classical observable

#Contents#
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## Idea

In [[physics]], an [[observable]] in the context of [[classical mechanics]]/[[classical field theory]] and in fact also in [[prequantum field theory]] is called a _classical observable_.

In [[symplectic geometry]]/[[geometric prequantum theory]], classical observables are [[functions]] on [[phase space]], often called [[Hamiltonian]] functions. They form the [[Poisson algebra]] of classical/prequantum observables, wih the [[Poisson bracket]] [[Lie algebra]] structure.

In [[higher prequantum field theory]] these observables are refined to [[Hamiltonian forms]]/[[local currents]] and form the [[Poisson-bracket Lie n-algebra of local observables]].

## Related concepts

[[!include classical-to-quantum notions - table]]

[[!redirects classical observables]]

[[!redirects prequantum observable]]
[[!redirects prequantum observables]]


classical physics

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The term __'classical physics'__ usually denotes the part of [[physics]] which is based on exploring fundamental laws of nature which do not involve [[quantum physics]], or from another point of view, currently accepted areas of physics whose theoretical foundations in an almost canonical form existed before the discovery of quantum mechanics. This includes [[classical mechanics]], [[continuum mechanics]], [[classical electrodynamics]], classical fluid mechanics (with its main part [[hydrodynamics]] (and enriched versions like magnetohydrodynamics), dynamics of classical plasma, dynamics of gaseous media...), aerodynamics (dynamics of gases and motion of bodies moving within fluids), [[thermodynamics]], classical [[statistical mechanics]], geometrical and wave [[optics]] and the areas of phenomena which can be explained using these. For example, due to high speeds quantum phenomena are surpressed in the study of plasma (e.g. research on fusion).

Depending on context, relativity theory (both special and general) may or may not be included in classical physics. In the Lagrangean and Hamiltonian approaches, special relativity can be easily introduced into a classical mechanical system; general relativity, however, brings up the [[problem of time]].  Classical electrodynamics (with the [[Maxwell equation]]) is implicitly also a relativistic theory, even if it was originally described in a way hiding this fact.

Early (pre-quantum) theories which are generally rejected even in the non-quantum regime, for example the theory of light based on the concept of the luminiferous ether, do not usually count as classical physics today. (In the case of the ether, the deciding role was played by the appearance of relativity theory and the experiments supporting it.)


classical r-matrix

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#Contents#
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## Idea
 {#Idea}

An *$R$-matrix* is a solition to a [[Yang-Baxter equation]]:

A __classical r-matrix__ is a solution of  [[classical Yang-Baxter equation]].

(A [[quantum R-matrix]] is a solution to the [[quantum Yang-Baxter equation]].)

## Related entries

* [[Lie bialgebra]]

* [[Poisson Lie group]]

[[!include Yang-Baxter equations -- contents]]

## References

* M. A. Semenov-Tyan-Shanskii, _Classical r-matrices and quantization_, J. Math. Sciences __31__:6 (1985), 3411-3416, [doi](http://dx.doi.org/10.1007/BF02107242); _Classical r-matrices and the method of orbits_, J. Math. Sciences __28__:4 (1985), 513-524, [doi](http://dx.doi.org/10.1007/BF02104981)

* [[V. G. Drinfel'd]], _Quantum groups_, Proceedings of the International Congress of Mathematicians 1986, Vol. 1, 798–820, AMS 1987, [djvu:1.3M](http://www.mathunion.org/ICM/ICM1986.1/Main/icm1986.1.0798.0820.ocr.djvu), [pdf:2.5M](http://www.mathunion.org/ICM/ICM1986.1/Main/icm1986.1.0798.0820.ocr.pdf)

* Victor Mouquin, _The Fock-Rosly Poisson structure as defined by a quasitriangular r-matrix_, [arxiv/1610.01782](https://arxiv.org/abs/1610.01782)

[[!redirects classical Yang-Baxter equation]]
[[!redirects classical r-matrices]]

[[!redirects R-matrix]]
[[!redirects R-matrices]]


classical state

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# Contents
* table of contents
{: toc}

## Idea

A _classical state_ is a [[state]] of a [[system of classical mechanics]].

In principle, a [[pure state]] in classical mechanics specifies completely all information about the state of the system, while a [[mixed state]] is a [[probability measure]] on the space of pure states.  This space of pure states may be identified with the [[state space]] in [[Lagrangian mechanics]] or with the [[phase space]] in [[Hamiltonian mechanics]].

## Definition

We give a definition in a very general context.

For $A$ a [[commutative algebra|commutative]] unital [[associative algebra]] that encodes a [[system of classical mechanics]] (say, the associative algebra underlying a [[Poisson algebra]]), a **classical state** is an $\mathbb{R}$-[[linear function]]

$$
  \rho\colon A \to \mathbb{R}
$$

that satisfies

* **normalization** $\rho(1) = 1$;

* **positivity** for all $a \in A$ we have $\rho(a^2) \geq 0$.

This is essentially the definition of [[quantum state]], but formulated for commutative algebras and over the [[real number]]s.

If we take $A$ to be a $*$-[[star-algebra|algebra]] over the [[complex numbers]], then we may take $\rho$ to be a $\mathbb{C}$-linear function from $A$ to $\mathbb{C}$ instead.

## Related concepts

[[!include classical-to-quantum notions - table]]

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

\linebreak

[[!redirects classical state]]
[[!redirects classical states]]


classical triangulation

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#Contents#
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##Classical Triangulations##
###Aim###
The aim of this entry is to describe some of the classical versions of important concepts which are needed elsewhere. This may serve as an entry point for someone versed in a more classical version of algebraic topology, or being adept at the nPOV, and its ramifications, needs to bridge the gap to more classical ideas to understand some more classically written source. (It may also be useful if some classically written source is not at hand when you need it!) The exposition will be fairly 'classical' with asides to explain the significance for later developments and for connections to the nPOV.

##Triangulations à l'ancienne##
Let $X$ be a [[polyhedron]] (in the sense of polyhedral space), i.e. a space homeomorphic to the [[geometric realisation]] of a [[simplicial complex]].

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######Definition
A **classical triangulation** of $X$ is a pair $(K,f)$, where $K$ is a simplicial complex and $f : |K|\to X$ is a homeomorphism.
=--
 (In this context we will often drop the term 'classical' referring to 'triangulation' if there is little risk of confusion.)

##Classical subdivision##

The older form of [[subdivision]] involved the geometric realisation in the following way:
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######Definition
If $K$ is a simplicial complex, a **(classical) subdivision** of $K$ is a simplicial complex, $K^\prime$, such that

a) the vertices of $K^\prime$ are (identified with)  points of $|K|$;

b) if $s^\prime$ is a simplex of $K^\prime$  there is a simplex, $s$ of $K$ such that $s^\prime \subset |s|$; and

c) the mapping from $|K^\prime|$ to $|K|$, that extends the mapping of vertices of $K^\prime$ to the corresponding points of $|K|$, is a homeomorphism.
=--

The interpretation, in [[simplicial complex]], of the points of $|K|$ as convex combinations of the  vertices, allows an interpretation to be ascribed to $K^\prime$.  The general question of the meaning of 'refinements' that  will be examined later may need a deeper  examination of this subdivision process as it is a simple case of such a refinement.

###Properties of Classical Subdivisions###

* Any subdivision of a subdivision of $K$ is a subdivision of $K$.

* If $K'$ and  $K''$ are subdivisions of $K$ then there is a subdivision $K'''$  of $K$ that is a subdivision of both $K'$ and $K''$.

These statements thus assert that the subdivisions of a simplicial complex $K$ form a directed set with respect to the partial ordering defined by the relation of subdivision. (We will return to this later in this entry.)

Given any simplex, $s\in K$ (or put more pedantically $s \in S(K)$), and using the [[simplicial complex|Canonical Construction]] of $|K|$, we can assign an _open simplex_, $\langle s \rangle$, to $s$. We first recall, from [[simplicial complex]], that $|K|$ is constructed as follows:

$|K|$ is the set of all functions from $V(K)$ to  the closed interval $[0,1]$ such that

* if $\alpha \in |K|$, the set

$$\{v \in V(K) \mid \alpha(v)  \neq  0\}$$

is a simplex of $K$;

 * for each $v\in V(K)$,
$$\sum_{\alpha \in V(K)} \alpha (v) = 1.$$

(We then give this set a topology (see [[simplicial complex]]).)

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######Definition
For $s\in K$, the **open simplex**, $\langle s \rangle\subset |K|$ is defined  by
$$\langle s \rangle = \{ \alpha \in |K| \mid \alpha(v) \neq 0 \Leftrightarrow v\in s\}.$$
=--
Beware although a closed simplex  will be a closed subset of $|K|$, an open simplex need not be open in $|K|$. However every $\langle s \rangle$ is and open set of $|s|$. (see Spanier, p. 112, for a discussion.)

