Cayley plane

#Contents#
* table of contents
{:toc}

## Idea

The _Cayley plane_ is the [[projective plane]] over the [[octonions]], i.e. the [[octonionic projective space]] $\mathbb{O} P^2$.

## Definition

The Cayley plane can't be constructed using homogeneous coordinates the way it works for projective planes over [[division rings]], since multiplication of octonions is not associative.  However, it can be constructed in several other ways that generalize the approach for division rings:

* One can start with $\mathbb{O}^2$ and give it the structure of an [[affine plane]] in the obvious way, and then add "points at infinity" in the usual way to obtain a projective plane.  This is the most straightforward approach, but as always it has the defect that it makes the line at infinity appear special.

* One can consider the space of $3\times 3$ matrices over $\mathbb{O}$ that are "Hermitian" and idempotent, hence can be imagined as "projections onto dimension-1 subspaces of $\mathbb{O}^3$".

* Writing out the components of such a matrix explicitly, one obtains a *Veronese vector* $(x_1,x_2,x_3;\xi_1,\xi_2,\xi_3)$ where $x_i\in \mathbb{O}$ and $\xi_i\in \mathbb{R}$, such that $\xi_i \overline{x_i} = x_j x_k$ and $\Vert x_i\Vert^2 = \xi_j \xi_k$ for all cyclic permutations $(i,j,k)$ of $(1,2,3)$.  The Cayley plane can be identified with the space of nonzero such vectors modulo the scalar action of $\mathbb{R}$.

## Properties

### General

The octonionic projective plane is a non-Desarguesian plane, that is, [[Desargues' theorem]] does not hold. See [[projective plane]].

### Isometries

* [[F4]] is the [[isometry group]] of $\mathbb{O} P^2$, with the [[stabilizer]] of a point being [[Spin(9)]]. Hence $\mathbb{O} P^2 \cong F_4/Spin(9)$.

### Cell structure

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

There is a [[homeomorphism]]

$$
  \mathbb{O}P^2 \,\simeq\, D^{16} \underset{h_{\mathbb{O}}}{\cup} \mathbb{O}P^1
$$

between the [[octonionic projective plane]] and the [[attaching space]] obtained from the [[octonionic projective line]] along the [[octonionic Hopf fibration]].

=--

([Lackmann 19, Lemma 3.4.](#Lackmann19))

([Mimura 67, page 166](#Mimura67))

See also at _[[cell structure of projective spaces]]_.

### Homotopy groups

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

The [[homotopy groups]] of octonionic projective plane are
\[
  \label{ListOfHomotopyGroupsOfOctonionicProjectivePlane}
    \pi_k
    \big(
      \mathbb{O}P^2
    \big)
    \;=\;
    \left\{
    \array{
      1 &\vert& k \leq 7
      \\
      \pi_k
      \big(
        S^8
      \big)
      &\vert&
      8 \leq k \leq 14
    }
    \right.
\]
Further homotopy groups are
$$\pi_{15}\big(\mathbb{O}P^2\big)
\cong\mathbb{Z}_{120}$$
$$\pi_{16}\big(\mathbb{O}P^2\big)
\cong\mathbb{Z}_2^3$$
$$\pi_{17}\big(\mathbb{O}P^2\big)
\cong\mathbb{Z}_2^4$$
$$\pi_{18}\big(\mathbb{O}P^2\big)
\cong\mathbb{Z}_{24}\times\mathbb{Z}_2$$
$$\pi_{19}\big(\mathbb{O}P^2\big)
\cong\mathbb{Z}_{504}\times\mathbb{Z}_2$$
$$\pi_{20}\big(\mathbb{O}P^2\big)
\cong 1$$
$$\pi_{21}\big(\mathbb{O}P^2\big)
\cong\mathbb{Z}_6$$
$$\pi_{22}\big(\mathbb{O}P^2\big)
\cong\mathbb{Z}_4$$
$$\pi_{23}\big(\mathbb{O}P^2\big)
\cong\mathbb{Z}\times\mathbb{Z}_{120}\times\mathbb{Z}_2^2\,.$$
(While $\pi_{15}\big(S^8\big)
\cong\mathbb{Z}\times\mathbb{Z}_{120}$, which includes the [[homotopy class]] of the [[octonionic Hopf fibration]].)

