcal{C} \to \mathcal{C}$ on a [[model category]] $\mathcal{C}$ behaves like an [[idempotent monad]] in an appropriate model category theoretic sense, then the [[left Bousfield localization]] [[model category]] structure of $\mathcal{C}$ at the $Q$-equivalences exists.

The original proof assumed that $\mathcal{C}$ is a [[proper model category|right-proper model category]], but it turns out that this condition is not necessary ([Stanculescu 08, theorem 1.1](#Stanculescu08)).

## Statement

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

Let $\mathcal{C}$ be a [[proper model category]].
Say that a **Quillen idempotent monad** on $\mathcal{C}$ is

1. an [[endofunctor]]

   $Q \;\colon\;  \mathcal{C} \longrightarrow \mathcal{C}$

1. a [[natural transformation]]

   $\eta \colon id_{\mathcal{C}} \longrightarrow Q$

such that

1. ([[homotopical functor]]) $Q$ preserves weak equivalences;

1. (idempotency) for all $X \in \mathcal{C}$ the morphisms

   $$
     Q(\eta_X) \;\colon\; Q(X) \overset{\in W}{\longrightarrow} Q(Q(X))
   $$

   and

   $$
     \eta_{Q(X)} \;\colon\; Q(X) \overset{\in W}{\longrightarrow} Q(Q(X))
   $$

   are weak equivalences;

1. (right-properness of the localization) if in a [[pullback]] square in $\mathcal{C}$

   $$
     \array{
       f^\ast W &\stackrel{f^\ast h}{\longrightarrow}& X
       \\
       \downarrow && \downarrow^{\mathrlap{f}}
       \\
       W &\stackrel{h}{\longrightarrow}& Y
     }
   $$

   we have that

   1. $f$ is a fibration;

   1. $\eta_X$, $\eta_Y$, and $Q(h)$ are weak equivalences

   then $Q(f^\ast h)$ is a weak equivalence.

=--

(Here the formulation of the third item follows [Bousfield 01, def. 9.2](#Bousfield01). By lemma \ref{SecondLemmaForBousfieldFriedlander} below this condition implies that $f$ is a $Q$-fibration, which is the condition required in [Bousfield-Friedlander 78 (A.6)](#BousfieldFriedlander78)).

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

For $Q \colon \mathcal{C} \longrightarrow \mathcal{C}$
a Quillen idempotent monad according to def. \ref{QuillenIdempotentMonad}, say that a morphism $f$ in $\mathcal{C}$ is

1. a **$Q$-weak equivalence** if $Q(f)$ is a weak equivalence;

1. **a $Q$-cofibation** if it is a cofibration.

1. **a $Q$-fibration** if it has the [[right lifting property]] against the morphisms that are both ($Q$-)cofibrations as well as $Q$-weak equivalences.

Write $\mathcal{C}_Q$ for $\mathcal{C}$ equipped with these classes of morphisms.

=--

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

In the situation of def. \ref{ClassesOfMorphismsInBousfieldLocalizationAtQuillenIdempotentMonad},
a morphism is an acyclic fibration in $\mathcal{C}_Q$ precisely if it is an acyclic fibration in $\mathcal{C}$.

=--

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

Let $f$ be a fibration and a weak equivalence. Since $Q$ preserves weak equivalences by condition 1 in def. \ref{QuillenIdempotentMonad}, $f$ is also a $Q$-weak equivalence. Since $Q$-cofibrations are cofibrations, the acyclic fibration $f$ has right lifting against $Q$-cofibrations, hence in particular against against $Q$-acyclic $Q$-cofibrations, hence is a $Q$-fibration.

In the other direction, let $f$ be a $Q$-acyclic $Q$-fibration. Consider its factorization into a cofibration followed by an acyclic fibration

$$
  f \colon \underoverset{\in Cof}{i}{\longrightarrow} \underoverset{\in W \cap Fib}{p}{\longrightarrow}
  \,.
$$

Now the fact that $Q$ preserves weak equivalences together with [[two-out-of-three]] implies that $i$ is a $Q$-weak equivalence, hence a $Q$-acyclic $Q$-cofibration. This means by assumption that $f$ has the [[right lifting property]] against $i$. Hence the [[retract argument]], implies that $f$ is a [[retract]] of the acyclic fibration $p$, and so is itself an acyclic fibration.

=--

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

In the situation of def. \ref{ClassesOfMorphismsInBousfieldLocalizationAtQuillenIdempotentMonad}, if a morphism $f \colon X \longrightarrow Y$ is a fibration, and $\eta_X, \eta_Y$ are weak equivalences, then $f$ is a $Q$-fibration.

=--

(e.g. [Goerss-Jardine 96, chapter X, lemma 4.4](#GoerssJardine96)).

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

We need to show that for every [[commuting square]] of the form

$$
  \array{
    A &\overset{\alpha}{\longrightarrow}& X
    \\
    {}^{\mathllap{i}}_{\mathllap{\in W_Q \cap Cof_Q}}\downarrow
     &&
    \downarrow^{\mathrlap{f}}
    \\
    B &\underset{\beta}{\longrightarrow}& Y
  }
$$

there exists a lifting.

To that end, first consider a factorization of the image under $Q$ of this square as follows:

$$
  \array{
    Q(A) &\overset{Q(\alpha)}{\longrightarrow}& Q(X)
    \\
    {}^{\mathllap{Q(i)}}\downarrow && \downarrow^{\mathrlap{Q(f)}}
    \\
    Q(B) &\underset{Q(\beta)}{\longrightarrow}& Q(Y)
  }
  \;\;\;\;\;\;
  \simeq
  \;\;\;\;\;\;
  \array{
    Q(A)
     &\underoverset{\in W \cap Cof}{j_\alpha}{\longrightarrow}&
    Z
     &\underoverset{\in Fib}{p_\alpha}{\longrightarrow}&
    Q(X)
    \\
    {}^{\mathllap{Q(i)}}\downarrow
      &&
    \downarrow^{\pi}
      &&
    \downarrow^{\mathrlap{Q(f)}}
    \\
    Q(B)
      &\underoverset{j_\beta}{\in W \cap Cof}{\longrightarrow}&
    W
      &\underoverset{p_\beta}{\in Fib}{\longrightarrow}&
    Q(Y)
  }
$$

(This exists even without assuming [[functorial factorization]]: factor the bottom morphism, form the pullback of the resulting $p_\beta$, observe that this is still a fibration, and then factor (through $j_\alpha$) the universal morpism from the outer square into this pullback.)

Now consider the pullback of the right square above along the naturality square of $\eta \colon id \to Q$, take this to be the right square in the following diagram

$$
  \array{
    \alpha \colon
    &
    A
      &\overset{(j_\alpha \circ \eta_A, \alpha)}{\longrightarrow}&
    Z \underset{Q(X)}{\times} X
      &\overset{}{\longrightarrow}&
    X
    \\
    & {}^{\mathllap{i}}\downarrow
      &&
    \downarrow^{\mathrlap{(\pi,f)}}
      &&
    \downarrow^{\mathrlap{f}}&
    \\
    \beta \colon
    &
    B
      &\underset{(j_\beta\circ\eta_B,\beta)}{\longrightarrow}&
    W \underset{Q(Y)}{\times} Y
      &\underset{}{\longrightarrow}&
    Y
  }
  \,,
$$

where the left square is the universal morphism into the pullback which is induced from the naturality squares of $\eta$ on $\alpha$ and $\beta$.

We claim that $(\pi,f)$ here is a weak equivalence. This implies that we find the desired lift by factoring $(\pi,f)$ into an acyclic cofibration followed by an acyclic fibration and then lifting consecutively as follows

$$
  \array{
    \alpha \colon
    &
    A
      &\overset{(j_\alpha \circ \eta_A, \alpha)}{\longrightarrow}&
    Z \underset{Q(X)}{\times} X
      &\overset{}{\longrightarrow}&
    X
    \\
    & {}^{\mathllap{id}}\downarrow
      &&
    {}^{\mathllap{\in W \cap Cof}}\downarrow
      &{}^{\mathllap{\exists}}\nearrow&
    \downarrow^{\mathrlap{f}}_{\mathrlap{\in Fib}}&
    \\
    &
    A
      &\longrightarrow&
      &\overset{\phantom{AAAAAAA}}{\longrightarrow}&
    Y
    \\
    & {}^{\mathllap{i}}_{\mathllap{\in Cof}}\downarrow
      &{}^{\mathllap{\exists}}\nearrow&
    \downarrow^{\mathrlap{\in W \cap Fib}}
      &&
    \downarrow^{\mathrlap{id}}&
    \\
    \beta \colon &
    B
      &\underset{(j_\beta\circ\eta_B,\beta)}{\longrightarrow}&
    W \underset{Q(Y)}{\times} Y
      &\longrightarrow&
    Y
  }
  \,.
$$

To see that $(\phi,f)$ indeed is a weak equivalence:

Consider the diagram

$$
  \array{
     Q(A)
       &\underoverset{\in W \cap Cof}{j_\alpha}{\longrightarrow}&
     Z
       &\underoverset{\in W}{pr_1}{\longleftarrow}&
     Z \underset{Q(X)}{\times} X
     \\
     {}^{\mathllap{Q(i)}}_{\mathllap{\in W}}\downarrow
       &&
     \downarrow^{\mathrlap{\pi}}
       &&
     \downarrow^{\mathrlap{(\pi,f)}}
     \\
     Q(B)
       &\underoverset{j_\beta}{\in W \cap Cof}{\longrightarrow}&
     Z
       &\underoverset{pr_2}{\in W}{\longleftarrow}&
     W \underset{Q(X)}{\times} X
  }
  \,.
$$

Here the projections are weak equivalences as shown, because by assumption in def. \ref{QuillenIdempotentMonad} the ambient model category is [[right proper model category|right proper]] and these projections are the pullbacks along the fibrations $p_\alpha$ and $p_\beta$ of the morphisms $\eta_X$ and $\eta_Y$, respectively, where the latter are weak equivalences by assumption. Moreover $Q(i)$ is a weak equivalence, since $i$ is a $Q$-weak equivalence.

