Brian Williams

* [personal page](https://sites.google.com/view/brianrwilliams/research)

## Selected writings

On holomorphic [[topological twist|twists]] of [[superconformal algebras]]:

* [[Ingmar Saberi]], [[Brian Williams]], *Superconformal algebras and holomorphic field theories* ([arXiv:1910.04120](https://arxiv.org/abs/1910.04120))

On the [[CS-WZW correspondence]] via [[factorization algebras]]:

* {#GwilliamRabinovichWilliams2022} [[Owen Gwilliam]], [[Eugene Rabinovich]], [[Brian R. Williams]], *Factorization algebras and abelian CS/WZW-type correspondences*, Pure and Applied Mathematics Quarterly **18** 4 (2022) 1485–1553  [[arXiv:2001.07888](https://arxiv.org/abs/2001.07888), [doi:10.4310/PAMQ.2022.v18.n4.a7](https://dx.doi.org/10.4310/PAMQ.2022.v18.n4.a7)]

category: people

[[!redirects Brian R. Williams]]


Briana Foster-Greenwood

* [personal page](https://www.cpp.edu/~brianaf/)

## Selected writings

On [[Cayley graph spectra]]:

* [[Briana Foster-Greenwood]], [[Cathy Kriloff]], *Spectra of Cayley graphs of complex reflection groups*, J. Algebraic Combin., 44(1):33–57, 2016 ([arXiv:1502.07392](https://arxiv.org/abs/1502.07392), [doi:10.1007/s10801-015-0652-8](https://doi.org/10.1007/s10801-015-0652-8))

category: people


Brice Halimi

Brice Halimi is an Associate Professor at the Université Paris Ouest Nanterre La Défense. He is a member of the Institut de Recherches Philosophiques (IREPh, EA 373) and an Associate Member of [[Laboratoire SPHERE]], (CNRS, UMR 7219).

* [Webpage](http://bhalimi.u-paris10.fr/)

##References##

* _Sur une application possible de la théorie de l'homotopie à la théorie des modèles, Annales de la Faculté des Sciences de Toulouse, Série 6, XXII/5 (2013) : 1017-1043._[pdf](https://bhalimi.u-paris10.fr/wp-content/uploads/2014/07/application_homotopie_theorie_des_modeles.pdf)

* _Homotopy model theory_, 15th Congress on Logic, Methodology, and Philosophy of Science (CLMPS 2015), Helsinki, August 4, 2015, [slides](https://bhalimi.u-paris10.fr/wp-content/uploads/2014/07/homotopy_model_theory_LC.pdf)

category: people


BRICS

**Basic Research in Computer Science** is an international research and PhD
school within the areas of computer and information sciences, hosted
by the Universities of Aarhus and Aalborg in Denmark, and funded
partially by the Danish Agency for Science, Technology, and Innovation.


bridge number

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

# Bridge number
* table of contents
{: toc}

## Definitions

A __bridge__ in a [[knot diagram]] is an arc that is the overpass in at least one crossing.

The __bridge number__, $b(K)$ of a [[knot]] $K$ is the minimum number of bridges occurring in a diagram of the knot.

By convention the [[unknot]] has bridge number equal to $1$.

Almost by default this gives a knot invariant as it takes the minimum over all [[knot diagram|knot diagrams]] that represent the isotopy class of the knot. (There is some work to do, but this is left for the moment!)

## Example

The usual picture of the trefoil knot, which we will write as $T_{2,3}$,
has three bridges, but the trefoil knot actually has bridge number 2, as shown:

[[!include trefoil knot - SVG]]

[[!include trefoil knot (2 bridge) - SVG]]

The first drawing corresponds to the picture as a (2,3)-[[torus knot]]. The knot thus can be represented on the surface of a solid torus $S^1\times D^2$ in $S^3$. The 3-sphere can be represented as the union of two solid tori with, in this case, the second one being $D^2\times S^1$, where the boundary of the $D^2$ in each case is the corresponding $S^1$ of the other case. Thinking of the trefoil on this other solid torus it is a $(3,2)$-torus knot .... (the second picture) and this has just two bridges, so $b(T_{2,3})\leq 2$.

*How does one know that the trefoil does not have $b(K) = 1$?*

For that we use
+-- {: .un_proposition}
###### Proposition

A knot $K$ has $b(K) = 1$ if and only if $K$ is the unknot.
=--

To complete the proof that $b(trefoil)= 2$, one needs to show or know that the trefoil is a non-trivial knot. That is most simply done using some invariant such as 3-[[n-colorability|colorability]].

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

The bridge number is a [[knot invariant]].
=--
This is more or less 'by default' as $b(K)$ is defined as a minimum over all [[knot diagram|knot diagrams]].

#####Remark:
Why is there the convention  that $b(unknot)= 1$? There is a neat result that means that if you know bridge number for [[prime knot|prime knots]] then you can (theoretically) calculate it for all composite knots. This states the following:
+-- {: .un_proposition}
###### Proposition

Let $K$ and $L$ be two (oriented) knots, and let $K+L$ be their oriented [[knot sum|sum]], then
$$b(K+L) = b(K) + b(L) -1.$$
=--
The proof is not hard. As '$K$ + unknot = $K$', the formula would be false if either knot was the unknot, if, that is, we had the other possible convention, (i.e. that the bridge number of the unknot should be 0). (There are other deeper reasons for the convention, but this is a neat one.)

(The pictures of the trefoil on this page are based on [[two-trefoils.pdf|this picture of two trefoils:file]] from the [Maths and Knots Exhibition website](http://www.popmath.org.uk/exhib/knotexhib.html), of the Centre for the Popularisation of Mathematics. Their assistance is gratefully acknowledged.)
[[!redirects bridge number]]
[[!redirects bridge numbers]]


Bridgeland stability condition

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

#Contents#
* table of contents
{:toc}

## Idea

### General

A _stability condition_ ([Bridgeland 02](#Bridgeland02)) on a [[triangulated category]] ([hence](triangulated+category#FromStableModelCategories) on a [[stable model category]]/[[stable (∞,1)-category]]) $\mathcal{D}$ (Def. \ref{StabilityCondition} below) singles out some of its [[objects]], to be called the "stable objects" (Def. \ref{StableObjects} below), that behave, to some extent, like [[indecomposable objects]] (whence "stable"), in that a version of [[Schur's lemma]] applies to them (Prop. \ref{SchurLemma} below).

Indeed, in the degenerate case that $\mathcal{D}$ is the [[derived category]] of a [[semisimple category]], the stable objects are just the [[irreducible objects]].

In the case where $\mathcal{D}$ is the bounded [[derived category]] of [[coherent sheaves]] on some [[complex manifold]], the general notion of stability reduces to the classical notion of [[slope of a coherent sheaf|slope]]-[[stable coherent sheaf|stability of coherent sheaves]] ("$\mu$-stability", see Example \ref{SlopeStability} below).

In other examples Bridgeland stability reproduces the derived analogue of [[David Mumford]]'s concept of [[GIT-stable point|stability]] in [[geometric invariant theory]] ([King 94](#King94)).

There are in general different stability conditions on one and the same [[triangulated category]], in fact their [[moduli space]] $Stab(\mathcal{D})$ forms a [[complex manifold]] (Prop. \ref{SpaceOfStabilityConditions} below). The collection of stable objects in $\mathcal{D}$ in general depends on the choice of stability conditions, hence of the choice of point in this [[moduli space]]: There are in general [[codimension]]-1 [[submanifolds]] of $Stab(\mathcal{D})$, called _walls_, such that the set of stable objects is constant close to both sides of these walls, but changes as the wall is crossed. This phenomenon is hence known as _[[wall crossing]]_.

The purely mathematical motivation for these definitions is, to a large extent, just their intrinsic richness. The concept finds its meaning in the concept of stability of [[D-branes]] in [[string theory]] and was abstracted from informal considerations about [[B-branes]] of the [[B-model]] [[topological string]] due to ([Douglas-Fiol-Romerlsberger 00](#DouglasFiolRomerlsberger00)) and followups ("$\Pi$-stability").

This [[string theory|string-theoretic]] interpretation also sheds light on the older notion of [[slope of a coherent sheaf|slope]]-[[stable coherent sheaf|stability of coherent sheaves]], we discuss this [below](#AsStabilityOfBPSDBranes).

### As stability of BPS D-branes
 {#AsStabilityOfBPSDBranes}

We explain here how to understand the mathematical formulation of stability conditions as a formalization of the concept of stable [[BPS states]] of [[D-branes]], hence of [[extremal black hole|extremal]] [[black branes]]. This yields a very simple and beautiful picture, which may be hard to extract from the original proposal due to [Douglas-Fiol-Romerlsberger 00](#DouglasFiolRomerlsberger00) (as witnessed by various reviews of the concept, see for instance [Stellari 15, slide 9](#Stellari15)).

A key point to notice is that _[[coherent sheaves]]_ and with them most of the kinds of [[objects]] in [[triangulated categories]] of relevance here, may be thought of as models for [[D-branes]] carrying [[D-brane charge]].

Specifically, for $X$ a [[space]], to be thought of as [[spacetime]], a [[coherent sheaf]] $E$ over $X$ may be thought of as an [[abelian sheaf]] of [[sections]] of a kind of  [[complex vector bundle]] over $X$, whose [[fiber]] [[dimension]] is allowed to jump in some controlled way. Hence coherent sheaves are a slight generalization of _[[complex vector bundles]]_.

When considering [[D-branes]] on $X$, such coherent sheaves/vector bundles $E$ appear as the [[Chan-Paton gauge fields]] on the D-brane.

$$
  \text{object}\, E
  \phantom{AA}\leftrightarrow\phantom{AA}
  \text{D-brane carrying mass and charge}
$$

From this perspective, the [[rank of a vector bundle|rank]]

$$
  rank(E)
  \;\in\;
  \mathbb{R}
$$

of $E$ is the number of coincident D-branes (there may be [[fractional D-branes]], and, once we pass to the [[derived category]], there may be [[anti D-branes]], so that this number need not be a [[positive integer]]).

The [[D-branes]] have a fixed [[tension]] and hence a fixed [[mass]]-density, so that the total [[mass]] of the D-branes corresponding to $E$ is proportional to this rank. Ignoring the constant proportionality factor, we make this explicit by re-writing the rank as

$$
  M(E)
  \;\coloneqq\;
  rank(E)
  \,.
$$

In addition to their mass, the D-branes carry [[charge]], called [[RR-field|RR-charge]]. This is a generalization of the classical [[magnetic charge]] known from [[Dirac charge quantization]]. As explained there, magnetic charge reflected in a [[complex line bundle]], as sourced by magnetic [[monopoles]] ([[D0-branes]]) is measured by the [[first Chern class]]  $c_1(E)$. In generalization of this, the total [[D-brane charge]] reflected in a [[Chan-Paton gauge field]] $E$ is proportional its [[Chern character]] $ch(E)$.

$$
  Q(E)
    \;\in\;
  \mathbb{R}
  \,.
$$

For coherent sheaves this is essentially what is classically called the _[[degree of a coherent sheaf]]_. Interpreting this as the [[charge]] carried by the D-brane, we may write

$$
  Q(E)
  \;=\;
  degree(E)
$$

Accordingly, [[D-branes]] $E$ have a _charge density_ proportional to

$$
  ChargeDensity(E)
    \;\coloneqq\;
  \frac{ Q(E) }{ M(E) }
    \;=\;
  \frac{ degree(E) }{ rank(E) }
    \;=\;
   slope(E)
  \,.
$$

This quotient is what is classically called the _[[slope of a coherent sheaf|slope]]_, as shown on the right. The terminology comes from thinking of the [[pair]] ([[mass]], [[charge]]), hence the pair ([[rank of a coherent sheaf|rank]], [[degree of a coherent sheaf|degree]]) as specifying a [[point]] in the [[plane]]

$$
  (M(E), \; Q(E))
   \;=\;
  ( rank(E), \; degree(E) )
  \;\in\;
  \mathbb{R}^2
$$

Under this identification the charge density is the [[slope of a line]] of the [[line]] in the [[Cartesian space|Cartesian]] [[plane]] which goes through [[zero]] and through $(M(E), Q(E))$, whence the term _[[slope of a coherent sheaf]]_. But understanding this not as a slope but as a _charge density_ reveals why this has anything to do with "stability", as we proceed to explain now.

First notice that, alternatively, we may identify the [[real numbers|real]] [[plane]] with the [[complex plane]] and thus unify the [[mass]] and [[charge]] of [[D-branes]] into a single [[complex number]]

\[
  \label{TheComplexifiedMassInIdeaSection}
  \begin{aligned}
    Z(E)
    & \coloneqq\;
    M(E) + i \, Q(E)
    \\
    & = m(E) \, \exp( i \pi \phi(E) )
    \;\in\;
    \mathbb{C}
  \end{aligned}
\]

with an [[absolute value]] $m(E)$ and a [[complex phase]] (modulo $\pi$) $\phi(E)$.

(Such unification of two different quantities into a single complex quantity appears all over [[supersymmetry|supersymmetric]] theory, for instance also in the definition of [[complex volume]] or the complex coupling constant appearing in the context of [[S-duality]].)

