geometric quantization of the 2-sphere

> beware, some prefactors may still need harmonizing...

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Spheres
+--{: .hide}
[[!include spheres -- contents]]
=--
#### Geometric quantization
+--{: .hide}
[[!include geometric quantization - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The 2-[[sphere]] $S^2$ is naturally equipped with the structure of a [[symplectic manifold]] with symplectic form its canonical [[volume form]], up to a constant factor. As such, one can consider its [[geometric quantization]].

From the point of view of [[physics]], this yields a [[quantum mechanical system]] that describes a rigid [[rotor]] or [[spinor]] in 3-dimenensional [[Euclidean space]] (and just the rotation/spin degrees of freedom, not the translational degrees of freedom). Accordingly, the coresponding [[Hilbert space|Hilbert]]-[[space of quantum states]] carries a natural [[Hamiltonian action]] by the [[angular momentum]] [[quantum operators]] that generate the rotation of the sphere.

More abstractly in [[mathematics]], the 2-sphere is naturally identified with a [[coadjoint orbit]] of the [[special unitary Lie algebra]]/[[special unitary group]] $\mathfrak{su}(2)/SU(2)$ (and hence of the [[spin group]] $Spin(3)$, by one of the [exceptional isomorphisms](spin+group#ExceptionalIsomorphisms)). As such the [[geometric quantization]] of the 2-sphere is a special case of the general [[orbit method]] for producing [[irreducible representations]] of suitable [[Lie groups]] by [[geometric quantization]]. In this case it identifies the quantization of the [[Hamiltonian action]] of $SU(2)$ on $S^2$ in terms of [[quantum operators]] acting on the [[space of quantum states]] with an [[irreducible representation]] of $SU(2) \simeq Spin(3)$.

## Details

The following walks through all the ingredients and steps of the geometric quantization of the 2-spere. Several of the ingredients/steps described are not specific to the 2-sphere but apply generally in [[geometric quantization]], or more specifically in [[Kähler polarization]] quantization, but are spelled out again for completeness. Similarly, much of the discussion is really about the [[complex geometry]] of the [[Riemann sphere]].

### The symplectic geometry

We introduce first the 2-sphere with its canonical structure of a [[symplectic manifold]] and [[Kähler manifold]].

#### Coordinate charts

For all of the following it is useful to choose an adapted [[atlas]] of [[coordinate charts]] on the 2-sphere, namely that induced by regarding it as the [[Riemann sphere]].

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

Choose any point $x_{\infty} \in S^2$. Then the complement $S^2 - \{x_\infty\}$ is [[diffeomorphism|diffeomorphic]] to the [[plane]] $\mathbb{R}^2$, which in turn we regard as the [[complex plane]]. Hence

$$
  \mathbb{C} \simeq U_{+} \coloneqq (S^2 - \{x_\infty\}) \hookrightarrow S^2
$$

is one [[coordinate chart]] on the 2-sphere. Accordingly, removing the antipodal point

$$
  x_0 \coloneqq 0 \in \mathbb{C} \simeq U_+
$$

yields another [[coordinate chart]]

$$
  \mathbb{C} \simeq U_{-} \coloneqq (S^2 - \{x_0\}) \hookrightarrow S^2
  \,.
$$

We denote the canonical complex coordinate function on $U_{+/-}$ by $z_{+/-}$, respectively. In terms of these the canonical identification of the two coordinate charts on their common overlap

$$
  (\mathbb{C} - \{0\})
  \simeq
  S^2 - \{x_0, x_\infty\} \simeq (U_+ - \{x_0\})
  \to
  (U_- - \{x_\infty\})
  \simeq
  (S^2 - \{x_0, x_\infty\})
  \simeq
  (\mathbb{C} - \{0\})
$$

is given by

$$
  z \mapsto z^{-1}
  \,.
$$

Hence on $S^2 - \{x_0, x_\infty\}$ the two coordinate functions are complex inverses of each other

$$
  z_+ = (z_-)^{-1}
  \,.
$$

This defines an [[atlas]] by two open charts which defines the 2-[[sphere]] as a [[smooth manifold]]. Since both patches are complex planes and the transition function is a [[holomorphic function]] this moreover gives $S^2$ the structure of a [[complex manifold]]. As such it is called the _[[Riemann sphere]]_.

=--

In the following we often just write "$z$" for "$z_+$" or "$z_-$" when the chart $U_+$ or $U_-$ is understood.

By the canonical embedding

$$
  S^2 \hookrightarrow \mathbb{R}^3
$$

into the 3-[[dimension|dimensional]] [[Cartesian space]] as the unit 2-sphere given by

$$
  S^2 = \left\{ (x_1, x_2, x_3) \in \mathbb{R}^3 \;|\; (x_1)^2 + (x_2)^2 + (x_3)^2 = 1 \right\}
$$

the 2-[[sphere]] inherits a [[Riemannian metric]] as the [[pullback of a differential form|pullback]] of the canonical metric $\sum_{i = 1} ^3  d x_i \otimes d x_i$  on $\mathbb{R}^3$.

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

In the [[coordinate charts]] of def. \ref{RiemannSphere}, the canonical functions

$$
  x_i \colon (S^2 - \{x_\infty\}) \hookrightarrow \mathbb{R}^3 \stackrel{x_i}{\to} \mathbb{R}
$$

have the expressions

* $x_1 = \frac{1}{2} \frac{z_+ + \overline{z}_+}{1 - {\vert z_+\vert}^2}$

* $x_2 = -\frac{i}{2} \frac{z_+ - \overline{z}_+}{1 - {\vert z_+\vert}^2}$

* $x_3 = 1 - \frac{1}{1 + {\vert z_+\vert}^2}$.

=--

#### Symplectic and Kähler structure

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

The canonical [[Riemannian metric]] on $S^2$ induces the [[volume form]] $\omega_1 \in \Omega^2(S^2)$ given in the [[coordinate charts]] of def. \ref{RiemannSphere} by

$$
  \omega_1
  =
  \frac{1}{(1 + {\vert z\vert}^2)^2}
  d z \wedge d \overline{z}
  \,.
$$

More generally, for $S^2 \hookrightarrow \mathbb{R}^3$ embedded as the 2-sphere of [[radius]] $k \in (0,\infty)$, the corresponding volume form is

$$
  \omega_k
  =
  \frac{k}{(1 + {\vert z\vert}^2)^2}
  d z \wedge d \overline{z}
  \,.
$$

=--

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

Equipped with this [[Riemannian metric]] for $k \in (0,\infty)$ and with the [[complex manifold]] structure of def. \ref{RiemannSphere} the 2-sphere is a [[Kähler manifold]] with

* [[Kähler potential]] on $U_\pm$ given by

  $$
    K_k(z_\pm,\overline{z}_\pm) = - k ln(1 + {\vert z_\pm^2\vert})
    \,.
  $$

* [[Kähler form]] on $U_\pm$ given by

  $$
   \begin{aligned}\omega_k & = \partial \overline{\partial} K_k \\  & = \frac{k}{(1 + {\vert z\vert}^2)^2}
  d z_{} \wedge d \overline{z} \end{aligned}
  \,.
  $$

=--

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

The above [[Kähler form]] indeed has the same coordinate expression on both patches $U_\pm$: on $U_+ \cap U_- \simeq (S^2 - \{x_0, x_\infty\})$ we have

$$
  \begin{aligned}
    \frac{k}{(1 + {\vert z_+\vert}^2)^2}
  d z_{+} \wedge d \overline{z}_+
    & =
    \frac{k}{(1 + {\vert z_-\vert}^{-2})^2}
  (-z_-^{-2} d z_{-}) \wedge (-\overline{z}_-^{-2} d \overline{z}_-)
    \\
    & =
    \frac{k}{{\vert z_-\vert^4}(1 + {\vert z_-\vert}^{-2})^2}
   d z_{-} \wedge d \overline{z}_-
    \\
    & =
    \frac{k}{(1 + {\vert z_-\vert}^{2})^2}
   d z_{-} \wedge d \overline{z}_-
  \end{aligned}
 \,.
$$

=--

In the following we leave the subscript $(-)_k$ implicit and write just "$\omega$" and "$K$" etc, the dependency on the parameter $k$ being understood.

### The prequantum geometry

We first discuss the [[complex line bundles]] on the 2-sphere

* _[Without connection](#PrequantumBundleWithoutConnection)_

and then consider genuine [[prequantum line bundles]]

* _[With connection](#PrequantumConnection)_ .