Hence now it follows by [[two-out-of-three]] ([def.](Introduction+to+Stable+homotopy+theory+--+P#CategoryWithWeakEquivalences)) that $\pi$ and then $(\pi,f)$ are weak equivalences.

=--

+-- {: .num_prop #BousfieldFriedlanderTheorem}
###### Proposition
**(Bousfield-Friedlander theorem)**

For $Q \colon \mathcal{C} \longrightarrow \mathcal{C}$
a Quillen idempotent monad according to def. \ref{QuillenIdempotentMonad},
then $\mathcal{C}_Q$, def. \ref{ClassesOfMorphismsInBousfieldLocalizationAtQuillenIdempotentMonad} is a [[model category]].

=--

([Bousfield-Friedlander 78, theorem 8.7](#BousfieldFriedlander78), [Bousfield 01, theorem 9.3 ](#Bousfield01), [Goerss-Jardine 96, chapter X lemma 4.5, lemma 4.6](#GoerssJardine96))

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

The existence of [[limits]] and [[colimits]] is guaranteed since $\mathcal{C}$ is already assumed to be a model category.
The [[two-out-of-three]] poperty for $Q$-weak equivalences is an immediate consequence of two-out-of-three for the original weak equivalences of $\mathcal{C}$.

Moreover, according to lemma \ref{FirstLemmaForBousfieldFriedlander} the pair of classes $(Cof_{Q}, W_Q \cap Fib_Q)$ equals the pair $(Cof, W \cap Fib)$, and this is a [[weak factorization system]] by the model structure $\mathcal{C}$.

Hence it remains to show that $(W_Q \cap Cof_Q, \; Fib_Q)$ is a [[weak factorization system]]. The condition $Fib_Q = RLP(W_Q \cap Cof_Q)$ holds by definition of $Fib_Q$. Once we show that every morphism factors as $W_Q \cap Cof_Q$ followed by $Fib_Q$, then the condition $W_Q \cap Cof_Q = LLP(Fib_Q)$ follows from the [[retract argument]] (and the fact that $W_Q$ and $Cof_Q$ are stable under retracts, because $W$ and $Cof = Cof_Q$ are).

So we may conclude by showing the existence of $(W_Q \cap Cof_Q, \; Fib_Q)$ factorizations:

First we consider the case of a morphism of the form $f \colon Q(Y) \to Q(Y)$. This may be factored with respect to $\mathcal{C}$ as

$$
  f
  \;\colon\;
  Q(X)
    \underoverset{\in W \cap Cof}{\in i}{\longrightarrow}
  Z
    \underoverset{\in Fib}{p}{\longrightarrow}
  Q(Y)
  \,.
$$

Here $i$ is already a $Q$-acyclic $Q$-cofibration. Now apply $Q$ to obtain

$$
  \array{
  f
  \colon
  &
  Q(X)
    &\underoverset{\in W \cap Cof}{i}{\longrightarrow}&
  Z
    &\underoverset{\in Fib}{p}{\longrightarrow}&
  Q(Y)
  \\
  &
  \downarrow^{\mathrlap{\eta_{Q(X)}}}_{\mathrlap{\in W}}
    &&
  \downarrow^{\mathrlap{\eta_Z}}
    &&
  \downarrow^{\mathrlap{\eta_{Q(Y)}}}_{\mathrlap{\in W}}
  \\
  &
  Q(Q(X))
    &\underoverset{Q(i)}{\in W}{\longrightarrow}&
  Q(Z)
    &\underset{}{\longrightarrow}&
  Q(Q(Y))
  }
  \,,
$$

where $\eta_{Q(X)}$ and $\eta_{Q(Y)}$ are weak equivalences by idempotency, and $Q(i)$ is a weak equivalence since $Q$ preserves weak equivalences. Hence by [[two-out-of-three]] also $\eta_Z$ is a weak equivalence. Therefore lemma \ref{SecondLemmaForBousfieldFriedlander} gives that $p$ is a $Q$-fibration, and hence the above factorization is already as desired

$$
  f
  \;\colon\;
  Q(X)
    \underoverset{\in W_Q \cap Cof_Q}{\in i}{\longrightarrow}
  Z
    \underoverset{\in Fib_Q}{p}{\longrightarrow}
  Q(Y)
  \,.
$$

Now for $g$ an arbitrary morphism $g \colon X \to Y$, form a factorization of $Q(g)$ as above and then decompose the naturality square for $\eta$ on $g$ into the pullback of the resulting $Q$-fibration along $\eta_Y$:

$$
  \array{
    g \colon
    &
    X
      &\overset{\tilde i}{\longrightarrow}&
    Z \underset{Q(Y)}{\times} Y
      &\overset{\tilde p}{\longrightarrow}&
    Y
    \\
    &
    {}^{\mathllap{\eta_X}}_{\mathllap{\in W_Q}}\downarrow
      &&
    \downarrow^{\mathrlap{\eta'}}_{\mathrlap{}}
      &(pb)&
    \downarrow^{\mathrlap{\eta_Y}}_{\mathrlap{\in W_Q}}
    \\
    Q(g)
    \colon
    &
    Q(X)
      &\underoverset{i}{\in W_Q}{\longrightarrow}&
    Z
      &\underoverset{p}{\in Fib_Q}{\longrightarrow}&
    Q(Y)
  }
  \,.
$$

This exhibits $\eta'$ as the pullback of a $Q$-weak equivalence along a $Q$-fibration, and hence itself as a $Q$-weak equivalence. This way, [[two-out-of-three]] implies that $\tilde i$ is a $Q$-weak equivalence.

Finally, apply factorization in $(Cof,\; W\cap Fib)$ to $\tilde i$ to obtain the desired factorization

$$
  f
    \;\colon\;
  \overset{W_Q \cap Cof}{\longrightarrow}
  \overset{W \cap Fib = W_Q \cap Fib_Q}{\longrightarrow}
  \overset{Fib_Q}{\longrightarrow}
  \,.
$$

=--

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

For $Q \colon \mathcal{C} \longrightarrow \mathcal{C}$
a Quillen idempotent monad according to def. \ref{QuillenIdempotentMonad},
then a morphism $f \colon X \to Y$ in $\mathcal{C}$ is a $Q$-fibration (def. \ref{ClassesOfMorphismsInBousfieldLocalizationAtQuillenIdempotentMonad}) precisely if

1. $f$ is a fibration;

1. the $\eta$-naturality square on $f$

   $$
     \array{
       X &\stackrel{\eta_X}{\longrightarrow}& Q(X)
       \\
       {}^{\mathllap{f}}\downarrow &{}^{(pb)^h}& \downarrow^{\mathrlap{Q(f)}}
       \\
       Y &\underset{\eta_Y}{\longrightarrow}& Q(Y)
     }
   $$

   exhibits a [[homotopy pullback]] in $\mathcal{C}$ ([def.](Introduction+to+Stable+homotopy+theory+--+P#HomotopyPullback)), in that for any factorization of $Q(f)$ through a weak equivalence followed by a fibration $p$, then the universally induced morphism

   $$
     X \longrightarrow p^\ast Y
   $$

   is weak equivalence (in $\mathcal{C}$).

=--

(e.g. [Goerss-Jardine 96, chapter X, theorem 4.8](#GoerssJardine96))

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

First consider the case that $f$ is a fibration and that the square is a homotopy pullback. We need to show that then $f$ is a $Q$-fibration.

Factor $Q(f)$ as

$$
  Q(f)
    \;\colon\;
  Q(X)
    \underoverset{\in W \cap Cof}{i}{\longrightarrow}
  Z
    \underoverset{\in Fib}{p}{\longrightarrow}
  Q(Y)
  \,.
$$

By the proof of prop. \ref{BousfieldFriedlanderTheorem}, the morphism $p$ is also a $Q$-fibration. Hence by the existence of the $Q$-local model structure, also due to prop. \ref{BousfieldFriedlanderTheorem}, its [[pullback]] $\tilde p$ is also a $Q$-fibration

$$
  \array{
    X &\overset{\eta_X}{\longrightarrow}& Q(X)
    \\
    {}^{\mathllap{\tilde i}}_{\mathllap{\in W}}\downarrow
      &&
    \downarrow^{\mathrlap{i}}_{\mathrlap{\in W}}
    \\
    Y \underset{Q(Y)}{\times} Z
      &\overset{p^\ast \eta_Y}{\longrightarrow}&
    Z
    \\
    {}^{\mathllap{\tilde p}}_{\mathllap{\in Fib_Q}}\downarrow
      &(pb)&
    \downarrow^{\mathrlap{p}}_{\mathrlap{\in Fib_Q}}
    \\
    Y &\underset{\eta_Y}{\longrightarrow}& Q(Y)
  }
  \,.
$$

Here $\tilde i$ is a weak equivalence by assumption that the diagram exhibits a [[homotopy pullback]]. Hence it factors as

$$
  \tilde i
  \;\colon\;
  X
    \underoverset{\in W \cap Cof}{j}{\longrightarrow}
  \hat X
    \underoverset{\in W \cap Fib = W_Q \cap Fib_Q}{\pi}{\longrightarrow}
  Y \underset{Q(Y)}{\times} Z
  \,.
$$

This yields the situation

$$
  \array{
    X &\overset{=}{\longrightarrow}& X
    \\
    {}^{\mathllap{j}}_{\mathllap{\in W \cap Cof}}\downarrow
      &{}^{\mathllap{\exists}}\nearrow&
    \downarrow^{\mathrlap{f}}_{\mathrlap{\in Fib}}
    \\
    \hat X
      &\underoverset{\tilde p \circ \pi}{\in Fib_Q}{\longrightarrow}&
    Y
  }
  \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
  \leftrightarrow
  \;\;\;\;\;\;\;\;\;\;\;\;
  \array{
    X
      &\overset{j}{\longrightarrow}&
    \hat X
      &\overset{\exists}{\longrightarrow}&
    X
    \\
    {}^{\mathllap{}f}\downarrow
      &&
    \downarrow^{\mathrlap{\tilde p \circ \pi}}
      &&
    \downarrow^{\mathrlap{f}}
    \\
    Y &=& Y &=& Y
  }
  \,.
$$

As in the [[retract argument]] ([prop.](Introduction+to+Stable+homotopy+theory+--+P#RetractArgument)) this diagram exhibits $f$ as a [[retract]] (in the [[arrow category]], [rmk.](Introduction+to+Stable+homotopy+theory+--+P#RetractsOfMorphisms)) of the $Q$-fibration $\tilde p \circ \pi$. Hence by the existence of the $Q$-model structure (prop. \ref{BousfieldFriedlanderTheorem}) and by the closure properties for fibrations ([prop.](Introduction+to+Stable+homotopy+theory+--+P#ClosurePropertiesOfInjectiveAndProjectiveMorphisms)), also $f$ is a $Q$-fibration.