Finally to see what all this has to do with "stability":

In a [[supersymmetry|supersymmetric]] theory such as [[superstring theory]] the stable states are supposed to be the [[BPS-states]]. When thinking of [[D-branes]] as [[black branes]], being higher dimensional generalizations of charged [[black holes]], the BPS states correspond to the higher dimensional analog of the _[[extremal black holes]]_, namely those that carry maximum [[charge]] for given [[mass]], hence that _maximize their charge density_

$$
  \text{stable D-brane}
  \;\;\Leftrightarrow\;\;
  \text{BPS/extremal black D-brane}
  \;\;\Leftrightarrow\;\;
  \text{ maximum charge density }\, Q/M
$$

Now it is plausible that a D-brane $E$ maximizes its potential charge density if removing any part $e$ of it reveals that $e$ by itself has lower charge density.

More formally, $E$ should maximize its charge density $Q(E)/M(E)$ and hence be _stable_ if for all sub-parts

$$
  e \subset E
$$

hence for all _[[subobjects]]_ in the relevant [[category]] (such as that of [[coherent sheaves]]) we have that the charge density of the part is smaller than the charge density of the whole:

\[
  \frac{Q(e)}{M(e)}
  \;\lt\;
   \frac{ Q(E) }{ M(E) }
\]

Conversely, this means that a D-brane state $e$ can increase its charge density, hence get close to being BPS and hence stable, by forming a [[bound state]] to be come an $E$.

In the case of coherent sheaves, where the D-brane charge density is called the _[[slope of a coherent sheaf]]_ $slope(E) = Q(E)/M(E)$, as above, this says that $E$ is a [[stable coherent sheaf]] precisely if for all [[subobjects]] $e \subset E$ we have

$$
  slope(e) \;\lt\; slope(E)
  \,.
$$

This is the classical formulation of _[[stable coherent sheaf|slope stability]]_ or _$\mu$-stability_ of [[coherent sheaves]].

Of course we may equivalently re-express in terms of the [[complex phase]] $\phi(E)$ induced by the complexified mass/charge from (eq:TheComplexifiedMassInIdeaSection). Since the slope of a [[complex number]] (regarded as a [[vector]] in the [[plane]]) equals the [[tan]] of its [[complex phase]] we have

$$
  \frac{Q(E)}{M(E)}
  \;=\;
  tan\big( \phi(E)/2 \big)
  \,.
$$

But since $tan((-)/2)$ is a [[monotone function]] on the [[domain]] $(-\pi/2, \pi/2)$, then if we agree to regard $\phi$ as taking values in that [[interval]], then the above stability condition becomes equivalently the statement that for all [[subobjects]] $e \hookrightarrow E$ we have

$$
  \phi(e) \;\lt\; \phi(E)
  \,.
$$

This is the form in which [Bridgeland 02](#Bridgeland02) phrases the stability condition, see Def. \ref{StableObjects} below.

But the upshot is that this is still equivalent to saying that $E$ is stable precisely if it maximizes its [[charge]]- over [[mass]]-density, as befits an [[extremal black hole]] and, more generally, a [[BPS state|BPS]] [[black brane]].

$\,$

## Definition

+-- {: .num_defn #StabilityFunction}
###### Definition
**(stability function)**

Let $\mathcal{A}$ be an [[additive category]].

A **stability function** (sometimes also called a _central charge_) is a  [[linear map]]

\[
  \label{StabilityFunction}
  Z
  \;\colon\;
  K(\mathcal{A})
  \longrightarrow
  \mathbb{C}
\]

from [[Grothendieck group]] $K(\mathcal{A})$ to the [[abelian group|additive group]] of [[complex numbers]],
such that for all non-[[zero objects]] $E \in \mathcal{A}$, the [[image]] $Z(E)$ lies in the [[upper half-plane|semi-upper half plane]]

$$
  H
  \;=\;
  \big\{
    r exp(i\pi \phi)
    \;\vert\;
    r \gt 0
    \;\text{and}\;
    0 \lt \phi \leq 1
  \big\}
$$

The _phase_

\[
  \label{PhaseOfAnObject}
  \phi(E)
  \;\coloneqq\;
  \tfrac{1}{\pi}
  arg\big(
    Z(E)
  \big)
  \;\in\;
  (0,1]
\]

of an non-[[zero object]] $E \in \mathcal{A}$ is just the [[complex phase]] $\phi$ that occurs in the representation from $H$. Alternatively, by plotting $Z(E)$ in the  [[complex plane]], the phase is the argument (slope) divided by $\pi$.

=--

([Bridgeland 02, Def. 2.1](#Bridgeland02))

+-- {: .num_defn #StableObjects}
###### Definition
**((semi-)stable objects)**

For $\mathcal{A}$ an [[abelian category]] equipped with a stability function $Z = r exp(i\pi \phi)$ (Def. \ref{StabilityFunction}). Then a non-[[zero object]] $E \in \mathcal{A}$ is called

1. a _semi-stable object_ if for all non-[[zero object|zero]] [[subobjects]] $F\subset E$ the phase (eq:PhaseOfAnObject) of $F$ is smaller or equal that of $E$

   \[
     \label{SemiStability}
     \phi(F)
       \;\leq\;
     \phi(E)
   \]

1. a _stable object_ if for all non-[[zero object|zero]], proper [[subobjects]] $F \subset E$ the phase (eq:PhaseOfAnObject) of $F$ is strictly smaller than that of $E$:

   \[
     \label{Stability}
     \phi(F)
       \;\lt\;
     \phi(E)
     \,.
   \]

=--

([Bridgeland 02, Def. 2.2](#Bridgeland02))

+-- {: .num_defn #HarderNarasimhanProperty}
###### Definition
**(Harder-Narasimhan property)**

A stability function $Z \;\colon\; K(\mathcal{A})\to \mathbb{C}$ (Def. \ref{StabilityFunction}) is said to have the _Harder-Narasimhan property_ if for any non-[[zero object]] $E$ there exists a finite [[filtered object|filtration]] by [[subobjects]]

$$
  0=E_0 \subset E_1 \subset \cdots \subset E_n =E
$$

such that the [[quotients]]

$$
  F_i
   =
  E_i/E_{i-1}
$$

are all semi-stable (eq:SemiStability) and satisfy

$$
  \phi(F_1) \gt \phi(F_2) \gt \cdots \gt\phi(F_n)
  \,.
$$

=--

([Bridgeland 02, Def. 2.3](#Bridgeland02))

+-- {: .num_defn #Slicing}
###### Definition
**(slicing)**

Let $\mathcal{D}$ be a [[triangulated category]] (usually arising as the [[derived category]] of some [[abelian category]]).

A _slicing_  $\mathcal{P}$ on $\mathcal{D}$ is a choice of [[additive category|additive]] [[full subcategories]] $\mathcal{P}(\phi) \subset \mathcal{D}$ for each $\phi \in \mathbb{R}$ satisfying

1. $\mathcal{P}(\phi +1)=\mathcal{P}(\phi)[1]$

2. If $\phi_1 \gt \phi_2$ and $A_j\in \mathcal{P}(\phi_j)$, then $Hom(A_1, A_2)=0$.

3. Any object has a finite [[filtration]] by the slicing:

   If $E\in \mathcal{D}$, then there exists $\phi_1 \gt \cdots \gt \phi_n$ and a sequence $0=E_0\to E_1 \to \cdots \to E_n = E$ such that the [[mapping cone]] $E_{j-1}\to E_j \to F_j \to E_{j-1}[1]$ satisfies $F_j\in \mathcal{P}(\phi_j)$.

=--

([Bridgeland 02, Def. 3.3](#Bridgeland02))

+-- {: .num_defn #StabilityCondition}
###### Definition
**(stability condition)**

A **stability condition** on a [[triangulated category]] $\mathcal{D}$ is a [[pair]] $\sigma = (Z, \mathcal{P})$ consisting of

1. a stability function (Def. \ref{StabilityFunction})

1. a slicing (Def. \ref{Slicing})

satisfying the relation that given a non-[[zero object]] $E\in \mathcal{P}(\phi)$, then there is a [[positive number|positive]] [[real number]] $m(E)$ such that the value $Z(E)$ of the stability function (Def. \ref{StabilityCondition}) is

$$
  Z(E)
  \;=\;
  m(E)
  \,
  exp(i\pi \phi)
  \,.
$$

=--

([Bridgeland 02, Def. 5.1](#Bridgeland02))

This justifies the repeated notation of $\phi$, since this says that if an object lies in a particular slice $\mathcal{P}(\phi)$, then it must also have [[complex phase]] $\phi$.

$\,$

## Properties

### In terms of hearts of t-structures

Bridgeland proves that an equivalent way to give a stability is to specify a heart of a bounded [[t-structure]] on $\mathcal{D}$ and give a stability function the heart that satisfies the Harder-Narasimhan property.

### Schur's lemma
 {#LemmaSchur}

+-- {: .num_prop #SchurLemma}
###### Proposition
**([[Schur's lemma]])**

Let $\mathcal{A}$ be an [[abelian category]] equipped with a stability condition (Def. \ref{StabilityCondition}).

For $E \in \mathcal{A}$ a stable object (Def \ref{StableObjects}), every [[endomorphism]] of $E$ is either the [[zero morphism]] or is an [[isomorphism]]; in particular if $\mathcal{A}$ is [[enriched category|enriched]] in [[vector spaces]] over an [[algebraically closed field]] $k$, then $End(E) \simeq k$.

More generally, for $E_1, E_2 \in \mathcal{A}$ two stable objects of the same slope/phase, $\phi(E_1) =\phi(E_2)$, any [[morphism]] $E_1 \to E_2$ is either the [[zero morphism]] or is an [[isomorphism]].

=--

e.g. ([Bayer 11, 2.3 Exercise 1](#Bayer11),  [Martinez 13, prop. 3](#Martinez13))

### Space of stability conditions
 {#SpaceOfStabilityConditions}

+-- {: .num_prop #SpaceOfStabilityConditions}
###### Proposition
**(space of stability conditions)**

Under reasonable hypotheses (...), one can put a natural [[topological manifold|topology]] on the space $Stab(\mathcal{D})$ of all stability conditions (Def. \ref{StabilityCondition}) , under which the space becomes a [[complex manifold]].
=--

([Bridgeland 02, theorem 1.2](#Bridgeland02), reviewed in [Bridgeland 09, 3.2](#Bridgeland09))

Most work using this fact has been done in the case of the [[bounded derived category of coherent sheaves]] $\mathcal{D}=D^b(Coh(X))$ where $X$ is a smooth, projective [[algebraic variety|variety]] over $\mathbb{C}$ so that $\mathcal{D}$ is $\mathbb{C}$-linear and $K(\mathcal{D})$ is finitely generated.

The space of stability conditions $Stab(X)$ has a locally finite wall and chamber decomposition. The philosophy is that if one fixes a numerical condition $v$ and considers the [[moduli space]] of $\sigma$-stable sheaves as $\sigma$ varies through $Stab(X)$, then the moduli spaces $M_\sigma(v)\simeq M_{\sigma '}(v)$ should be isomorphic if $\sigma$ and $\sigma'$ are in the same chamber. Bayer and Macri have shown this is true for K3 surfaces, and upon [[wall crossing|crossing a wall]] the moduli spaces are related by a [[birational map]].

## Examples

### Slope-stability of vector bundles / coherent sheaves
 {#SlopeStabilityOfVectorBundles}

A motivating example for the concept of Bridgeland stability is the following classical notion.

+-- {: .num_example #SlopeStability}
###### Example
**([[slope of a coherent sheaf|slope]]-[[stable coherent sheaf|stability of coherent sheaves]])**

Let $X$ be a non-singular, [[algebraic curve|projective curve]] over $\mathbb{C}$. Let $\mathcal{A}=Coh(X)$ be the [[category]] of [[coherent sheaves]] on $X$.

In this case the standard stability function (Def. \ref{StabilityFunction}) is

\[
  \label{StandardStabilityFunctionForVectorBundles}
  Z(E)
    \;\coloneqq\;
  -deg(E) + i rk(E)
\]

where $deg$ is the _[[degree of a coherent sheaf|degree]]_ and $rk$ the _[[rank of a coherent sheaf|rank]]_ of a [[coherent sheaf]] $E$.

The classical notion of the _[[slope of a coherent sheaf|slope]] of a vector bundle is

$$
  \mu(E)
  \;\coloneqq\;
  \frac{deg(E)}{rk(E)}
$$

When constructing a [[moduli space]] of vector bundles using [[geometric invariant theory|GIT]] one needs to consider only [[slope of a coherent sheaf|slope]] (semi-)[[stable vector bundles]] (see e.g. [Reineke 08, sections 3 and 4](#Reineke08)).

One can immediately see that a [[vector bundle]]/[[coherent sheaf]] is [[slope of a vector bundle|slope]] (semi-)[[stable vector bundle|stable]] if and only if it is (semi-)stable with respect to this stability function (eq:StandardStabilityFunctionForVectorBundles).

=--

Thus Bridgeland stability generalizes the classical notions of [[stable vector bundle|stability of vector bundles]].

### Over resolutions of ADE-singularities
 {#OverResolutionsOfADESingularities}

For $G_{ADE} \subset SU(2)$ a [[finite subgroup of SU(2)]],
let $\tilde X$ be the [[resolution of singularities|resolution]] of the corresponding [[ADE-singularity]].