#### Prequantum bundle without connection
 {#PrequantumBundleWithoutConnection}

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

Hermitian [[complex line bundles]] are classified via their [[first Chern class]] (just as [[circle group]]-[[principal bundles]]) by second [[ordinary cohomology]] with [[integer]] [[coefficients]], which for the 2-sphere is

$$
  H^2(S^2, \mathbb{Z}) \simeq \mathbb{Z}
  \,.
$$

By the [[clutching construction]] a line bundle given by two trivializing sections $\sigma_\pm$ on $U_{\pm}$, respectively, of the trivial line bundle on the coordinate patch $\mathbb{C}$ has class the [[winding number]] of the transition function

$$
  S^1 \hookrightarrow \mathbb{C} - \{0\}
  \stackrel{s_2 s_1^{-1}}{\to} \mathbb{C}^\times
  \,.
$$

=--

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

For $k \in \mathbb{Z} \simeq H^2(S^2, \mathbb{Z})$ we take the standard [[Cech cohomology|Cech cocycle]] representing this class to be that given on $U_+ \cap U_- \simeq S^2 - \{x_0, x_\infty\}$ by the transition function

$$
  f(z_+) \coloneqq z_+^{-k} = z_-^{k}
  \,.
$$

=--

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

By the discussion there, a [[spin structure]] on $S^2$ is equivalently a [[square root]] $\sqrt{\Omega^{1,0}} \in \mathbf{Line}_{\mathbb{C}}(S^2)$ of the [[canonical line bundle]] $\Omega^{1,0}$, which here is simply the holomorphic 1-form bundle.

=--

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

The [[first Chern class]] of the [[canonical line bundle]] on the 2-sphere is

$$
  c_1\left(\Omega^{1,0}\left(S^2\right)\right)
  =
  2
  \in
  \mathbb{Z} \simeq H^2(S^2, \mathbb{Z})
$$

(up to a unique choice of sign, which determines the [[isomorphism]] on the right). Accordingly the 2-sphere has an essentially unique [[spin structure]] whose [[theta characteristic]] $\sqrt{\Omega^{1,0}}$ is the complex line bundle with unit [[first Chern class]]

$$
  c_1\left(\sqrt{\Omega^{1,0}\left(S^2\right)}\right)
  =
  1
  \in
  \mathbb{Z} \simeq H^2(S^2, \mathbb{Z})
  \,,
$$

the [[basic line bundle on the 2-sphere]].

=--

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

The canonical section of the holomorphic 1-form bundle on $\mathbb{C}$ is simply the canonical 1-form $d z$ itself. By the [[coordinate charts]] of def. \ref{RiemannSphere} we have on $\mathbb{C} - \{0\}$

$$
  d z_- = d (z_+^{-1}) = - z_+^{-2} d z_+
$$

and so the transition function of the canonical bundle in this local trivialization is

$$
  - z^{-2} \colon (\mathbb{C}-\{0\}) \to \mathbb{C}^\times
$$

as in def. \ref{StandardCechCocycle}.

This has [[winding number]] $2$. Therefore the [[first Chern class]] of the holomorphic 1-form bundle $\Omega^{1,0}$ is $c_1(\Omega^{1,0}) =  \pm 2$ (the sign being an arbitrary convention, determined by the identification $H^2(S^2, \mathbb{Z}) \simeq \mathbb{Z}$).

And so it follows that there is a unique spin structure, namely given by choosing $\sqrt{\Omega^{1,0}}$ to be the line bundle on $S^2$ with first Chern class $\pm 1$.

=--

#### Prequantum bundle with connection
 {#PrequantumConnection}

We discuss equipping the above complex line bundles on $S^2$ with [[connection on a bundle|connections]].

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

For $k \in  H^2(S^2, \mathbb{Z}) \simeq \mathbb{Z} \hookrightarrow \mathbb{R}$,
refine the [[Cech cohomology|Cech cocycle]] of def. \ref{StandardCechCocycle} to a Cech-[[Deligne cohomology|Deligne cocycle]] with [[curvature]] [[differential 2-form]] the [[Kähler form]] of prop. \ref{KaehlerStructure}

$$
  \omega = \partial \overline{\partial} K
$$

by taking the connection 1-form on $U_{\pm}$ to be

$$
  \begin{aligned}
    \theta
      & \coloneqq
    \exp\left(-K\right)
     {\partial}
    \exp\left(K\right)
    \\
    & =
    {\partial} K
    \\
    & = k \frac{\overline{z} }{ 1 + {\vert z\vert}^2} d {z}
  \end{aligned}
  \,.
$$

=--

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

The Cech-Deligne cocycle in def. \ref{DeligneCocycle} is indeed well defined.

=--

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

We need to check that the connection 1-form satisfies on $U_+ \cap U_- \simeq (S^2 - \{x_0, x_\infty\})$ the transition law

$$
  \theta_+ \;+\; z^{-k} d z^k \;=\; \theta_-
  \,.
$$

To see this first notice that on $U_+ \cap U_- \simeq (\mathbb{C} - \{0\})$

$$
  \begin{aligned}
   \theta_+ & = k\frac{\overline{z}_-^{-1}}{1 + {\vert z_-\vert^{-2}}} (-){z}_-^{-2} d {z}_-
   \\
   & =
   -k \frac{{z}_-^{-1} d{z}_-}{1 + {\vert z_-\vert^2}}
   \\
   & =
   -\frac{{z}_-^{-k} d {z}_-^k}{1  + {\vert z_-\vert}^2}
  \end{aligned}
  \,.
$$

With this it follows that on $U_+ \cap U_-$ indeed we have

$$
  \begin{aligned}
    z_-^{-k} d z_-^k + \theta_+
    & =
    k
    \left(
      z_-^{-1} -\frac{z_-^{-1}}{1 + {\vert z_-\vert}^2}
    \right) d z_-
    \\
    & =
    k \frac{\overline{z}_-}{1 + {\vert z_-\vert}^2}
    d z_-
    \\
    & = \theta_-
  \end{aligned}
  \,.
$$

=--

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

Compare this invariance of the form of the connection to the invariance of the form of its [[curvature]] in remark \ref{KaehlerFormIsInvariant} above.

=--

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

For $\epsilon \in (0,1)$ let $D_{(\epsilon, \infty)}$ be the [[disk]] of [[radius]] $\epsilon$ around $x_\infty \in S^2$. Then by the [[Stokes theorem]] with def. \ref{DeligneCocycle} we have
the [[integral]] equality

$$
  \underset{S^2 - D_{(\epsilon,\infty)}}{\int} \omega
  \;=\;
  \underset{\partial D_{(\epsilon, \infty)}}{\int} \theta
  \;=\;
  \frac{k}{1 + \epsilon^2}
  \,.
$$

In the [[limit of a sequence|limit]] that $\epsilon \to 0$ this shows that for integral $k$ as above, $\omega$ has integral [[periods]]

$$
  \begin{aligned}
  \int_{S^2} \omega
  =
  \underset{\epsilon \to 0}{\lim}
  \underset{S^2 - D_{(\epsilon,\infty)}}{\int} \omega
  =
  \underset{\epsilon \to 0}{\lim}
  \frac{k}{1 + \epsilon^2}
  =
  k
  \in \mathbb{Z}
  \end{aligned}
  \,.
$$

This is essentially the relation that historically was first recognized by [[Paul Dirac]] in the context of what today is known as the _[[Dirac charge quantization]]_ argument. In more modern language this expresses the defining [[homotopy pullback]] property of [[ordinary differential cohomology]].

=--

### The space of quantum states
 {#TheSpaceOfQuantumStates}

#### Kähler polarization

The [[Kähler manifold]] structure on $S^2$ of prop. \ref{KaehlerStructure} which lifts its [[symplectic structure]] given by the [[volume form]] canonically induces a [[polarization]] of the symplectic structure: the corresponding [[Kähler polarization]].

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

Given a [[prequantum line bundle]] $(L,\nabla)$ for $(S^2, \omega)$, a [[section]] $\Psi \in \Gamma(L)$ is polarized if for all $v \in \Gamma(T^{0,1}S^2)$ the [[covariant derivative]] of $\psi$ along $v$ vanishes

$$
  \nabla_v \Psi = 0
  \,.
$$

=--

By prop. \ref{DeligneCocycle} the connection $\nabla$ is what is called "well adapted" in that in the complex coordinate patch $U_\pm \simeq \mathbb{C}$ the covariant derivative along anti-holomorphic vector fields is given simply by the ordinary anti-holomorphic derivative:

$$
  \nabla_{\partial/\partial \overline{z}} \Psi =
  \frac{\partial}{\partial \overline{z}} \Psi
  \,.
$$

This means that:

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

The Kähler-polarized sections are precisely the holomorphic sections.