Now for the converse. Assume that $f$ is a $Q$-fibration. Since $\mathcal{C}_Q$ is a [[Bousfield localization of model categories|left Bousfield localization]] of $\mathcal{C}$ (prop. \ref{BousfieldFriedlanderTheorem}), $f$ is also a fibration. We need to show that the $\eta$-naturality square on $f$ exhibits a homotopy pullback.

So factor $Q(f)$ as before, and consider the pasting composite of the factorization of the given square with the naturality squares of $\eta$:

$$
  \array{
    X
      &\underoverset{\in W_Q}{\eta_X}{\longrightarrow}&
    Q(X)
      &\underoverset{\in W \subset W_Q}{\eta_{Q(X)}}{\longrightarrow}&
    Q(Q(X))
    \\
    {}^{\mathllap{\tilde i}}_{\mathllap{\in W_Q}}\downarrow
      &&
    {}^{\mathllap{i}}_{\mathllap{\in W\subset W_Q}}\downarrow
      &&
    \downarrow^{\mathrlap{Q(i)}}_{\mathrlap{\in W}}
    \\
    Y \underset{Q(Y)}{\times} Z
      &\underoverset{\in W_Q}{p^\ast \eta_Y}{\longrightarrow}&
    Z
      &\underoverset{\in W}{\eta_Z}{\longrightarrow}&
    Q(Z)
    \\
    {}^{\mathllap{\tilde p}}_{\mathllap{\in Fib_Q}}\downarrow
      &(pb)&
    \downarrow^{\mathrlap{p}}_{\mathrlap{\in Fib_Q \subset Fib}}
      &&
    \downarrow^{\mathrlap{Q(p)}}
    \\
    Y
      &\underoverset{\eta_Y}{\in W_Q}{\longrightarrow}&
    Q(Y)
      &\underoverset{\eta_{Q(Y)}}{\in W \subset W_Q}{\longrightarrow}&
    Q(Q(Y))
  }
  \,.
$$

Here the top and bottom horizontal morphisms are weak ($Q$-)equivalences by the idempotency of $Q$, and $Q(i)$ is a weak equivalence since $Q$ preserves weak equivalences (first and second clause in def. \ref{QuillenIdempotentMonad}). Hence by [[two-out-of-three]] also $\eta_Z$ is a weak equivalence. From this, lemma \ref{SecondLemmaForBousfieldFriedlander} gives that $p$ is a $Q$-fibration.
Then $p^\ast \eta_Y$ is a $Q$-weak equivalence since it is the pullback of a $Q$-weak equivalence along a fibration between objects whose $\eta$ is a weak equivalence, via the third clause in def. \ref{QuillenIdempotentMonad}. Finally [[two-out-of-three]] implies that $\tilde i$ is a $Q$-weak equivalence.

In particular, the bottom right square is a homotopy pullback (since two opposite edges are weak equivalences, by [this prop.](Introduction+to+Stable+homotopy+theory+--+P#HomotopyPullbackOfWeakEquivalences)), and since the left square is a genuine pullback of a fibration, hence a homotopy pullback, the total bottom rectangle here exhibits a homotopy pullback by the [[pasting law]] for homotopy pullbacks ([prop.](Introduction+to+Stable+homotopy+theory+--+P#ClosurePropertiesOfHomotopyPullbacks)).

Now by [[natural transformation|naturality]] of $\eta$, that total bottom rectangle is the same as the following rectangle

$$
  \array{
    Y \underset{Q(Y)}{\times} Z
      &\overset{\eta_{\left(Q \underset{Q(Y)}{\times} Z\right)}}{\longrightarrow}&
    Q(Y \underset{Q(Y)}{\times} Z)
      &\underoverset{\in W}{Q(p^\ast \eta_Y)}{\longrightarrow}&
    Q(Z)
    \\
    {}^{\mathllap{\tilde p}}_{\mathllap{\in Fib_Q}}\downarrow
      &&
    \downarrow^{\mathrlap{Q(\tilde p)}}_{\mathrlap{}}
      &&
    \downarrow^{\mathrlap{Q(p)}}
    \\
    Y
      &\underset{\eta_Y}{\longrightarrow}&
    Q(Y)
      &\underoverset{Q(\eta_Y)}{\in W}{\longrightarrow}&
    Q(Q(Y))
  }
  \,,
$$

where now $Q(p^\ast \eta_Y) \in W$ since $p^\ast \eta_Y \in W_Q$, as we had just established. This means again that the right square is a homotopy pullback ([prop.](Introduction+to+Stable+homotopy+theory+--+P#HomotopyPullbackOfWeakEquivalences)), and since the total rectangle still is a homotopy pullback itself, by the previous remark, so is now also the left square, by the other direction of the [[pasting law]] for homotopy pullbacks ([prop.](Introduction+to+Stable+homotopy+theory+--+P#ClosurePropertiesOfHomotopyPullbacks)).

So far this establishes that the $\eta$-naturality square of $\tilde p$ is a homotopy pullback. We still need to show that also the $\eta$-naturality square of $f$ is a homotopy pullback.

Factor $\tilde i$ as a cofibration followed by an acyclic fibration. Since $\tilde i$ is also a $Q$-weak equivalence, by the above, [[two-out-of-three]] for $Q$-fibrations gives that this factorization is of the form

$$
  \array{
    X
      &\underoverset{\in W_Q \cap Cof = W_Q \cap Cof_Q}{j}{\longrightarrow}&
   \hat X
      &\underoverset{\in W \cap Fib = W_Q \cap Fib_Q }{\pi}{\longrightarrow}&
   Y\underset{Q(Y)}{\times} Z
  }
  \,.
$$

As in the first part of the proof, but now with $(W \cap Cof, Fib)$ replaced by $(W_Q \cap Cof_Q, Fib_Q)$ and using lifting in the $Q$-model structure, this yields the situation

$$
  \array{
    X &\overset{=}{\longrightarrow}& X
    \\
    {}^{\mathllap{j}}_{\mathllap{\in W_Q \cap Cof_Q}}\downarrow
      &{}^{\mathllap{\exists}}\nearrow&
    \downarrow^{\mathrlap{f}}_{\mathrlap{\in Fib_Q}}
    \\
    \hat X
      &\underoverset{\tilde p \circ \pi}{}{\longrightarrow}&
    Y
  }
  \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
  \leftrightarrow
  \;\;\;\;\;\;\;\;\;\;\;\;
  \array{
    X
      &\overset{j}{\longrightarrow}&
    \hat X
      &\overset{\exists}{\longrightarrow}&
    X
    \\
    {}^{\mathllap{}f}\downarrow
      &&
    \downarrow^{\mathrlap{\tilde p \circ \pi}}
      &&
    \downarrow^{\mathrlap{f}}
    \\
    Y &=& Y &=& Y
  }
  \,.
$$

As in the [[retract argument]] ([prop.](Introduction+to+Stable+homotopy+theory+--+P#RetractArgument)) this diagram exhibits $f$ as a [[retract]] (in the [[arrow category]], [rmk.](Introduction+to+Stable+homotopy+theory+--+P#RetractsOfMorphisms)) of $\tilde p \circ \pi$.

Observe that the $\eta$-naturality square of the weak equivalence $\pi$ is a [[homotopy pullback]], since $Q$ preserves weak equivalences (first clause of def. \ref{QuillenIdempotentMonad}) and since a square with two weak equivalences on opposite sides is a homotopy pullback ([prop.](Introduction+to+Stable+homotopy+theory+--+P#HomotopyPullbackOfWeakEquivalences)). It follows that also the $\eta$-naturality square of $\tilde p \circ \pi$ is a homotopy pullback, by the [[pasting law]] for homotopy pullbacks ([prop.](Introduction+to+Stable+homotopy+theory+--+P#ClosurePropertiesOfHomotopyPullbacks)).

In conclusion, we have exhibited $f$ as a [[retract]] (in the [[arrow category]], [rmk.](Introduction+to+Stable+homotopy+theory+--+P#RetractsOfMorphisms)) of a morphism $\tilde p \circ \pi$ whose $\eta$-naturality square is a homotopy pullback. By [[natural transformation|naturality]] of $\eta$, this means that the whole $\eta$-naturality square of $f$ is a retract (in the category of commuting squares in $\mathcal{C}$) of a homotopy pullback square. This means that it is itself a homotopy pullback square ([prop.](Introduction+to+Stable+homotopy+theory+--+P#ClosurePropertiesOfHomotopyPullbacks)).

=--

## Examples

### Stable model structure on sequential spectra

The [[Bousfield-Friedlander model structure]] $SeqSpectra_{stable}$ on [[sequential spectra]] (in any [[proper model category|proper]], [[pointed category|pointed]] [[simplicial model category]]), modelling [[stable homotopy theory]], arises via the Bousfield-Friedlander theorem from localizing the strict model structure $SeqSpectra_{strict}$ [[transferred model structure|transferred]] from the model structure on sequences (in the [[classical model structure on simplicial sets]]/[[classical model structure on topological spaces|on topological spaces]]) at $Q$ being the [[spectrification]] endofunctor.

(For pre-spectra in the [[classical model structure on simplicial sets]], [[spectrification]] is readily defined, more generally one needs to prooceed as in [Schwede 97, section 2.1](spectrification#Schwede97).)