##Barycentric subdivision (classical geometric forms)
The barycentric subdivision is one of the best known and most useful natural subdivisions available in general.  (Other are also used, for instance, the middle edge or [[ordinal subdivision]].)  The barycentric subdivision has the good property that it exists without recourse to the realisation process, although usually introduced via that process. It is in that form that it is  discussed in [[subdivision]].  Here we give the 'classical' form and go from that towards the other functorial form.

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######Definition
If $\sigma = \{ v_0, \ldots, v_q\} \in K_q$, the set of $q$-simplices of a simplicial complex, $K$, then its **barycentre**, $b(\sigma)$, is the point

$$b(\sigma) = \sum_{0\leq i \leq q}\frac{1}{q + 1} v_i \in |K|.$$

The **barycentric subdivision**, $sd K$, of $K$ is the simplicial complex whose vertices are the barycentres of the simplices of $K$ and whose simplices are finite non-empty collections of barycentres of simplices, which are totally ordered by  the face relation of $K$, i.e., by inclusion when considered as subsets of $V(K)$.
=--
As $b(\sigma)$ is completely determined by $\sigma$, this can be rephrased as:

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######Definition: (Alternative form)
The **barycentric subdivision**, $sd K$, of $K$ is the simplicial complex specified by

* the vertices of $sd K$ are the simplices of $K$;
* the subset $\{\sigma_0, \ldots \sigma_q\}$ is a simplex of $sd K$ if there is a total order on its elements (which will be assumed to be the given order, as written) such that

$$\sigma_0\subset  \ldots \subset\sigma_q.$$
=--

##Triangulations and Covers##
We now need a bit more terminology:

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######Definition
 Given any vertex $v$ of $K$, its **star** is defined by

$$st( v) = \{ \alpha \in |K| \mid \alpha(v)\neq 0\}.$$

=--

The set, $st(v)$, is open in $|K|$. We have

$$st(v) = \bigcup \{\langle s \rangle\mid v is a vertex of s\},$$

the union of the interiors of those simplices that have $s$ as a vertex.  These vertex stars give an [[open cover]], $\mathcal{U}$, of $|K|$ and the following classical result tells us that the nerve $N(\mathcal{U})$ of this covering is $K$ itself (up to isomorphism):

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######Proposition (cf. Spanier, p. 114)
Let $\mathcal{U} = \{st(v)\mid v \in V(K)\}$.  The vertex map $\phi$ from $K$ to $N(\mathcal{U})$ defined by

$$\phi (v) = \langle st(v)\rangle$$

is a simplicial isomorphism

$$\phi : K \cong N(\mathcal{U}).$$
=--

category:motivation


classical-to-quantum notions - table

| | [[classical mechanics]] | [[semiclassical approximation]] |  ... | [[formal deformation quantization]] | [[quantum mechanics]] |
|--|--|--|--|--|--|
| order of [[Planck's constant]] $\hbar$ | $\mathcal{O}(\hbar^0)$ | $\mathcal{O}(\hbar^1)$ | $\mathcal{O}(\hbar^n)$ | $\mathcal{O}(\hbar^\infty)$  |  |
| [[states]] | [[classical state]] | [[semiclassical state]]  |  | | [[quantum state]] |
| [[observables]] | [[classical observable]] |  |  | | [[quantum observable]] |


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This entry is about the articles

* {#Pontrjagin38a} [[Lev Pontrjagin]]:

  **Classification of continuous maps of a complex into a sphere**, **Communication I**

  Doklady Akademii Nauk SSSR 19(3) (1938), 147-149

* {#Pontrjagin38b} [[Lev Pontrjagin]]:

  **Classification of continuous maps of a complex into a sphere**,

  **Communication II**

  Doklady Akademii Nauk SSSR 19(5) (1938), 361-363

  > (this article contains a famous mistake, see also [p. 6](https://arxiv.org/pdf/math/0211216.pdf#page=6) of [[Michael Hopkins|Hopkins]], [[Isadore Singer|Singer]]'s _[[Quadratic Functions in Geometry, Topology, and M-Theory]]_ and [[Michael Hopkins|Hopkins]]'s talk at [Atiyah's 80th Birthday conference](https://www.maths.ed.ac.uk/~v1ranick/atiyah80.htm), [slide 8](https://www.maths.ed.ac.uk/~v1ranick/hopkins.pdf#page=8), [9:45](https://empg.maths.ed.ac.uk/Videos/Atiyah80/Hopkins.mov))

* [[Lev Pontryagin]],

  **Homotopy classification of mappings of an $(n+2)$-dimensional sphere on an $n$-dimensional one**

  Doklady Akad. Nauk SSSR (N.S.) 19 (1950), 957–959 ([pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/pont3.pdf))

  > (this article fixes the mistake)

(all three of which are available in English translation in [Gamkrelidze 86](Revaz+Gamkrelidze#Gamkrelidze86))

or rather about their joint comprehensive exposition in

* {#Pontrjagin55} [[Lev Pontrjagin]]:

  **Smooth manifolds and their applications in homotopy theory**

  Trudy Mat. Inst. im Steklov, No 45, Izdat. Akad. Nauk. USSR, Moscow, 1955

  AMS Translation Series 2, Vol. 11, 1959

  [doi:10.1142/9789812772107_0001](https://www.worldscientific.com/doi/abs/10.1142/9789812772107_0001)

  [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/pont001.pdf))

on [[differential topology]], establishing the [[Pontryagin isomorphism]] between unstable [[Cohomotopy]] and [[cobordism classes]] of [[normally framed submanifolds]], and applying it to the computation of the lowest couple of [[stable homotopy groups of spheres]] (the [[first stable homotopy group of spheres|first]] and the [[second stable homotopy group of spheres|second]]).

Together with

* {#Thom54} [[René Thom]], _[[Quelques propriétés globales des variétés différentiables]]_, Comment. Math. Helv. 28, (1954). 17-86 ([doi:10.1007/BF02566923](https://doi.org/10.1007/BF02566923), [dml:139072](https://eudml.org/doc/139072), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/thomcob.pdf)),

due to which Pontryagin's construction came to be mainly known as the _[[Pontryagin-Thom construction]]_, this lays the foundations of [[cobordism theory]] as such and as a tool in [[stable homotopy theory]].

#Contents#
* table of contents
{:toc}

## Chapter I -- Smooth manifolds and smooth maps

### 1. Smooth manifolds

* [[differentiable manifold]], [[differentiable map]]

* [[smooth manifold]], [[smooth map]]

### 2. Embedding of smooth manifolds in Euclidean space

* [[submanifold]]

* [[Whitney embedding theorem]]

### 3. Improper points of smooth maps

* [[regular point]], [[critical point]]

### 4. Non-degenerate singular points of smooth maps

## Chapter II -- Normally-framed manifolds

### 5. Smooth approximations to continuous maps and deformations

### 6. The basic method

* [[normally framed submanifold]]

* [[cobordism]], [[cobordism class]]

* [continuous homotopy implies smooth homotopy](smooth+homotopy#ContinuousHomotopyImpliesSmoothHomotopy)

* [[Pontryagin's theorem]]

### 7. Homology groups of framed manifolds

### 8. The suspension operation

## Chapter III -- The Hopf invariant

### 9. The homotopy classification of maps of $n$-dimensional manifolds into the $n$-sphere

* [[Hopf degree theorem]]

### 10. The Hopf invariant of maps of $\Sigma^{2k+1}$ into $S^{k+1}$

### 11. Framed manifolds with zero Hopf invariant

## Chapter IV -- Classification of maps between spheres

* [[homotopy groups of spheres]]

### 12. The rotation group of Euclidean space

* [[orthogonal group]]

### 13. Classification of maps to the 3-sphere into the 2-sphere

* [[Hopf fibration]]

### 14. Classification of maps of the $(n+1)$-sphere into the $n$-sphere

* [[first stable homotopy group of spheres]]

### 15. Classification of maps of the $(n+2)$-sphere into the $n$-sphere

* [[second stable homotopy group of spheres]]

category: reference

[[!redirects Smooth manifolds and their applications in homotopy theory]][[!redirects Smooth manifolds and their applications in Homotopy theory]]

[[!redirects Homotopy classification of mappings of an (n+2)-dimensional sphere on an n-dimensional one]]


classification of finite simple groups

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There are 18 countably infinite families and 26 sporadic [[finite group|finite]] [[simple group|simple]] [[group]]s.
In slightly more detail, a finite simple group is one of the following

1. A group of prime order
1. An alternating group $A_n$ for $n\geq 5$
1. A [[group of Lie type]] over a finite field
1. One of the 26 [[sporadic finite simple groups]].