=--

([Lackmann 19, page 7](#Lackmann19))

([Mimura 67, Theorem 7.2.](#Mimura67))

### Cohomology

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

For $A \in $ [[Ab]] any [[abelian group]], then the [[ordinary cohomology]] [[cohomology groups|groups]] of octionionic projective plane $\mathbb{O}P^2$ with [[coefficients]] in $A$ are

$$
  H^k(\mathbb{O}P^2,A)
   \simeq
  \left\{
    \array{
       A & for \; k=0,8,16
       \\
       0 & otherwise
    }
  \right.
  \,.
$$

=--

([Lackmann 19, Corollary 4.1.](#Lackmann19))

### AKM-Theorem

+-- {: .num_prop}
###### Proposition
**(octonionic [[AKM-theorem]])**

The [[13-sphere]] is the [[quotient space]] of the (right-)[[octonionic projective plane]] by the left multiplication [[action]] by [[Sp(1)]]:

$$
  \mathbb{O}P^2 / \mathrm{Sp}(1)
  \simeq
  S^{13}
$$

=--

([Atiyah-Berndt 02](#AtiyahBerndt02))

## Related concepts

* [[Cayley 4-form]]

## References

### General

* {#Lackmann19} [[Malte Lackmann]], _The octonionic projective plane_, in MATRIX Book Series **4**, Springer (2021) [[doi:10.1007/978-3-030-62497-2_6](https://doi.org/10.1007/978-3-030-62497-2_6), [arXiv:1909.07047](https://arxiv.org/abs/1909.07047)]

See also:

* Wikipedia, _[Cayley plane](http://en.wikipedia.org/wiki/Cayley_plane)_

* Salzmann et. al., *Compact Projective Planes, with an introduction to Octonion Geometry*

The [[AKM-theorem]] for $\mathbb{O}P^2$:

* {#AtiyahBerndt02} [[Michael Atiyah]], [[Jürgen Berndt]], in:  Surv. Differ. Geom. VIII, Papers in Honor of Calabi, Lawson, Siu and Uhlenbeck (International Press, Somerville, MA, 2003) 1-27 ([arXiv:math/0206135](https://arxiv.org/abs/math/0206135), [doi:10.4310/SDG.2003.v8.n1.a1](https://dx.doi.org/10.4310/SDG.2003.v8.n1.a1))

### In string theory

Discussion of the [[Witten genus]] of Cayley plane-[[fiber bundles]] is in

* {#McTague10} [[Carl McTague]], _The Cayley Plane and the Witten Genus_ ([arXiv:1006.0728](http://arxiv.org/abs/1006.0728))

Indications that [[M-theory]] in 10+1 dimensions may be understood as the [[KK-compactification]] on Cayley-plane [[fibers]] of some kind of [[bosonic M-theory]] in 26+1 dimensions:

* [[Hisham Sati]], _$\mathbb{O}P^2$ bundles in M-theory_, Commun. Num. Theor. Phys 3:495-530,2009 ([arXiv:0807.4899](https://arxiv.org/abs/0807.4899))

* [[Hisham Sati]], _On the geometry of the supermultiplet in M-theory_, Int. J. Geom. Meth. Mod. Phys. 8 (2011) 1-33 ([arXiv:0909.4737](https://arxiv.org/abs/0909.4737))

* [[Michael Rios]], [[Alessio Marrani]], [[David Chester]], _Exceptional Super Yang-Mills in $D=27+3$ and Worldvolume M-Theory_, Phys. Lett. B, 808, (2020)
([arXiv:1906.10709](https://arxiv.org/abs/1906.10709))

* {#Mimura67} Mamoru Mimura, _The Homotopy groups of Lie groups of low rank_, J. Math. Kyoto Univ. 6 (2) 131 - 176, 1967. [https://doi.org/10.1215/kjm/1250524375](https://doi.org/10.1215/kjm/1250524375)

[[!redirects Cayley planes]]

[[!redirects octonion projective plane]]
[[!redirects octonion projective planes]]

[[!redirects octonionic projective plane]]
[[!redirects octonionic projective planes]]


Cayley transform

#Contents#
* table  of contents
{:toc}

##  Idea

The  classical __Cayley transform__ is an involutive [[bijection]] between the [[vector space]] of [[skew-symmetric matrices]]  to the [[special orthogonal group|group of special orthogonal matrices]], given by the formula

$$
  X \mapsto \frac{I + X}{I - X}
  \,.
$$

There are many generalizations (some non-involutive), say for [[J-orthogonal matrix|J-orthogonal matrices]], to shifted center, for operators, in complex analysis and so on.