## References

The theorem is due to

* {#BousfieldFriedlander78} [[Aldridge Bousfield]], [[Eric Friedlander]], section A.3 of  _Homotopy theory of $\Gamma$-spaces, spectra, and bisimplicial sets_, Springer Lecture Notes in Math., Vol. 658, Springer, Berlin, 1978, pp. 80-130. ([pdf](https://ncatlab.org/nlab/files/BousfieldFriedlanderSpectra.pdf))

and in improved form due to

* {#Bousfield01} [[Aldridge Bousfield]], section 9 of _On the telescopic homotopy theory of spaces_, Trans. Amer. Math. Soc. 353 (2001), no. 6, 2391–2426 ([AMS](http://www.ams.org/journals/tran/2001-353-06/S0002-9947-00-02649-0/), [jstor](http://www.jstor.org/stable/221952))

The right-properness condition is shown to be unnecessary in

* {#Stanculescu} [[Alexandru E. Stanculescu]], *Note on a theorem of Bousfield and Friedlander*, Topology and its Applications **155** 13 (2008) 1434-1438 [[arxiv:0806.4547](https://arxiv.org/abs/0806.4547), [doi:10.1016/j.topol.2008.05.003](https://doi.org/10.1016/j.topol.2008.05.003)]

Textbook accounts include

* {#GoerssJardine96} [[Paul Goerss]], [[Rick Jardine]], section X.4 of _[[Simplicial homotopy theory]]_, (1996)


Bousfield-Kan map

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{:toc}

## Idea

The Bousfield--Kan map(s) are comparison morphisms in a [[simplicial model category]] between two different "puffed up" versions of (co)limits over (co-)simplicial objects: one close to a [[homotopy limit|homotopy (co)limit]] and the other a version of nerve/geometric realization.

## Definition

Let $C$ be an ([[SimpSet|SSet]],$\otimes = \times$)-[[enriched category]] and write

$$
  \Delta : \Delta \to Set^{\Delta^{op}}
$$

for the canonical cosimplicial [[simplicial set]] (the [[adjunct]] of the [[hom-functor]] $\Delta^{op} \times \Delta \to Set$).

Write furthermore $N(\Delta/-) : \Delta \to Set^{\Delta^{op}}$ for the [[fat simplex]], the  [[cosimplicial object|cosimplicial]] [[simplicial set]] which assigns to $[n]$ the [[nerve]] of the [[overcategory]] $\Delta / [n]$.

The **Bousfield--Kan map of cosimplicial simplicial maps** is a canonical morphism

$$
  \varphi
  \,\colon\,
  N\big(\Delta/(-)\big)
   \longrightarrow
  \Delta
$$

of cosimplicial simplicial sets.

This can also be regarded as a morphism

$$
  \varphi
    \,\colon\,
  N\big(
    (-)/\Delta^{op}
  \big)^{op}
    \longrightarrow
  \Delta
  \,.
$$

This morphism induces the following morphisms between (co)[[simplicial objects]] in $C$.

### Bousfield--Kan for simplicial objects

For $X \,\colon\, \Delta^\op \to C$ any [[simplicial object]] in $C$, the **realization** of $X$ is the [[end|coend]]

$$
  {|X|}
    \;\coloneqq\;
  X \otimes_{\Delta^{op}} \Delta
    \;\coloneqq\;
  \int^{[n] \in \Delta}
    X_n \otimes \Delta^n
  \,,
$$

where in the integrand we have the *[[copower]]* (or *[[tensoring]]*) of $C$ by [[sSet]].

Here the **Bousfield--Kan** map is the morphism

$$
  X \otimes_{\Delta^{op}}
   N\big((-)/\Delta^{op}\big)^{op}
  \stackrel
    {Id_X \otimes_{\Delta^{op}} \phi }
    {\longrightarrow}
  X \otimes_{\Delta^{op}} \Delta
  \,.
$$

### Bousfield--Kan for cosimplicial objects

For $X \,\colon\, \Delta \to C$ any cosimplicial object, its **totalization** is the $\Delta$-[[weighted limit]]

$$
  Tot X
    \;\coloneqq\;
  lim^\Delta X
    \;\simeq\;
  \int_{[n \in \Delta]} X_n^{\Delta^n}
  \,,
$$

where in the integrand we have the [[power]] or cotensor $X_n^{\Delta^n} = \pitchfork(\Delta, X_n)$ of $C$ by [[SimpSet|SSet]].

Here the **Bousfield--Kan** morphism is the morphism

$$
  Tot X \simeq hom(\Delta,X)
    \stackrel
      {hom(\phi,Id_X)}
      {\longrightarrow}
  hom\Big(
    N\big(\Delta/(-)\ig)
    ,\,
    X
    \Big)
  \,.
$$

## Properties

+-- {: .un_theorem}
###### Theorem

If the [[simplicial object]] $X$ is [[Reedy category|Reedy]]-[[cofibrant objects|cofibrant]] then its Bousfield--Kan map is a natural [[weak equivalence]].

If the co-[[simplicial object]] $X$ is [[Reedy category|Reedy fibrant]] then its Bousfield--Kan map is a natural weak equivalence.

=--

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

This can be proven for instance using [[homotopy colimits]] in the [[Reedy model structure]]. Details are at *[Reedy model structure -- over the simplex category](Reedy+model+structure#SimplexCategory)*.

=--

## Relation to homotopy limits

When the co[[simplicial object]]  $X$ is degreewise fibrant,  then

$$
  lim^{N(\Delta/(-))} X
  \simeq holim X
$$

computes the [[homotopy limit]] of $X$ as a [[weighted limit]] (as explained there). Then the above theorem says that the homotopy limit is already computed by the totalization

$$
  holim X \simeq lim^\Delta X
  \,.
$$

## References

The original reference is

* [[Aldridge Bousfield]] and [[Dan Kan]], _Homotopy limits, completions and localizations_, Springer-Verlag, Berlin, 1972. Lecture Notes in Mathematics, Vol. 304.
 {#BousfieldKan}

Reviews include

* Hirschhorn, _Simplicial model categories and their localization_.

The Bousfield--Kan map(s) are on p. 397, def. 18.7.1 and def. 18.7.3.

 _Realization_ and _totalization_ are defs 18.6.2 and 18.6.3 on p. 395.

Notice that this book writes $B$ for the nerve!

[[!redirects Bousfield-Kan formula]]


Bousfield-Kuhn functor

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#### Higher algebra
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=--
=--
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* table of contents
{:toc}

## Idea

For $p$ a [[prime number]], given $n \in \mathbb{N}$ write $K(n)$ for the coresponding [[Morava K-theory]] [[spectrum]] and write $L_{K(n)}(-)$ for [[K(n)-local spectrum|K(n)-localization]] of spectra.

The _Bousfield-Kuhn functor_ $\Phi$ is a [[functor]] from [[pointed object|pointed]] [[homotopy types]] to [[spectra]]

$$
  \Phi \;\colon\; \in \infty Grpd_\ast \longrightarrow Spectra
$$

such that there is a [[natural equivalence]]

$$
  \Phi(\Omega^\infty Y ) \simeq L_{K(n)}(Y)
$$

and

$$
  \pi_\bullet \Phi(X) \simeq v_n^{-1} \pi_\bullet(X)
  \,.
$$

(...)

## Related concepts

* [[power operation]]

* [[logarithmic cohomology operation]]

## References

The original articles are

* [[Aldridge Bousfield]], _Uniqueness of infinite deloopings for K-theoretic spaces_, Pacific J. Math. 129 (1987), no. 1, 1–31. MR 89g:55017

* [[Nicholas Kuhn]], _Morava K-theories and infinite loop spaces, Algebraic topology (Arcata, CA, 1986) (Berlin), Lecture Notes in Math., vol. 1370, Springer, 1989, pp. 243–257. MR
MR1000381 (90d:55014)

See also

* [[Nat Stapleton]], _Power operations and the Bousfield-Kuhn functro_, in _Report of $E$-theory conjectures seminar_ (2013) ([pdf](http://math.berkeley.edu/~ericp/latex/misc/mit-ethy.pdf))

The relation to [[topological André-Quillen cohomology]] is discussed in

* [[Mark Behrens]], [[Charles Rezk]], _The Bousfield-Kuhn functor and topological André-Quillen cohomology_ ([pdf](http://math.mit.edu/~mbehrens/papers/BKTAQ4.pdf))

Review and further discussion of [[logarithmic cohomology operations]] is in

* {#Rezk06} [[Charles Rezk]], _The units of a ring spectrum and a logarithmic cohomology operation_, J. Amer. Math. Soc. 19 (2006), 969-1014 ([arXiv:math/0407022](http://arxiv.org/abs/math/0407022))

* {#AndoHopkinsRezk10} [[Matthew Ando]], [[Mike Hopkins]], [[Charles Rezk]], section 4.2 of  _Multiplicative orientations of KO-theory and the spectrum of topological modular forms_, 2010 ([pdf](http://www.math.uiuc.edu/~mando/papers/koandtmf.pdf))

[[!redirects Bousfield-Kuhn functors]]

[[!redirects Bousfield-Kuhn construction]]
[[!redirects Bousfield-Kuhn constructions]]


Bousfield–Kan spectral sequence

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* table of contents
{:toc}

## Idea

The classical and motivating examples of a [[homotopy spectral sequence]] is the [[Bousfield-Kan spectral sequence]] which computes [[homotopy groups]] of a [[topological space]]/[[simplicial set]]/[[homotopy type]] realized as the [[totalization]] of a [[cosimplicial homotopy type]]. It may be regarded as the unstable analog of the [[Adams spectral sequence]], which computes homotopy groups of certain [[spectra]]. The corresponding [[spectral sequence]] for [[homology groups]] of the totalization of a [[cosimplicial homotopy type]] is the [[Eilenberg-Moore spectral sequence]].

## References

The [[Bousfield-Kan spectral sequence]] was introduced and originally studied in

* [[Aldridge Bousfield]], [[Daniel Kan]], _The homotopy spectral sequence of a space with coefficients in a ring_. Topology, 11, pp. 79–106, 1972.

* [[Aldridge Bousfield]], [[Daniel Kan]], _A second quadrant homotopy spectral sequence_, Transactions of the American Mathematical Society Vol. 177, Mar., 1973 ([pdf](http://www.ams.org/journals/tran/1973-177-00/S0002-9947-1973-0372859-6/S0002-9947-1973-0372859-6.pdf))

* [[Aldridge Bousfield]], [[Daniel Kan]], _Pairings and products in the homotopy
spectral sequence_. Transactions of the American Mathematical Society, 177,
pp. 319–343, 1973.