The original 'proof' fills 500 journal articles. An updated, self-contained proof is in the process of being written, and it is estimated that it will be 5000 pages long. As of 2018 seven volumes had been published, out of an expected 11.

For now see the [Wikipedia page](http://en.wikipedia.org/wiki/Classification_of_finite_simple_groups).

An original conceptual insight into the classification
of finite groups from the point of algebraic geometry (involving embeddings into algebraic groups) has been recently achieved in an [award](http://newsinfo.iu.edu/news/page/normal/23661.html)-winning article

* Michael J. Larsen, Richard Pink, _Finite subgroups of algebraic groups_, J. Amer. Math. Soc. 24 (2011), 1105-1158 [doi](http://dx.doi.org/10.1090/S0894-0347-2011-00695-4)

category: algebra


classification of long-range forces -- references

### Classification of long-range forces
 {#ClassificationOfLongRangeForces}

Classification of possible long-range [[forces]], hence of [[scattering]] processes of [[mass|massless]] [[field (physics)|fields]], by classification of suitably factorizing and decaying [[Poincaré group|Poincaré]]-[[invariant]] [[S-matrices]] depending on particle spin, leading to uniqueness statements about [[Maxwell theory|Maxwell]]/[[photon]]-, [[Yang-Mills theory|Yang-Mills]]/[[gluon]]-, [[gravity]]/[[graviton]]- and [[supergravity]]/[[gravitino]]-[[interactions]]:

* [[Steven Weinberg]], _Feynman Rules for Any Spin. 2. Massless Particles_, Phys. Rev. 134 (1964) B882 ([doi:10.1103/PhysRev.134.B882](https://doi.org/10.1103/PhysRev.134.B882))

* [[Steven Weinberg]], _Photons and Gravitons in $S$-Matrix Theory: Derivation of Charge Conservationand Equality of Gravitational and Inertial Mass_, Phys. Rev. 135 (1964) B1049 ([doi:10.1103/PhysRev.135.B1049](https://doi.org/10.1103/PhysRev.135.B1049))

* [[Steven Weinberg]], _Photons and Gravitons in Perturbation Theory: Derivation of Maxwell’s and Einstein’s Equations,” Phys. Rev. 138 (1965) B988 ([doi:10.1103/PhysRev.138.B988](https://doi.org/10.1103/PhysRev.138.B988))

* Paolo Benincasa, Freddy Cachazo, _Consistency Conditions on the S-Matrix of Massless Particles_ ([arXiv:0705.4305](https://arxiv.org/abs/0705.4305))

* David A. McGady, Laurentiu Rodina, _Higher-spin massless S-matrices in four-dimensions_, Phys. Rev. D 90, 084048 (2014) ([arXiv:1311.2938](https://arxiv.org/abs/1311.2938), [doi:10.1103/PhysRevD.90.084048](https://doi.org/10.1103/PhysRevD.90.084048))

Review:

* {#Kiefer07} Claus Kiefer, section 2.1.3 of: _Quantum Gravity_, Oxford University Press 2007 ([doi:10.1093/acprof:oso/9780199585205.001.0001](https://www.oxfordscholarship.com/view/10.1093/acprof:oso/9780199585205.001.0001/acprof-9780199585205), [cds:1509512](https://cds.cern.ch/record/1509512))

* {#Baumann19} [[Daniel Baumann]], _What long-range forces are allowed?_, 2019 ([[BaumannLongRange19.pdf:file]])


classifying (infinity,1)-topos

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## Idea

The notion of a _classifying (∞,1)-topos_ is the [[vertical categorification]] of the notion of [[classifying topos]] to the context of [[(∞,1)-category]] theory.

Any (∞,1)-topos $K$ by definition classifies the ∞-geometric morphisms into it in that it is the representing object of $geom(-,K)$.

## Examples

### For objects and pointed objects
 {#ForObjectsAndPointedObjects}

Write $\infty Grpd_{fin}$ for the [[(∞,1)-category]] of [[finite homotopy types]] and write $\infty Grpd_{fin}^{\ast/}$ for its [[pointed objects]].

For $\mathbf{H}$ any [[base (∞,1)-topos]], the [[(∞,1)-presheaf (∞,1)-topos]]

$$
  \mathbf{H}[X] \coloneqq PSh(\infty Grpd_{fin}^{op}, \mathbf{H})
$$

is the classifying $\infty$-topos for [[objects]], and

$$
  \mathbf{H}[X_\ast] \coloneqq PSh((\infty Grpd_{fin}^{\ast/})^{op}, \mathbf{H})
$$

is the classifying $\infty$-topos for [[pointed objects]].

For $K \in \infty Grpd_{fin}^{\ast/}$, write $R(K) \in (\infty Grpd_{fin}^{\ast/})^{op} \hookrightarrow \mathbf{H}[X_\ast]$ for its [[formal dual]] under [[(∞,1)-Yoneda embedding]].  The generic pointed object in $\mathbf{H}[X_\ast]$ is that [[representable functor|represented]] by the [[0-sphere]]:

$$
  X_\ast = R(S^0)
  \,.
$$

### For local structures

A special case of this is the notion of a classifying (∞,1)-topos for a geometry in the sense of [[geometry (for structured (∞,1)-toposes)|structured spaces]]:

The geometry $\mathcal{G}$ is the [[(∞,1)-category]] that plays role of the syntactic theory. For $\mathcal{X}$ an [[(∞,1)-topos]], a model of this theory is a limits and covering-preserving [[(∞,1)-functor]]

$$
  \mathcal{G} \to \mathcal{X}
  \,.
$$

The [[Yoneda embedding]] followed by [[∞-stackification]]

$$
  \mathcal{G} \stackrel{Y}{\to} PSh_{(\infty,1)}(\mathcal{G})
  \stackrel{\bar(-)}{\to} Sh_{(\infty,1)}(\mathcal{G})
$$

constitutes a model of $\mathcal{G}$ in the (Cech) [[∞-stack]] [[(∞,1)-topos]] $Sh_{(\infty,1)}(\mathcal{G})$ and exhibits it as the classifying topos for such models (geometries):

This is _[[Structured Spaces]]_ [prop 1.4.2](http://arxiv.org/PS_cache/arxiv/pdf/0905/0905.0459v1.pdf#page=26).

[[!redirects classifying (∞,1)-topos]]
[[!redirects classifying (∞,1)-toposes]]

[[!redirects classifying (∞,1)-topoi]]

[[!redirects classifying (infinity,1)-topos]]
[[!redirects classifying (infinity,1)-toposes]]
[[!redirects classifying (infinity,1)-topoi]]

category: higher topos theory


classifying (∞,1)-topos > history


classifying morphism

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## Idea

A **classifying map** or **classifying morphism** for a given [[object]] is a [[morphism]] into a [[classifying space]] or [[moduli space]] that classified this object.

Typically this means that the map characterizes the object (only) up to [[equivalence]]. There is also the more refined concept of morphisms into a [[moduli stack]] that characterizes the object itself. For emphasis this case might be called a _[[modulating morphism]]_.

## Examples

* For [[subobjects]] one typically speaks of **[[characteristic maps]]** or **characteristic functions**. The corresponding classifiyng space is a _[[subobject classifier]]_ .

* More generally, in an [[(infinity,1)-topos]] every "small" (see there) object in a [[slice (infinity,1)-topos]] is given by a classifying morphism into the [[object classifier]];

* In [[dependent type theory|dependent]]/[[homotopy type theory|homotopy]] [[type theory]] these classifying morphisms are the [[categorical semantics]] for [[functions]] into a [[type of types]] that classify [[dependent types]]. See at _[[categorical model of dependent types]]_ for more on this.

## Related concepts

* [[classifying space]], [[classifying stack]], [[moduli]], [[moduli space]], [[moduli stack]], [[derived moduli space]], [[modulating morphism]]

* [[display map]], [[substitution]]

[[!redirects classifying morphisms]]

[[!redirects classifying map]]
[[!redirects classifying function]]

[[!redirects classifying morphisms]]
[[!redirects classifying maps]]
[[!redirects classifying functions]]


classifying space

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#### Geometry
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## Idea

A **classifying space** for some sort of data refers to a [[space]] (or a more general object), usually written $\mathcal{B}(data)$, such that maps $X\to \mathcal{B}(data)$ correspond to data over $X$.

The classical example is the classifying space $\mathcal{B}G$ of a [[group]] $G$, which has the property that there is a bijection between [[homotopy]] classes of maps $X\to \mathcal{B}G$ and isomorphism classes of $G$-[[bundle|bundles]] over $X$.  (In fact, one can jack this up to an equivalence of [[groupoid|groupoids]] or $\infty$-[[infinity-groupoid|groupoids]].)  Various improvements of this are possible which classify bundles with extra structure or [[fibration|fibrations]].