Then the [[connected component]] of the space of stability conditions (Def. \ref{StabilityCondition}) on the bounded [[derived category]] of [[coherent sheaves]] over $\tilde X$ can be described explicitly ([Thomas 02](#Thomas02), [Bridgeland 05](#Bridgeland05), [Brav-Thomas 09](#BravThomas09)).

Specifically for type-A singularities the space of stability conditions (Prop. \ref{SpaceOfStabilityConditions}) is in fact [[connected topological space|connected]] and [[simply-connected topological space]] ([Ishii-Ueda-Uehara 10](#IshiiUedaUehara10)). In fact spaces of stability structures over [[Dynkin quivers]] are [[contractible space|contractible]] ([Qiu-Woold 14](#QiuWoold14))

Brief review is in [Bridgeland 09, section 6.3](#Bridgeland09).

## Related concepts

* [[BPS states]]

* [[wall crossing]]

* [[geometric invariant theory]]

$\,$

## References

### General

The general definition is due to

* {#Bridgeland02} [[Tom Bridgeland]], _Stability conditions on triangulated categories_, Ann. of Math. 166 (2007) 317–345 ([math.AG/0212237](http://arxiv.org/abs/math/0212237))

generalizing the classical concept of [[slope of a vector bundle|slope]]-[[stable vector bundle|stability of vector bundles]] and of [[modules]] as in

* {#King94} [[Alastair King]], _Moduli of representations of finite dimensional algebras_, The Quarterly Journal of Mathematics 45.4 (1994): 515-530 ([pdf](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.623.649&rep=rep1&type=pdf))

and motivated by informal arguments in [[string theory]] about the "$\Pi$-stability" for [[B-branes]] of the [[B-model]] [[topological string]], due to [Douglas-Fiol-Römerlsberger 00](#DouglasFiolRomerlsberger00) and expanded on in [Douglas 01](#Douglas01), [Douglas 02](#Douglas02), [Aspinwall 04](#Aspinwall04) and other articles.

Further developments include

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

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

* Rina Anno, Roman Bezrukavnikov, Ivan Mirković, _A thin stringy moduli space for Slodowy slices_, [arxiv/1108.1563](http://arxiv.org/abs/1108.1563)

* Arend Bayer, Emaneule Macri, _Projectivity and Birational Geometry of Bridgeland Moduli Spaces_ ([arXiv:1203.4613](http://arxiv.org/abs/1203.4613))

* [[Tom Bridgeland]], Ivan Smith, _Quadratic differentials as stability conditions_, [arxiv/1302.7030](http://arxiv.org/abs/1302.7030)

* {#Martinez13} Cristian Martinez, _Duality, Bridgeland wall-crossing and flips of secant varieties_ ([arXiv:1311.1183](https://arxiv.org/abs/1311.1183))

### Introduction and review

* {#Bridgeland09} [[Tom Bridgeland]], _Spaces of stability conditions_, Proc. of symposia in pure math. __80__, 2009 ([math/0611510](http://arxiv.org/abs/math/0611510))

* {#Reineke08} Markus Reineke, _Moduli of representations of quivers_ ([arXiv:0802.2147](https://arxiv.org/abs/0802.2147))

* {#Bayer11} Arend Bayer, _A tour to stability conditions on derived categories_, 2011 ([pdf](https://www.maths.ed.ac.uk/~abayer/dc-lecture-notes.pdf))

* [[Daniel Huybrechts]], _Introduction to stability conditions_ ([arXiv:1111.1745](https://arxiv.org/abs/1111.1745))

* Jan Engenhorst, _Bridgeland Stability Conditions in Algebra, Geometry and Physics_, 2014 ([pdf](https://www.freidok.uni-freiburg.de/fedora/objects/freidok:9595/datastreams/FILE1/content))

* {#Stellari15} Paolo Stellari, _A tour on Bridgeland stability_, 2015 ([pdf](https://users.unimi.it/stellari/Research/Slides/LucidiStabCondBeamer.pdf))

* [[Franco Rota]], _An introduction to stability conditions_, lecture notes [[pdf](https://drive.google.com/file/d/1sJSwN-JTOr77fpIgRWfDXkidYZmrJ0jC/view), [[Rota-StabilityConditions.pdf:file]]]

### Examples

Discussion of examples of stability conditions

over [[resolution of singularities|resolutions]] of [[ADE-singularities]]:

* {#Thomas02} R. P. Thomas, _Stability conditions and the braid group_, Communications in Analysis and Geometry 14, 135-161, 2006 ([arXiv:math/0212214](https://arxiv.org/abs/math/0212214))

* {#Bridgeland05} [[Tom Bridgeland]], _Stability conditions and Kleinian singularities_, International Mathematics Research Notices 2009.21 (2009): 4142-4157 ([arXiv:0508257](https://arxiv.org/abs/math/0508257))

* {#IshiiUedaUehara10} Akira Ishii, Kazushi Ueda, Hokuto Uehara, _Stability conditions on $A_n$-singularities_, Journal of Differential Geometry 84 (2010) 87-126 ([arXiv:math/0609551](https://arxiv.org/abs/math/0609551))

* {#BravThomas09} [[Christopher Brav]], Hugh Thomas, _Braid groups and Kleinian singularities_ ([arXiv:0910.2521](https://arxiv.org/abs/0910.2521))

* {#QiuWoold14} [[Yu Qiu]], Jon Woolf, _Contractible stability spaces and faithful braid group actions_ ([arXiv:1407.5986](https://arxiv.org/abs/1407.5986))

* {#Qiu15} [[Yu Qiu]], Def. 2.1 _Stability conditions and quantum dilogarithm identities for Dynkin quivers_, Adv. Math., 269 (2015), pp 220-264 ([arXiv:1111.1010](https://arxiv.org/abs/1111.1010))

* [[Tom Bridgeland]], [[Yu Qiu]], Tom Sutherland, _Stability conditions and the $A_2$ quiver_ ([arXiv:1406.2566](https://arxiv.org/abs/1406.2566))

### Relation to stable branes in string theory
 {#RelationToStableBranesInStringTheory}

The proposal that slope-stability of vector bundles should generalize to a notion of stability ("$\Pi$-stability") of [[B-branes]]/[[D-branes]] originates with

* {#DouglasFiolRomerlsberger00} [[Michael Douglas]], Bartomeu Fiol, Christian Römelsberger, _Stability and BPS branes_, JHEP 0509:006, 2005 ([arXiv:hep-th/0002037](https://arxiv.org/abs/hep-th/0002037) [doi](https://doi.org/10.1088/1126-6708/2005/09/006))

In terms of stability ($\Pi$-stability) of [[B-branes]] of the [[B-model]] [[topological string]]:

* {#Douglas01} [[Michael Douglas]], _D-branes, categories and $N=1$ supersymmetry, J.Math.Phys. __42__ (2001) 2818–2843;

* {#Douglas02} [[Michael Douglas]],  _Dirichlet branes, homological mirror symmetry, and stability_, Proc. ICM, Vol. III (Beijing, 2002), 395–408, Higher Ed. Press, Beijing, 2002

* {#Aspinwall04} [[Paul Aspinwall]], _D-Branes on Calabi-Yau Manifolds_ ([arXiv:hep-th/0403166](https://arxiv.org/abs/hep-th/0403166))

* [[Aaron Bergman]], _Stability Conditions and Branes at Singularities_,  Journal of High Energy Physics 2008.10 (2008): 07 ([arXiv:hep-th/0702092](http://arxiv.org/abs/hep-th/0702092))

* {#MalyshevVerlinde07} Dmitry Malyshev, [[Herman Verlinde]], _D-branes at Singularities and String Phenomenology_, Nucl.Phys.Proc.Suppl.171:139-163, 2007 ([arXiv:0711.2451](https://arxiv.org/abs/0711.2451))

On marginally stable branes:

* [[Paul Aspinwall]], Alexander Maloney, Aaron Simons, _Black Hole Entropy, Marginal Stability and Mirror Symmetry_, JHEP0707:034, 2007 ([arxiv:hep-th/0610033](https://arxiv.org/abs/hep-th/0610033))

[[!redirects Bridgeland stability conditions]]

[[!redirects Bridgeland stability]]

[[!redirects stability condition]]
[[!redirects stability conditions]]

[[!redirects stability function]]
[[!redirects stability functions]]


Brie Finegold

Teaching postdoc at University of Arizona, writer for AMS Math in the Media website, former AMS-AAAS Mass Media Fellow, [current website](http://www.math.arizona.edu/~bfinegold)


Brillouin torus

+-- {: .rightHandSide}
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### Context
#### Solid state physics
+-- {: .hide}
[[!include solid state physics -- contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Given a [[crystal|crystalline]] material, its *Brillouin torus* is the [[space]] of distinguishable [[momenta]]/[[wave vectors]] of its [[electron|electronic]] excitations. Due to the periodicity of the crystal, also these momenta/wave vectors are (dually) periodically identified, which makes this space an [[n-torus]].

Mathematically, the [[discrete space]] underlying a [[crystal]] is a [[subset]] of a [[Euclidean space]] $S \;\subset\;E^n \simeq \mathbb{R}^n$ (the set of *atomic sites*) which is preserved (as a subset) by the [[group action|action]] of a [[crystallographic group]] $\Lambda \rtimes G \,\subset\, Iso(E)$, where $\Lambda$ is a full [[lattice in a vector space|lattice]] (and $G$ the corresponding [[point group]]). Then its Brillouin torus is the [[Pontrjagin dual group]] of this lattice (eg. [Freed-Moore 13](#FreedMoore13)):

$$
  \mathbb{T}^n_{Br} \,=\, Hom_{Grp}\big(\Lambda, \, S^1 \big)
  \,.
$$

## Related concepts

* [[Brillouin zone]]

* [[Bravais lattice]]

* [[Bloch-Floquet theory]]

* [[valence bundle]], [[conduction bundle]]

## References
 {#References}

The concept is named after:

* {#Brillouin30} [[Léon Brillouin]], *Les électrons libres dans les métaux et le role des réflexions de Bragg*, J. Phys. Radium **1** (1930) 377-400 [[doi:10.1051/jphysrad:01930001011037700](https://doi.org/10.1051/jphysrad:01930001011037700), [hal:jpa-00233038](https://hal.archives-ouvertes.fr/jpa-00233038), [pdf](https://hal.archives-ouvertes.fr/jpa-00233038/document)]

Textbooks on [[solid state physics]] traditionally speak of the "reciprocal lattice" (e.g. [Kittel 1953](solid+state+physics#Kittel53), [p. 27](http://metal.elte.hu/~groma/Anyagtudomany/kittel.pdf#page=47)) which is the dual lattice $Hom_{Grp}\big(\Lambda, \, \mathbb{Z} \big)$, e.g.:

* [[Michael Reed]], [[Barry Simon]], p. 311 of: Sec. XIII.16 *Schrödinger operators with periodic potentials*, of:  *[[Methods of Modern Mathematical Physics]] -- IV: Analysis of Operators*, Academic Press (1978)
([ISBN:9780080570457](https://www.elsevier.com/books/iv-analysis-of-operators/reed/978-0-08-057045-7))

* {#Tong2017} [[David Tong]], [§2.2.2](http://www.damtp.cam.ac.uk/user/tong/aqm/solidstate.pdf#page=57) in: *Lectures on solid state physics* (2017) $[$[pdf](http://www.damtp.cam.ac.uk/user/tong/aqm/solidstate.pdf), [webpage](http://www.damtp.cam.ac.uk/user/tong/solidstate.html)$]$

The resulting Brillouin torus $Hom_{Grp}\big(\Lambda, \, \mathbb{R} \big)/Hom_{Grp}\big(\Lambda, \, \mathbb{Z} \big)$ is often left implicit. It is made explicit in:

* {#FreedMoore13} [[Daniel Freed]], [[Gregory Moore]], [p. 52](https://arxiv.org/pdf/1208.5055.pdf#page=52) of: _Twisted equivariant matter_, Ann. Henri Poincaré (2013) 14: 1927 $[$[arXiv:1208.5055](https://arxiv.org/abs/1208.5055), [doi:10.1007/s00023-013-0236-x](https://doi.org/10.1007/s00023-013-0236-x)$]$

[[!redirects Brillouin tori]]

[[!redirects Brillouin zone]]
[[!redirects Brillouin zones]]


Brillouin zone

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### Context
#### Solid state physics
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[[!include solid state physics -- contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

(...)