## Literature

* [[M M Postnikov]], _Lectures on geometry, Semester V, Lie groups and Lie algebras_
* wikipedia/[Cayley transform](https://en.wikipedia.org/wiki/Cayley_transform)
* <https://encyclopediaofmath.org/wiki/Cayley_transform> (for operators)
* [[D. Quillen]], _Superconnection character forms and the Cayley transform_, Topology 27 (1988), no. 2, 211–238
* A. Gómez-Tato, E. Macías-Virgós, M. J.  Pereira-Sáez, _Trace map, Cayley transform and LS category of Lie groups_, Ann Glob Anal Geom 39, 325–335 (2011) [doi](https://doi.org/10.1007/s10455-010-9239-8)
* E. Macías-Virgós, M. José Pereira-Sáez, D. Tanré, _Cayley transform on Stiefel manifolds_, J. Geom. Phys. 123, 53-60 (2018) [doi](https://doi.org/10.1016/j.geomphys.2017.08.011)
* Michael Jauch, Peter D. Hoff, David B. Dunson, _Random orthogonal matrices and the Cayley transform_, Bernoulli 26:2 (2020) 1560-1586 [doi](https://doi.org/10.3150/19-BEJ1176)

The general case related to symmetric spaces:

* F Astengo, M Cowling, B Di Blasio, _The Cayley transform and uniformly bounded representations_, J. Funct. Analysis __213__:2 (2004) 241-269 [doi](https://doi.org/10.1016/j.jfa.2003.12.009)


Cayley's theorem

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

#Contents#
* table of contents
{:toc}

## Idea

In [[group theory]], *Cayley's theorem* is the statement that every *[[discrete group]]* is [[isomorphism|isomorphic]] to a [[subgroup]] of the [[symmetric group]] $Sym(G)$ on its [[underlying set]], namely as the group of [[permutations]] of elements of $G$ by left-multiplication with $G$ on itself (the "[[regular representation|regular]] [[permutation representations]]" of $G$).

[[Wagner-Preston theorem]] is a generalization of Cayley's theorem to [[inverse semigroup]]s (every inverse semigroup is a subsemigroup of a semigroup of partial bijections of a set).

## References

Named after (cf. eg. [HMM18, p. 103](#HMM18)):

* [[Arthur Cayley]], *On the theory of groups, as depending of the symbolic equation $\theta^n = 1$. Philos.
Mag. VII: (1854) 40–47. Reprinted in *The Collected Mathematical Papers of Arthur Cayley*, Vol. II., Cambridge Univeristy Press (1898)

* [[Arthur Cayley]], *Desiderata and suggestions. No. 1. The theory of groups*, Amer. J. Math. I: (1878) 50–52,
Reprinted in: *The Collected Mathematical Papers of Arthur Cayley, Vol. X, Cambridge University Prerss (1898)

Textbooks and lecture notes:

* [[Joseph J. Rotman]], Thm. 3.12 in: *An Introduction to the Theory of Groups*, Springer (1995) [[doi:10.1007/978-1-4612-4176-8](https://doi.org/10.1007/978-1-4612-4176-8), [pdf](https://eclass.uoa.gr/modules/document/file.php/MATH676/Rotman%20An%20introduction%20to%20the%20theory%20of%20groups.pdf)]

* [[James Milne]], Thm. 1.22 in: *Group theory* (2021) [[web](https://www.jmilne.org/math/CourseNotes/gt.html), [pdf](https://www.jmilne.org/math/CourseNotes/GT.pdf)]

See also:

* Wikipedia, *[Cayley's theorem](https://en.wikipedia.org/wiki/Cayley%27s_theorem)*

More on the case of [[finite groups]]:

* {#HMM18} R. Heffernan, Des Machale, Brendan McCann, *Cayley’s Theorem Revisited: Embeddings of Small Finite Groups*, Mathematics Magazine **91** 2 (2018) 103-111 [[jstor:48665514](https://www.jstor.org/stable/48665514)]

category: algebra


Cayley-Dickson construction

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Algebra
+-- {: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

# Cayley--Dickson construction
* table of contents
{: toc}

## Idea

The _Cayley-Dickson construction_ or _Cayley-Dickson double_ ([Dickson 1919, (6)](#Dickson1919)) takes a [[real numbers|real]] [[star-algebra]] $A$ to a new real [[star-algebra]] whose elements are [[pairs]] of elements of $A$, in generalization of how the [[complex numbers]] arise as a doubling of the [[real numbers]].