Categorically, the corresponding statement is that [[Grothendieck fibrations]] over a [[category]] $X$ correspond to [[pseudofunctors]] $X^{op}\to Cat$.  Thus $Cat$ is the "classifying space for categories."  Similarly, discrete fibrations over $X$ correspond to [[functors]] $X^{op}\to Set$.

To see the connection between the two, consider the case when $X$ is a groupoid and we restrict the fibers of the fibration to be isomorphic to a given set $F$.   Then the functor $X^{op}\to Set$ must land in the subcategory of $Set$ consisting of just the [[automorphisms]] of $F$, which is the one-object groupoid corresponding to the [[automorphism group]] $Aut(F)$.  If we further restrict the automorphisms appearing to preserve some given structure on $F$, so that they lie in some smaller group $G$, then the "classifying space" will be the one-object groupoid corresponding to $G$.  Under the [[homotopy hypothesis]], groupoids correspond to [[homotopy 1-types]], and the one-object groupoid of a group $G$ corresponds precisely to the usual topological classifying space $\mathcal{B}G$ (in fact, this is one _construction_ of $\mathcal{B}G$).  For this reason, $\mathbf{B}G$ is often used to denote that one-object groupoid; see the [[delooping hypothesis]] and the discussion at [[category algebra]].

The phrase "classifying space" is also sometimes used for the realization of the nerve of any category, although it is more complicated to say what exactly this space "classifies."  (One answer is "torsors modulo concordance.")

## Examples

### For principal bundles

> A good account is in [Rudolf & Schmidt 17, Sec. 3, esp. Thm. 3.5.1 & Prop. 3.6.2](#RudolfSchmidt17).

For $G$ a [[topological group]] there is a classifying space $B G \in$ [[Top]] for topological $G$-[[principal bundle]]s, hence a space such that for $X$ any sufficiently nice topological space there is a [[natural isomorphism]]

$$
  G Bund(X)_0 \simeq \pi_0 Top(X, B G)
$$

between the set of [[isomorphism]] classes of $G$-[[principal bundle]]s on $X$ and the set of [[homotopy]]-classes of [[continuous function]]s $X \to B G$.

This space may be constructed as follows:

write $\mathbf{B}G \in Top^{\Delta^{op}}$ for the [[simplicial topological space]] obtained as the [[nerve]] of the one-object [[topological groupoid]] associated to $G$, the simplicial space given by

$$
  (\mathbf{B}G)_n = G^{\times n}
$$

whose face maps are induced by the product operation on $G$ and whose degeneracy maps are induced from the unit map.

If $G$ is [[well-pointed simplicial topological group|well-pointed]], then the [[geometric realization of simplicial topological spaces]] of $G$ is a model for the [[homotopy type]] of the classifying space

$$
  B G \simeq \vert \mathbf{B}G\vert
  \,.
$$

For more details on this construction see the section classifying spaces at [[geometric realization of simplicial topological spaces]].

As discussed there, too, this construction generalizes to more general [[simplicial topological group]]s and classifying spaces for their [[principal ∞-bundle]]s.

\linebreak

\begin{theorem}
  If $G \,\in\, Grp(HausSp)$ is a [[Hausdorff space|Hausdorff]] [[topological group]], then the Milnor join construction classifies topological $G$-principal bundles over all [[paracompact Hausdorff spaces]].
\end{theorem}
(due to Milnor, laid out in [Rudolf & Schmidt 17, Thm. 3.5.1](#RudolfSchmidt17))

### For orthogonal and unitary principal bundles
 {#ForOrthogonalAndUnitaryPrincipalBundles}

#### Idea

For $G = O(n)$ the [[orthogonal group]] and
$G = U(n)$ the [[unitary group]], there are standard realizations of the corresponding classifying spaces as [[direct limits]] of [[Grassmannian]] spaces.

Let $V_n(\mathbb{R}^q)$ be the [[Stiefel manifold]] of [[orthonormal basis|orthonormal]] $n$-frames in the [[Cartesian space]] $\mathbb{R}^q$. Its points are $n$-[[tuples]] of orthonormal [[vectors]] in $\mathbb{R}^q$, and it is topologized as a [[subspace topology|subspace]] of $(\mathbb{R}^q)^n$,
or, equivalently, as a subspace of $(S^{q-1})^n$. It is a [[compact space|compact]] [[manifold]].

Let $G_n(\mathbb{R}^q)$ be the [[Grassmannian]] of $n$-planes in $\mathbb{R}^q$. Its points are the n-dimensional subspaces
of $\mathbb{R}^q$. Sending an $n$-tuple of orthonormal vectors to the $n$-plane they span gives
a surjective function $V_n(\mathbb{R}^q) \to G_n(\mathbb{R}^q)$, and we topologize $G_n(\mathbb{R}^q)$ as a [[quotient topology|quotient]] space of $V_n(\mathbb{R}^q)$. It too is a compact manifold.

The standard inclusion of $\mathbb{R}^q$ in $\mathbb{R}^{q+1}$ induces inclusions
$V_n(\mathbb{R}^q) \hookrightarrow V_n(\mathbb{R}^{q+1})$ and $G_n(\mathbb{R}^q) \hookrightarrow G_n(\mathbb{R}^{q+1})$. We define $V_n(\mathbb{R}^\infty)$ and $G_n(\mathbb{R}^\infty)$ to be the unions of the $V_n(\mathbb{R}^q)$ and $G_n(\mathbb{R}^q)$, with the topology of the union.

Then $G_n(\mathbb{R}^\infty)$ is a model for the classifying space $B O(n)$.

#### Definitions

In the following we take [[Top]] to denote [[compactly generated topological spaces]]. For these the [[Cartesian product]] $X \times (-)$ is a [[left adjoint]] and hence preserves [[colimits]].

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

For $n, k \in \mathbb{N}$ and $n \leq k$, then the $n$th **real [[Stiefel manifold]]** of $\mathbb{R}^k$ is the [[coset]] [[topological space]].

$$
  V_n(\mathbb{R}^k) \coloneqq O(k)/O(k-n)
  \,,
$$

where the [[action]] of $O(k-n)$ is via its canonical embedding $O(k-n)\hookrightarrow O(k)$.

Similarly the $n$th **complex Stiefel manifold** of $\mathbb{C}^k$ is

$$
  V_n(\mathbb{C}^k) \coloneqq U(k)/U(k-n)
  \,,
$$

here the [[action]] of $U(k-n)$ is via its canonical embedding $U(k-n)\hookrightarrow U(k)$.

=--

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

For $n, k \in \mathbb{N}$ and $n \leq k$, then the $n$th **real [[Grassmannian]]** of $\mathbb{R}^k$ is the [[coset]] [[topological space]].

$$
  Gr_n(\mathbb{R}^k) \coloneqq O(k)/(O(n) \times O(k-n))
  \,,
$$

where the [[action]] of the [[product group]] is via its canonical embedding $O(n)\times O(k-n) \hookrightarrow O(n)$ into the [[orthogonal group]].

Similarly the $n$th **complex [[Grassmannian]]** of $\mathbb{C}^k$ is the [[coset]] [[topological space]].

$$
  Gr_n(\mathbb{C}^k) \coloneqq U(k)/(U(n) \times U(k-n))
  \,,
$$

where the [[action]] of the [[product group]] is via its canonical embedding $U(n)\times U(k-n) \hookrightarrow U(n)$ into the [[unitary group]].

=--

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

* $Gr_1(\mathbb{R}^{n+1}) \simeq \mathbb{R}P^n$ is [[real projective space]] of [[dimension]] $n$.

* $Gr_1(\mathbb{C}^{n+1}) \simeq \mathbb{C}P^n$ is [[complex projective space]] of [[dimension]] $n$.

=--

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

For all $n \leq k \in \mathbb{N}$, the canonical [[projection]] from the real [[Stiefel manifold]] (def. \ref{StiefelManifold}) to the [[Grassmannian]] is a $O(n)$-[[principal bundle]]

$$
  \array{
    O(n) &\hookrightarrow& V_n(\mathbb{R}^k)
    \\
    && \downarrow
    \\
    && Gr_n(\mathbb{R}^k)
  }
$$

and the projection from the complex Stiefel manifold to the Grassmannian us a $U(n)$-[[principal bundle]]:

$$
  \array{
    U(n) &\hookrightarrow& V_n(\mathbb{C}^k)
    \\
    && \downarrow
    \\
    && Gr_n(\mathbb{C}^k)
  }
  \,.
$$

=--

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

By ([this cor.](coset#QuotientProjectionForCompactLieSubgroupIsPrincipal) and [this prop.](coset#ProjectionOfCosetsIsFiberBundleForClosedSubgroupsOfCompactLieGroup)).