## Related concepts

* [[Brillouin torus]]

* [[solid state physics]]

* [[K-theory classification of topological phases of matter]]

* [[noncommutative topology of quasiperiodicity]]

## References

Named after:

* [[Léon Brillouin]], *Les électrons libres dans les métaux et le role des réflexions de Bragg*, J. Phys. Radium **1** (1930) 377-400 [[doi:10.1051/jphysrad:01930001011037700](https://doi.org/10.1051/jphysrad:01930001011037700), [hal:jpa-00233038](https://hal.archives-ouvertes.fr/jpa-00233038), [pdf](https://hal.archives-ouvertes.fr/jpa-00233038/document)]

See also:

* Wikipedia, *[Brillouin zone](https://en.wikipedia.org/wiki/Brillouin_zone)*

[[!redirects Brillouin zone]]


Brooke Shipley

* [website](http://www.math.uic.edu/~bshipley/)

## Selected writings

On the [[stable Dold-Kan correspondence]]:

* {#SchwedeShipley03} [[Stefan Schwede]], [[Brooke Shipley]], _Stable model categories are categories of modules_, Topology 42 (2003), 103-153 ([pdf](http://www.math.uic.edu/~bshipley/classTopFinal.pdf), doi:10.1016/S0040-9383(02)00006-X)

* {#Shipley02} [[Brooke Shipley]], _$H \mathbb{Z}$-algebra spectra are differential graded algebras_, Amer. Jour. of Math. 129 (2007) 351-379. [[arXiv:math/0209215](http://arxiv.org/abs/math/0209215), [jstor:40068065](https://www.jstor.org/stable/40068065)]

On [[model categories]] of [[coalgebras over a comonad]] via left-[[transferred model category|left transfer]] (left-induction):

* [[Kathryn Hess]], [[Brooke Shipley]], *The homotopy theory of coalgebras over a comonad*, Proceedings of the London Mathematical Society **108** 2 (2014) [[arXiv:1205.3979](https://arxiv.org/abs/1205.3979), [doi:10.1112/plms/pdt038](https://doi.org/10.1112/plms/pdt038)]

On left-[[transferred model structures]] and [[model structures on functors]]:

* [[Marzieh Bayeh]], [[Kathryn Hess]], [[Varvara Karpova]], [[Magdalena Kedziorek]], [[Emily Riehl]], [[Brooke Shipley]], *Left-induced model structures and diagram categories*, in: *Women in Topology: Collaborations in Homotopy Theory*, Contemporary Mathematics **641** American Mathematical Society (2015) [[arXiv:1401.3651](http://arxiv.org/abs/1401.3651), [ISBN:978-1-4704-2495-4](https://bookstore.ams.org/conm-641)]

* {#HKRS15} [[Kathryn Hess]], [[Magdalena Kedziorek]], [[Emily Riehl]], [[Brooke Shipley]], *A necessary and sufficient condition for induced model structures*, J. Topology **10** 2  (2017) 324-369 [[arXiv:1509.08154](http://arxiv.org/abs/1509.08154), [doi:10.1112/topo.12011](https://doi.org/10.1112/topo.12011)]

  > (this contains an error which is corrected in [Garner, Kedziorek & Riehl 2018](transferred+model+structure#GKR18))

## Related $n$Lab entries

* [[stable model category]]

* [[stable Dold-Kan correspondence]]

* [[monoidal Dold-Kan correspondence]]

* [[model structure for ring spectra]]

* [[algebra spectrum]]

* [[free spectrum]]

category: people


Brouwer's fixed point theorem

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###Context###
#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

[[Brouwer]]'s fixed point theorem says that every [[continuous function]] from a [[compact topological space|compact]] [[convex set]] to itself has at least one [[fixed point]].

This is also a special case of the [[Lefschetz fixed point theorem]], see [there](Lefschetz+trace+formula#ForContractibleCompactPolyhedra).

## Related theorems

* [[fixed point]]

* [[topological invariance of dimension]]

* [[Tietze extension theorem]]

* [[Lefschetz fixed point theorem]]

* [[Lawvere's fixed point theorem]]

* [[Kleene's fixed point theorem]]

* [[Atiyah-Bott fixed point formula]]

## References

Textbook account

* {#DubrovinNovikovFomenko85} B. A. Dubrovin, [[S. P. Novikov]], A. T. Fomenko, corollary 15.3.4 of _Modern Geometry — Methods and Applications: Part II: The Geometry and Topology of Manifolds_, Graduate Texts in Mathematics 104, Springer-Verlag New York, 1985

See also

* Wikipedia, _[Brouwer fixed-point theorem](https://en.wikipedia.org/wiki/Brouwer_fixed-point_theorem)_

Discussion in [[cohesive homotopy type theory]] is in

* {#Shulman15} [[Mike Shulman]], _Brouwer's fixed-point theorem in real-cohesive homotopy type theory_, Mathematical Structures in Computer Science Vol 28 (6) (2018): 856-941 ([arXiv:1509.07584](https://arxiv.org/abs/1509.07584), [doi:10.1017/S0960129517000147](https://doi.org/10.1017/S0960129517000147))

[[!redirects Brouwer fixed point theorem]]

[[!redirects Brouwer fixed-point theorem]]
[[!redirects Brouwer's fixed-point theorem]]


Brown -- Abstract Homotopy Theory and Generalized Sheaf Cohomology > history

< [[Brown -- Abstract Homotopy Theory and Generalized Sheaf Cohomology]]
[[!redirects Brown -- Abstract Homotopy Theory and Generalized Sheaf Cohomology -- history]]


Brown functor

#Contents#
* table of contents
{:toc}

## In the context of Brown representability

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

Let $\mathcal{C}$ be a [[locally presentable (∞,1)-category]]. A [[functor]]

$$
  F \;\colon\; Ho(\mathcal{C})^{op} \longrightarrow Set
$$

(from the [[opposite category|opposite]] of the [[homotopy category of an (infinity,1)-category|homotopy category]] of $\mathcal{C}$ to [[Set]])

is called a **[[Brown functor]]** if

1. it sends small [[coproducts]] to [[products]];

1. it sends [[(∞,1)-pushouts]] in $\mathcal{C}\to Ho(\mathcal{C})$ to [[weak pullbacks]] in [[Set]].

=--

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

A _[[weak pullback]]_ is a diagram that satisfies the existence clause of a [[pullback]], but not necessarily the uniqueness condition. Hence the second clause in def. \ref{BrownFunctorOnInfinityCategory} says that for a [[(∞,1)-pushout]] square

$$
  \array{
    Z &\longrightarrow& X
    \\
    \downarrow &\swArrow& \downarrow
    \\
    Y &\longrightarrow& X \underset{Z}{\sqcup}Y
  }
$$

in $\mathcal{C}$, then the induced universal morphism

$$
  F\left(X \underset{Z}{\sqcup}Y\right)
  \stackrel{epi}{\longrightarrow}
  F(X) \underset{F(Z)}{\times} F(Y)
$$

into the actual [[pullback]] is an [[epimorphism]].

=--

+-- {: .num_prop}
###### Proposition
**([[Brown representability theorem]])**

Let $\mathcal{C}$ be a [[locally presentable (∞,1)-category]].

If

1. $\mathcal{C}$ is [[compactly generated (∞,1)-category|generated]] by a set

   $$
    \{S_i \in \mathcal{C}\}_{i \in I}
   $$

   of [[compact object in an (infinity,1)-category|compact objects]] (i.e. every object of $\mathcal{C}$ is an [[(∞,1)-colimit]] of the objects $S_i$.)

1. each $S_i$ admits the structure of a [[cogroup]] object in the [[homotopy category of an (infinity,1)-category|homotopy category]] $Ho(\mathcal{C})$,

then a [[functor]]

$$
  F \;\colon\; Ho(\mathcal{C})^{op} \longrightarrow Set
$$

(from the [[opposite category|opposite]] of the [[homotopy category of an (infinity,1)-category|homotopy category]] of $\mathcal{C}$ to [[Set]])

is [[representable functor|representable]] precisely if it is a Brown functor, def. \ref{BrownFunctorOnInfinityCategory}.

=--

## In the context of categories of fibrant objects

A **right Brown functor** between [[categories of fibrant objects]] is a [[functor]] that preserves [[weak equivalences]] between [[fibrant objects]].

Dually for a _left Brown functor_ (for instance between [[Waldhausen categories]]).

[[!redirects Brown functors]]


Brown representability theorem

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#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### $(\infty,1)$-Topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The classical _Brown representability theorem_ ([Brown 62](#Brown62), [Adams 71](#Adams71)) says contravariant [[functors]] on the (pointed) [[classical homotopy category]] satisfying two conditions ("[[Brown functors]]") are [[representable functor|representable]].

This is used notably to show that every additive reduced [[generalized (Eilenberg-Steenrod) cohomology]] theory $E^\bullet$ is representable by a [[spectrum]] $E$ as

$$
  E^n(X) \simeq [X,E_n]
  \,.
$$

(But beware that the cohomology theory in general contains less information than the spectrum, due to [[phantom maps]] (see also this [MO discussion](http://mathoverflow.net/q/117684/381)).)

The Brown representability theorem as such, with the two conditions on a [[Brown functor]] understood, only applies to [[contravariant functors]], not to covariant functors.  But by way of [[Spanier-Whitehead duality]] it implies at least over [[finite CW-complexes]] that dually every additive [[generalized homology theory]] $E_\bullet$ is representable by a spectrum $E$ via

$$
  E_n(X) \simeq \pi_n(E_n \wedge X)
  \,.
$$

There are various generalizations, such as to [[triangulated category|triangulated categories]] ([Neeman 96](#Neeman96)), to [[homotopy category of a model category|homotopy categories of]] various [[model categories]] ([Jardine 09](#Jardine09)) and [[homotopy category of an (infinity,1)-category|homotopy categories of]]  [[(∞,1)-categories]] ([Lurie, theorem 1.4.1.2](#LurieHigherAlgebra)).  But in any case there are some crucial conditions for the theorem to apply, such as

* [[group]] structure on the values of the functor, as is the case for an (abelian) cohomology theory, and as would be the case for a represented functor in any [[additive category]],

OR

* the existence of a [[strong generator]] in the homotopy category in question.

In particular, there is no Brown representability theorem for functors from the homotopy category of pointed not-necessarily-connected spaces to pointed sets, or for functors from the homotopy category of unpointed spaces to sets.  In fact, there are counterexamples in these two cases ([Freyd-Heller 93](#FreydHeller93), see remark \ref{Counterexamples}).

## Classical formulation for homotopy functors on topological spaces
 {#ClassicalFormulation}

Let $Ho(Top_*^c)$ denote the [[homotopy category]] of connected [[pointed topological spaces]] under [[weak homotopy equivalence]] --- or equivalently the homotopy category of pointed connected [[CW complexes]] under [[homotopy equivalence]].  Then $Ho(Top_*^c)$ has [[coproducts]] (given by [[wedge sums]]) and also [[weak limit|weak pushouts]] (namely, [[homotopy pushouts]], see at [weak limit](weak+limit#RelationToHomotopyLimits)).

+-- {: .num_theorem #ClassicalBrownRepresentabilityTheorem}
###### Theorem
**(Brown-Adams)**

A [[functor]] $F:Ho(Top_*^c)^{op} \to Set_*$ is [[representable functor|representable]] precisely if

1. it [[preserved limit|takes]] [[coproducts]] to [[products]],

1. it takes  [[weak pushouts]] to  [[weak pullbacks]].

=--

(e.g. [Aguilar-Gitler-Prito 02, theorem 12.2.18](#AguilarGitlerPrito02)).

Note that it is immediate that every representable functor has the given properties; the nontrivial statement is that these properties already characterize representable functors.

When the theorem is stated in terms of CW complexes, the second property (taking [[weak pushouts]] to [[weak pullbacks]]) is often phrased equivalently as:

* The **Mayer-Vietoris axiom**: For every [[CW-pair|CW-triple]] $(X; A_1, A_2)$ (with $A_1\cup A_2 = X$) and any elements $x_1\in F(A_1)$, $x_2\in F(A_2)$ such that $x_1|A_1\cap A_2 = x_2|A_1\cap A_2$, there exists $y\in F(X)$ such that $y|A_1 = x_1$ and $y|A_2 = x_2$.

+-- {: .num_remark #Counterexamples}
###### Remark
**(counterexamples)**

The statement of theorem \ref{ClassicalBrownRepresentabilityTheorem} without the restriction to *connected* pointed spaces is false, as is the analogous statement for unpointed spaces.

In ([Freyd-Heller 93](#FreydHeller93)), it is shown that if $G$ is [[Thompson's group F]], with $g:G\to G$ its canonical endomorphism, then $g$ does not [[split idempotent|split]] in the quotient of [[Grp]] by conjugacy.  Since the quotient of [[Grp]] by conjugacy embeds as the full subcategory of the unpointed homotopy category $Ho(Top)$ on connected [[homotopy 1-types]], we have an endomorphism $B g:B G \to B G$ of the [[classifying space]] of $G$ which does not split in $Ho(Top)$.

Thus, if $F:Ho(Top)^{op} \to Set$ splits the idempotent $[-,B g]$ of $[-,B G]$, then $F$ satisfies the hypotheses of the Brown representability theorem (being a [[retract]] of a representable functor), but is not representable.  A similar argument using $B G_+$ applies to $Ho(Top_*)$.

There is also another example due to Heller, which fails to be representable for cardinality reasons.

* MO questions [non-connected spaces](http://mathoverflow.net/questions/104866/brown-representability-for-non-connected-spaces), [unpointed spaces](http://mathoverflow.net/questions/11456/unpointed-brown-representability-theorem)
* [blog post](http://golem.ph.utexas.edu/category/2012/08/brown_representability.html)

=--

## For homotopy functors on presentable $(\infty,1)$-categories
 {#InInfinity1Categories}

The [classical formulation](#ClassicalFormulation) of Brown representability is only superficially concerned with [[topological spaces]]. Instead, via the [[classical model structure on topological spaces]], one finds that the statement only concerns the [[simplicial localization]] of [[Top]] at the [[weak homotopy equivalences]], hence the [[(∞,1)-category]] $L_{whe} Top\simeq $ [[∞Grpd]].