=--

#### Wavefunctions

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

Given a [[prequantum line bundle]] $(L,\omega)$ prequantizing $(S^2, \omega)$, the _[[space of quantum states]]_ is the [[vector space]] of [[polarization|polarized]] [[sections]] of $L \otimes \sqrt{\Omega^{1,0}}$, the _[[wave functions]]_:

$$
  \mathcal{H} \coloneqq \left\{ \Psi \in \Gamma\left(L\otimes \sqrt{\Omega^{1,0}}\right) | \overline{\partial} \Psi = 0 \right\}
$$

=--

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

Here the [[tensor product]] with $\sqrt{\Omega^{1,0}}$ is the [[metaplectic correction]], remark \ref{SpinStructureByThetaCharacteristic}, which by prop. \ref{CanonicalBundleAndSquareRoot} is the essentially unique line bundle of unit [[first Chern class]].

Since by prop. \ref{CanonicalBundleAndSquareRoot} this simply shifts the class of the prequantum bundle by one, and since in much of the traditional literature it is customary to ignore the metaplectic correction (which in the present case is indeed harmless), we will in the following regard $L \otimes \sqrt{\Omega^{1,0}}$ as the prequantum line bundle, by which we mean that we keep referring by "$k$" to the [[first Chern class]] of that line bundle on $S^2$ of which we consider sections etc. Strictly speaking the prequantum line bundle hence has class $k-1$ from now on, but this shift collides with all established literature and is not worth bothering with.

It is however maybe worth noticing that the shift of the class by +1 induced by the metaplectic correction can be regarded as lifting the possibly non-regular [[coadjoint orbits]] with $k \geq 0$ to the regular ones, with $k \geq 1$.

=--

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

The [[dimension]] of the [[space of quantum states]] on the 2-sphere is

$$
  dim(\mathcal{H})
  =
  \left\{
   \array{
      k+1 & if \, k \geq 0;
      \\
      0 & otherwise
   }
  \right.
  \,,
$$

where $k \coloneqq c_1(L \otimes \sqrt{\Omega^{1,0}})$ is the [[first Chern class]] of the [[prequantum line bundle]] with [[metaplectic correction]].

=--

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

By prop. \ref{KaehlerPolarizedIsHolomorphic} the polarized sections are precisely the holomorphic sections. By the construction in def. \ref{StandardCechCocycle} a holomorphic section of $L \otimes \sqrt{\Omega^{1,0}}$ is equivalently a pair of holomorphic functions $\Psi_\pm$ on $\mathbb{C}$ such that on $\mathbb{C} - \{0\}$ they are related by

$$
  \Psi_-(z_-) = \Psi_+(z_-^{-1}) z_-^{k}
  \,.
$$

A linear [[basis]] for the space of unconstrained holomorphbic functions on $\mathbb{Z}$ is of course given by the monomials $\{z^n\}$. Therefore a [[basis]] for the admissible pairs as above is given by pairs $\Psi_\pm(z_-) = z_-^{n_\pm}$ such that

$$
  z_-^{n_-} = z_-^{-n_+} z_-^{k}
$$

hence such that

$$
  n_+ + n_- = k
  \,.
$$

If $k \geq 0$ then this equation has $(k+1)$ solutions. If $k \lt 0$ it has no solution.

=--

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

For $k = 2$ (by prop. \ref{CanonicalBundleAndSquareRoot} the class of the [[canonical line bundle]]) the [[space of quantum states]] is 3-dimensional and canonically identified with the [[vector space]] underlying the [[special unitary Lie algebra]] $\mathfrak{su}(2)$.

=--

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

For $k = 1$ (by prop. \ref{CanonicalBundleAndSquareRoot} the class of the unique [[theta characteristic]]) it is 2-dimensional, being the space of states of a bare [[spinor]] in 3-dimensional Euclidean space. This is the standard model for a [[qbit]]. Below in example \ref{PauliMatrices} we see how the canonical [[quantum observables]] act on this space of states as the [[Pauli matrices]].

=--

### The quantum observables

#### The Hamiltonian $SU(2)$-action

We discuss the [[Hamiltonian action]] of the [[special unitary group]]/[[spin group]] $SU(2) \simeq Spin(3)$ on $(S^2, \omega)$.

First we state the abstract situation in terms of the [[orbit method]], then we unwind what this implies in detail.

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

The [[Lie algebra]] $\mathfrak{su}(2)$ as a [[matrix Lie algebra]] is the sub Lie algebra on those matrices of the form

$$
  \left(
    \array{
      i z & x + i y
      \\
      - x + i y & - i z
    }
  \right)
  \;\;\;
  with
  \;\;
  x,y,z \in \mathbb{R}
  \,.
$$

=--

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

The standard [[basis]] elements of $\mathfrak{su}(2)$ given by the above presentation are

$$
  \sigma_1
   \coloneqq
  \frac{1}{\sqrt{2}}
  \left(
   \array{
     0 & 1
     \\
     -1 & 0
   }
  \right)
$$

$$
  \sigma_2
   \coloneqq
  \frac{1}{\sqrt{2}}
  \left(
   \array{
     0 & i
     \\
     i & 0
   }
  \right)
$$

$$
  \sigma_3
   \coloneqq
  \frac{1}{\sqrt{2}}
  \left(
   \array{
     i & 0
     \\
     0 & -i
   }
  \right)
  \,.
$$

These are called the _[[Pauli matrices]]_.

=--

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

The [[Pauli matrices]] satisfy the [[commutator]] relations

$$
  [\sigma_1, \sigma_2] = \sigma_3
$$

$$
  [\sigma_2, \sigma_3] = \sigma_1
$$

$$
  [\sigma_3, \sigma_1] = \sigma_2
  \,.
$$

=--

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

The [[maximal torus]] of $SU(2)$ is the [[circle group]] $U(1)$. In the above [[matrix group]] presentation this is naturally identified with the subgroup of matrices of the form

$$
  \left(
    \array{
      t & 0
      \\
      0 & t^{-1}
    }
  \right)
  \;\;
  with
  t \in U(1) \hookrightarrow \mathbb{C}
  \,.
$$

=--

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

The [[coadjoint orbits]] of the [[coadjoint action]] of $SU(2)$ on $\mathfrak{su}(2)$ are equivalent to the subset of the above matrices with $x^2 + y^2 + z^2 = k^2$ for some $k \geq 0$.

These are _regular_ coadjoint orbits for $k \gt 0$.

=--

By the discussion at _[orbit method -- The coadjoint orbit and the flag manifold](orbit%20method#TheCoadjointOrbitAndTheCosetSpaceAndFlagManifold)_ it follows that

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

For $k \gt 0$ the corresponding coadjoint orbit and hence the symplectic 2-sphere is equivalent to the [[quotient]]/[[coset space]]

$$
  SU(2)/U(1) \stackrel{\simeq}{\to} \mathcal{O}_\lambda \simeq S^2
  \,,
$$

where the equivalence is induced by the map

$$
  g U(1) \mapsto Ad_g^\ast\langle \lambda, - \rangle
  \,.
$$

=--

#### The quantum operators

Recall the canonical [[coordinate]] functions $x_1, x_2, x_2 \colon S^2 \to \mathbb{R}$ from prop. \ref{TheThreeCanonicalFunctions}.

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

For $(L,\nabla)$ a [[prequantization]] of $(S,\omega)$ equipped with the above [[Kähler polarization]], the
[[prequantum operators]] associated to the canonical [[coordinate]] functions

$$
  x_1, x_2, x_2 \colon S^2 \to \mathbb{R}
$$

from prop. \ref{TheThreeCanonicalFunctions} are [[quantum operators]] (respect the [[polarization]]) and in the canonical trivializaton of $L$ on $U_-$ their action on sections is that of the following [[vector fields]]:

$$
  Q(x_1 + i x_2) = z_-^2 \frac{\partial}{\partial z_-} - k z_-
$$

$$
  Q(x_1 - i x_2) = - \frac{\partial}{\partial z_-}
$$

$$
  Q(x_3) = z_- \frac{\partial}{\partial z_-} - k/2
  \,.
$$

=--

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

For $k = 1$ these act as the [[Pauli matrices]] on the [[space of quantum states]] $\mathcal{H} \simeq \mathbb{C}^2$.