When iteratively applied to the [[real numbers]], regarded as a [[star-algebra]] with [[identity morphism|trivial]] [[involution]], the Cayley-Dickson construction yields, consecutively, the [[complex numbers]], then the [[quaternions]], then the [[octonions]] (thus all four [[real normed division algebras]]), then the [[sedenions]], ...

## Definition

Let $A$ be an [[nonassociative algebra|possibly nonassociative]] [[star-algebra]] over the [[field]] $\mathbb{R}$ of [[real numbers]]: an algebra equipped with an [[involution]] $\overline(-) \colon x \mapsto \overline{x}$ which is an [[antiautomorphism]].  (Actually, $\mathbb{R}$ could be replaced by any [[commutative ring]] in the definitions, although some properties may depend on this ring.)

### Definition in components

+-- {: .num_defn #CayleyDicksonDoubleInComponents}
###### Definition
**(Cayley-Dickson construction in components)**

The  _Cayley--Dickson double_ of the [[real numbers|real]] [[star-algebra]] $A$ is the [[real numbers|real]] [[star-algebra]] $CD(A)$

* whose underlying [[real vector space]] is the [[direct sum]] $A \oplus A$,

* whose multiplication is given by

  \[
    \label{ProductInComponents}
    (a,b)
    (c, d)
     \;\coloneqq\;
    (a c - d \overline{b}, \overline{a} d + c b)
  \]

* whose star-[[involution]] is given by

  \[
    \label{InvolutionInComponents}
    \widebar{(a,b)}
    \;\coloneqq\;
    (\overline{a},-b)
   \,.
  \]

=--

([Dickson 1919, (6)](#Dickson1919))

### Definition by generators and relations
 {#DefinitionByGeneratorsAndRelations}

+-- {: .num_defn #CayleyDicksonDoubleByAdjoiningFurtherGenerator}
###### Definition
**(Cayley-Dickson double by [[generators and relations]])

The _Cayley-Dickson double_ of a [[real numbers|real]] [[star-algebra]] $A$ is the [[real numbers|real]] [[star-algebra]] $\widetilde{CD}(A)$ obtained by adjoining one generator $\ell$ to $A$ subject to the following [[generators and relations|relations]]:

\[
  \label{TheNewGenerator}
  \ell^2 \;=\; -1
  \,,
  \phantom{AAA}
  \overline{\ell}
  \;=\;
  - \ell
\]

and

\[
  \label{DicksonRelations}
  a (\ell b)
  =
  \ell (\overline{a} b)
  \,,
  \phantom{AA}
  (a \ell) b
  =
  (a \overline{b}) \ell
  \,,
  \phantom{AA}
  (\ell a) (b \ell)
  =
  - \overline{a b}
\]

for all $a, b \in A$.

=--

([Baez 02, second half of 2.2](#Baez02))

+-- {: .num_lemma #InducedRelations}
###### Lemma
(**induced relations**)

The relation in Def. \ref{CayleyDicksonDoubleByAdjoiningFurtherGenerator} imply the following further relations:

\[
  \label{al}
  a \ell \;=\; \ell \overline{a}
  \,,
  \phantom{AA}
  \ell a \;=\; \overline{a} \ell
\]

\linebreak

\[
  (\ell a) b
  \;=\;
  \ell (b a)
  \,,
  \phantom{AA}
  a (b \ell)
  \;=\;
  (b a) \ell
\]

\linebreak

\[
  \label{laTimeslb}
  (\ell a) (\ell b)
  \;=\;
  - b \overline{a}
  \,,
  \phantom{AA}
  (a \ell) (b \ell)
  \;=\;
  - \overline{b} a
\]

for all $a, b \in A$.