We discuss now the natural formulation of the Brown representability theorem for functors out of [[homotopy categories of (∞,1)-categories]] following ([Lurie, section 1.4.1](#LurieHigherAlgebra)). See also the exposition in ([Mathew 11](#Mathew11)).

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

Let $\mathcal{C}$ be a [[locally presentable (∞,1)-category]]. A [[functor]]

$$
  F \;\colon\; Ho(\mathcal{C})^{op} \longrightarrow Set
$$

(from the [[opposite category|opposite]] of the [[homotopy category of an (infinity,1)-category|homotopy category]] of $\mathcal{C}$ to [[Set]])

is called a **[[Brown functor]]** if

1. it sends small [[coproducts]] to [[products]];

1. it sends [[(∞,1)-pushouts]] in $\mathcal{C}\to Ho(\mathcal{C})$ to [[weak pullbacks]] in [[Set]].

=--

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

A _[[weak pullback]]_ is a diagram that satisfies the existence clause of a [[pullback]], but not necessarily the uniqueness condition. Hence the second clause in def. \ref{BrownFunctorOnInfinityCategory} says that for a [[(∞,1)-pushout]] square

$$
  \array{
    Z &\longrightarrow& X
    \\
    \downarrow &\swArrow& \downarrow
    \\
    Y &\longrightarrow& X \underset{Z}{\sqcup}Y
  }
$$

in $\mathcal{C}$, then the induced universal morphism

$$
  F\left(X \underset{Z}{\sqcup}Y\right)
  \stackrel{epi}{\longrightarrow}
  F(X) \underset{F(Z)}{\times} F(Y)
$$

into the actual [[pullback]] is an [[epimorphism]].

=--

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

Say that  a [[locally presentable (∞,1)-category]] $\mathcal{C}$
is **compactly generated by cogroup objects closed under suspensions** if

1. $\mathcal{C}$ is [[compactly generated (∞,1)-category|generated]] by a set

   $$
    \{S_i \in \mathcal{C}\}_{i \in I}
   $$

   of [[compact object in an (infinity,1)-category|compact objects]] (i.e. every object of $\mathcal{C}$ is an [[(∞,1)-colimit]] of the objects $S_i$.)

1. each $S_i$ admits the structure of a [[cogroup]] object in the [[homotopy category of an (infinity,1)-category|homotopy category]] $Ho(\mathcal{C})$;

1. the set $\{S_i\}$ is closed under forming [[reduced suspensions]].

=--

+-- {: .num_example #SuspensionsAreHCogroupObjects}
###### Example
**([[suspensions are H-cogroup objects]])**

Let $\mathcal{C}$ be an [[(∞,1)-category]] with [[finite (∞,1)-colimits]] and with a [[zero object]]. Write $\Sigma \colon X \mapsto 0 \underset{X}{\coprod} 0$ for the [[reduced suspension]] functor.

Then the [[fold map]]

$$
  \Sigma X \coprod \Sigma X
    \simeq
  0 \underset{X}{\sqcup} 0 \underset{X}{\sqcup} 0
    \longrightarrow
  0 \underset{X}{\sqcup} X \underset{X}{\sqcup} 0
  \simeq
  0 \underset{X}{\sqcup} 0
  \simeq
  \Sigma X
$$

exhibits [[cogroup]] structure on the image of any [[suspension object]] $\Sigma X$ in the [[homotopy category of an (∞,1)-category|homotopy category]].

This is equivalently the [[group]]-structure of the first ([[fundamental group|fundamental]]) [[homotopy group]] of the values of [[representable functor|functor co-represented]] by $\Sigma X$:

$$
  Ho(\mathcal{C})(\Sigma X, -)
   \;\colon\;
  Y
   \mapsto
  Ho(\mathcal{C})(\Sigma X, Y)
   \simeq
  Ho(\mathcal{C})(X, \Omega Y)
   \simeq
  \pi_1 Ho(\mathcal{C})(X, Y)
  \,.
$$

=--

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

In bare [[pointed homotopy types]] $\mathcal{C} = $[[∞Grpd]]${}^{\ast/}$, the ([[homotopy types]] of) [[n-spheres]] $S^n$ are [[cogroup]] objects for $n \geq 1$, but not for $n = 0$, by example \ref{SuspensionsAreHCogroupObjects}. And of course they are [[compact object in an (∞,1)-category|compact objects]].

So while $\{S^n\}_{n \in \mathbb{N}}$ generates all of $\infty Grpd^{\ast/}$, the latter is _not_ an example of def. \ref{CompactGenerationByCogroupObjects} due to the failure of $S^0$ to have [[cogroup]] structure.

Removing that generator, the $(\infty,1)$-category generated by $\{S^n\}_{{n \in \mathbb{N}} \atop {n \geq 1}}$ is $\infty Grpd^{\ast/}_{\geq 1}$, that of _[[connected object|connected]]_ [[pointed homotopy types]]. This is one way to see how the connectedness condition in the classical version of Brown representability theorem arises.

=--

See also ([Lurie, example 1.4.1.4](#LurieHigherAlgebra))

In $\infty$-categories compactly generated by cogroup objects closed under forming suspensions, the following strenghtening of the [[Whitehead theorem]] holds.

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

In an $(\infty,1)$-category compactly generated by cogroup objects $\{S_i\}_{i \in I}$ closed under forming suspensions, according to def. \ref{CompactGenerationByCogroupObjects}, a morphism $f\colon X \longrightarrow Y$ is an [[equivalence in an (infinity,1)-category|equivalence]] precisely if for each $i \in I$ the induced function of maps in the [[homotopy category of an (infinity,1)-category|homotopy category]]

$$
  Ho(\mathcal{C})(S_i,f)
  \;\colon\;
  Ho(\mathcal{C})(S_i,X)
   \longrightarrow
  Ho(\mathcal{C})(S_i,Y)
$$

is an [[isomorphism]] (a [[bijection]]).

=--

([Lurie, p. 114, Lemma star](#LurieHigherAlgebra))

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

By the [[(∞,1)-Yoneda lemma]], the morphism $f$ is an [[equivalence in an (∞,1)-category|equivalence]] precisely if for all objects $A \in \mathcal{C}$  the induced morphism

$$
  \mathcal{C}(A,f)
  \;\colon\;
  \mathcal{C}(A,X)
   \longrightarrow
  \mathcal{C}(A,Y)
$$

is an equivalence in [[∞Grpd]]. By assumption of compact generation and since the hom-functor $\mathcal{C}(-,-)$ sends $\infty$-colimits in the first argument to $\infty$-limits, this is the case precisely already if it is the case for $A \in \{S_i\}_{i \in I}$.

Now by the standard [[Whitehead theorem]] in [[∞Grpd]] (being a [[hypercomplete (∞,1)-topos]]), the morphisms

$$
  \mathcal{C}(S_i,f)
  \;\colon\;
  \mathcal{C}(S_i,X)
   \longrightarrow
  \mathcal{C}(S_i,Y)
$$

in [[∞Grpd]] are  [[equivalence in an (∞,1)-category|equivalences]] precisely if they are [[weak homotopy equivalences]], hence precisely if they induce [[isomorphisms]] on all [[homotopy groups]] $\pi_n$ for **all basepoints**.

It is this last condition of testing on all basepoints that the assumed [[cogroup]] structure on the $S_i$ allows to do away with: this cogroup structure implies that $\mathcal{C}(S_i,-)$ has the structure of an $H$-group, and this implies (by group multiplication), that all [[connected components]] have the same homotopy groups, hence that all homotopy groups are independent of the choice of basepoint, up to isomorphism.

Therefore the above morphisms are equivalences precisely if they are so under applying $\pi_n$ based on the connected component of the [[zero morphism]]

$$
  \pi_n\mathcal{C}(S_i,f)
    \;\colon\;
  \pi_n \mathcal{C}(S_i,X)
     \longrightarrow
  \pi_n\mathcal{C}(S_i,Y)
  \,.
$$

Now in this pointed situation we may use that

$$
  \begin{aligned}
    \pi_n \mathcal{C}(-,-) & \simeq \pi_0 \mathcal{C}(-,\Omega^n(-))
    \\
    & \simeq \pi_0\mathcal{C}(\Sigma^n(-),-)
    \\
    & \simeq Ho(\mathcal{C})(\Sigma^n(-),-)
  \end{aligned}
$$

to find that $f$ is an equivalence in $\mathcal{C}$ precisely if the induced morphisms

$$
  Ho(\mathcal{C})(\Sigma^n S_i, f)
    \;\colon\;
  Ho(\mathcal{C})(\Sigma^n S_i,X)
    \longrightarrow
  Ho(\mathcal{C})(\Sigma^n S_i,Y)
$$

are isomorphisms for all $i \in I$ and $n \in \mathbb{N}$.

Finally by the assumption that each suspension $\Sigma^n S_i$ of a generator is itself among the set of generators, the claim follows.

=--

+-- {: .num_theorem #BrownRepresentabilityOnPresentableInfinityCategories}
###### Theorem
**(Brown representability)**

Let $\mathcal{C}$ be an $(\infty,1)$-category compactly generated by cogroup objects closed under forming suspensions, according to def. \ref{CompactGenerationByCogroupObjects}. Then a [[functor]]

$$
  F \;\colon\; Ho(\mathcal{C})^{op} \longrightarrow Set
$$

(from the [[opposite category|opposite]] of the [[homotopy category of an (infinity,1)-category|homotopy category]] of $\mathcal{C}$ to [[Set]])
is [[representable functor|representable]] precisely if it is a [[Brown functor]], def. \ref{BrownFunctorOnInfinityCategory}.

=--

([Lurie, theorem 1.4.1.2](#LurieHigherAlgebra))

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

Due to the version of the Whitehead theorem of prop. \ref{WhiteheadTheoremForCompactGenerationByCogroupObjects} we are essentially reduced to showing that [[Brown functors]] $F$ are representable on the $S_i$. To that end consider the following lemma.
(In the following we notationally identify, via the [[Yoneda lemma]], objects of $\mathcal{C}$, hence of $Ho(\mathcal{C})$, with the functors they [[representable functor|represent]].)

Lemma ($\star$): _Given $X \in \mathcal{C}$ and $\eta \in F(X)$, hence $\eta \colon X \to F$, then there exists a morphism $f \colon X \to X'$ and an [[extension]] $\eta' \colon X' \to F$ of $\eta$ which induces for each $S_i$ a [[bijection]] $\eta'\circ (-) \colon Ho(\mathcal{C})(S_i,X') \stackrel{\simeq}{\longrightarrow} PSh(Ho(\mathcal{C}))(S_i,F) \simeq F(S_i)$._

To see this, first notice that we may directly find an extension $\eta_0$ along a map $X\to X_o$ such as to make a [[surjection]]: simply take $X_0$ to be the [[coproduct]] of **all** possible elements in the codomain and take

$$
  \eta_0
  \;\colon\;
   X
     \sqcup
   \left(
     \underset{{i \in I,} \atop {\gamma \colon S_i \stackrel{}{\to} F}}{\coprod}
     S_i
   \right)
   \longrightarrow
   F
$$

to be the canonical map. (Using that $F$, by assumption, turns coproducts into products, we may indeed treat the coproduct in $\mathcal{C}$ on the left as the coproduct of the corresponding functors.)