=--

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

By def. \ref{RealAndComplexGrassmannian} there are canonical inclusions

$$
  Gr_n(\mathbb{R}^k) \hookrightarrow Gr_n(\mathbb{R}^{k+1})
$$

and

$$
  Gr_n(\mathbb{C}^k) \hookrightarrow Gr_n(\mathbb{C}^{k+1})
$$

for all $k \in \mathbb{N}$. The [[colimit]] (in [[Top]], see [there](Top#UniversalConstructions)) over these inclusions is denoted

$$
  B O(n) \coloneqq \underset{\longrightarrow}{\lim}_k Gr_n(\mathbb{R}^k)
$$

and

$$
  B U(n) \coloneqq \underset{\longrightarrow}{\lim}_k Gr_n(\mathbb{C}^k)
  \,,
$$

respectively.

Moreover, by def. \ref{StiefelManifold} there are canonical inclusions

$$
   V_n(\mathbb{R}^k) \hookrightarrow V_n(\mathbb{R}^{k+1})
$$

and

$$
  V_n(\mathbb{C}^k) \hookrightarrow V_n(\mathbb{C}^{k+1})
  \,,
$$

respectively, that are compatible with the $O(n)$-[[action]] and the $U(n)$-action, respectively. The [[colimit]] (in [[Top]], see [there](Top#UniversalConstructions)) over these inclusions, regarded as equipped with the induced [[action]], is denoted

$$
  E O(n) \coloneqq \underset{\longrightarrow}{\lim}_k V_n(\mathbb{R}^k)
$$

and

$$
  E U(n) \coloneqq \underset{\longrightarrow}{\lim}_k V_n(\mathbb{C}^k)
  \,,
$$

respectively. The inclusions are in fact compatible with the bundle structure from prop. \ref{ProjectionFromStiefelManifoldToGrassmannianIsFiberBundle}, so that there are induced projections

$$
  \left(
  \array{
    E O(n)
    \\
    \downarrow
    \\
    B O(n)
  }
  \right)
  \;\;
   \simeq
  \;\;
  \underset{\longrightarrow}{\lim}_k
  \left(
    \array{
       V_n(\mathbb{R}^k)
       \\
       \downarrow
       \\
       Gr_n(\mathbb{R}^k)
    }
  \right)
$$

and

$$
  \left(
  \array{
    E U(n)
    \\
    \downarrow
    \\
    B U(n)
  }
  \right)
  \;\;
   \simeq
  \;\;
  \underset{\longrightarrow}{\lim}_k
  \left(
    \array{
       V_n(\mathbb{C}^k)
       \\
       \downarrow
       \\
       Gr_n(\mathbb{C}^k)
    }
  \right)
  \,,
$$

respectively. These are the standard models for the **[[universal principal bundles]]** for $O$ and $U$, respectively. The corresponding [[associated bundles|associated]] [[vector bundles]]

$$
 E O(n) \underset{O(n)}{\times} \mathbb{R}^n
$$

and

$$
 E U(n) \underset{U(n)}{\times} \mathbb{C}^n
$$

are the corresponding **[[universal vector bundles]]**.

=--

Since the [[Cartesian product]] $O(n)\times (-)$ in [[compactly generated topological spaces]] preserves [[colimits]], it follows that the colimiting bundle is still an $O(n)$-[[principal bundle]]

$$
  \begin{aligned}
    (E O(n))/O(n)
    &
    \simeq
    (\underset{\longrightarrow}{\lim}_k V_{n}(\mathbb{R}^k))/O(n)
    \\
    & \simeq
    \underset{\longrightarrow}{\lim}_k (V_n(\mathbb{R}^k)/O(n))
    \\
    & \simeq
    \underset{\longrightarrow}{\lim}_k Gr_n(\mathbb{R}^k)
    \\
    & \simeq B O(n)
  \end{aligned}
$$

and anlogously for $E U(n)$.

As such this is the standard presentation for the $O(n)$-[[universal principal bundle]]. Its base space $B O(n)$ is the corresponding _classifying space_.

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

There are canonical inclusions

$$
  Gr_n(\mathbb{R}^k)
  \hookrightarrow
  Gr_{n+1}(\mathbb{R}^{k+1})
$$

and

$$
  Gr_n(\mathbb{C}^k)
  \hookrightarrow
  Gr_{n+1}(\mathbb{C}^{k+1})
$$

given by adjoining one coordinate to the ambient space and to any subspace. Under the colimit of def. \ref{EOn} these induce maps of classifying spaces

$$
  B O(n) \longrightarrow B O(n+1)
$$

and

$$
  B U(n) \longrightarrow B U(n+1)
  \,.
$$

=--

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

There are canonical maps

$$
  Gr_{n_1}(\mathbb{R}^{k_1})
  \times
  Gr_{n_2}(\mathbb{R}^{k_2})
  \longrightarrow
  Gr_{n_1 + n_2}(\mathbb{R}^{k_1 + k_2})
$$

and

$$
  Gr_{n_1}(\mathbb{C}^{k_1})
  \times
  Gr_{n_2}(\mathbb{C}^{k_2})
  \longrightarrow
  Gr_{n_1 + n_2}(\mathbb{C}^{k_1 + k_2})
$$

given by sending ambient spaces and subspaces to their [[direct sum]].

Under the colimit of def. \ref{EOn} these induce maps of classifying spaces

$$
  B O(n_1) \times B O(n_2)
  \longrightarrow
  B O(n_1 + n_2)
$$

and

$$
  B U(n_1) \times B U(n_2)
  \longrightarrow
  B U(n_1 + n_2)
$$

=--

#### Properties

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

The real [[Grassmannians]] $Gr_n(\mathbb{R}^k)$ and the complex Grassmannians $Gr_n(\mathbb{C}^k)$ of def. \ref{RealAndComplexGrassmannian} admit the structure of [[CW-complexes]]. Moreover the canonical inclusions

$$
  Gr_n(\mathbb{R}^k)
  \hookrightarrow
  Gr_n(\mathbb{R}^{k+1})
$$

and

$$
  Gr_n(\mathbb{C}^k)
  \hookrightarrow
  Gr_n(\mathbb{C}^{k+1})
$$

are subcomplex incusions (hence [[relative cell complex]] inclusions).

Accordingly there is an induced CW-complex structure on the [[classifying spaces]] $B O(n)$ and $B U(n)$ (def. \ref{EOn}).

=--

A general proof is spelled out in [Hatcher, section 1.2 (pages 31-34)](#Hatcher). For the case of [[real vector bundle|real]]- , [[complex vector bundle|complex]]- and [[quaternionic vector bundles]] see at _[[cell structure on projective space]]_.

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

The [[Stiefel manifold]] $V_n(\mathbb{R}^k)$ from def. \ref{StiefelManifold} admits the structure of a [[CW-complex]].

=--

e.g. ([James 59, p. 3](Stiefel+manifold#James59), [James 76, p. 5 with p. 21](Stiefel+manifold#James76), [Blaszczyk 07](Stiefel+manifold#Blaszczyk07))

(And I suppose with that cell structure the inclusions $V_n(\mathbb{R}^k) \hookrightarrow V_n(\mathbb{R}^{k+1})$ are subcomplex inclusions.)

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

The [[Stiefel manifold]] $V_n(\mathbb{R}^k)$ (def. \ref{StiefelManifold}) is [[n-connected topological space|(k-n-1)-connected]].

=--

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

Consider the [[coset]] [[quotient]] [[projection]]

$$
  O(k-n)
    \longrightarrow
  O(k)
    \longrightarrow
  O(k)/O(k-n)
    =
  V_n(\mathbb{R}^k)
  \,.
$$

Since the orthogonal groups is compact ([prop.](orthogonal+group#OrthogonalGroupIsCompact)) and by [this corollary](coset#QuotientProjectionForCompactLieSubgroupIsPrincipal) the projection $O(k)\to O(k)/O(k-n)$ is a [[Serre fibration]]. Therefore there is induced the [[long exact sequence of homotopy groups]] of this [[fiber sequence]], and by [this prop.](orthogonal+group#InclusionOfOnIntoOkIsnMinus1Equivalence) it has the following form in degrees bounded by $n$:

$$
  \cdots
    \to
  \pi_{\bullet \leq k-n-1}(O(k-n))
    \overset{epi}{\longrightarrow}
  \pi_{\bullet \leq k-n-1}(O(k))
    \overset{0}{\longrightarrow}
  \pi_{\bullet \leq k-n-1}(V_n(\mathbb{R}^k))
    \overset{0}{\longrightarrow}
  \pi_{\bullet-1 \lt k-n-1}(O(k))
    \overset{\simeq}{\longrightarrow}
  \pi_{\bullet-1 \lt k-n-1}(O(k-n))
    \to
  \cdots
  \,.
$$

This implies the claim. (Exactness of the sequence says that every element in $\pi_{\bullet \leq n-1}(V_n(\mathbb{R}^k))$ is in the kernel of zero, hence in the image of 0, hence is 0 itself.)