By the proof of prop. \ref{DimensionOfSpaceOfQuantumStates} the [[space of quantum states]] is spanned by those [[wavefunctions]] which on $U_-$ are of the form

$$
  \vert -\rangle \coloneqq z_-^0
$$

$$
  \vert +\rangle \coloneqq z_-^1
  \,.
$$

These are [[eigenvectors]] for $Q(x_3)$:

$$
  Q(x_3) \vert - \rangle = -\frac{1}{2} \vert - \rangle
$$

$$
  Q(x_3) \vert + \rangle = \frac{1}{2} \vert + \rangle
  \,.
$$

It is clear that $Q(x_1 - i x_2)$ acts as a lowering operator on these states. Notice that also $Q(x_1 + i x_2)$ indeed acts as a raising operator in that for the special value of $k = 1$ we have indeed

$$
  \begin{aligned}
    Q(x_1 + i x_2) \vert + \rangle
    & =
    z_-^2 \frac{\partial z_-}{\partial z_-} + z_- z_-
    \\
    & = 0
  \end{aligned}
  \,.
$$

=--
## Related concepts

* [[quantization of the 2-sphere]]

  * **geometric quantization of the 2-sphere**

  * [[deformation quantization of the 2-sphere]]

## References

For general references see at _[[orbit method]]_ and at _[[geometric quantization]]_.

Reviews with an emphasis on the quantization of the 2-sphere include

section 3.1 of

* [[Alexander Kirillov]], _[[Lectures on the Orbit Method]]_

section 7 of

* J. Maes, _An introduction to the orbit method_, Master thesis (2011) ([pdf](http://web.science.uu.nl/itf/Teaching/2011/JeroenMaes.pdf), [pdf slides](http://www.imus.us.es/FSMYT12/Talk_Jeroen_Maes.pdf), [web](https://dspace.library.uu.nl/handle/1874/205802))

Discussion in terms of [[Dirac induction]] is in

* [[Daniel Freed]], [[Michael Hopkins]], [[Constantin Teleman]], part II, example 1.37 of _[[Loop Groups and Twisted K-Theory]]_
 {#FHT}


geometric realization

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Category theory
+--{: .hide}
[[!include category theory - contents]]
=--
#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

What is commonly called _geometric realization_ (or, less commonly but more accurately: *topological realization*,see Rem. \ref{OnTerminology}) is the operation that builds from a [[simplicial set]] $X$ a [[topological space]] $|X|$ obtained by interpreting each
element in $X_n$ -- each abstract [[n-simplex|$n$-simplex]] in $X$ -- as one copy of the standard topological $n$-simplex $\Delta^n_{Top}$ and then gluing together all these along their [[boundaries]] to a big topological space, using the information encoded in the [[face map|face and degeneracy maps]] of $X$ on how these simplices are supposed to be stuck together. This procedure generalises the geometric realization of [[simplicial complexes]] as described at that entry.

Geometric realization is the special case of the general notion of [[nerve and realization]] that is induced from the standard [[simplicial set|cosimplicial]] [[topological space]] $[n] \mapsto \Delta^n_{Top}$. (N.B.: in this article, $[n]$ denotes the ordinal with $n+1$ elements. The corresponding contravariant representable is denoted $\Delta(-, n)$.)  Analogous constrtuctions yield [[cubical geometric realization]], etc.

In the context of [[homotopy theory]] geometric realization plays a notable role in the [[homotopy hypothesis]], where it is part of the [[Quillen equivalence]] between the [[classical model structure on topological spaces]] and the [[classical model structure on simplicial sets]].

The construction generalizes naturally to a map from [[simplicial object|simplicial]] [[topological spaces]] to plain topological spaces. For more on that see *[[geometric realization of simplicial spaces]]*.

The dual concept is _[[totalization]]_ .

\begin{remark}\label{OnTerminology}
**(alternative terminology: "topological realization")**
\linebreak
While the term *geometric realization* is classical ([Quillen 1968](#Quillen68)) and in common use by [[algebraic topology|algebraic topologists]] (e.g. [Goerss & Jardine 2009, §VII.3](#GoerssJardine09)), it is somewhat inaccurate or at least misleading, in that the bare [[topological spaces]] produced by the realization construction are not yet what [[geometry|geometers]] would commonly regard as reflecting [[geometry|geometric]] [[mathematical structure|structure]]: [[geometry|Geometry]] on a topological space is instead typically taken to be the [[extra structure]] of a [[differentiable manifold]] (e.g. [[smooth structure]]) together with some kind of [[G-structure]] (such as [[Riemannian structure]], [[conformal structure]], etc.). This terminological mismatch is quite stark in the main application to [[homotopy theory]], where one cares only about the ([[weak homotopy type|weak]]) [[homotopy type]] of the realization, which is the most non-geometric aspect of any notion of "[[space]]". This becomes a real issue in contexts of [[differential topology]] and more generally of [[geometric homotopy theory]] (such as for [[differential cohomology theories]]) where realizations of simplicial sets to genuine geometric spaces richer than topological spaces do play a role (e.g. already for the notion of smooth/[[equivariant triangulation|equivariant]] [[triangulations]] of [[smooth manifolds]]).

In conclusion, a more accurate and more descriptive term for the realization operation discussed here would be **topological realization**. References which use this terminology include [Simpson 1996](#Simpson96), [Jardine 2004](simplicial+approximation+theorem#Jardine04), [Lackenby 2008, §I.2](#Lackenby08). In [Fresse 2017](operad#FresseHOGTG) the term "topological realization" is used in the pdf draft but is replaced by "geometric realization" in the published version.
\end{remark}

## Definition

There are various levels of generality in which the notion of (topological) geometric realization makes sense. The basic definition is

* [For cell complexes such as simplicial sets](#OfSimplicialSets).

A generalization of this of central importance is the

* [Geometric realization of simplicial topological spaces](#OfSimplicialTopologicalSpaces)

Up to homotopy, this is a special case of a general notion of

* [Geometric realization of cohesive ∞-groupoids](#OfCohesiveInfinityGroupoids).

At the point-set level, it is also a special case of a general notion of

* [Geometric realization of simplicial objects](#OfSimplicialObjects).

### Of cell complexes such as simplicial sets
 {#OfSimplicialSets}

Let $S$ be one of the categories of [[geometric shapes for higher structures]], such as the [[globe category]] or the [[simplex category]] or the [[cube category]].

There is an obvious functor

 $ st : S \to $ [[Top]]

which sends the standard _cellular_ shape $[n]$ (the standard cellular [[globe]], [[simplex]] or [[cube]], respectively) to the corresponding standard _topological_ shape (for instance  the standard $n$-simplex $ st([n]) := \{ (x_1, \cdots, x_n) | \sum_{i=1}^n x_i = 1, x_i \geq 0 \} \subset \mathbb{R}^{n}$ ) with the obvious induced face and boundary maps.

Using this, in cases where $Top$ can be regarded as [[enriched category|enriched]] over and [[copower|tensored]] over a base category $V$, the **geometric realization** of a [[presheaf]] $K^\bullet : S^{op} \to V$ on $S$ -- e.g., of a [[globular set]], a [[simplicial set]] or a [[cubical set]], respectively  (when $V= Set$) -- is the [[topological space]] given by the [[end|coend]], [[weighted limit|weighted colimit]], or [[tensor product of functors]]

$$
  |K^\bullet| = \int^{[n] \in S} st([n]) \cdot K^n
  \,.
$$

In the case of [[simplicial sets]], see for more discussion also

* [[nerve]], [[homotopy hypothesis]].

Via simplicial [[nerve]] functors geometric realization of simplicial sets induces geometric realizations of many other structures, for instance

* [[geometric realization of categories]]

For the case of [[cubical set|cubical sets]], see [[cubical geometric realisation]].

### Of simplicial topological spaces
 {#OfSimplicialTopologicalSpaces}

See

* [[geometric realization of simplicial topological spaces]]

### Of cohesive $\infty$-groupoids
 {#OfCohesiveInfinityGroupoids}

Every [[cohesive (∞,1)-topos]] $\mathbf{H}$ (in fact every [[locally ∞-connected (∞,1)-topos]]) comes with its intrinsic notion of geometric realization.

The general abstract definition is at [[cohesive (∞,1)-topos]] in the section
Geometric homotopy.

For the choice $\mathbf{H} = $ [[∞Grpd]] this reproduces the [geometric realization of simplicial sets](#OfSimplicialSets), see at [[discrete ∞-groupoid]] the section

For the choice $\mathbf{H} = $  [[ETop∞Grpd]] and [[Smooth∞Grpd]] this reproduces [geometric realization of simplicial topological spaces](#OfSimplicialTopologicalSpaces). See the sections ETop∞Grpd -- Geometric homotopy and Smooth ∞-groupoid --  Geometric homotopy

### Of simplicial objects in a category
 {#OfSimplicialObjects}

Let $M$ be a [[cocomplete category|cocomplete]] [[simplicially enriched category]] with [[copowers]].  A **[[simplicial object]]** in $M$ is a functor $X:\Delta^{op}\to M$, where $\Delta$ is the [[simplex category]].  Its **geometric realization** is defined similarly to the classical case as a [[coend]]:

$$ |X| = \int^{[n]\in\Delta} \Delta[n] \odot X_n $$

where $\odot$ denotes the [[copower]] in $M$.  This operation is a left adjoint which is even a simplicially enriched functor; see [[simplicial object]] for more details.