To turn the surjection thus constructed into a bijection, we now successively form quotients of $X_0$. To that end proceed by [[induction]] and suppose that $\eta_n \colon X_n \to F$ has been constructed. Then for $i \in I$ let

$$
  K_i
    \coloneqq
  ker
  \left(
     Ho(\mathcal{C})(S_i, X_n)
        \stackrel{\eta_n \circ (-)}{\longrightarrow}
     F(S_i)
   \right)
$$

be the [[kernel]] of  $\eta_n$ evaluated on $S_i$. These $K_i$ are the pieces that need to go away in order to make a bijection. Hence define $X_{n+1}$ to be their joint [[homotopy cofiber]]

$$
  X_{n+1}
     \coloneqq
   coker\left(
     \left(
        \underset{{i \in I,} \atop {\gamma \in K_i}}{\sqcup} S_i
     \right)
     \overset{(\gamma)_{{i \in I} \atop {\gamma\in K_i}}}{\longrightarrow}
     X_n
  \right)
  \,.
$$

Then by the assumption that $F$ takes this homotopy cokernel to a [[weak limit|weak]] [[fiber]] (as in remark \ref{WeakPullbacks}), there exists an extension $\eta_{n+1}$ of $\eta_n$ along $X_n \to X_{n+1}$:

$$
  \array{
    \left(
      \underset{{i \in I}\atop {\gamma \in K_i}}{\sqcup} S_i
    \right)
    &\overset{(\gamma)_{{i \in I}\atop \gamma \in K_i}}{\longrightarrow}&
    X_n
    &\overset{\eta_n}{\longrightarrow}& F
    \\
    \downarrow &(po^{h})& \downarrow & \nearrow_{\mathrlap{\exists \eta_{n+1}}}
    \\
    \ast &\longrightarrow& X_{n+1}
  }
  \;\;\;\;\;\;\;\;\;\;\;
  \Leftrightarrow
  \;\;\;\;\;\;\;\;\;\;\;
  \array{
    && F(X_{n+1}) &\longrightarrow& \ast
    \\
    &{}^{\mathllap{\exists \eta_{n+1}}}\nearrow& \downarrow^{\mathrlap{epi}} && \downarrow
    \\
    \ast &\overset{\eta_n}{\longrightarrow}& ker\left((\gamma^\ast\right)_{{i \in I} \atop {\gamma \in K_i}}) &\longrightarrow& \ast
    \\
    &{}_{\mathllap{\eta_n}}\searrow& \downarrow &(pb)& \downarrow
    \\
    && F(X_n)
    &\underset{(\gamma^\ast)_{{i \in I} \atop {\gamma \in K_i}} }{\longrightarrow}&
    \underset{{i \in I}\atop {\gamma\in K_i}}{\prod}F(S_i)
  }
  \,.
$$

It is now clear that we want to take

$$
  X' \coloneqq \underset{\rightarrow}{\lim}_n X_n
$$

and extend all the $\eta_n$ to that colimit. Since we have no condition for evaluating $F$ on colimits other than pushouts, observe that this [[sequential colimit]] is equivalent to the following pushout:

$$
  \array{
    \underset{n}{\sqcup} X_n
    &\longrightarrow& \underset{n}{\sqcup} X_{2n}
    \\
    \downarrow &(po^h)& \downarrow
    \\
    \underset{n}{\sqcup} X_{2n+1} &\longrightarrow& X'
  }
  \,,
$$

where the components of the top and left map alternate between the identity on $X_n$ and the above successor maps $X_n \to X_{n+1}$.
Now the excision property of $F$ applies to this pushout, and we conclude the desired  extension $\eta' \colon X' \to F$:

$$
  \array{
    && \underset{n}{\sqcup} X_n
    \\
    & \swarrow && \searrow
    \\
    \underset{n}{\sqcup} X_{2n+1} &\longrightarrow& X' &\longleftarrow& \underset{n}{\sqcup} X_{2n}
    \\
    & {}_{\mathllap{(\eta_{2n+1})_{n}}}\searrow& \downarrow^{\mathrlap{\exists \eta'}} & \swarrow_{\mathrlap{(\eta_{2n})_n}}
    \\
    && F
  }
  \;\;\;\;\;\;\;\;\;
  \Leftrightarrow
  \;\;\;\;\;\;\;\;\;
  \array{
    && F(X')
    \\
    &{}^{\mathllap{\exists \eta'}}\nearrow& \downarrow^{\mathrlap{epi}}
    \\
    &\ast \overset{(\eta_n)_n}{\longrightarrow}& \underset{\longleftarrow}{\lim}_n F(X_n)
    \\
    & \swarrow && \searrow
    \\
    \underset{n}{\prod}F(X_{2n+1})
    && &&
    \underset{n}{\prod}(X_{2n})
    \\
    & \searrow && \swarrow
    \\
    && \underset{n}{\prod}F(X_n)
  }
  \,,
$$

It remains to confirm that this indeed  gives the desired bijection. Surjectivity is clear. For injectivity use that all the $S_i$ are, by assumption, [[compact object|compact]], hence they may be taken inside the [[sequential colimit]]:

$$
  \array{
    && X_{n(\gamma)}
    \\
    &{}^{\mathllap{ \exists \hat \gamma}}\nearrow& \downarrow
    \\
    S_i
      &\overset{\gamma}{\longrightarrow}&
    X'
      =
    \underset{\longrightarrow}{\lim}_n X_n
  }
  \,.
$$

With this, injectivity follows because by construction we quotiented out the kernel at each stage. Because suppose that $\gamma$ is taken to zero in $F(S_i)$, then by the definition of $X_{n+1}$ above there is a factorization of $\gamma$ through the point:

$$
  \array{
    0 \colon & S_i
    &\overset{\hat \gamma}{\longrightarrow}&
    X_{n(\gamma)}
    &\overset{\eta_n}{\longrightarrow}& F
    \\
    & \downarrow && \downarrow &
    \\
    & \ast &\longrightarrow& X_{n(\gamma)+1}
    \\
    & && \downarrow
    \\
    & && X'
  }
$$

This concludes the proof of Lemma ($\star$).

Now apply the construction given by this lemma to the case
$X_0 \coloneqq 0$ and the unique $\eta_0 \colon 0 \stackrel{\exists !}{\to} F$. Lemma $(\star)$ then produces an object $X'$ which represents $F$ on all the $S_i$, and we want to show that this $X'$ actually represents $F$ generally, hence that for every $Y \in \mathcal{C}$ the function

$$
  \theta \coloneqq \eta'\circ (-)
  \;\colon\;
  Ho(\mathcal{C})(Y,X')
   \stackrel{}{\longrightarrow}
  F(Y)
$$

is a [[bijection]].

First, to see that $\theta$ is surjective, we need to find a preimage of any $\rho \colon Y \to F$. Applying Lemma $(\star)$ to $(\eta',\rho)\colon X'\sqcup Y \longrightarrow F$ we get an extension $\kappa$ of this through some $X' \sqcup Y \longrightarrow Z$ and the morphism on the right of the following commuting diagram:

$$
  \array{
    Ho(\mathcal{C})(-,X') && \longrightarrow && Ho(\mathcal{C})(-, Z)
    \\
    & {}_{\mathllap{\eta'\circ(-)}}\searrow && \swarrow_{\mathrlap{\kappa \circ (-)}}
    \\
    && F(-)
  }
  \,.
$$

Moreover, Lemma $(\star)$ gives that evaluated on all $S_i$, the two diagonal morphisms here become isomorphisms. But then prop. \ref{WhiteheadTheoremForCompactGenerationByCogroupObjects} implies that $X' \longrightarrow Z$ is in fact an equivalence. Hence the component map $Y \to Z \simeq Z$ is a lift of $\kappa$ through $\theta$.

Second, to see that $\theta$ is injective, suppose $f,g \colon Y \to X'$ have the same image under $\theta$. Then consider their [[homotopy pushout]]

$$
  \array{
    Y \sqcup Y &\stackrel{(f,g)}{\longrightarrow}& X'
    \\
    \downarrow && \downarrow
    \\
    Y &\longrightarrow& Z
  }
$$

along the [[codiagonal]] of $Y$. Using that $F$ sends this to a [[weak pullback]] by assumption, we obtain an extension $\bar \eta$ of $\eta'$ along $X' \to Z$. Applying Lemma $(\star)$ to this gives a further extension $\bar \eta' \colon Z' \to Z$ which now makes the following diagram

$$
  \array{
    Ho(\mathcal{C})(-,X') && \longrightarrow && Ho(\mathcal{C})(-, Z)
    \\
    & {}_{\mathllap{\eta'\circ(-)}}\searrow && \swarrow_{\mathrlap{\bar \eta' \circ (-)}}
    \\
    && F(-)
  }
$$

such that the diagonal maps become isomorphisms when evaluated on the $S_i$. As before, it follows via prop. \ref{WhiteheadTheoremForCompactGenerationByCogroupObjects} that the morphism
$h \colon X' \longrightarrow Z'$ is an equivalence.

Since by this construction $h\circ f$ and $h\circ g$ are homotopic

$$
  \array{
    Y \sqcup Y &\stackrel{(f,g)}{\longrightarrow}& X'
    \\
    \downarrow && \downarrow & \searrow^{\mathrlap{\stackrel{h}{\simeq}}}
    \\
    Y &\longrightarrow& Z &\longrightarrow& Z'
  }
$$

it follows with $h$ being an equivalence that already $f$ and $g$ were homotopic, hence that they represented the same element.

=--

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

Let $\mathcal{C}$ be an [[(∞,1)-category]] with [[(∞,1)-pushouts]], and with a [[zero object]] $0 \in \mathcal{C}$. Write $\Sigma \colon \mathcal{C} \to \mathcal{C}\colon X\mapsto 0 \underset{X}{\sqcup} 0$ for the corresponding [[suspension]] [[(∞,1)-functor]].

A **reduced additive [[generalized (Eilenberg-Steenrod) cohomology]] theory** on $\mathcal{C}$ is

1. a [[functor]]

   $$
     H^\bullet \;\colon \; Ho(\mathcal{C})^{op} \longrightarrow Ab^{\mathbb{Z}}
   $$

   (from the [[opposite category|opposite]] of the [[homotopy category of an (infinity,1)-category|homotopy category]] of $\mathcal{C}$ into $\mathbb{Z}$-[[graded abelian groups]]);

1. a [[natural isomorphisms]] ("[[suspension isomorphisms]]") of degree +1

   $$
     \delta \; \colon \; H^\bullet \longrightarrow H^{\bullet+1} \circ \Sigma
   $$

such that $H^\bullet$

1. **(exactness)** takes [[homotopy cofiber sequences]] to [[exact sequences]].

1. **(additivity)** takes small [[coproducts]] to [[products]];

=--

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

Given a generalized cohomology theory $(H^\bullet,\delta)$ on some $\mathcal{C}$ as in def. \ref{GeneralizedCohomologyOnGeneralInfinityCategory}, and given a [[homotopy cofiber sequence]] in $\mathcal{C}$

$$
  X \stackrel{f}{\longrightarrow} Y \stackrel{g}{\longrightarrow} Z
  \stackrel{coker(g)}{\longrightarrow}
  \Sigma X
  \,,
$$

then the corresponding **[[connecting homomorphism]]** is the composite

$$
  \partial
    \;\colon\;
  H^\bullet(X)
   \stackrel{\delta}{\longrightarrow}
  H^{\bullet+1}(\Sigma X)
   \stackrel{coker(g)^\ast}{\longrightarrow}
  H^{\bullet+1}(Z)
  \,.
$$

=--

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

The [[connecting homomorphisms]] of def. \ref{ConnectinHomomorphismForCohomologyTheoryOnInfinityCategory} are parts of [[long exact sequences]]

$$
  \cdots
   \stackrel{\partial}{\longrightarrow}
  H^{\bullet}(Z) \longrightarrow H^\bullet(Y) \longrightarrow H^\bullet(X)
  \stackrel{\partial}{\longrightarrow}
  H^{\bullet+1}(Z)
   \to \cdots
  \,.
$$

=--

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

By the defining exactness of $H^\bullet$, def. \ref{GeneralizedCohomologyOnGeneralInfinityCategory}, and the way this appears in def. \ref{ConnectinHomomorphismForCohomologyTheoryOnInfinityCategory}, using that $\delta$ is by definition an isomorphism.

=--

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

Given a reduced addive generalized cohomology functor $H^\bullet \colon Ho(\mathcal{C})^{op}\to Ab^{\mathbb{Z}}$, def. \ref{GeneralizedCohomologyOnGeneralInfinityCategory}, its underlying [[Set]]-valued functors $H^n \colon Ho(\mathcal{C})^{op}\to Ab\to Set$ are [[Brown functors]], def. \ref{BrownFunctorOnInfinityCategory}.

=--

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

The first condition on a [[Brown functor]] holds by additivity. For the second condition, given a homotopy pushout square

$$
  \array{
    X_1 &\stackrel{f_1}{\longrightarrow}& Y_1
    \\
    \downarrow^{} && \downarrow
    \\
    X_2 &\stackrel{f_2}{\longrightarrow}& Y_2
  }
$$

in $\mathcal{C}$, consider the induced morphism of the [[long exact sequences]] given by prop. \ref{LongExactSequenceOfACohomologyTheoryOnAnInfinityCategory}

$$
  \array{
    H^\bullet(coker(f_2))
      &\longrightarrow&
    H^\bullet(Y_2)
      &\stackrel{f^\ast_2}{\longrightarrow}&
    H^\bullet(X_2)
      &\stackrel{}{\longrightarrow}&
    H^{\bullet+1}(\Sigma coker(f_2))
    \\
    {}^{\mathllap{\simeq}}\downarrow && \downarrow && \downarrow && \downarrow^{\mathrlap{\simeq}}
    \\
    H^\bullet(coker(f_1))
      &\longrightarrow&
    H^\bullet(Y_1)
      &\stackrel{f^\ast_1}{\longrightarrow}&
    H^\bullet(X_1)
      &\stackrel{}{\longrightarrow}&
    H^{\bullet+1}(\Sigma coker(f_1))
  }
$$

Here the outer vertical morphisms are [[isomorphisms]], as shown, due to the [[pasting law]] (see also at _[fiberwise recognition of stable homotopy pushouts](homotopy+pullback#FiberwiseRecognitionInStableCase)_).
This means that the [[four lemma]] applies to this diagram. Inspection shows that this implies the claim.