=--

Similarly:

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

The complex [[Stiefel manifold]] $V_n(\mathbb{C}^k)$ (def. \ref{StiefelManifold}) is [[n-connected topological space|2(k-n)-connected]].

=--

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

Consider the [[coset]] [[quotient]] [[projection]]

$$
  U(k-n)
    \longrightarrow
  U(k)
    \longrightarrow
  U(k)/U(k-n)
    =
  V_n(\mathbb{C}^k)
  \,.
$$

By prop. \ref{UnitaryGroupIsCompact} and by [this corollary](coset#QuotientProjectionForCompactLieSubgroupIsPrincipal) the projection $U(k)\to U(k)/U(k-n)$ is a [[Serre fibration]]. Therefore there is induced the [[long exact sequence of homotopy groups]] of this [[fiber sequence]], and by prop. \ref{InclusionOfUnitaryGroupnIntoUnitaryGroupnPlusIneIsnMinus1Equivalence} it has the following form in degrees bounded by $n$:

$$
  \cdots
    \to
  \pi_{\bullet \leq 2(k-n)}(U(k-n))
    \overset{epi}{\longrightarrow}
  \pi_{\bullet \leq 2(k-n)}(U(k))
    \overset{0}{\longrightarrow}
  \pi_{\bullet \leq 2(k-n)}(V_n(\mathbb{C}^k))
    \overset{0}{\longrightarrow}
  \pi_{\bullet-1 \lt 2(k-n)}(U(k))
    \overset{\simeq}{\longrightarrow}
  \pi_{\bullet-1 \lt 2(k-n)}(U(k-n))
    \to
  \cdots
  \,.
$$

This implies the claim.

=--

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

The colimiting space $E O(n) = \underset{\longrightarrow}{\lim}_k V_n(\mathbb{R}^k)$ from def. \ref{EOn} is [[weakly contractible topological space|weakly contractible]].

The colimiting space $E U(n) = \underset{\longrightarrow}{\lim}_k V_n(\mathbb{C}^k)$ from def. \ref{EOn} is [[weakly contractible topological space|weakly contractible]].

=--

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

The [[homotopy groups]] of the classifying spaces $B O(n)$ and $B U(n)$ (def. \ref{EOn}) are those of the [[orthogonal group]] $O(n)$ and of the [[unitary group]] $U(n)$, respectively, shifted up in degree: there are [[isomorphisms]]

$$
  \pi_{\bullet+1}(B O(n))
    \simeq
  \pi_\bullet O(n)
$$

and

$$
  \pi_{\bullet+1}(B U(n))
    \simeq
  \pi_\bullet U(n)
$$

(for homotopy groups based at the canonical basepoint).

=--

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

Consider the sequence

$$
  O(n)
    \longrightarrow
  E O(n)
    \longrightarrow
  B O(n)
$$

from def. \ref{EOn}, with $O(n)$ the [[fiber]]. Since (by [this prop.](coset#ProjectionOfCosetsIsFiberBundleForClosedSubgroupsOfCompactLieGroup)) the second map is a [[Serre fibration]], this is a [[fiber sequence]] and so it induces a [[long exact sequence of homotopy groups]] of the form

$$
  \cdots
    \to
  \pi_\bullet(O(n))
    \longrightarrow
  \pi_\bullet(E O(n))
    \longrightarrow
  \pi_\bullet(B O(n))
    \longrightarrow
  \pi_{\bullet-1}(O (n))
   \longrightarrow
  \pi_{\bullet-1}(E O(n))
    \to
  \cdots
  \,.
$$

Since by cor. \ref{EOnIsWeaklyContractible} $\pi_\bullet(E O(n))= 0$, exactness of the sequence implies that

$$
  \pi_\bullet(B O(n))
   \overset{\simeq}{\longrightarrow}
  \pi_{\bullet-1}(O (n))
$$

is an isomorphism.

The same kind of argument applies to the complex case.

=--

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

For $n \in \mathbb{N}$ there are [[homotopy fiber sequences]]

$$
  S^n
    \longrightarrow
  B O(n)
    \longrightarrow
  B O(n+1)
$$

and

$$
  S^{2n+1}
    \longrightarrow
  B U(n)
    \longrightarrow
  B U(n+1)
  \,,
$$

exhibiting the [[n-sphere]] ($(2n+1)$-sphere) as the [[homotopy fiber]] of the canonical maps from def. \ref{InclusionOfBOnIntoBOnPlusOne}.

This means that there is a replacement of the canonical inclusion $B O(n) \hookrightarrow B O(n+1)$ (induced via def. \ref{EOn}) by a [[Serre fibration]]

$$
  \array{
     B O(n) &\hookrightarrow& B O(n+1)
     \\
     {}^{\mathllap{{weak \, homotopy} \atop equivalence}}\downarrow  & \nearrow_{\mathrlap{Serre \, fib.}}
     \\
     \tilde B O(n)
  }
$$

such that $S^n$ is the ordinary [[fiber]] of $B O(n)\to \tilde B O(n+1)$, and analogously for the complex case.

=--

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

Take $\tilde B O(n) \coloneqq (E O(n+1))/O(n)$.

To see that the canonical map $B O(n)\longrightarrow (E O(n+1))/O(n)$ is a [[weak homotopy equivalence]] consider the [[commuting diagram]]

$$
  \array{
    O(n) &\overset{id}{\longrightarrow}& O(n)
    \\
    \downarrow && \downarrow
    \\
    E O(n) &\longrightarrow& E O(n+1)
    \\
    \downarrow && \downarrow
    \\
    B O(n) &\longrightarrow& (E O(n+1))/O(n)
  }
  \,.
$$

By [this prop.](coset#ProjectionOfCosetsIsFiberBundleForClosedSubgroupsOfCompactLieGroup) both bottom vertical maps are [[Serre fibrations]] and so both vertical sequences are [[fiber sequences]]. By prop. \ref{HomotopyGroupsOfBOnThoseOfOnShifted} part of the induced morphisms of [[long exact sequences of homotopy groups]] looks like this

$$
  \array{
    \pi_\bullet(B O(n))
    &\overset{}{\longrightarrow}&
    \pi_\bullet( (E O(n+1))/O(n) )
    \\
    {}^{\mathllap{\simeq}}\downarrow && \downarrow^{\mathrlap{\simeq}}
    \\
    \pi_{\bullet-1}(O(n))
    &\overset{=}{\longrightarrow}&
    \pi_{\bullet-1}(O(n))
  }
  \,,
$$

where the vertical and the bottom morphism are isomorphisms. Hence also the to morphisms is an isomorphism.

That $B O(n)\to \tilde B O(n+1)$ is indeed a [[Serre fibration]] follows again with [this prop.](coset#ProjectionOfCosetsIsFiberBundleForClosedSubgroupsOfCompactLieGroup), which gives the [[fiber sequence]]

$$
  O(n+1)/O(n)
    \longrightarrow
  (E O(n+1))/O(n)
    \longrightarrow
  (E O(n+1))/O(n+1)
  \,.
$$

The claim in then follows since ([this exmpl.](coset#nSphereAsCosetSpace))

$$
  O(n+1)/O(n) \simeq S^n
  \,.
$$

The argument for the complex case is of the same form, concluding now with the identification ([this exmpl.](unitary+group#nSphereAsUnitaryCosetSpace))

$$
  U(n+1)/U(n) \simeq S^{2n+1}
  \,.
$$

=--

#### Classification of bundles

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

For $X$ a [[paracompact topological space]], the operation of [[pullback]] of the [[universal principal bundle]] $E O(n) \to B O(n)$ from def. \ref{EOn} along [[continuous functions]] $f \colon X \to B O(n)$ establishes a [[bijection]]

$$
  [X, B O(n)]
   \underoverset{iso}{f \mapsto f^\ast E O(n)}{\longrightarrow}
  O(n) Bund/_\sim
$$

between [[homotopy classes]] of functions from $X$ to $B O(n)$ and isomorphism classes of $O(n)$-[[principal bundles]] on $X$.

=--

A full proof is spelled out in [Hatcher, section 1.2, theorem 1.16](#Hatcher).

### For symmetric principal bundles

* the _unordered_ [[Fadell's configuration space]] of $n$ points in $\mathbb{R}^\infty$ is a model for the classifying space $B \Sigma(n)$ of the [[symmetric group]] $\Sigma(n)$;

* the _ordered_ configuration space of $n$ points, equipped with the canonical $\Sigma(n)$-[[action]], is a model for the $\Sigma(n)$-[[universal principal bundle]].

### For crossed complexes

We discuss here classifying spaces of [[crossed complex]]es.