* [[Aldridge Bousfield]], _Homotopy Spectral Sequences and Obstructions_, Isr. J. Math. 66 (1989), 54-104.

An early textbook account of this work is in

* {#BousfieldKan72} [[Aldridge Bousfield]], [[Daniel Kan]], _Homotopy limits, completions and localization_, Springer 1972

Lecture notes include

* [[Bertrand Guillou]], _The Bousfield-Kan spectral sequence_ [pdf](http://www.ms.uky.edu/~guillou/BKss.pdf)

Discussion of computations using [[effective homology]] includes

* Romero Ibanez, _Effective homology and spectral sequences_, 2007 ([pdf](http://www.unirioja.es/cu/anromero/zgz07.pdf))

[[!redirects Bousfield-Kan spectral sequence]]


Bowen Chen

## Selected writings

On [[quantum information theory]] in view of [[AdS-CFT duality]] ([[holographic entanglement entropy]]):

* [[Bowen Chen]], [[Bartlomiej Czech]], [[Zi-zhi Wang]], *Quantum Information in Holographic Duality*, Rept. Prog. Phys. **85** (2022) 4, 046001 [[arXiv:2108.09188](https://arxiv.org/abs/2108.09188), [doi:10.1088/1361-6633/ac51b5](https://doi.org/10.1088/1361-6633/ac51b5)]

category: people


box topology

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## Definition

Given a [[set]] $\{X_i\}_{i \in I}$ of [[topological spaces]], then the _box topology_ on the [[Cartesian product]] $\underset{i \in I}{\prod} X_i$ of the underlying sets of these spaces is the [[topological space|topology]] which is generated from the [[topological base]] whose elements are the [[Cartesian products]]

$$
  \underset{i \in I}{\prod} U_i
  \;\subset\;
  \underset{i\in I}{\prod} X_i
$$

of [[open subsets]] $U_i \subset X_i$ of each of the factor spaces.

## Properties

### Relation to Tychonoff topology

If the index set $I$ is a [[finite set]], then this box topology coincides with the [[Tychonoff topology]] on the Cartesian product. For general $I$ however the Tychnoff topology has as base open subsets only those products $\underset{i \in I}{\prod} U_i$ for which all but a finite number of factors are in fact the corresponding total space $X_i$.

Hence for non-finite index set $I$ the box topology is strictly [[finer topology|finer]] that the Tychonoff topology. Beware that it is the Tychonoff toopology which yields the actual [[Cartesian product]] in the [[category]] [[Top]] of [[topological spaces]]. Accordingly, for non-finite $I$ the box topology _fails_ to possess the properties that one expects from a categorical product.

For example, there may be a family of continuous functions, $f_i: X \to X_i$, for which the associated map by components, $X \to \underset{i\in I}{\prod} X_i$, is not continuous for the box topology.

## References

* Wikipedia, _[Box topology](https://en.wikipedia.org/wiki/Box_topology)_

[[!redirects box topologies]]

[[!redirects box product]]
[[!redirects box products]]


BPR-theory

## Idea

The [[real oriented cohomology]]-version of the [[cohomology theory]] represented by the [[BP-spectrum]].

## Related concepts

* [[KR-theory]]

* [[MR-theory]]

* [[ER-theory]]

## References

* {#Araki78} S. Araki, _$\tau$-cohomology theories_, Japan Journal of Mathematics (N.S.) 4 (2) (1978) 363-416.

* {#Araki79} S. Araki, _Forgetful spectral sequences_, Osaka Journal of Mathematics 16 (1) (1979) 173-199.

* S. Araki, _Orientations in $\tau$-cohomology theories_, Japan Journal of Mathematics (N.S.) 5 (2) (1979) 403-430.

[[!redirects BPR theory]]

[[!redirects BPR spectrum]]


BPS state

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* table of contents
{:toc}

## Idea

### General
 {#General}

In [[supersymmetry|supersymmetric]] [[quantum field theory]] with [[extended supersymmetry]], certain extremal [[supermultiplets]] have some of the [[supersymmetry|supersymmetries]] retained (have 0-[[eigenvalue]] under some of the [[supersymmetry]] generators). These are called __Bogomol'nyi--Prasad--Sommerfield saturated solutions__.

More in detail, where in a plain [[supersymmetry]] [[super Lie algebra]] a suitable [[basis]] $\{Q_A\}$ of supersymmetry generators has odd bracket proportional to the spacetime translation and hence to an [[energy]]/[[mass]] operator $E$ (with terminology as at [[unitary representation of the Poincaré group]])

$$
 \{Q_A, Q_B\} = E \delta_{A B}
$$

for [[extended supersymmetry]] there are further bosonic super Lie algebra generators $K_{A B}$ (charges) such that

$$
 \{Q_A, Q_B\} = E \delta_{A B} -  K_{A B}
 \,.
$$

It follows from the supersymmetry algebra that $(E \delta_{A B} -  K_{A B})$ is a positive definite [[bilinear form]], which puts a lower bound on the [[energy]] given the values of these extra charges. This is called the _BPS bound_. See also at _[[Bridgeland stability condition]]_.

In particular when this bound is satisfied in that some of the [[eigenvalues]] of the [[matrix]] $(K_{A B})$ are actually equal to the energy/mass, then the corresponding component of the right hand side in the above equation vanishes and hence then the corresponding supersymmetry generators may annihilate the given state, then called a _BPS state_. This way enhanced supersymmetry of states goes along with certain charges taken extremal values.

States with similar behaviour are also considered also in some models of [[soliton]] theory
(English Wikipedia: [Bogomol'nyi–Prasad–Sommerfield bound](http://en.wikipedia.org/wiki/Bogomol'nyi%E2%80%93Prasad%E2%80%93Sommerfield_bound)).

BPS states play a central role in the investigation of [[moduli spaces]] of classical [[vacua]] as they form part of the moduli problem which is often the most tractable.

Several mathematical theories in [[geometry]] are interpreted as counting BPS-states in the sense of integration on appropriate compactification of the [[moduli space]] of BPS-states in a related physical model attached to the underlying geometry: most notably the [[Gromov-Witten invariants]], [[Donaldson-Thomas invariants]] and the [[Thomas-Pandharipande invariants]]; all the three seem to be deeply interrelated though they are defined in rather very different terms. The compactification of the moduli space involves various [[stability conditions]].

### In supergravity

In the context of [[supergravity]] BPS states correspond to [[super spacetimes]] admitting [[Killing vectors]]. These notably include extremal [[black brane]] solutions.

### In superstring theory

Specifically in [[superstring theory]] BPS states in [[target space]] correspond to string states on the worldsheet which are annihilated by the left-moving (say) half of the [[Dirac-Ramond operator]]. These are counted by the [[Witten genus]], see at _[Witten genus -- Relation to BPS states](Witten+genus#RelationToBPSStateCounting)_.

The degeneracy of BPS states in string theory has been used to provide a microscopic interpretation of [[Bekenstein-Hawking entropy]] of [[black holes]], see at _[[black holes in string theory]]_.

## Formalization in higher differential geometry
 {#InTermsOfHigherDifferentialGeometry}

> The following are some observations on the formalization of BPS states from the [[nPOV]], in [[higher differential geometry]], following ([Sati-Schreiber 15](#SatiSchreiber15)).

Let $\mathbb{R}^{d-1,1|N}$ be a [[super-Minkowski spacetime]], let $(d,N,p)$ be in the [[brane scan]] and write

$$
  \phi \coloneqq \bar{\psi} \wedge E^{\wedge p} \wedge \psi \in \Omega^{p+2}(\mathbb{R}^{d-1,1|N})
$$

for the correspoding [[super Lie algebra]] [[Lie algebra cohomology|cocycle]], as discussed at _[[Green-Schwarz action functional]]_, see ([FSS 13](#FSS13)) for the perspective invoked here.

Consider then $X$ a [[super-spacetime]] locally modeled on $\mathbb{R}^{d-1,1|N}$ as a [[Cartan geometry]], solving the relevant [[supergravity]] [[equations of motion]] (e.g. [[11-dimensional supergravity]] for $d= 11$, [[heterotic supergravity]] for $d = 10$ and $N = (1,0)$, [[type IIA supergravity]] for $d = 10$ and $N= (1,1)$ or [[type IIB supergravity]] for $d = 10$ $N= (2,0)$).

This means in particular that $X$ carries a [[super differential form]]

$$
  \omega \in \Omega^{p+2}(X)
$$

which is [[definite form|definite]] on $\phi$. This is the [[curvature]] of the [[WZW-term]] which defines the relevant [[super p-brane sigma-model]] with [[target space]] $X$.

By ([AGIT 89](#AGIT89)) $X$ is a BPS state to the extent that it carries [[Killing spinors]] which form a central [[Lie algebra extension]] of a sub-algebra of the [[supersymmetry]] algebra (i.e. of the [[super translation Lie algebra]]) by $H^p_{dR}(X)$ which is classified by the [[Lie algebra cohomology|cocycle]] given by

$$
  (\epsilon_1, \epsilon_2) \mapsto
  \omega(\epsilon_1,\epsilon_2)
  \in
  \Omega^p(X)_{cl}/im(\mathbf{d}_{dR})
  \,.
$$

Now we observe that by ([hgpII, theorem 3.3.1](#hgpII)) this is precisely the [[0-truncation]] of the [[super L-infinity algebra|super]]-[[Poisson bracket Lie n-algebra]] $\mathfrak{Pois}(X,\omega)$ induced by regarding $(X,\omega)$ as an [[pre-n-plectic manifold|pre-n-plectic]] [[supermanifold]] and restricting along the inclusion of the [[Killing vectors]]/[[Killing spinors]] into all the [[Hamiltonian vector fields]].

$$
  H^p_{dR}(X) \to \tau_0 \mathfrak{Pois}(X,\omega) \to Vect_{Ham}(X)
$$

(Here we are using that if an [[n-type]] is an extension of a [[0-type]], then its 0-truncation is still an extension by the 0-truncation of the original homotopy fiber.)