## Properties ##

In this section we consider topological [geometric realization of simplicial sets](#OfSimplicialSets), which is the best studied and perhaps most significant case.

### Realizations as CW complexes ###

Each ${|X|}$ is a CW complex (see lemma \ref{mono} below), and so geometric realization ${|(-)|}: Set^{\Delta^{op}} \to Top$ takes values in the full subcategory of CW complexes, and therefore in any [[convenient category of topological spaces]], for example in the category $CGHaus$ of compactly generated Hausdorff spaces. Let $Space$ be any convenient category of topological spaces, and let $i \colon Space \to Top$ denote the inclusion.

+-- {: .un_prop}
######Proposition
For any simplicial set $X$, there is a natural isomorphism $i(\int^{n: \Delta} X(n) \cdot \sigma(n)) \cong {|X|}$, where the coend on the left is computed in $Space$.
=--

This is obvious: more generally, if $F: J \to A$ is a diagram and $i: A \hookrightarrow B$ is a full replete subcategory, and if the colimit in $B$ of $i \circ F$ lands in $A$, then this is also the colimit of $F$ in $A$.
(The dual statement also holds, with limits instead of colimits.)

Below, we let $R: Set^{\Delta^{op}} \to Space$ denote the geometric realization when considered as landing in $Space$.

### Theorem: Geometric realization is left exact ###
 {#GeometricRealizationIsLeftExact}

We continue to assume $Space$ is any [[convenient category of topological spaces]]. In this section we prove that geometric realization

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

is a left [[exact functor]] in that it preserves [[finite limits]].

It is important that we use some such "convenience" assumption, because for example

$${|(-)|}: Set^{\Delta^{op}} \to Top,$$

valued in general [[topological spaces]], does not preserve products. (To get a correct statement, one usual procedure is to "kelley-fy" products by applying the coreflection $k \colon Haus \to CGHaus$ to [[Hausdorff space|Hausdorff]] and [[compactly generated topological spaces]]. This gives the correct isomorphism in the case $Space = CGHaus$, where we have that ${|X \times Y|} \cong {|X|} \times_k {|Y|} \coloneqq k({|X|} \times {|Y|})$; the product on the right has been "kelleyfied" to the product appropriate for $CGHaus$.)

We reiterate that $R$ denotes the geometric realization functor considered as valued in a convenient category of spaces, whereas ${|(-)|}$ is geometric realization viewed as taking values in $Top$.

+-- {: .num_theorem #leftexact}
###### Theorem
Let $U = \hom(1, -): Space \to Set$ be the underlying-set functor. Then the composite $U R: Set^{\Delta^{op}} \to Set$ is left exact.
=--

+-- {: .proof}
###### Proof
As described at the nLab article on triangulation [here](triangulation#StandardAffineSimplexFunctor), the composite
$$\Delta \stackrel{\sigma}{\to} Space \stackrel{U}{\to} Set$$
can be described as the functor
$$\Delta \cong FinInt^{op} \hookrightarrow Int^{op} \stackrel{Int(-, I)}{\to} Set$$
where $Int$ is the category of intervals (linearly ordered sets with distinct top and bottom). Because every interval, in particular $I$, is a [[filtered colimit]] of finite intervals, and because finite intervals are finitely presentable intervals, it follows that $U \sigma \colon \Delta \to Set$ is a [[flat functor]] (a [[filtered colimit]] of representables). But on general grounds, [[copower|tensoring]] with a flat functor is left exact, which in this case means
$$U R = - \otimes_\Delta U \sigma: Set^{\Delta^{op}} \to Set$$
is left exact.
=--

Obviously the preceding proof is not sensitive to whether we use $Space$ or $Top$.

#### Geometric realization preserves equalizers

+-- {: .num_lemma #mono}
###### Lemma
If $i: X \to Y$ is a [[monomorphism]] of simplicial sets, then $R(i): R(X) \to R(Y)$ is a [[closed subspace]] inclusion, in fact a [[relative CW-complex]]. In particular, taking $X = \emptyset$, $R(Y)$ is a $CW$-complex.
=--

+-- {: .proof}
###### Proof
Any monomorphism $i \colon X \to Y$ in $Set^{\Delta^{op}}$ can be seen as the result of iteratively adjoining nondegenerate $n$-simplices. In other words, there is a chain of inclusions $X = F(0) \hookrightarrow F(1) \hookrightarrow \ldots Y = colim_i F(i)$, where $F: \kappa \to Set^{\Delta^{op}}$ is a functor from some ordinal $\kappa = \{0 \leq 1\leq \ldots\}$ (as [[preorder]]) that preserves directed colimits, and each inclusion $F(\alpha \leq \alpha + 1): F(\alpha) \to F(\alpha + 1)$ fits into a pushout diagram

$$\array{
\partial \Delta(-, n) & \to & F(\alpha) \\
\mathllap{j} \downarrow & & \downarrow \\
\Delta(-, n) & \to & F(\alpha+1)
}$$

where $j$ is the inclusion. Now $R(j)$ is identifiable as the inclusion $S^{n-1} \to D^n$, and since $R$ preserves pushouts (which are calculated as they are in $Top$), we see by [this lemma](/nlab/show/subspace+topology#pushout) that $R F(\alpha) \to R F(\alpha+1)$ is a closed subspace inclusion and evidently a [[relative CW-complex]]. By [another lemma](/nlab/show/subspace+topology#transfinite), it follows that $X \to Y$ is also a closed inclusion and indeed a relative CW-complex.
=--

+-- {: .num_cor}
###### Corollary
$R: Set^{\Delta^{op}} \to Space$ preserves [[equalizers]].
=--

+-- {: .proof}
###### Proof
The equalizer of a pair of maps in $Top$ is computed as the equalizer on the level of underlying sets, equipped with the subspace topology. So if
$$E \stackrel{i}{\to} X \stackrel{\overset{f}{\longrightarrow}}{\underset{g}{\longrightarrow}} Y$$
is an equalizer diagram in $Set^{\Delta^{op}}$, then ${|i|}$ is the equalizer of the pair ${|f|}$, ${|g|}$, because the underlying function $U({|i|})$ is the equalizer of $U({|f|})$, $U({|g|})$ on the underlying set level by the preceding theorem, and because ${|i|}$ is a (closed) subspace inclusion by lemma \ref{mono}. But this $Top$-equalizer ${{|i|}}: {{|E|}} \to {{|X|}}$ lives in the full subcategory $Space$, and therefore $R(i) = {|i|}$ is the equalizer of the pair $R(f) = {|f|}$, $R(g) = {|g|}$.
=--

As the proof indicates, that realization preserves equalizers is not at all sensitive to whether we use $Top$ or a convenient category of spaces $Space$.

#### Geometric realization preserves finite products

That geometric realization preserves products _is_ sensitive to whether we think of it as valued in $Top$ or in a convenient category $Space$. In particular, the proof uses [[cartesian closed category|cartesian closure]] of $Space$ in an essential way (in the form that [[finite products]] distribute over arbitrary [[colimits]]).

First, an easy result on products of simplices.

+-- {: .num_lemma #product}
###### Lemma
The realization of a product of two representables $\Delta(-, m) \times \Delta(-, n)$ is compact.
=--

+-- {: .proof}
###### Proof
It suffices to observe that $\Delta[m] \times \Delta[n]$ has finitely many non-degenerate simplices. That is clear since non-degenerate $k$-simplices in the nerve of a poset $P$ are exactly injective order preserving maps $[k] \to P$.
=--

+-- {: .num_lemma #canonical}
###### Lemma
The canonical map
$${|\Delta(-, m) \times \Delta(-, n)|} \to {|\Delta(-, m)|} \times {|\Delta(-, n)|}$$
is a homeomorphism.
=--

+-- {: .proof}
###### Proof
The canonical map is continuous, and a bijection at the underlying set level by theorem \ref{leftexact}. The codomain is the compact Hausdorff space $\sigma(m) \times \sigma(n)$, and the domain is compact by Lemma \ref{product}. But a continuous bijection from a compact space to a Hausdorff space is a homeomorphism.
=--

+-- {: .num_remark}
###### Remark
The key properties of $I$ needed for this subsection are (1) the fact it is compact Hausdorff, and (2) the order relation $\leq$ on the interval $I$ defines a closed subset of $I \times I$. These properties ensure that the affine $n$-simplex $\{(x_1, \ldots, x_n) \in I^n: x_1 \leq \ldots \leq x_n\}$ is itself compact Hausdorff, so that the proof of lemma \ref{canonical} goes through. The point is that in place of $I$, we can really use any interval $L$ that satisfies these properties, thus defining an $L$-based geometric realization instead of the standard ($I$-based) geometric realization being developed here.
=--