=--

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

Using (eq:DicksonRelations) we have:

\[
  \begin{aligned}
  a \ell
  & =
  a (\ell 1)
  \\
  & =
  \ell (\overline{a} 1)
  \\
  & =
  \ell \overline{a}
  \end{aligned}
  \phantom{AAA}
  ,
  \phantom{AAA}
  \begin{aligned}
  \ell a
  & =
  (1 \ell) a
  \\
  & =
  (1 \overline{a}) \ell
  \\
  & =
  \overline{a} \ell
  \end{aligned}
\]

Using this and (eq:DicksonRelations) we have:

\[
  \label{lab}
  \begin{aligned}
    (\ell a) b
    & =
    (\overline{a} \ell ) b
    \\
    & =
    (\overline{a} \overline{b}) \ell
    \\
    & =
    \ell \overline{
      \overline{a} \overline{b}
    }
    \\
    & =
    \ell (b a)
  \end{aligned}
  \phantom{AAA}
  ,
  \phantom{AAA}
  \begin{aligned}
    a (b \ell)
    & =
    a (\ell \overline{b})
    \\
    & = \ell( \overline{a} \overline{b} )
    \\
    & =

    \overline{
      \overline{a} \overline{b}
    }
    \ell
    \\
    & =
    (b a) \ell
  \end{aligned}
\]

and

\[
  \label{lalb}
  \begin{aligned}
  (\ell a) (\ell b)
  & =
  (\ell a) (\overline{b} \ell)
  \\
  & =
  - \overline{ a \overline{b} }
  \\
  & =
  - b \overline{a}
  \end{aligned}
  \phantom{AAA}
  ,
  \phantom{AAA}
  \begin{aligned}
    (a \ell) (b \ell)
    & =
    (\ell \overline{a}) (b \ell)
    \\
    & =
    - \overline{ \overline{a} b }
    \\
    & =
    - \overline{b} a
  \end{aligned}
\]

=--

### Equivalence of the definitions

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

Definition \ref{CayleyDicksonDoubleInComponents} and Definition \ref{CayleyDicksonDoubleByAdjoiningFurtherGenerator} are equivalent, in that we have an [[isomorphism]] of [[real numbers|real]] [[star-algebras]]:

$$
  \array{
    \phi
    &
    DK(A)
      &\overset{\simeq}{\longrightarrow}&
    \widetilde{DK}(A)
    \\
    &
    (a,b)
    &\mapsto&
    a + \ell b
  }
$$

=--

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

It is clear from Def. \ref{CayleyDicksonDoubleByAdjoiningFurtherGenerator} that for every element $x \in \widetilde{DK}(A)$ there is a _unique_ [[pair]] of elements $a,b \in A$ such that

$$
  x = a + \ell b \;=\; \phi(a,b)
  \,.
$$

This means that $\phi$ is a [[linear isomorphism]] of the underlying [[real vector spaces]]. Hence it only remains to check that $\phi$ is indeed an [[algebra homomorphism]] and that it respects the [[involution]].

To see that $\phi$ is an [[algebra homomorphism]], we multiply out and then use the relations (eq:lab) and (eq:lalb) from Lemma \ref{InducedRelations}:

$$
  \begin{aligned}
    \phi(a,b) \phi(c,d)
    & =
    (a + \ell b)
    (c + \ell d)
    \\
    & =
    a c + (\ell b)(\ell d)
    +
    a (\ell d) + (\ell b) c
    \\
    & =
    a c - d \overline{b}
    +
    \ell (
      \overline{a} d
      +
      c b
    )
    \\
    &
    =
    \phi(a c - d \overline{b}, \overline{a} d + c b)
  \end{aligned}
$$

Here in the last line we indeed find the component formula (eq:ProductInComponents).

To see that $\phi$ respects the involution we use (eq:TheNewGenerator) from Def. \ref{CayleyDicksonDoubleByAdjoiningFurtherGenerator} and (eq:al) from Lemma \ref{InducedRelations}:

$$
  \begin{aligned}
    \overline{\phi(a,b)}
    & =
    \overline{ a + \ell b }
    \\
    & =
    \overline{a} + \overline{\ell b}
    \\
    & =
    \overline{a} - \overline{b} \ell
    \\
    & =
    \overline{a} - \ell b
    \\
    & = \phi( \overline{a}, -b )
    \,.
  \end{aligned}
$$

Here in the last line we indeed find the component formula (eq:InvolutionInComponents).

=--

\linebreak

## Properties

The map $a\mapsto (a,0)$ is a [[monomorphism]] $A\to A^2$. If $A$ is [[unital algebra|unital]] with unit $1$ then $A^2$ is unital with unit $(1,0)$. In the unital case, the element $\mathrm{i} \coloneqq (0,1)$ has the property $\mathrm{i}^2 = -1 \coloneqq (-1,0)$, and we may write $(a,b)$ as $a + b \mathrm{i}$ (while $a + \mathrm{i} b = (a,\overline{b})$).  For this reason, we may write $A[\mathrm{i}]$ in place of $A^2$, at least when $A$ is unital.