The elements in $H^p_{dR}(X)$ here are precisely the $p$-brane charges, as discussed in ([AGIT 89, p. 8](#AGIT89)).

Hence $X$ is the more BPS the more odd-graded elements there are in $\tau_0 \mathfrak{Pois}(X,\omega)$ (or its restriction to super-isometries). Hence $X$ is a 1/2 BPS state of supergravity if the odd dimension of this is half that of $\mathbb{R}^{d-1,d|N}$, it is 1/4 BPS if the odd dimension is one fourth of that of $\mathbb{R}^{d-1,d|N}$, etc.

Notice that if

$$
  \array{
    &&
     \mathbf{B}^{p+1} (\mathbb{R}/\Gamma)_{conn}
    \\
    &  {}^{\mathllap{\mathbf{L}_{WZW}}}\nearrow & \downarrow^{\mathrlap{F_{(-)}}}
    \\
    X
    &\stackrel{\omega}{\longrightarrow}&
    \mathbf{\Omega}^{p+2}_{cl}
  }
$$

is a [[prequantum line n-bundle|prequantization]] of $\omega$, i.e. an actual [[WZW term]] with [[curvature]] $\omega$, then $\mathfrak{Pois}(X,\omega)$ is supposed to be the [[Lie differentiation]] of the [[stabilizer group]] of $\mathbf{L}_{WZW}$, which is the [[quantomorphism n-group]] $QuantMorph(\mathbf{L}_{WZW})$. (This Lie differentiation statement is strictly shown only for $p = 0$ and $p = 1$ in [[schreiber:differential cohomology in a cohesive topos|dcct]] but clearly should hold generally.)

Hence we may regard $\mathbf{QuantMorph}(\mathbf{L}_{WZW})$ (or its restriction to super-[[isometries]]) as the [[Lie integration]] of the brane-charge extended supersymmetry algebra. By the discussion at _[conserved current -- In higher differential geometry](conserved+current#InHigherPrequantumGeometry)_ this is indeed the [[n-group]] of conserved currents of $\mathbf{L}_{WZW}$ regraded as a [[local Lagrangian]], and so this conceptually connects back to the considerations in ([AGIT 89](#AGIT89)).

## Examples

### In 11d Supergravity

In [[11-dimensional supergravity]] ([[M-theory]]) there are four kinds of 1/2 BPS states (the [[black brane|black]] [[M-branes]]) (e.g. [Stelle 98, section 3](#Stelle98) [EHKNT 07](#EHKNT07)):

* the [[M2-brane]];

* the [[M5-brane]];

* the [[M-wave]];

* the [[Kaluza-Klein monopole]].

## Related concepts

* [[Bridgeland stability condition]]

* [[Killing spinor]]

* [[supermultiplet]]

* [[wall crossing]]

* [[protection from quantum corrections]]

* [[positive energy theorem]]

* [[enhanced gauge symmetry]]

* [[geometry of physics -- BPS charges]]

* [[intersecting branes]]

  [[membrane triple junction]]

## References

### General

The BPS bound derives its name from the discussion of [[magnetic monopoles]] in 4-dimensional [[Yang-Mills theory]] in

* (E. B. Bogomol'nyj) [[Е. Б. Богомольный]], _Устойчивость классических решений_, Яд. Физ. __24__ (1976) 449--454 (_Stability of slassical solutions_, Sov. J. Nucl. Phys. 24 (1976), 449); _Stability of slassical solutions_, Yad. Fiz. 24 (1976) 861

* M. K. Prasad, [[Charles Sommerfield]], _Exact classical solution for 't Hooft monopole and the Julia-Zee dyon_, Phys. Rev. Lett. __35__ (1975) 760--762.

The extension of the term "BPS-saturated state" from this case to situations in [[string theory]] seems to have happened in

* {#Witten95} [[Edward Witten]], around (2.5) of _[[String Theory Dynamics In Various Dimensions]]_, Nucl.Phys.B443:85-126,1995 ([arXiv:hep-th/9503124](https://arxiv.org/abs/hep-th/9503124))

The original article identifying the role of BPS states in supersymmetric field theory is

* [[Edward Witten]], [[David Olive]], _Supersymmetry algebras that include topological charges_ (2002) ([journal](http://www.sciencedirect.com/science/article/pii/037026937890357X))

Exposition and review includes

* [[Andrew Neitzke]], _What is a BPS state?_, 2012 ([pdf](http://www.ma.utexas.edu/users/neitzke/expos/bps-expos.pdf))

* {#Dimofte10} [[Tudor Dimofte]], _Refined BPS invariants, Chern-Simons theory, and the quantum dilogarithm_, 2010 ([pdf](http://thesis.library.caltech.edu/5808/1/DimofteTDofficial.pdf), [web](http://thesis.library.caltech.edu/5808/))

Further developments are in

*  [[Jeffrey Harvey]], [[Greg Moore]], _Algebras, BPS states, and strings_, Nucl.Phys.B463:315--368 (1996) ([doi](http://doi.org/10.1016/0550-3213%2895%2900605-2)); [hep-th/9510182](https://arxiv.org/abs/hep-th/9510182);

*  [[Jeffrey Harvey]], [[Greg Moore]], _On the algebras of BPS states_, Comm. Math. Phys. __197__ (1998), 489–-519, [doi](https://doi.org/10.1007/s002200050461), [hep-th/9609017](https://arxiv.org/abs/hep-th/9609017).

* [[Ali Chamseddine]], M. S. Volkov, _Non-abelian BPS
monopoles in $N=4$ gauged supergravity_, Physical Review Letters 79: 3343–3346 (1997) [hep-th/9707176](http://arxiv.org/abs/hep-th/9707176).

* [[Steven Weinberg]], _The quantum theory of fields_, vol. II

* [[Tudor Dimofte]], [[Sergei Gukov]], _Refined, Motivic, and Quantum_, [arXiv:0904.1420](http://arXiv.org/abs/0904.1420)

* [[Davide Gaiotto]], [[Gregory Moore]], [[Andrew Neitzke]], _Wall-crossing, Hitchin systems, and the WKB approximation_, [arxiv:0907.3987](https://arxiv.org/abs/0907.3987)

* R. Pandharipande, R.P. Thomas, _Stable pairs and BPS invariants_, [arXiv:0711.3899](https://arxiv.org/abs/0711.3899)

* [[Markus Reineke]], _Cohomology of quiver moduli, functional equations, and integrality of Donaldson-Thomas type invariants_, Compositio Mathematica __147__:3 (2011) 943--964 [doi](https://doi.org/10.1112/S0010437X1000521X) [arXiv:0903.0261](https://arXiv.org/abs/0903.0261)

* Duiliu-Emanuel Diaconescu, _Moduli of ADHM sheaves and local Donaldson-Thomas theory_, J. Geom. & Physics __62__:4 (2012) 763--799 [arXiv:0801.0820](https://arXiv.org/abs/0801.0820) f[doi](https://doi.org/10.1016/j.geomphys.2011.12.018)

* [[Tom Bridgeland]], _Stability conditions on triangulated categories_, Ann. of Math. 166 (2007) 317–345,[math.AG/0212237](https://arxiv.org/abs/math/0212237)

* [[Maxim Kontsevich]], [[Yan Soibelman]], _Stability structures, motivic Donaldson-Thomas invariants and cluster transformations_, [arXiv:0811.2435](https://arxiv.org/abs/0811.2435)

* [[Maxim Kontsevich]], [[Yan Soibelman]], _Motivic Donaldson-Thomas invariants: summary of results_, in: Mirror Symmetry and Tropical Geometry, Contemp. Math. __527__ (2010) [doi](https://doi.org/10.1090/conm/527/10400) [arxiv/0910.4315](https://arxiv.org/abs/0910.4315)

* [[Dominic Joyce]], Y. Song, _A theory of generalized Donaldson-Thomas invariants_,  [arxiv/0810.5645](https://arxiv.org/abs/0810.5645)

### Introductions, surveys and lectures
 {#IntroductoryReferences}

An introduction that starts at the beginning and then covers much of the ground in some detail is

* [[Greg Moore]], _PiTP Lectures on BPS states and wall-crossing in $d = 4$, $\mathcal{N} = 2$ theories_ ([pdf](http://www.physics.rutgers.edu/~gmoore/PiTP_July26_2010.pdf))

A survey of progress on the most general picture is in

* Katzutoshi Ohta, _BPS state counting and related physics_ (2005) ([pdf](http://www2.yukawa.kyoto-u.ac.jp/~qft/2005/slides/ohta.pdf))

### In supergravity

Discussion of extremal/BPS [[black branes]] in [[supergravity]] (especially in [[11-dimensional supergravity]] and 10d [[type II supergravity]]) includes

* {#Stelle98} [[Kellogg Stelle]], _BPS Branes in Supergravity_ ([arXiv:hep-th/9803116](http://arxiv.org/abs/hep-th/9803116))

* {#GauntlettHull99} [[Jerome Gauntlett]], [[Chris Hull]], _BPS states with extra supersymmetry_, JHEP 0001 (2000) 004 ([arXiv:hep-th/9909098](https://arxiv.org/abs/hep-th/9909098) [doi](https://doi.org/10.1088/1126-6708/2000/01/004))

* {#EHKNT07} [[Francois Englert]], [[Laurent Houart]], [[Axel Kleinschmidt]], [[Hermann Nicolai]], Nassiba Tabti, _An $E_9$ multiplet of BPS states_, JHEP 0705:065 (2007) [[arXiv:hep-th/0703285](https://arxiv.org/abs/hep-th/0703285)]

* Andrew Callister, [[Douglas Smith]], _Topological BPS charges in 10 and 11-dimensional supergravity_, Phys. Rev. D78:065042,2008 ([arXiv:0712.3235](https://arxiv.org/abs/0712.3235))

* Andrew Callister, [[Douglas Smith]], _Topological charges in $SL(2,\mathbb{R})$ covariant massive 11-dimensional and Type IIB SUGRA_, Phys.Rev.D80:125035,2009 ([arXiv:0907.3614](http://arxiv.org/abs/0907.3614))

* Andrew Callister, _Topological BPS charges in 10- and 11-dimensional supergravity_, thesis 2010 ([spire](http://inspirehep.net/record/1221591?ln=en))