+-- {: .num_theorem}
###### Theorem
The functor $R: Set^{\Delta^{op}} \to Space$ preserves products.
=--

+-- {: .proof}
###### Proof
The proof is purely formal. Let $X$ and $Y$ be simplicial sets. By the [[co-Yoneda lemma]], we have isomorphisms
$$X \cong \int^m X(m) \cdot \Delta(-, m)  \qquad Y \cong \int^n Y(n) \cdot \Delta(-, n)$$
and so we calculate
$$\array{
R(X \times Y) & \cong & R((\int^m X(m) \cdot \Delta(-, m)) \times (\int^n Y(n) \cdot \Delta(-, n))) \\
 & \cong & R(\int^m \int^n X(m) \cdot Y(n) \cdot (\Delta(-, m) \times \Delta(-, n))) \\
 & \cong & \int^m \int^n X(m) \cdot Y(n) \cdot R(\Delta(-, m) \times \Delta(-, n)) \\
 & \cong & \int^m \int^n X(m) \cdot Y(n) \cdot (R(\Delta(-, m)) \times R(\Delta(-, n)) \\
 & \cong & \int^m \int^n X(m) \cdot Y(n) \cdot (\sigma(m) \times \sigma(n)) \\
 & \cong & (\int^m X(m) \cdot \sigma(m)) \times (\int^n Y(n) \cdot \sigma(n)) \\
 & \cong & R(X) \times R(Y)
}$$
where in each of the second and penultimate lines, we twice used the fact that $- \times -$ preserves colimits in its separate arguments (i.e., the fact that the nice category $Space$ is cartesian closed), and the remaining lines used the fact that $R$ preserves colimits, and also products of representables by lemma \ref{canonical}.
=--

* A slightly higher-level rendition of the proof might look like this:
$$\array{
R(X \times Y) & \cong & R((X \otimes_{\Delta} \hom) \times (Y \otimes_{\Delta} \hom)) \\
 & \cong & R((X \times Y) \otimes_{\Delta \times \Delta} (\hom \times \hom)) \\
 & \cong & (X \times Y) \otimes_{\Delta \times \Delta} R(\hom \times \hom) \\
 & \cong & (X \times Y) \otimes_{\Delta \times \Delta} (R(\hom) \times R(\hom)) \\
 & \cong & (X \otimes_{\Delta} R(\hom)) \times (Y \otimes_{\Delta} R(\hom)) \\
 & \cong & R(X \otimes_{\Delta} \hom) \times R(Y \otimes_{\Delta} \hom) \\
 & \cong & R(X) \times R(Y)
}$$

### A construction of Drinfeld of geometric realization as Hom([0,1],-)

[Drinfeld, On the notion of geometric realization](https://arxiv.org/abs/math/0304064) provides a conceptual explanation of preserving finite limits, and
"reformulates the definitions so that the following facts become obvious:

* geometric realization commutes with finite projective limits (e.g., with Cartesian products);

* the geometric realization of a simplicial set ... (resp. cyclic set) is
equipped with an action of the group of orientation preserving homeomorphisms
of the segment $I := [0, 1]$ ... (resp. the circle $S^ 1$)."
(quote from the paper)

A draft [M.Gavrilovich, K.Pimenov. Geometric realisation as a Skorokhod semi-continuous path space endofunctor](http://mishap.sdf.org/Skorokhod_Geometric_Realisation.pdf)
attempts to further reformulate this by showing that, in a certain precise sense,
geometric realisation is an endofunctor of a certain category $sF$ of simplicial sets equipped
with extra structure of topological nature (a notion of smallness). The underlying
endofunctor of $sSets$ is
$$HHom ( Hom_{preorders}(-, [0,1]_\leq), Y_.) $$

The category $sF$ contains simplicial sets, topological and uniform spaces as full subcategories,
and has forgetful functors $sF\to sSets$, $sF\to Top$, and $sF\to UniformSpaces$ such that
the following compositions are identity:
$sSets\to sF\to sSets$, $Top\to sF\to Top$, and $UniformSpaces\to sF\to UniformSpaces$.
Moreover, this endofunctor seems to
have the right adjoint, defined by the usual construction.

Here are some details. The category $sF$ may be thought as the category of simplicial sets with
 extra structure of topological nature, a notion of smallness.
 Formally it is just the category of simplicial objects in the category of [[filters]].
 The endofunctor $HHom ( Hom_{preorders}(-, [0,1]_\leq), Y_.):sF\to sF $
 above is the inner hom of sSets equipped with an extra structure motivated by Skorokhod/Levi-Prokhorov convergence.
 The precise claim is that the geometric realisation of sSets factors as
$$ sSets  \to sF \to sF \to Top $$

 To gain some intuition, consider $Y_.=\Delta_n=Hom(-,[n])$ the standard simplex. Then
 by Remark  2.4.1-2 of [Grayson](http://www.math.uiuc.edu/~dan/Courses/2003/Spring/416/GraysonKtheory.pdf)
 the standard geometric simplex is the set of monotone functions $[0.1]\to [n]$
 $$Hom_{preorders}([0,1]_\leq), [n] ) = Hom ( Hom_{preorders}(-, [0,1]_\leq), \Delta_n )$$
 equipped with a metric
 $$\dist(f,g)=\inf \{ \epsilon: \forall x \exists y ( f(x)=f(y) ) \wedge  \forall y \exists x ( f(x)=g(y) ) \}$$
 reminiscent of Skorokhod metric in probability theory.
Now instead of $\Delta_n$ take an arbitrary simplicial set, and rewrite the definition of Skorokhod convergence in terms of the notion of smallness in $sF$.

### Geometric realization preserves fibrations

\begin{theorem}
The geometric realization of a [[Kan fibration]] is a [[Serre fibration]].
\end{theorem}
\begin{proof}
This is shown in [Quillen 68](#Quillen68).
\end{proof}

This result implies that the geometric realization functor preserves all five classes
of maps in a [[model category]]: [[weak equivalences]], [[cofibrations]], [[acyclic cofibrations]],
[[fibrations]], and [[acyclic fibrations]].

In fact the geometric realization of a Kan fibration is even a [[Hurewicz fibration]] (at least relative to a [[convenient category of spaces]] in which it lives).  This follows from the fact that [[a Serre fibration between CW-complexes is a Hurewicz fibration]]; a direct proof along the lines of Quillen's can be found in [Fritch and Piccinini, Theorem 4.5.25](#FP90).

### Induced properties of the fibrant replacement

The previous two sections show that the geometric realization preserves finite limits and fibrations.  Since its right adjoint, the singular complex functor $Top \to sSet$, also preserves both (much more trivially), and since all objects of $Top$ are fibrant and the adjunction is simplicially enriched, it follows that the composite $sSet \to Top \to sSet$ is a simplicially enriched [[fibrant replacement functor]] on $sSet$ that additionally preserves both finite limits and fibrations.

### Geometric realization of barycentric subdivisions

\begin{theorem}
([Fritsch–Puppe, 1967](#FP1967); [Fritsch 1974](#F1974).)
There is a [[homeomorphism]]
$$|Sd X| \to |X|$$
from the [[geometric realization]] of the [[barycentric subdivision]] of a [[simplicial set]] $X$
to the geometric realization of $X$.
This homeomorphism is homotopic to the geometric realization of the last vertex map.
The homeomorphism turns the [[CW-complex]] $|Sd X|$ into a subdivision of the [[CW-complex]] $|X|$.
The statement also holds relative a simplicial subset $A\subset X$.
\end{theorem}

For an expository account, see [Fritsch–Piccinini](#FP1990).

## Examples

* For $G$ a [[group]], $\mathbf{B}G$, its one-object [[groupoid]] obtained by [[delooping]], $N(\mathbf{B}G)$ the corresponding simplicial [[nerve]] [[Kan complex]], we have that the geometric realization
  $$
    \mathcal{B}G = |N\mathbf{B}G|
  $$
  is the [[topological space]] that is the [[classifying space]] for $G$-[[principal bundle]]s ([[covering space]]s), as long as we give $G$ the [[discrete topology]].

## Related concepts

* **geometric realization**

  * of [[geometric realization of categories|categories]], [[geometric realization of simplicial topological spaces|simplicial topological spaces]], [[geometric realization of cohesive ∞-groupoids|cohesive ∞-groupoids]], [[geometric realisation of cubical sets|cubical sets]].