The notion of classifying space should be regarded in general terms as giving a functor

$$  \mathcal{B} :(algebraic data) \to (topological data).$$

Composition with a forgetful functor $U: (topological data) \to (topological spaces) $ gives a [[classifying space]]. In such cases one would also like a homotopically defined functor

$$ \Xi: (topological data) \to (algebraic data) $$

such that

1. $\Xi \circ \mathcal{B}$ is equivalent to the identity;

1. $\Xi$ preserves certain colimits (Generalised [[van Kampen theorem]]) allowing some calculation;

1. there are notions of homotopy for both types of data leading to a bijection of homotopy classes for some $X$

$$[X,U\mathcal{B}C] \cong [\Xi X_*, C].$$

This happens for the algebraic data of crossed complexes and the topological data of filtered spaces, when $X$ is a CW-complex, and $\Xi$ is the fundamental crossed complex of a filtered space. Thus in this case the classifying space does classify homotopy classes of maps, and more work is needed to sort out the data over $X$ which this classifies (gerbes?).

However $\mathcal{B}C$ is in this case defined by a nerve construction which generalises that for groupoids, and can also be applied to topological crossed crossed complexes, giving a simplicial space.

+--{: .query}
[[Mike Shulman|Mike]]: I don't really get any intuition from that.  There might be lots of functors from "algebraic data" to "topological data" but it seems to me that only particular sorts of them deserve the name "classifying space."  Can you say more specifically what sorts of functors you have in mind, and relate it to the more basic ideas that I am familiar with?  What do these classifying spaces classify?

[[Ronnie Brown|Ronnie]] What I am trying to characterise is that higher categories carry structure such as a filtration by lower dimensional higher categories, or, for multiple structures, a multiple filtration. Thus one expects a classifying space to inherit this extra structure. Conversely, the construction of an infinity-groupoid from a space might depend on this extra structure.

So I spent 9 years trying to construct a _strict_ homotopy double groupoid of a space, yet Philip Higgins and I did this overnight in 1974 when we tried the simplest relative example we could think of: take homotopy classes of maps from a square to $X$ which take the edges to a subspace $X_1$ and the vertices to a base point $x_0$. Then the filtered case took another 4 years or so to complete.

Then Loday constructed a [[cat-n-group]] from an n-cube of spaces, published in 1982. Its multi-nerve  is an $(n+1)$-simplicial set, whose realisation is $(n+1)$-filtered.

A strict homotopy double groupoid of a Hausdorff space has been constructed but this needs a subtle notion of [[thin element|thin]] homotopy.

Of course the filtration for a group is not so apparent, but it is more clear that  [[groupoids]]  carry structure in dimension 0 and 1, and hence are useful for representing non connected homotopy 1-types, and their identifications in dimension 0, as explained in the first edition (1968) of my Topology book.

The intuition for the [[higher homotopy van Kampen theorem]] is that you need _structure in all  dimensions from 0 to n _to get colimit theorems in dimension n, because in homotopy, low dimensional identifications, even in dimension 0,  usually effect high dimensional homotopy information. In effect, the [[higher homotopy van Kampen theorem]] is about gluing [[homotopy n-types]].

_Mike_: Thanks, that is helpful.
=--

Some such constructions arise from generalisations of the [[Dold-Kan correspondence]], with values in [[simplicial sets]]. For example, from a [[crossed complex]] $C$ one obtains a simplicial  set $Nerve(C)$ which in dimension $n$ is $Crs(\Pi(\Delta^n_*),C)$. The geometric realisation $\mathcal{B}C$ of this is canonically filtered by the skeleta of $C$, so $\mathcal{B}$ is really a functor to [[filtered spaces]]. This ties in with the functor $\Pi$ which goes in the opposite direction. But note that there is a different filtration of the space $\mathcal{B}C$ since it is a CW-complex, and so $\Pi$ of this filtration gives a free crossed complex.

Special cases of crossed complexes are [[groupoids]], and so we get the classifying space of a groupoid; and similarly of a [[crossed module]].

A crossed module is equivalent to a category object in groups, and so a nerve of this can be constructed as a bisimplicial set. The geometric realisation of this is naturally bifiltered, in several ways!

In considering what is desirable for a [[fundamental infinity-groupoid]] one should bring the notion of classifying space, and its inherited structure, into account.

### For simplicial groups
 {#ForSimplicialGroups}

The [[simplicial classifying space]] $\overline{W}(-)$-construction (see [[simplicial group]] and [[groupoid object in an (∞,1)-category]]) which gives the classifying space functor for [[simplicial group]]s and simplicially enriched groupoids is given in the entry on [[simplicial group|simplicial groups]].  It provides a good example of the above as the W-bar functor is right adjoint to the [[Dwyer-Kan loop groupoid]] functor and induces an equivalence of homotopy categories between that of simplicial sets and that of simplicially enriched groupoids.
The simplicial sets here are playing the role of 'topological data'.

## Properties

### For classical Lie groups

Let $O(n)$ be the [[orthogonal group]] and $U(n)$ the [[unitary group]] in real/complex [[dimension]] $n$, respectively.

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

The real [[Grassmannians]] $Gr_n(\mathbb{R}^k)$ and the complex Grassmannians $Gr_n(\mathbb{C}^k)$ admit the structure of [[CW-complexes]]. Moreover the canonical inclusions

$$
  Gr_n(\mathbb{R}^k)
  \hookrightarrow
  Gr_n(\mathbb{R}^{k+1})
$$

are subcomplex incusion (hence [[relative cell complex]] inclusions).

Accordingly there is an induced CW-complex structure on the classifying space

$$
  B O(n) \simeq \underset{\longrightarrow}{\lim}_k Gr_n(\mathbb{R}^k)
  \,.
$$
=--

A proof is spelled out in ([Hatcher, section 1.2 (pages 31-34)](#Hatcher)).

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

The classifying spaces $B O(n)$ are [[paracompact spaces]].
=--

An early source of this statement is ([Cartan-Schwartz 63, exposé 5](#CartanSchwartz63)). It follows for instance by prop. \ref{CWComplexStructure} the fact that every [[CW-complex]] is paracompact.

### Cohomology

[[!include Segal completion -- table]]

## Moduli spaces

The notion of **[[moduli space]]** is closely related to that of classifying space, but has some subtle differences. See there for more on this.

## Related concepts

* **classifying space**, [[classifying stack]], [[moduli space]], [[moduli stack]], [[derived moduli space]], [[Kan-Thurston theorem]]

  * [[Milnor classifying space]]

  * [[numerable bundle]]

  * [[acyclic group]]

  * [[Sullivan model of classifying space]]

  * [[free loop space of classifying space]]

* [[classifying topos]]

* [[universal principal bundle]], [[universal principal ∞-bundle]]

* [[subobject classifier]]

* [[classifying morphism]]

## References

### General

Original discussion of classifying spaces in topological homotopy theory:

* [[John Milnor]], _Construction of Universal Bundles, II_, Annals of Mathematics Second Series, Vol. 63, No. 3 (May, 1956), pp. 430-436 ([jstor:1970012](https://www.jstor.org/stable/1970012))

* [[Richard Palais]], Section 2.6 of: *The classification of $G$-spaces*, Memoirs of the AMS **36**, 1960 ([ISBN:978-0-8218-9979-3](https://bookstore.ams.org/memo-1-36) [pdf](http://vmm.math.uci.edu/PalaisPapers/ClassificationOfG-Spaces.pdf), [[Palais_ClassificationOfGSpaces.pdf:file]])

* {#Milgram67} [[R. James Milgram]], *The bar construction and abelian H-spaces*, Illinois J. Math. 11(2): 242-250 (1967) ([doi:10.1215/ijm/1256054662](https://projecteuclid.org/journals/illinois-journal-of-mathematics/volume-11/issue-2/The-bar-construction-and-abelian-H-spaces/10.1215/ijm/1256054662.full))

* [[Michael C. McCord]], *Classifying Spaces and Infinite Symmetric Products*, Transactions of the American Mathematical Society, Vol. 146 (Dec., 1969), pp. 273-298  ([jstor:1995173](https://www.jstor.org/stable/1995173), [pdf](https://www.ams.org/journals/tran/1969-146-00/S0002-9947-1969-0251719-4/S0002-9947-1969-0251719-4.pdf))

  > (using -- and introducing -- [[compactly generated weakly Hausdorff topological spaces]])

* {#MacLane} [[Saunders MacLane]], Section 6 of: _The Milgram bar construction as a tensor product of functors_,  In: F.P. Peterson  (eds.) *The Steenrod Algebra and Its Applications: A Conference to Celebrate N.E. Steenrod's Sixtieth Birthday* Lecture Notes in Mathematics *168*,  Springer, 1970 ([doi:10.1007/BFb0058523](https://doi.org/10.1007/BFb0058523), [pdf](https://link.springer.com/content/pdf/10.1007/BFb0058523.pdf))

  > (understanding  [[geometric realization of simplicial topological spaces]] via a [[coend]]-formula)