=--

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

Let $\mathcal{C}$ be an [[(∞,1)-category]] which satisfies the conditions of theorem \ref{BrownRepresentabilityOnPresentableInfinityCategories}, and let $(H^\bullet, \delta)$ be a [[generalized cohomology]] functor on $\mathcal{C}$, def. \ref{GeneralizedCohomologyOnGeneralInfinityCategory}. Then there exists a  [[spectrum object]] $E \in Stab(\mathcal{C})$ such that

1. $H^\bullet$ is degreewise [[representable functor|represented]] by $E$:

   $$
     H^\bullet \simeq Ho(\mathcal{C})(-,E_\bullet)
     \,,
   $$

1. the [[suspension isomorphism]] $\delta$ is given by the structure morphisms $\tilde \sigma_n \colon E_n \to \Omega E_{n+1}$ of the spectrum, in that

   $$
     \delta
       \colon
     H^n(-)
      \simeq
     Ho(\mathcal{C})(-,E_n)
      \stackrel{Ho(\mathcal{C})(-,\tilde\sigma_n) }{\longrightarrow}
     Ho(\mathcal{C})(-,\Omega E_{n+1})
       \simeq
     Ho(\mathcal{C})(\Sigma (-), E_{n+1})
       \simeq
     H^{n+1}(\Sigma(-))
     \,.
   $$

=--

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

Via prop. \ref{CohomologyFunctorOnInfinityCategoryIsBrownFunctor}, theorem \ref{BrownRepresentabilityOnPresentableInfinityCategories} gives the first clause. With this, the second clause follows by the [[(∞,1)-Yoneda lemma]] (in fact just with the [[Yoneda lemma]]).

=--

## Further variants

### For triangulated categories and model categories

For [[triangulated categories]]  a discussion is in ([Neeman 96](#Neeman96)).

For [[model categories]] a discussion is in ([Jardine 09](#Jardine09)):

Let $C$ be a [[model category]] with a [[zero object]] and such that there is a [[set]] $S$ of [[cofibrant object|cofibrant]] [[compact objects]] such that the [[weak equivalences]] in $C$ are precisely the $S$-[[local morphisms]].

([Jardine 09, p. 20](#Jardine09))

+-- {: .num_theorem}
###### Theorem
**(Jardine)**

Let $F : C^{op} \to Set_{*}$ be a pointed [[homotopical functor]] from $C^{op}$ to the category of [[pointed sets]] such that

1. $F$ preserves small [[coproducts]] of cofibrant objects (including preserving the [[zero object]]).

1. **(Mayer-Vietoris property)** $F$ takes any [[pushout]] diagram

   $$
     \array{
       A &\to& X
       \\
       \downarrow^i && \downarrow
       \\
       B &\to& B \coprod_A X,
     }
   $$

   with all objects cofibrant and $i$ a cofibration, to a [[weak pullback]].

([Jardine 09, p. 21](#Jardine09))

Then $F$ is [[representable functor|representable]].

=--

([Jardine 09, theorem 24](#Jardine09))

### For $RO(G)$-graded equivariant cohomology
 {#ForEquivariantCohomology}

The generalization of the Brown representability theorem to [[equivariant cohomology]] says that [[RO(G)-grading|RO(G)-graded]] equivariant cohomology is represented by [[genuine G-spectra]] (e.g. [May 96, chapter XIII.3](#May96)).

## Properties

### Multiplicative cohomology and ring spectra

The spectrum Brown-representing a [[multiplicative cohomology theory]] inherits (at least) the structure of an [[H-ring spectrum]]. See [there](multiplicative+cohomology+theory#BrownRepresentabilityByRingSpectra).

## Related concepts

* [[Landweber exact functor theorem]]

## References

The original version of the theorem:

* {#Brown62} [[Edgar Brown]], _Cohomology theories_, Annals of Mathematics, Second Series 75: 467–484 (1962) ([jstor:1970209](https://www.jstor.org/stable/1970209))

The category-theoretic generalization:

* {#Brown65} [[Edgar Brown]], _Abstract homotopy theory_, Trans. AMS 119 no. 1 (1965) ([doi:10.1090/S0002-9947-1965-0182970-6](https://doi.org/10.1090/S0002-9947-1965-0182970-6))

A simplified version of the proof was spelled out in

* {#Spanier66} [[Edwin Spanier]], section 7.7 of _Algebraic topology_, McGraw-Hill, 1966

and a strengthening in

* {#Adams71} [[John Frank Adams]], _A variant of E. H. Brown's representability theorem_, Topology, 10:185-198, 1971

Textbook accounts:

* {#Adams74} [[Frank Adams]], part III, section 6 of _[[Stable homotopy and generalised homology]]_, 1974

* {#Switzer75} [[Robert Switzer]], theorem 8.42, theorem 9.27 in _Algebraic Topology - Homotopy and Homology_, Die  Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, Vol. 212, Springer-Verlag, New York, N. Y., 1975.

* {#Kochmann96} [[Stanley Kochmann]], section 3.4 of _[[Bordism, Stable Homotopy and Adams Spectral Sequences]]_, AMS 1996

* {#AguilarGitlerPrito02} Marcelo Aguilar, [[Samuel Gitler]], Carlos Prieto, section 12 of _Algebraic topology from a homotopical viewpoint_, Springer (2002) ([toc pdf](http://tocs.ulb.tu-darmstadt.de/106999419.pdf))

* {#TamakiKono06} [[Dai Tamaki]], [[Akira Kono]], Sections 2.4, 2.5 in: _Generalized Cohomology_, Translations of Mathematical Monographs, American Mathematical Society, 2006 ([ISBN: 978-0-8218-3514-2](https://bookstore.ams.org/mmono-230))

Quick reviews include

* {#Muro10} [[Fernando Muro]], _Representability of cohomology theories_, 2010 ([pdf](http://personal.us.es/fmuro/praha.pdf))

* {#Lurie}  [[Jacob Lurie]], _[[Chromatic Homotopy Theory]]_, Lecture series 2010, Lecture 17 _Phantom maps_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture17.pdf))

Generalization to [[triangulated categories]] is discussed in

* {#Neeman96} [[Amnon Neeman]], _The Grothendieck duality theorem via Bousfield's techniques and Brown representability_, J. Amer. Math. Soc. 9 (1996), 205-236 ([AMS](http://www.ams.org/journals/jams/1996-9-01/S0894-0347-96-00174-9/))

A [[model category]] version, with applications to [[∞-stacks]] -- or rather to the standard  [[models for ∞-stack (∞,1)-toposes]] in terms of the standard [[model structure on simplicial presheaves]] -- is given in

* {#Jardine09} [[Rick Jardine]], _Representability theorems for simplicial
presheaves_, 2009 ([pdf](http://ncatlab.org/nlab/files/JardineBrownrep.pdf))

A version in [[(infinity,1)-category theory]] is in

* {#LurieHigherAlgebra} [[Jacob Lurie]], section 1.4.1 of _[[Higher Algebra]]_

Exposition of this is in

* {#Mathew11} [[Akhil Mathew]], _[Brown representability and infinity-categories](https://amathew.wordpress.com/2011/10/10/brown-representability-and-infinity-categories/#more-2907)_, 2011

The case of [[RO(G)-grading|RO(G)-graded]] [[equivariant cohomology]] theory is discussed for instance in

* {#May96} [[Peter May]], chapter XIII.3 of _Equivariant homotopy and cohomology theory_, CBMS Regional Conference Series in Mathematics, vol. 91, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1996. ([pdf](http://www.math.uchicago.edu/~may/BOOKS/alaska.pdf))

The counterexamples to nonconnected and unpointed Brown representability are from

* {#FreydHeller93} [[Peter Freyd]], [[Alex Heller]], _Splitting homotopy idempotents. II._ J. Pure App#l. Algebra 89 (1993), no. 1-2, 93–106.

The relation to [[Grothendieck contexts]] ([[six operations]]) is highlighted in

* {#Neeman96} [[Amnon Neeman]], _The Grothendieck duality theorem via Bousfield's techniques and Brown representability_, J. Amer. Math. Soc. 9 (1996), 205-236 ([web](http://www.ams.org/journals/jams/1996-9-01/S0894-0347-96-00174-9/))

[[!redirects Brown representability]]

[[!redirects Brown's representability theorem]]


Brown-Comenetz duality

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

#Contents#
* table of contents
{:toc}

## Idea

[Brown-Comenetz 76](#BrownComenetz76) introduced the [[spectrum]] $I_{\mathbb{Q}/\mathbb{Z}}$ which [[Brown representability theorem|represents]] the [[cohomology theory]] that assigns to a [[spectrum]] $X$ the [[algebraic dual|algebraic]] [[Pontrjagin duals]] $Hom_{\mathbb{Z}}(\pi_{- \ast} X, \mathbb{Q}/\mathbb{Z})$ of its [[homotopy groups]] with respect to [[Q/Z]].

If $X$ is a spectrum whose [[stable homotopy groups]] are [[finite groups]], then the natural "double-duality" map $X \to I_{\mathbb{Q}/\mathbb{Z}} I_{\mathbb{Q}/\mathbb{Z}} X$ (the [[unit of a monad|unit]] of the corresponding [[continuation monad]]) is an [[equivalence in an (infinity,1)-category|equivalence]] (see at _[[dualizing object in a closed category]]_).

The Brown-Comenetz spectrum $I_{\mathbb{Q}/\mathbb{Z}}$ has an "integral" refinement which is the [[Anderson spectrum]] $I_{\mathbb{Z}}$ inducing [[Anderson duality]].

## Properties

* [[Eilenberg-Mac Lane spectra]] are Brown-Comenetz self-dual: $I_{\mathbb{Q}/\mathbb{Z}} H \mathbb{F}_p \cong H \mathbb{F}_p$.

## Related concepts

* [[Anderson duality]]

* [[Spanier-Whitehead duality]]

## References

* E. H. Brown, Jr. and M. Comenetz, _Pontrjagin duality for generalized homology and cohomology theories_, Amer. J. Math. 98 (1976), no. 1, 1–27.
 {#BrownComenetz76}

* [[Vesna Stojanoska]], _Duality for Topological Modular Forms_ ([arXiv:1105.3968](http://arxiv.org/abs/1105.3968))


Brown-Gersten property

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Locality and descent
+--{: .hide}
[[!include descent and locality - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The *Brown-Gersten property* is the statement that for certain [[Grothendieck topologies]] the [[homotopy descent]]-property for [[simplicial presheaves]] (i.e. their [[infinity-stack|$\infty$-stack]]-property) follows essentially (up to a simple condition on connected components) already as soon as the given [[simplicial presheaf]] $\mathcal{F}$ satisfies a [Mayer-Vietoris-type property](Whitehead-generalized+cohomology#MayerVietorisSequenceInGeneralizedCohomology) in that for [[covering|covers]] by a [[pair]] of patches $U_1, U_2$ their [[fiber product]]-square is taken by $\mathcal{F}$ to a [[homotopy pullback]]-square.

This has traditionally been discussed over [[schemes]] (see [below](#OverSchemes)) but an analogous statement holds also over [[topological spaces]] and [[smooth manifolds]] (see [further below](#OverSmoothmanifolds)).

## Details

### Over schemes
 {#OverSchemes}

Detailed review in in [Jardine 2015, §5.4](#Jardine15).

### Over smooth manifolds
 {#OverSmoothmanifolds}

Write

* [[SmthMfd]] for the [[category]] of [[smooth manifolds]], regarded as a [[site]] via its [[Grothendieck topology]] of [[open covers]];

* $PSh\big(SmthMfd, sSet)\big)$ for the [[category of simplicial presheaves]] over smooth manifolds.

Recall (e.g. from *[[model structure on simplicial presheaves]]*) that a [[simplicial presheaf]] $\mathcal{F} \,\in\, Psh(SmthMfd, sSet)$ is said to be a [[local object]] or *satisfy [[homotopy descent]]* if for

* all [[smooth manifolds]] $X \in SmthMfd$;

* all its [[open covers]] $\big\{ U_i \hookrightarrow X \big\}_{i \in I}$

the canonical morphism of [[simplicial sets]] from the value of $\mathcal{F}$ on $X$ into the [[homotopy limit]] of the [[simplicial diagram]] over values of $\mathcal{F}$ on the [[Čech nerve]] of the cover
\[
  \label{GeneralDescentProperty}
  \mathcal{F}(X)
  \xrightarrow{\phantom{----}}
  \underset{}{holim}
  \bigg(
    \underset{i \in I}{\prod}
    \mathcal{F}(U_i)
    \rightrightarrows
    \underset{i_1, i_2 \in I}{\prod}
    \mathcal{F}(U_{i_1} \cap U_{i_2})
    \cdots
  \bigg)
\]
is a [[simplicial weak equivalence]].

\begin{theorem}\label{BGPropertyForSmoothManifolds}
**(Brown-Gersten property over smooth manifolds, [Pavlov 2022, Thm. 1.1](#Pavlov22))**
\linebreak
  A simplicial presheaf $\mathcal{F} \,in\, PSh(SmthMfd, sSet)$ satisfies [[homotopy descent]] (eq:GeneralDescentProperty) for all open covers, as soon as it does so

1. for open covers consisting of a [[pair]] of open subsets;

1. for open covers by pairwise [[disjoint]] patches (hence by [[connected components]]).

\end{theorem}

Notice here that

1. in the case of an open cover by pair of patches
   $$
     X \,\simeq\, U_1 \cup U_2
     \,,
   $$
   the [[homotopy descent]]-condition (eq:GeneralDescentProperty) reduces to saying that the   [[commuting square]]

   $$
     \array{
       \mathcal{F}(X)
       &\longrightarrow&
       \mathcal{F}(U_1)
       \\
       \big\downarrow
       &&
       \big\downarrow
       \\
       \mathcal{F}(U_2)
       &\longrightarrow&
       \mathcal{F}(U_1 \cap U_2)
     }
   $$
   is a [[homotopy cartesian square]] (a [[homotopy pullback]]).