* [[totalization]]

* [[singular complex functor]]

## References

### General

Original articles:

* {#Quillen68} [[Daniel Quillen]], *The geometric realization of a Kan fibration is a Serre fibration.  Proc. Amer. Math. Soc. 19 1968 1499--1500.  [pdf](https://www.ams.org/journals/proc/1968-019-06/S0002-9939-1968-0238322-1/S0002-9939-1968-0238322-1.pdf)

Textbook accounts:

* {#GoerssJardine09} [[Paul Goerss]], [[J. F. Jardine]], Section VII.3 of: *[[Simplicial homotopy theory]]*, Progress in Mathematics, Birkhäuser (1999)
Modern Birkhäuser Classics (2009)
[[doi:10.1007/978-3-0346-0189-4](https://link.springer.com/book/10.1007/978-3-0346-0189-4)]

* {#FP90}  [[Rudolf Fritsch]], [[Renzo A. Piccinini]], _Cellular structures in topology_, Cambridge studies in advanced mathematics Vol. 19, Cambridge University Press (1990). ([doi:10.1017/CBO9780511983948](https://doi.org/10.1017/CBO9780511983948), [pdf](https://epub.ub.uni-muenchen.de/4493/1/4493.pdf))

Lecture notes:

* {#Lackenby08} [[Marc Lackenby]], Section I.2 of: *Topology and Groups* (2008, 2018) [[pdf](http://people.maths.ox.ac.uk/lackenby/tg050908.pdf), [[Lackenby-TopologyAndGroups.pdf:file]]]

Generalization to realization of [[simplicial presheaves]]:

* {#Simpson96} [[Carlos Simpson]], *The topological realization of a simplicial presheaf* [[arXiv:q-alg/9609004](https://arxiv.org/abs/q-alg/9609004)]

### Compatibility with homotopy limits

Discussion of sufficient conditions for homotopy geometric realization to be compatible with [[homotopy pullback]] (see also at _[[geometric realization of simplicial topological spaces]]_):

* D. Anderson, _Fibrations and geometric realization_ ,
Bull. Amer. Math. Soc. Volume 84, Number 5 (1978), 765-788. ([euclid:1183541139](http://projecteuclid.org/euclid.bams/1183541139))

* {#Rezk14} [[Charles Rezk]], _When are homotopy colimits compatible with homotopy base change?_, 2014 ([pdf](https://faculty.math.illinois.edu/~rezk/i-hate-the-pi-star-kan-condition.pdf), [[RezkHomotopyColimitsBaseChange.pdf:file]])

* Edoardo Lanari, _Compatibility of homotopy colimits and homotopy pullbacks of simplicial presheaves_ ([pdf](http://algant.eu/documents/theses/lanari.pdf), [[LanariHomotopyColimitsBaseChange.pdf:file]])

  (expanded version of [Rezk 14](#Rezk14))

### Geometric realization of barycentric subdivisions

* {#FP1967} [[Rudolf Fritsch]], [[Dieter Puppe]], _Die Homöomorphie der geometrischen Realisierungen einer semisimplizialen Menge und ihrer Normalunterteilung_, Arch. Math. (Basel) 18 (1967), 508–512.

* {#F1974} Relative semisimpliziale Approximation, Arch. Math. (Basel) 25 (1974), 75–78.

* {#FP1990} [[Rudolf Fritsch]], [[Renzo A. Piccinini]], _Cellular structures in topology_, CUP, 1990.

[[!redirects geometric realisation]]

[[!redirects geometric realizations]]

[[!redirects the geometric realization of a Kan fibration is a Serre fibration]]
[[!redirects the geometric realization of a Kan fibration is a Hurewicz fibration]]

[[!redirects topological realization]]
[[!redirects topological realizations]]


geometric realization of categories

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

#Contents#
* table of contents
{:toc}

## Idea

What is called _geometric realization of categories_ is a [[functor]] that sends [[categories]] to [[topological spaces]], namely the functor which first forms the [[simplicial set]] $N(\mathcal{C})$ that is the [[nerve]] of the category $\mathcal{C}$, and then forms the [[geometric realization]] ${\vert N(\mathcal{C})\vert}$ of this simplical set. Typically one is interested in this geometric realization up to [[weak homotopy equivalence]].

By the [[homotopy hypothesis]]-theorem the [[geometric realization]] of simplicial sets constitutes a ([[Quillen equivalence|Quillen]])[[equivalence of (infinity,1)-categories|equivalence]] between the [[classical model structure on simplicial sets|classical homotopy theory of simplicial sets]] and the [[classical model structure on topological spaces|classical homotopy theory of topological spaces]].
This means that inasmuch as one is interested in geometric realization of categories up to weak homotopy equivalence, then the key part of the operation is in forming the simplicial nerve $N(\mathcal{C})$ of a category, with the latter regarded as a model for an [[∞-groupoid]]. Indeed, equivalently one could consider the [[Kan fibrant replacement]] of the nerve $N(\mathcal{C})$ (which still has the same geometric realization, up to weak homotopy equivalence).

Therefore an equivalent perspective on geometric realization of categories is that it universally turns a category into an [[infinity-groupoid]] by freely turning all its morphisms into [[equivalence in an (infinity,1)-category|homotopy equivalences]].

Geometric realization of categories has various good properties:

It sends [[equivalences of categories]] to [[weak homotopy equivalences]] (corollary \ref{RealizationOfEquivalenceIsHomotopyEquivalence} below).  A more general sufficient criterion for the geometric realization of a functor is given  by the seminal theorem known as _Quillen's theorem A_ (theorem \ref{QuillenTheoremA} below.)

The existence of the [[Thomason model structure]] ([below](#ThomasonModelStructure)) implies that every [[homotopy type]] arises as the geometric realization of some category. In fact it already arises as the geometric realization of some [[poset]] ([[(0,1)-category]]).

## Definition

Write
$$
  N \colon Cat \to sSet
$$
for the [nerve functor](nerve#NerveOfACategory) from [[Cat]] to [[sSet]]. Write
$$
  {\vert - \vert} : sSet \to Top
$$
for the [[geometric realization]] of [[simplicial sets]] from [[sSet]] to [[Top]].

The _geometric realization of categories_ is the [[composition|composite]] of these two operations:

$$
  {\vert - \vert} \coloneqq {\vert N(-)\vert} \;\colon\; Cat \to Top
$$

## Properties

### Thomason model structure
  {#ThomasonModelStructure}

There is a [[model category]] structure on [[Cat]] whose weak equivalences are those [[functors]] $F \colon \mathcal{C} \to \mathcal{D}$ which under [[geometric realization]] become weak equivalences in the [[classical model structure on topological spaces]], hence which become [[weak homotopy equivalences]]. This is called the _[[Thomason model structure]]_.

The existence of the Thomas model structure implies that every [[homotopy type]] arises as the geometric realization of some category, in fact already as the realization of some [[poset]]/[[(0,1)-category]]:

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

For $C$ a [[category]], let $\nabla C$ be the [[poset]] of [[simplices]] in the [[nerve]] $N C$, ordered by inclusion.

=--

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

For every category $\mathcal{C}$ the poset $\nabla \mathcal{C}$ from def. \ref{PosetOfSimplicesInNerveOfCategory} has [[weak homotopy equivalence|weakly homotopy equivalent]] geometric realization

$$
  {\vert N(\nabla \mathcal{C}) \vert} \simeq_{wh} {\vert \mathcal{C} \vert}
  \,.
$$

=--

### Recognizing weak equivalences: Quillen's theorem A and B

Let $\mathcal{C}, \mathcal{D}$ be two [[categories]] and let

$$
  F \;\colon\; \mathcal{C} \longrightarrow \mathcal{D}
$$

be a [[functor]] between them.

+-- {: .num_theorem #QuillenTheoremA}
###### Theorem
**([Quillen 72](#Quillen72), theorem A)**

If for all [[objects]] $d \in \mathcal{D}$ the [[geometric realization]] ${\vert N(F/d)\vert}$ of the [[comma category]] $F/d$ is [[contractible]] (meaning that $F$ is a "homotopy [[cofinal functor]]", hence a [[cofinal (∞,1)-functor]]), then

$$
  {\vert N(F) \vert}
   \;\colon\;
  {\vert N(\mathcal{C}) \vert}
    \longrightarrow
  {\vert N(\mathcal{D}) \vert}
$$

is a [[weak homotopy equivalence]].