Generally speaking, the double $A^2$ of an algebra $A$ has a nice property iff $A$ is one level nicer.  For simplicity, assume that $A$ is unital (so that $\mathbb{R}$ is a subalgebra).  Since $\overline{\mathrm{i}} = -\mathrm{i}$, we see that the involution on $A^2$ is trivial iff the involution on $A$ is trivial and $A$ further has $2 = 0$.  Since $\mathrm{i} a = \overline{a} \mathrm{i}$, $A^2$ is [[commutative algebra|commutative]] iff $A$ is commutative and the involution in $A$ is trivial.  Since $a (b \mathrm{i}) = (b a) \mathrm{i}$, $A^2$ is [[associative algebra|associative]] iff $A$ is associative and commutative. Finally, $A^2$ is [[alternative algebra|alternative]] iff $A$ is associative (and hence also alternative).

## Examples

\begin{imagefromfile}
    "file_name": "QuaternionOctonionMultiplicationTable.jpg",
    "float": "right",
    "width": 400,
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 10
    },
    "caption": ""
\end{imagefromfile}

The standard example is the sequence of consecutive doubles starting with $\mathbb{R}$ itself (with the [[identity map]] as involution); these are the __Cayley--Dickson algebras__: the [[real numbers]] $\mathbb{R}$, the [[complex numbers]] $\mathbb{C}$, the [[quaternions]] $\mathbb{H}$, the [[octonions]] (or Cayley numbers) $\mathbb{O}$,

These are the [[real normed division algebras]]

The diagram on the right shows a basis of [[imaginary number|imaginary]] [[octonions]] obtained via Cayley-Dickson doubling from a standard basis $\{i,j,k\}$ of [[imaginary number|imaginary]] [[quaternions]].

\linebreak

\linebreak

\linebreak

Next, the CD-double of the [[octonions]] is the [[sedenions]] $\mathbb{S}$, etc. , followed by further algebras which are not [[division algebras]].
All of these algebras are [[power-associative algebra|power-associative]], [[flexible algebra|flexible]], and unital, and have all [[inverse elements]]; the subalgebra with $\overline{x} = x$ is always just $\mathbb{R}$.

## Cayley-Dickson-Albert construction

The Cayley-Dickson construction described above was slightly generalized in [Section 5 of Albert (1942)](#Alb42CDA), where the multiplication is further modified by a parameter $\gamma$ in the [[ground field]]. This process constructs an algebra $A^{\gamma}$ starting from a unital algebra $A$ of order $n$ over a field $F$ endowed with an [[involution]] $J:A\to A$ over $F$ such that for all $x\in A$:

* $x+ J(x) \in F$,

* $x J(x) = J(x) x \in F$.

To construct $A^{\gamma}$, one chooses a parameter $\gamma\in F$, and on the vector space $A\oplus A$ one defines a product

$$
(g_1 ,g_2 )\cdot (h_1, h_2 ) = (g_1 h_1 + \gamma J(h_2) g_2 , h_2 g_1 +g_2 J(h_1 ))
$$

and an involution $K$ on $A^{\gamma}$ as

$$
K((g_1, g_2 ))= (J(g_1) , -g_2)
$$

In particular, for $F=\mathbb{R}$ this process allows to get the [*split-*](composition+algebra#split_composition_algebras) variants of the real division algebra, cf.:

* [[split-complex numbers]]

* [[split quaternions]]

* [[split octonions]]

## Higher algebra

[Albuquerque & Majid (1999)](#AM99quasi) describe the Cayley-Dickson process from a different point of view. The quaternions, and octonions are realized as $\mathbb{Z}_2^n$ [[group algebra|group algebras]] over the reals *twisted* by a group $n$-cocycle $\omega\in Z^n(\mathbb{Z}_2^n,\mathbb{R}^{\times})$, for $n=2,3$, respectively.