1. in the case of a cover by [[disjoint]] patches, hence $X \,\simeq\, \underset{i \in I}{\coprod} \, U_i$, the [[homotopy descent]]-condition (eq:GeneralDescentProperty) reduces to saying that

   $$
     \mathcal{F}(X)
     \;\;\simeq\;\;
     \underset{i \in I}{\prod}
     \mathcal{F}(U_i)
   $$

   is a [[homotopy product]].

   In particular, if $X = \varnothing$ is the [[empty topological space|empty space]] covered by itself, this means that

   $$
     \mathcal{F}(\varnothing)
     \;\;\simeq\;\;
     \ast
   $$

   is the [[terminal object]] (i.e. the simplicial presheaf which is constant on the [[singleton set]]).

## In condensed mathematics

A Brown–Gersten-type property holds
for [[condensed sets]]; as explained in the linked
article, it suffices to verify the descent
property for disjoint covers with zero or two elements,
as well as singleton families given by surjections of [[profinite sets]].

This property no longer holds for [[condensed ∞-groupoids]], and [[hypercovers]] are now necessary.
However, one can pass to the equivalent site
of [[compact extremally disconnected Hausdorff spaces]],
where finite disjoint covers suffice.

## References

The original discussion for the [[Zariski topology]]:

* {#BrownGersten73} [[Kenneth S. Brown]], [[Stephen Gersten]], *Algebraic K-theory as generalized sheaf cohomology*, in: *Algebraic K-theory, I: Higher K-theories* (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), pp. 266-292. Lecture Notes in Math. **341** Springer (1973) [[doi:10.1007/BFb0067062](https://doi.org/10.1007/BFb0067062)]

Lecture notes on this case:

* [[J. F. Jardine]],  *Brown-Gersten and Nisnevich descent, motivic descent*, Lecture 13 of: [Lectures on Local Homotopy Theory](https://math.sci.uwo.ca/~jardine/LocalHom/)  [[pdf](https://www.uwo.ca/math/faculty/jardine/courses/localhom/localhom-lecture13.pdf)]

The version for the [[Nisnevich topology]]:

* [[Fabien Morel]], [[Vladimir Voevodsky]], *$\mathbb{A}^1$-homotopy theory of schemes*, Publications Mathématiques de l'Institut des Hautes Études Scientifiques **90**  (1999) 45–143 [[doi:10.1007/BF02698831](https://doi.org/10.1007/BF02698831)]

Detailed review of these two cases:

* {#Jardine15} [[John F. Jardine]], Section 5.4 of: *Local homotopy theory*, Springer Monographs in Mathematics (2015) [[doi:10.1007/978-1-4939-2300-7](https://doi.org/10.1007/978-1-4939-2300-7)]

Generalization of these two cases to [[completely decomposable topology|completely decomposable topologies]]:

* {#Voevodsky10} [[Vladimir Voevodsky]], *Homotopy theory of simplicial presheaves in completely decomposable topologies*, Journal of Pure and Applied Algebra **214** 8 (2010) 1384-1398  [[arXiv:0805.4578](https://arxiv.org/abs/0805.4578), [doi:10.1016/j.jpaa.2009.11.004](https://doi.org/10.1016/j.jpaa.2009.11.004)]

Discussion for [[numerable open covers]] of [[topological spaces]] and of [[smooth manifolds]] with application to [[smooth infinity-groupoids|smooth $\infty$-groupoids]] and their [[shape via cohesive path ∞-groupoids]]:

* {#Pavlov22} [[Dmitri Pavlov]], *Numerable open covers and representability of topological stacks*, Topology and its Applications **318** (2022) 108203 [[arXiv:2203.03120](https://arxiv.org/abs/2203.03120), [doi:10.1016/j.topol.2022.108203](https://doi.org/10.1016/j.topol.2022.108203)]

[[!redirects Brown-Gersten properties]]

[[!redirects Brown-Gersten condition]]
[[!redirects Brown-Gersten conditions]]


Brown-Gitler spectrum

#Contents#
* table of contents
{:toc}

## Idea

A [[spectrum]] whose [[cohomology]] is a certain cyclic [[module]] over the [[Steenrod algebra]].

## References

* [[Edgar Brown]], [[Samuel Gitler]], _A spectrum whose cohomology is a certain cyclic module over the Steenrod algebra_, Topology 12 (1973), 283-295

* [[Paul Goerss]], _Brown-Gitler spectra_ ([[GoerssOnBrownGitler.pdf:file]])

[[!redirects Brown-Gitler spectra]]


Brown-Grossman homotopy group

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
=--
=--
#Contents#
* automatic table of contents goes here
{:toc}

## Idea: use a string of spheres not just one!

Within the context of [[proper homotopy theory]], what might be the analogues of the classical homotopy groups?  We might hope that Čech style homotopy groups would work, but they don't, or rather, 'not exactly'!

Given a locally compact and $\sigma$-compact space $X$, and with a base ray $* : [0,\infty) \to X$, with end $\varepsilon(X)$, the Čech homotopy groups of $\varepsilon(X)$ relative to $*$ are defined by $lim \pi_n(\varepsilon(X))$.  These 'make sense' but do not behave well, since the limit will destroy exactness in situations where we would hope for, and expect, long exact sequences. The problem is related to the fact that, depending how you view this, we are using just one sphere, or alternatively an infinite sequence of unrelated spheres.

E.M. Brown (1974) suggested a different construction based on an infinite string of spheres and this helps reveal extra structure that helps:

##Strings of spheres
+--{: .un_defn}
######Definition######
Let $\underline{S}^n = ([0,\infty) \cup \bigcup_{k=0}^\infty(S^n \times \{k\})/(k\sim (\underline{1},k)$. This is called a **string of $n$-spheres.**
=--

For the string of circles, it looks something like this:

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 </g>
</svg>

###Properties###
1. $\varepsilon(\underline{S}^n)$ is a pro-space isomorphic to an $\mathbb{N}$-indexed one with $k^{th}$ level, $\bigvee_{j\geq k}S^n_j$, where each $S^n_j$ is a copy of $S^n$.  The structural/ bonding maps from the $(k+1)^{st}$ level to the $k^{th}$ level are the obvious inclusions.

1. The space $\underline{S}^n$ is, of course, considered as being based at the ray (given by the right vertical map in the pushout square).

1. The space $\underline{S}^n$ is a [[spherical object]] and the family $\mathcal{A} = \{\underline{S}^n\}^\infty_{n=0}$ defines a theory.

##The Brown-Grossman homotopy groups##
+--{: .un_defn}
######Definition######
The $n^{th}$ _Brown-Grossman homotopy group_ of $(X,*)$ is given by the  group of based proper homotopy classes of based proper maps from $\underline{S}^n$ to $(X,*)$;

$$\underline{\pi}_n(X,*) = Ho(Proper)(\underline{S}^n, (X,*)).$$
=--

###Properties###
1. As was pointed out above, the $\underline{S}^n$ are cogroups so the $\underline{\pi}_n(X,*)$ are groups, which are abelian for $n\geq 2$.

1. In fact the $\underline{\pi}_n(X,*)$ form a $\Pi_\mathcal{A}$-[[spherical object|algebra]] for $\mathcal{A}$ as above.  Not only are there the sort of morphisms in $\Pi_\mathcal{A}$ that will induce analogues of [[Whitehead products]], [[composition operations]] etc. but there are other interesting morphisms there, for instance:

1. There is a proper **shift map** $S: \underline{S}^n\to \underline{S}^n$, which  shifts all the spheres one place to the right.   This induces, by composition, a morphism $S^*(X): \underline{\pi}_n(X,*)\to \underline{\pi}_n(X,*)$. This means that each individual $\underline{\pi}_n(X,*)$ has more structure than simply being a group.

##The variant 'at the end'##
###The Brown-Grossman homotopy groups at $\infty$###
+--{: .un_defn}
######Definition######
The $n^{th}$ _Brown-Grossman homotopy group of $(X,*)$ at $\infty$_ is given by the  group of based germ homotopy classes of based proper germs from $\underline{S}^n$ to $(X,*)$;

$$\underline{\pi}^\infty_n(X,*) = Ho(Proper_\infty)(\underline{S}^n, (X,*)).$$
=--
With this description, it is clear that $\underline{\pi}^\infty_n$ is functorial on base rayed spaces and that it only depends on the choice of the class of $*$ within $e(X)$ (cf. [[proper homotopy theory]]).

##A proper Whitehead theorem##
There are several different types of generalisation of Whitehead's theorem to the proper homotopy setting.  The following is due to Ed Brown (1974):
+--{: .un_thm}
######Theorem######
**(Brown, 1974)**
Let $K$, $L$ be finite dimensional connected locally finite [[simplicial complexes]], then a proper map $f : K \to L$ is a proper homotopy equivalence if, and only if:

1. $e(f) : e(K)\to e(L)$ is a homeomorphism;

1. for each $n$ $\pi_n(f) : \pi_n(K,*(0))\to \pi_n(L,f(*(0)))$ is an isomorphism;

1. for each $n$, and for each base ray, $*$, in $K$, $\underline{\pi}^\infty_n(K,*)\to \underline{\pi}^\infty_n(L,f(*))$ is an isomorphism.

=--
If one removes the condition of finite dimensionality, the result no longer holds. (There is an error in Brown's subsequent reasoning in the quoted paper, for which one needs to consult Edwards and Hastings, (1976).)

##Brown's $\mathcal{P}$-functor##
It would be useful to have a construction of the groups $\underline{\pi}^\infty_n(X,*)$ from the [[pro-group]] $\pi_n(\varepsilon(X),*(k))$. Such a construction was given by Brown in the same article (1974).  (An alternative construction due to Grossman will be discussed in a separate entry.)

Let $\underline{G} = \{G_n,p^m_n\}$ be an inverse sequence of groups (aka _tower of groups_), that is a pro-group that is indexed by the ordered set of positive integers. We assume $G_0=1$. Consider all sequences $\{g_{k(n)}\}$ with $g_{k(n)} \in G_{k(n)}$, where $k(n)$ is a sequence of natural numbers such that $k(n)\to \infty$ as $n\to \infty$. Given two such sequences $\{g_{k(n)}\}$ and $\{g\prime_{l(n)}\}$, we say they are equivalent if there is a third sequence $m(n)$, $m(n)\to \infty$ as $n \to \infty$, with $m(n)\leq min(k(n),l(n)))$ and $p^{k(n)}_{m(n)}(g_{k(n)}) =  p^{l(n)}_{m(n)}(g\prime_{l(n)})$.  We let $\mathcal{P}(\underline{G})$ be the set of equivalence classes.

* This has a natural group structure;

* If $X = \bigcup_n K_n$, $U_n = X- K_n$, and $* : [0,\infty) \to X$ is chosen so that $*[n,\infty)\subset U_n$, then setting $G_n = \pi_k(U_n,*(n))$ with $G_n\to G_{n-1}$ induced by the inclusion of $U_n$ into $U_{n-1}$ and the change of base point along $*([n-1,n])$, then $\underline{\pi}^\infty_n(X,*) \cong \mathcal{P}(\underline{G}).$

* For any tower of groups, $\underline{G}$ there is an action of the group $F = \underline{\pi}_1(\underline{S}^1,[0,\infty)$ on $\mathcal{P}(\underline{G})$.

* (Chipman) Let $\underline{G}$, $\underline{H}$ be towers of finitely generated groups, then $\underline{G}$ is isomorphic to $\underline{H}$ if, and only if, there is an isomorphism from $\mathcal{P}(\underline{G})$ to $\mathcal{P}(\underline{H})$ commuting with the operation of $F$.  (What is remarkable here is that initially no morphism between $\underline{G}$ and $\underline{H}$ is given.  It is constructed from the ones on the images under $\mathcal{P}$.)

###Grossman's reduced product construction###
Grossman gave an alternative construction of $\mathcal{P}(\underline{G})$ for any tower of groups (or, in fact,  any tower of sets, $\underline{X}$).  This involves the [[reduced product]].

##Relationship with the Milnor sequence

##References##
 General references include: the survey article:

* [[Tim Porter]], _Proper Homotopy Theory_,  in the _Handbook on Algebraic Topology_, Ed. I.M.James, Elsevier, 1995, p. 127-167,

and for a slightly different approach:

* [[Hans-Joachim Baues|Hans Baues]] and Antonio Quintero, _Infinite homotopy theory_, Volume 6 of K-monographs in mathematics,	Springer, 2001

A specific reference for the Brown Whitehead theorem is

*  E. M. Brown, _Proper homotopy theory in simplicial complexes,_ Topology Conference (Virginia Polytechnic Institute and State University, Dickmann and Fletcher, eds.) SLNM 375, Springer (1974), pp. 41-46,

and further

*  [[D. A. Edwards]] and [[H. M. Hastings]], _Čech and Steenrod Homotopy Theory with Applications,_ SLNM 542, Springer (1976).

[[!redirects Brown-Grossman homotopy groups]]