=--

+-- {: .num_theorem #QuillenTheoremB}
###### Theorem
**([Quillen 72](#Quillen72) theorem B)**

If for all [[objects]] $d \in \mathcal{D}$ we have that ${\vert N(F/d)\vert}$ is [[weak homotopy equivalence|weakly homotopy equivalent]] to a given [[topological space]] $X$ and all morphisms $f \colon d_1 \to d_2$ induce [[weak homotopy equivalences]] between these, then $X$ is the [[homotopy fiber]] of ${\vert N(F) \vert}$, hence we have a [[homotopy fiber sequence]] (in the [[classical model structure on topological spaces]]) of the form

$$
  X
    \longrightarrow
  {\vert N(\mathcal{C}) \vert}
    \overset{\vert N(F) \vert }{\longrightarrow}
  {\vert N(\mathcal{D}) \vert}
  \,.
$$

=--

As a consequence:

+-- {: .num_prop}
###### Proposition
**([McCord 66, theorem 6](#McCord66), [Quillen 78, prop. 1.6](#Quillen78))**

Let $\mathcal{C}, \mathcal{D}$ be [[finite set|finite]] [[posets]] and consider $F \colon  \mathcal{C} \to \mathcal{D}$ be a [[functor]].

If for each element/object $y \in \mathcal{D}$ its [[preimage]] $f^{-1}( \{ y' \in Y \vert  y' \leq y \})$ has [[contractible topological space|contractible]] geometric realization, then ${\vert N(F)\vert}$ is a [[homotopy equivalence]].

=--

An alternative proof is given in ([Barmak 10](#Barmak10)).

### Natural transformations and homotopies

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

A [[natural transformation]] $\eta : F \Rightarrow G$ between two [[functors]] $F, G : \mathcal{C} \to \mathcal{D}$ induces under geometric realization a [[homotopy]]

$$
  {|N(\eta)|} \colon {\vert N(F)\vert} \longrightarrow {\vert N(G) \vert}
  \,.
$$

=--

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

The natural transformation is equivalently a functor of the form

$$
  \eta \;\colon\; \mathcal{C} \times \{0 \to 1\} \to \mathcal{D}
$$

out of the [[product category]] of $\mathcal{C}$ with the [[interval category]].

Since [[geometric realization]] of [[simplicial sets]] preserves [[Cartesian products]] (see there) we have that

$$
  {\vert N( \mathcal{C} \times \{0,1\} ) \vert}
    \;\simeq_{iso}\;
  {\vert N(\mathcal{C}) \vert} \times {\vert N(\{0 \to 1\}) \vert}
$$

But this is a [[cylinder object]] in topological spaces, hence ${\vert N(\eta) \vert}$ is a [[left homotopy]].

=--

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

An [[equivalence of categories]] $\mathcal{C} \simeq \mathcal{D}$ induces a [[homotopy equivalence]] between their geometric realizations.

=--

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

The statement still remains true for a pair of [[adjoint functor]]s $\mathcal{C} \leftrightarrows \mathcal{D}$.

=--

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

Notice that the converse is far from true: Very different categories can have geometric realizations that are (weakly) homotopy equivalent. This is because geometric realization implicitly involves [[Kan fibrant replacement]]: it freely turns morphisms into [[equivalence in an (infinity,1)-category|equivalences]].

=--

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

If a [[category]] $\mathcal{C}$ has an [[initial object]] or a [[terminal object]], then its geometric realization is [[contractible]].

=--

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

Assume the case of a terminal object, the other case works [[formal dual|formally dually]]. Write $*$ for the terminal category.

Then we have an equality of functors

$$
  Id_* = (* \stackrel{\bottom}{\to} C \to *)
  \,,
$$

where the first functor on the right picks the terminal object, and we have a [[natural transformation]]

$$
  Id_C \Rightarrow (C \to * \stackrel{\bottom}{\to} C)
$$

whose components are the unique morphisms into the terminal object.

By prop. \ref{NaturalTrafoMapsToHomotopy} it follows that we have a [[homotopy equivalence]] $\vert N(\mathcal{C}) \vert \to \vert N(\ast) \vert = \ast$.

=--

### Behaviour under homotopy colimits

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

For $F \colon \mathcal{D} \to Cat$ a [[functor]], let

$$
  {\vert N(F(-))\vert}
   \;\colon\;
  \mathcal{D}
    \overset{F}{\longrightarrow}
  Cat
   \stackrel{\vert N(-) \vert}{\to}
  Top
$$

be its postcomposition with geometric realization of categories

Then we have a [[weak homotopy equivalence]]

$$
  {\left\vert N\left(\int F \right) \right\vert}
    \simeq
  hocolim {\vert F(N(-)) \vert}
$$

exhibiting the [[homotopy colimit]] in [[Top]] over $\vert N(F (-)) \vert$ as the geometric realization of the [[Grothendieck construction]] $\int F$ of $F$.

=--

This is due to ([Thomason 79](#Thomason79)).

## Related concepts

* [[geometric realization]]

  * **of categories**, [[geometric realization of simplicial topological spaces|of simplicial topological spaces]], [[geometric realization of cohesive ∞-groupoids|of cohesive ∞-groupoids]]

## References

### General

For general references see also _[[nerve]]_ and _[[geometric realization]]_.

### Quillen's theorems A and B

The original articles are

* {#McCord66} [[Michael C. McCord]], _Singular homology groups and homotopy groups of finite topological spaces_, Duke Math. J. 33 (1966), 465-474 ([pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/mccord1.pdf))

* {#Quillen72} [[Daniel Quillen]], _Higher algebraic K-theory, I: Higher K-theories_ Lect. Notes in Math. 341 (1972), 85-1 ([pdf](http://math.mit.edu/~hrm/kansem/quillen-higher-algebraic-k-theory.pdf))

* {#Quillen78} [[Daniel Quillen]], _Homotopy  properties  of  the  poset  of  nontrivial p-subgroups  of a group_, Adv. Math. 28 (1978), 101-128.

The geometric realization of [[Grothendieck constructions]] has been analyzed in

* {#Thomason79} [[R. W. Thomason]], _Homotopy colimits in the category of small categories_ , Math. Proc. Cambridge Philos. Soc. 85 (1979), no. 1, 91109.([pdf](https://www.cambridge.org/core/services/aop-cambridge-core/content/view/32ACB453B263410CE1643E682D7DE335/S0305004100055535a.pdf/homotopy-colimits-in-the-category-of-small-categories.pdf))

Review is in

* {#Barmak10} [[Jonathan Barmak]], _On Quillen's Theorem A for posets_, Journal of Combinatorial Theory Series A, Volume 118 Issue 8, November, 2011
Pages 2445-2453  ([arXiv:1005.0538](http://arxiv.org/abs/1005.0538))

Further development includes

* [[Clark Barwick]], [[Daniel Kan]], _A Quillen theorem $B_n$ for homotopy pullbacks_ ([arXiv:1101.4879](http://arxiv.org/abs/1101.4879))

* [[David Roberts]], _[[davidroberts:Theorem A for topological categories]]_

[[!redirects geometric realization of a category]]

[[!redirects Quillen's theorem A]]
[[!redirects Quillen's theorem B]]
[[!redirects Quillen's Theorem A]]
[[!redirects Quillen's Theorem B]]
[[!redirects Quillen theorem A]]
[[!redirects Quillen theorem B]]
[[!redirects Quillen Theorem A]]
[[!redirects Quillen Theorem B]]


geometric realization of cohesive infinity-groupoids

[[!redirects geometric realization of cohesive ∞-groupoids]]

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohesive $\infty$-Toposes
+--{: .hide}
[[!include cohesive infinity-toposes - contents]]
=--
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

For $(\Pi \dashv Disc \dashv \Gamma \dashv coDisc) : \mathbf{H} \to \infty Grpd$ a [[cohesive (∞,1)-topos]], we call the action of the [[shape modality]]

$$
  {\vert \Pi (- )\vert} : \mathbf{H} \stackrel{\Pi}{\to} \infty Grpd \stackrel{\vert - \vert}{\to} Top
$$

the [[geometric realization]] functor. For $X \in \mathbf{H}$ any [[object]], hence any [[cohesive ∞-groupoid]], $\vert \Pi(X)\vert$ is its geometric realization.

Notice that $\Pi(X)$ is the [[fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos]] and $\vert - \vert : $ [[∞Grpd]] $\to$ [[Top]] is the "[[homotopy hypothesis]]" [[equivalence of (∞,1)-categories]].

## Properties

See at [[cohesive (∞,1)-topos -- structures]] the section Geometric homotopy and Galois theory.

## Examples

In $\mathbf{H} = $ [[ETop∞Grpd]] the geometric realization of cohesive $\infty$-groupoids subsumes the [[geometric realization of simplicial topological spaces]] (see there for details).

## Related concepts

* [[geometric realization]]

  * [[geometric realization of categories|of categories]], [[geometric realization of simplicial topological spaces|of simplicial topological spaces]], **of cohesive ∞-groupoids**

  * [[shape]], [[shape modality]]

[[!redirects geometric realization in a locally ∞-connected (∞,1)-topos]]

[[!redirects geometric realization of a cohesive homotopy type]]
[[!redirects geometric realization of cohesive homotopy types]]