* [[Graeme Segal]], _Classifying spaces and spectral sequences_,  Publications Mathématiques de l'IHÉS, Volume 34  (1968), p. 105-112  ([numdam:PMIHES_1968__34__105_0](http://www.numdam.org/item/PMIHES_1968__34__105_0/))

* [[Norman Steenrod]], _Milgram's classifying space of a topological group_, Topology Volume 7, Issue 4, November 1968, Pages 349-368 (doi:10.1016/0040-9383(68)90012-8)

* [[Jim Stasheff]], _H-spaces and classifying spaces: foundations and recent developments_, Algebraic topology (Proc. Sympos. Pure Math., Vol. XXII, Univ. Wisconsin, Madison, Wis., 1970), Providence, R.I.: American Mathematical Society (1971) pp. 247–272

* {#tomDieck74} [[Tammo tom Dieck]], *On the homotopy type of classifying spaces*, Manuscripta Math 11, 41–49 (1974) ([doi:10.1007/BF01189090](https://doi.org/10.1007/BF01189090))

  > (relating the plain and fat [[geometric realization of simplicial topological spaces]] appearing in Milnor's and Milgram's constructions)

Review:

* {#Dwyer2008} [[William Dwyer]], _Homotopy theory of classifying spaces_, Lecture notes, Copenhagen 2008, ([pdf](http://www.math.ku.dk/~jg/homotopical2008/Dwyer.CopenhagenNotes.pdf), [[Dwyer_HomotopyTheoryOfClassifyingSpaces.pdf:file]])

* {#Mitchell11} [[Stephen Mitchell]], _Notes on principal bundles and classifying spaces_, Lecture Notes. University of Washington, 2011 ([pdf](https://sites.math.washington.edu/~mitchell/Notes/prin.pdf), [[MitchellPrincipalBundles.pdf:file]])

Textbook accounts:

* [[Dale Husemoeller]], Section 4.12, 4.13 of: _Fibre bundles_, McGraw-Hill 1966 (300 p.); Springer Graduate Texts in Math. __20__, 2nd ed. 1975 (327 p.), 3rd. ed. 1994 (353 p.) ([gBooks](http://books.google.hr/books?id=DPr_BSH89cAC), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/husemoller))

* {#Steenrod51} [[Norman Steenrod]], section II.19 of: _The topology of fibre bundles_, Princeton Mathematical Series 14, Princeton Univ. Press, 1951 ([jstor:j.ctt1bpm9t5](https://www.jstor.org/stable/j.ctt1bpm9t5))

* {#Kochmann96} [[Stanley Kochmann]], section 1.3 of of _[[Bordism, Stable Homotopy and Adams Spectral Sequences]]_, AMS 1996

* {#May} [[Peter May]], Chapter 23 of: _A concise course of algebraic topology_ ([pdf](http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf))

* {#Hatcher} [[Allen Hatcher]], section 1.2 of _Vector bundles and K-theory_ ([web](https://www.math.cornell.edu/~hatcher/VBKT/VBpage.html))

* [[Dale Husemoeller]], [[Michael Joachim]], [[Branislav Jurco]], [[Martin Schottenloher]], _[[Basic Bundle Theory and K-Cohomology Invariants]]_,
Lecture Notes in Physics, Springer 2008 ([pdf](http://www.mathematik.uni-muenchen.de/~schotten/Texte/978-3-540-74955-4_Book_LNP726corr1.pdf))

* {#RudolfSchmidt17} [[Gerd Rudolph]], [[Matthias Schmidt]], Thm. 3.5.1 of: *Differential Geometry and Mathematical Physics Part II. Fibre Bundles, Topology and Gauge Fields*, Springer 2017 ([doi:10.1007/978-94-024-0959-8](https://link.springer.com/book/10.1007/978-94-024-0959-8))

* [[Loring Tu]], Section I.5 in: *Introductory Lectures on Equivariant Cohomology*, Annals of Mathematics Studies **204**, AMS 2020 ([ISBN:9780691191744](https://press.princeton.edu/books/hardcover/9780691191744/introductory-lectures-on-equivariant-cohomology))

* [[Dai Tamaki]], *Fiber Bundles and Homotopy*, World Scientific 2021 ([doi:10.1142/12308](https://doi.org/10.1142/12308))

  > (in the context of [[homotopy theory]])

Discussion in [[simplicial homotopy theory]]:

* [[Paul Goerss]], [[Rick Jardine]], Section V.3 of: _[[Simplicial homotopy theory]]_, Progress in Mathematics, Birkhäuser (1996) ([doi:10.1007/978-3-0346-0189-4)]([doi:10.1007/978-3-0346-0189-4](https://link.springer.com/book/10.1007/978-3-0346-0189-4)))

* [[David Roberts]], [[Danny Stevenson]], _Simplicial principal bundles in parametrized spaces_, New York Journal of Mathematics Volume 22 (2016) 405-440 ([arXiv:1203.2460](https://arxiv.org/abs/1203.2460))

* [[Danny Stevenson]], _Classifying theory for simplicial parametrized groups_ ([arXiv:1203.2461](https://arxiv.org/abs/1203.2461))

* [[Thomas Nikolaus]], [[Urs Schreiber]], [[Danny Stevenson]], _[[schreiber:Principal ∞-bundles -- theory, presentations and applications|Principal ∞-bundles -- Presentations]]_, Journal of Homotopy and Related Structures, Volume 10, Issue 3 (2015), pages 565-622 ([doi:10.1007/s40062-014-0077-4](http://link.springer.com/article/10.1007/s40062-014-0077-4), [arXiv:1207.0249](http://arxiv.org/abs/1207.0249))

Discussion in [[topos theory]] relating to [[classifying toposes]]:

* [[Ieke Moerdijk]], _Classifying spaces and classifying topoi_, Lecture Notes in Math. 1616, Springer-Verlag, New York, 1995.

Discussion of characterization of principal bundles by rational [[universal characteristic classes]] and torsion information is in the appendices of

* Igor Belegradek, Vitali Kapovitch, _Obstructions to nonnegative curvature and rational homotopy theory_ ([arXiv:math/0007007](http://arxiv.org/abs/math/0007007))

* Igor Belegradek, _Pinching, Pontrjagin classes, and negatively curved vector bundles_ ([arXiv:math/0001132](http://arxiv.org/abs/math/0001132))

Discussion of classifying spaces in the context of [[measure theory]] is in

* {#Marin17} Ivan Marin, _Measure theory and classifying spaces_ ([arXiv:1702.01889](https://arxiv.org/abs/1702.01889))

### For equivariant bundles
 {#ReferencesForEquivariantBundles}

Discussion of classifying [[G-spaces]] for $G$-[[equivariant principal bundles]]:

* {#tomDieck69} [[Tammo tom Dieck]], _Faserbündel mit Gruppenoperation_, Arch. Math 20, 136–143 (1969) ([doi:10.1007/BF01899003](https://doi.org/10.1007/BF01899003))

* {#Lashof82} [[Richard Lashof]], _Equivariant bundles_, Illinois J. Math. 26(2): 257-271, 1982 ([doi:10.1215/ijm/1256046796.full](https://projecteuclid.org/journals/illinois-journal-of-mathematics/volume-26/issue-2/Equivariant-bundles/10.1215/ijm/1256046796.full))

* {#May90} [[Peter May]], _Some remarks on equivariant bundles and classifying spaces_, Théorie de l'homotopie, Astérisque, no. 191 (1990), 15 p. ([numdam:AST_1990__191__239_0](http://www.numdam.org/item/?id=AST_1990__191__239_0))

* {#MurayamaShimakawa95} [[Mitutaka Murayama]], [[Kazuhisa Shimakawa]], _Universal equivariant bundles_, Proc. Amer. Math. Soc. 123 (1995), 1289-1295 ([doi:10.1090/S0002-9939-1995-1231040-9](https://doi.org/10.1090/S0002-9939-1995-1231040-9))

* {#UribeLueck14} [[Bernardo Uribe]], [[Wolfgang Lück]], _Equivariant principal bundles and their classifying spaces_, Algebr. Geom. Topol. 14 (2014) 1925-1995 ([arXiv:1304.4862](https://arxiv.org/abs/1304.4862), [doi:10.2140/agt.2014.14.1925](http://dx.doi.org/10.2140/agt.2014.14.1925))

* {#GuillouMayMerling17} [[Bertrand Guillou]], [[Peter May]], [[Mona Merling]] _Categorical models for equivariant classifying spaces_, Algebr. Geom. Topol. 17 (2017) 2565-2602 ([arXiv:1201.5178](https://arxiv.org/abs/1201.5178), [doi:10.2140/agt.2017.17.2565](https://doi.org/10.2140/agt.2017.17.2565))

[[!redirects classifying spaces]]