* Cristine N. Ferreira, _BPS solution for eleven-dimensional supergravity with a conical defect configuration_ ([arXiv:1312.0578](https://arxiv.org/abs/1312.0578))

Specifically for $1/2^n$-BPS states of intersecting [[M-branes]] in 11d there is discussion in

* {#Tseytlin96} [[Arkady Tseytlin]], _Harmonic superpositions of M-branes_, Nucl.Phys.B475:149-163,1996 ([arXiv:hep-th/9604035](https://arxiv.org/abs/hep-th/9604035)) also in [[Mike Duff]] (ed.) chapter 5 of _[[The World in Eleven Dimensions]]_

see also

* [[Jerome Gauntlett]], _Intersecting Branes_ ([hep-th/9705011](https://arxiv.org/abs/hep-th/9705011))

* [[Igor Bandos]], [[José de Azcarraga]], [[José Izquierdo]], [[Jerzy Lukierski]], _BPS states in M-theory and twistorial constituents_, Phys.Rev.Lett. __86__:4451-4454,2001 ([arXiv:hep-th/0101113](https://arxiv.org/abs/hep-th/0101113))

* U. Gran, [[Jan Gutowski]], [[George Papadopoulos]], _Classification, geometry and applications of supersymmetric backgrounds_ ([arXiv:1808.07879](https://arxiv.org/abs/1808.07879))

Semiclassical approach

* T. Daniel Brennan, [[Gregory W. Moore]], _A note on the semiclassical formulation of BPS states in four-dimensional N = 2 theories_, Prog. Theor. Exp. Phys. 2016, 12C110 (19 pages) [doi](https://doi.org/10.1093/ptep/ptw159)

Discussion in the context of multiple [[M2-branes]] in the  [[BLG model]] is in

* {#BaggerLambert12} [[Jonathan Bagger]], [[Neil Lambert]], Sunil Mukhi, Constantinos Papageorgakis, section 1.6 of _Multiple Membranes in M-theory_ ([arXiv:1203.3546](https://arxiv.org/abs/1203.3546))

Discussion for [[4d supergravity]], hence in [[KK-compactification]] of [[type II supergravity]] on a [[Calabi-Yau manifold]] is due to

* {#Denef00} [[Frederik Denef]], _Supergravity flows and D-brane stability_, JHEP 0008:050, 2000 ([arXiv:hep-th/0005049](http://arxiv.org/abs/hep-th/0005049))

* [[Frederik Denef]], _Quantum Quivers and Hall/Hole Halos_, JHEP 0210:023, 2002 ([arXiv:hep-th/0206072](http://arxiv.org/abs/hep-th/0206072))

Discussion of more general classification of solutions to [[supergravity]] preserving some [[supersymmetry]], i.e. admitting some [[Killing spinors]] includes

* [[Jerome Gauntlett]], Stathis Pakis, _The Geometry of $D=11$ Killing Spinors_, JHEP 0304 (2003) 039 ([arXiv:hep-th/0212008](http://arxiv.org/abs/hep-th/0212008))

* {#HEGKS08} [[Eric D'Hoker]], John Estes, Michael Gutperle, Darya Krym, Paul Sorba, _Half-BPS supergravity solutions and superalgebras_, JHEP0812:047, 2008 ([arXiv:0810.1484](http://arxiv.org/abs/0810.1484))

The conceptual identification of the relevant brane-charge extension of the [[supersymmetry]] algebra as that of the [[conserved currents]] of the [[Green-Schwarz super p-brane sigma models]] for branes is due to

* {#AGIT89} [[José de Azcárraga]], [[Jerome Gauntlett]], [[José Izquierdo|J. M. Izquierdo, [[Paul Townsend]], _Topological extensions of the supersymmetry algebra for extended objects_, Phys. Rev. Lett. __63__ (1989) 2443 ([spire](https://inspirehep.net/record/26393?ln=en))

reviewed in

* [[José de Azcárraga]], [[José Izquierdo|José M. Izquierdo]], section 8.8. of _Lie groups, Lie algebras, cohomology and some applications in physics_ , Cambridge monographs of mathematical physics (1995)

This is for branes in the old [[brane scan]] ([[strings]], [[membranes]], [[NS5-branes]]), excluding [[D-branes]] and [[M5-brane]].

The generalization oft this perspective to the [[M5-brane]] is discussed in

* {#SorokinTownsend97} [[Dmitri Sorokin]], [[Paul Townsend]], _M-theory superalgebra from the M-5-brane_, Phys.Lett. B412 (1997) 265-273 ([arXiv:hep-th/9708003](http://arxiv.org/abs/hep-th/9708003))

and the generalization to [[D-branes]] is discussed in

* {#Hammer97} Hanno Hammer, _Topological Extensions of Noether Charge Algebras carried by D-p-branes_, Nucl.Phys. B521 (1998) 503-546 ([arXiv:hep-th/9711009](http://arxiv.org/abs/hep-th/9711009))

Detailed discussion of examples for various backgrounds is in

* Takeshi Sato, _Superalgebras in Many Types of M-Brane Backgrounds and Various Supersymmetric Brane Configurations_, Nucl.Phys. B548 (1999) 231-257 ([arXiv:hep-th/9812014](https://arxiv.org/abs/hep-th/9812014))

Discussion of this in [[higher differential geometry]] via the [[Poisson bracket Lie n-algebra]] is in

* {#SatiSchreiber15} [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:Lie n-algebras of BPS charges]]_ ([arXiv:1507.08692](http://arxiv.org/abs/1507.08692))

Discussion of relation of [[M5-brane]] BPS states to [[knot invariants]] includes

* [[Edward Witten]], _Fivebranes and Knots_ ([arXiv:1101.3216](http://arxiv.org/abs/1101.3216))

* [[Sergei Gukov]], Marko Stošić, _Homological algebra of knots and BPS states_, GTM __18__ (2012) 309--367 ([arXiv:1112.0030](https://arxiv.org/abs/1112.0030) [doi](https://doi.org/10.2140/gtm.2012.18.309))

* Ross Elliot, [[Sergei Gukov]], _Exceptional knot homology_ ([arXiv:1505.01635](https://arxiv.org/abs/1505.01635))

### Spectral networks
 {#SpectralNetworksReferences}

* [[Davide Gaiotto]], [[Greg Moore]], [[Andrew Neitzke]], _Spectral networks_ ([arXiv:1204.4824](http://arxiv.org/abs/1204.4824),
[illustrating animations](http://www.ma.utexas.edu/users/neitzke/spectral-network-movies/))

[[!redirects BPS state]]
[[!redirects BPS states]]
[[!redirects BPS-states]]
[[!redirects BPS-state]]

[[!redirects BPS invariant]]
[[!redirects BPS invariants]]

[[!redirects BPS charge]]
[[!redirects BPS charges]]

[[!redirects BPS brane]]
[[!redirects BPS branes]]

[[!redirects BPS solution]]
[[!redirects BPS solutions]]


BPST-instanton

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## Idea

The name for an  [[instanton]] [[gauge field]] configuration in $SU(2)$-[[Yang-Mills theory]] (describing the [[weak nuclear force]]).

The BPST construction considers -- on a 4-dimensional [[Minkowski spacetime]] [[Wick rotation|Wick rotated]] to the Euclidean $\mathbb{R}^4$ -- [[gauge field]] configurations for [[gauge group]] the [[special unitary group]] $SU(2)$ that have vanishing [[field strength]] outside some finite radius. These are then equivalently configurations on the 4-[[sphere]]. The BPST instanton is the $SU(2)$-gauge field whose underlying $SU(2)$-[[principal bundle]]  has [[second Chern class]]=instanton number equal to $\pm1 \in \mathbb{Z} \simeq H^4(S^4, \mathbb{Z})$.

The physics literature typically focuses on describing this $SU(2)$-bundle in terms of the [[Cech cohomology|Cech cocycle]] which after [[covering]] the 4-sphere with two 4-[[balls]] (two "hemispheres") is given by an $SU(2)$-valued transition function on the intersection of these two balls, which has the [[homotopy type]] of the 3-sphere. Since also the [[manifold]] underlying the [[special unitary group|special unitary]] [[Lie group]] $SU(2)$ is [[diffeomorphism|diffeomorphic]] to $S^3$, this allows to encode the classes of $SU(2)$-[[principal bundles]]/$SU(2)$-instantons on $S^4$ in terms of [[homotopy classes]] of maps $S^3 \to S^3$, and this is what much of the literature focuses on.

## $SU(2)$-instantons from the correct maths to the traditional physics story
 {#FromTheMathsToThePhysicsStory}

[[!include SU2-instantons from the correct maths to the traditional physics story]]

## Related concepts

* [[Yang-Mills instanton]]

  * [[instanton in QCD]]

* at [[thermal field theory|finite temperature]]: [[caloron]]

  * [[caloron correspondence]]

* [[NS5-brane]] in [[heterotic string theory]]

## References

The original articles are

* {#BelavinPolyakovSchwartzTyupkin75} A. A. Belavin, [[Alexander Polyakov]], A.S. Schwartz, Yu.S. Tyupkin, _Pseudoparticle solutions of the Yang-Mills equations_, Phys. Lett. B 59 (1), 85-87 (1975) doi

* A. A. Belavin, V.A. Fateev, A.S. Schwarz, Yu.S. Tyupkin, _Quantum fluctuations of multi-instanton solutions_, Phys. Lett. B 83 (3-4), 317-320 (1979) doi

For surveys and introductions see the references at _[[Yang-Mills instanton]]_.

[[!redirects BPST-instanton]]

[[!redirects BPST instanton]]

[[!redirects BPST-instantons]]

[[!redirects BPST instantons]]


BQP

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## Idea

In [[quantum complexity theory]], "BQP" stands for *bounded-error quantum polynomial time*, denoting a fundamental [[complexity class]] of [[quantum computation]].

## Related concepts

* [[PostBQP]]

* [[DQC1]]

## References

See also:

* Wikipedia, *[BQP](https://en.wikipedia.org/wiki/BQP)*


bra-ket

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#### Linear algebra
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## Idea

Traditional notation in [[physics]] [[Dirac 1939](#Dirac39)] for writing down [[pure state|pure]] [[quantum states]] (elements of [[Hilbert spaces]]), their [[hermitian adjoints]] and [[Hermitian form|Hermtian]] [[inner products]]:

With the [[Hermitian form|Hermitian]] [[inner product]] on the [[underlying]] [[vector space]] of a [[Hilbert space]] $H$ denoted $\langle-\vert-\rangle$, element $\psi \,\in\, H$ are turned into elements of the [[linear dual space]] $H^\ast$ by $\langle \psi \vert -\rangle \,\in\, H^\ast$, whose [[evaluation]] on another $\phi \in H$ is the given [[Hermitian form]] $\langle \psi \vert \phi\rangle$. Hence if one declares notation such that

$$
  \array{
     \vert \phi \rangle \;\equiv\; \phi \,\in\, H
     \\
     \langle \psi \vert \;\equiv\; \langle \psi \vert -\rangle \,\in\, H^\ast
  }
$$

then the [[evaluation]]-pairing in the [[Hermitian form]] is essentially juxtaposition

$$
  \array{
    H^\ast \otimes H &\longrightarrow& \mathbb{C}
    \\
    \langle \psi \vert
    ,
    \vert \phi \rangle
    &\mapsto&
    \langle \psi \vert \phi \rangle
  }
$$

With the inner product $\langle-\vert-\rangle$ referred to as a *bracket* this suggest to refer to "$\langle \psi \vert$" as a "bra" and "$\vert \phi \rangle$" as a "ket" [[Dirac 1939, last line](#Dirac39)].

The notation may be udnerstood as a lightweight precursor to the [[string diagram]]-calculus in [[dagger-compact categories]] [[Abramsky & Coecke 2004 §7.2](#AbramskyCoecke04), [2007 pp. 6](#AbramskyCoecke2007), [2008 §4.4](#AbramskyCoecke08), [Coecke 2010 §3.3](#Coecke10)].

For instance, if $\mathscr{H}$ is a [[finite-dimensional Hilbert space]] with [[orthonormal basis]] $\big(\left\vert w \right\rangle\big)_{w \colon W}$, then the [[compact closed category|compact closure]] is witnessed by the following [[isomorphism]] between the vector space of linear maps out of $\mathscr{H}$ and a vector space of [[matrices]]:

$$
  \array{
    \Big(
      \mathscr{H}
      \multimap
      \mathscr{H}'
    \Big)
    &\longrightarrow&
    \mathscr{H}' \otimes \mathscr{H}^\ast
    \\
    \Big(
      \left\vert w \right\rangle
      \,\mapsto\,
      \underset{w'}{\sum}
      \left\vert w' \right\rangle
      A_{w', w}
    \Big)
    &\mapsto&
      \underset{w,w'}{\sum}
      \left\vert w' \right\rangle
      A_{w', w}
      \left\langle w \right\vert
    \mathrlap{\,.}
  }
$$

## Related concepts

[[!include states and observables -- content]]

## References

The bra-ket notation is due to:

* {#Dirac39} [[Paul A. M. Dirac]], *A new notation for quantum mechanics*, Mathematical Proceedings of the Cambridge Philosophical Society **35** 3 (1939) 416-418 [[doi:10.1017/S0305004100021162](https://doi.org/10.1017/S0305004100021162), [pdf](https://www.ifsc.usp.br/~lattice/wp-content/uploads/2014/02/Dirac_notation.pdf)]

Textbook accounts:

* [[Jun John Sakurai]], Jim Napolitano, §1.2 in: *Modern Quantum Mechanics*, Cambridge University Press (1985, 1994, 2020) [[doi:10.1017/9781108587280](https://doi.org/10.1017/9781108587280), [Wikipedia](https://en.wikipedia.org/wiki/Modern_Quantum_Mechanics)]

* [[Michael A. Nielsen]], [[Isaac L. Chuang]], Fig. 2.1 in: *Quantum computation and quantum information*, Cambridge University Press (2000) [[doi:10.1017/CBO9780511976667](https://doi.org/10.1017/CBO9780511976667), [pdf](http://csis.pace.edu/~ctappert/cs837-19spring/QC-textbook.pdf), [[NielsenChuangQuantumComputation.pdf:file]]]

* [[Robert B. Griffiths]], *Linear Algebra in Dirac notation*, Section 3 in: *Consistent Quantum Theory*, Cambridge University Press (2002) [[doi:10.1017/CBO9780511606052](https://doi.org/10.1017/CBO9780511606052), [webpage](https://quantum.phys.cmu.edu/CQT)]

See also:

* Wikipedia, _[Bra-ket notation](http://en.wikipedia.org/wiki/Bra–ket_notation)_

Discussion in the (broader) context of [[string diagram]]-calculus for [[dagger compact categories]] (cf. *[[quantum information theory via dagger-compact categories]]*):

* {#AbramskyCoecke04} [[Samson Abramsky]], [[Bob Coecke]], §7.2 in: _A categorical semantics of quantum protocols_ , Proceedings of the 19th IEEE conference on Logic in Computer Science (LiCS'04). IEEE Computer Science Press (2004) [[arXiv:quant-ph/0402130](http://arxiv.org/abs/quant-ph/0402130)]

* {#AbramskyCoecke2007} [[Samson Abramsky]], [[Bob Coecke]], pp. 6 in: *Physics from Computer Science: a Position Statement*, [International Journal of Unconventional Computing **3** 3 (2007)](https://www.oldcitypublishing.com/journals/ijuc-home/ijuc-issue-contents/ijuc-volume-3-number-3-2007/) $[$[pdf](https://www.cs.ox.ac.uk/files/349/YORKIJUC.pdf), [ijuc-3-3-p-179-197](https://www.oldcitypublishing.com/journals/ijuc-home/ijuc-issue-contents/ijuc-volume-3-number-3-2007/ijuc-3-3-p-179-197/)$]$

* {#AbramskyCoecke08} [[Samson Abramsky]], [[Bob Coecke]], §4.4 in: *Categorical quantum mechanics*, in *[[Handbook of Quantum Logic and Quantum Structures]]*, Elsevier (2008) [[arXiv:0808.1023](http://arxiv.org/abs/0808.1023), [ISBN:9780080931661](https://www.sciencedirect.com/book/9780444528698/), [doi:10.1109/LICS.2004.1319636](https://doi.org/10.1109/LICS.2004.1319636)]

* [[Bob Coecke]], §3.e in: _Kindergarten quantum mechanics_ [[arXiv:quant-ph/0510032](http://arxiv.org/abs/quant-ph/0510032)]

* {#Coecke10} [[Bob Coecke]], §3.3 in: *Quantum Picturalism*, Contemporary Physics **51** 1 (2010) [[arXiv:0908.1787](http://arxiv.org/abs/0908.1787), [doi:10.1080/00107510903257624](https://doi.org/10.1080/00107510903257624)]

[[!redirects bra]]
[[!redirects ket]]
[[!redirects bras]]
[[!redirects kets]]

[[!redirects bra-ket notation]]


brace

This entry is about an algebraic structure of _brace_ stemming from the study of set-theoretic Yang-Baxter equation. See entry [[brace algebra]] for another algebraic structure, with $n$-ary operations, "braces".

### Definition

A __left brace__ is left [[truss]] which is an Abelian group under its multiplicative operation.

### Literature

* W. Rump, _Braces, radical rings, and the quantum Yang-Baxter equation_, J. Algebra __307__ (2007), no. 1, 153-170
* Wolfgang Rump, _Generalized radical rings, unknotted biquandles, and quantum groups_, Colloquium Mathematicum __109__ (2007), 85-100 [doi](https://doi.org/10.4064/cm109-1-7)
* W. Rump, _The brace of a classical group_, Note di Matematica __34__ (2014) n.1, 115-144 [pdf](https://core.ac.uk/download/pdf/187088098.pdf)
* W. Rump, _Construction of finite braces_, Ann. Comb. 23, 391–416 (2019) [doi](https://doi.org/10.1007/s00026-019-00430-1)
* F. Cedó, E. Jespers, J. Okniński, _Braces and the Yang–Baxter equation_, Commun. Math. Phys. __327__, 101–116 (2014). [doi](https://doi.org/10.1007/s00220-014-1935-y)

### Related notions

* [[truss]]
* [[quantum Yang-Baxter equation]]
* [[Jacobson radical ring]]

category: algebra


brace algebra

__Brace algebra__ involves higher-order operations related to the study of $A_\infty$-[[A-infinity-algebra|algebra]]s. There is a sequence of canonical functors

dendriform algebras --> brace algebras --> pre-Lie algebras

Hence, in some sense, brace algebras are intermediate between [[dendriform algebra]]s and [[pre-Lie algebra]]s.

There is also a notion of a __symmetric brace algebra__ which has analogous role for $L_\infty$-[[L-infinity-algebra|algebra]]s instead of $A_\infty$. They are related to post-Lie algebras.

## Literature

#### Brace algebras

* [[Tornike Kadeishvili|Tornike V. Kadeishvili]], _The structure of the A(∞)-algebra, and the Hochschild and Harrison cohomologies_, (Russian)
Trudy Tbiliss. Mat. Inst. Razmadze Akad. Nauk Gruzin. SSR 91 (1988) 19--27. MR91a:18016
* M. Gerstenkhaber, A. A.Voronov, Higher-order operations on the Hochschild complex, Funktsional. Anal. i
Prilozhen. __29__:1 (1995) 1--6, 96; translation in Funct. Anal. Appl. 29 (1995), no. 1, 1--5. MR96g:18006
* [[Ezra Getzler]], _Cartan homotopy formulas and the Gauss-Manin connection in cyclic homology_, Quantum deformations of algebras and their representations (Ramat-Gan, 1991/1992; Rehovot, 1991/1992) 65--78, Israel Math. Conf. Proc. 7, 1993 MR95c:19002
* [[Marcelo Aguiar]], _Infinitesimal bialgebras, pre-Lie and dendriform algebras_, [pdf]

#### Symmetric brace algebras

* [[Tom Lada]], [[Martin Markl]], _Symmetric brace algebras_, Appl Categor Struct 13, 351–370 (2005) [doi](https://doi.org/10.1007/s10485-005-0911-2)

category: algebra