In more detail, the quaternions are described as the group algebra $\mathbb{R}[\mathbb{Z}_2\times\mathbb{Z}_2]$ where the group multiplication is twisted by a cocycle $\omega(g,h)$. The 2-cocycle condition of such a function implies that the quaternions are associative. On the other hand, the octonions are realized as a group algebra $\mathbb{R}[\mathbb{Z}_2\times\mathbb{Z}_2]$ where the function $\omega(g,h)$ twisting the multiplication is no longer a 2-cocycle. One of the effects of this is that the octonions are no longer associative or, rather, that they satisfy a nontrivial associativity condition, in the sense of a [[quasi-Hopf algebra]]. This nontrivial associativity condition is witnessed by a group $3$-cocycle in $Z^3(\mathbb{Z}_2\times \mathbb{Z}_2\times \mathbb{Z}_2, \mathbb{R}^{\times})$. Notably, this $3$-cocycle turns out to be a $3$-coboundary, so that its cohomology class is trivial.

This point of view also clarifies the common claim that algebras in the Cayley-Dickson process lose "nice" properties after more iterations. Rather, a more precise claim is that these nice properties are weakened. In the case of the octonions, these are better regarded as associative *up to* a nontrivial cocycle.

This highlights that a more suitable point of view to study the algebras arising in the Cayley-Dickson process is [[higher algebra]]. As noted in e.g. ([p.8 of Baez (2002)](#Baez02)), the octonions are indeed associative, when viewed as a [[monoid object]] [[internalization|internal]] to the [[tensor category]] which is the [[fusion category]] of $\mathbb{Z}_2^3$-graded real vector spaces with nontrivial [[associator]] defined by the twisting $3$-cocycle.

This seems to suggest a pattern regarding the rest of the $2^n$-gons, as of now unproven(?). For instance, the sedenions could be realized as a twist of the group algebra $\mathbb{R}[\mathbb{Z}_2^4]$ by a 4-cocycle $Z^4(\mathbb{Z}_2^4,\mathbb{R}^{\times})$, and as such would have fewer nice properties *as a 1-algebra*. Yet as a higher algebra it would retain its nice properties: it would correspond to the [[fusion 2-category]] of $\mathbb{Z}_2^4$-graded [[2-vector space|2-vector spaces]] twisted by a group 4-cocycle.

## Related concepts

* [[composition algebra]]

## References

Named after [[Arthur Cayley]] and [[Leonard Dickson]].

The original article:

* {#Dickson1919} [[Leonard Dickson]], _On Quaternions and Their Generalization and the History of the Eight Square Theorem_,
Annals of Mathematics, Second Series, Vol. 20, No. 3 (Mar., 1919), pp. 155-171 ([jstor:1967865](https://www.jstor.org/stable/1967865))

Review and introduction:

*  [[M M Postnikov]], _Lectures on geometry, Semester V: Lie groups and Lie algebras_, Lec. 14 (russian and english editions)

* {#Baez02} [[John Baez]], _[The Cayley--Dickson construction](http://math.ucr.edu/home/baez/octonions/node5.html)_, in _[The octonions](http://math.ucr.edu/home/baez/octonions/)_, Bull. Amer. Math. Soc. 39 (2002), 145-205, [doi](http://dx.doi.org/10.1090/S0273-0979-01-00934-X)

* [[John Baez]], _[This Week's Finds --- Week 59](http://math.ucr.edu/home/baez/week59.html)_

* [[Tevian Dray]], [[Corinne Manogue]], Section 5.1 of: _The Geomety of Octonions_, World Scientific 2015 ([doi:10.1142/8456](https://doi.org/10.1142/8456))

See also

* Wikipedia, _[Cayley–Dickson construction](https://en.wikipedia.org/wiki/Cayley%E2%80%93Dickson_construction)_

More:

* Daniel K. Biss, [[Daniel Dugger]], [[Daniel Isaksen]], _Large annihilators in Cayley-Dickson algebras_,
 Communications in Algebra 36 (2), 632-664, 2008 ([arxiv:math/0511691](https://arxiv.org/abs/math/0511691))

* Daniel K. Biss, [[Daniel Christensen]], [[Daniel Dugger]], [[Daniel Isaksen]], _Large annihilators in Cayley-Dickson algebras II_, 	Boletin de la Sociedad Matematica Mexicana (3) 13(2) (2007), 269-292 ([arxiv:math/0702075](https://arxiv.org/abs/math/0702075))

* Daniel K. Biss, [[Daniel Christensen]], [[Daniel Dugger]], [[Daniel Isaksen]], _Eigentheory of Cayley-Dickson algebras_,
Forum Mathematicum 21(5) (2009), 833-851 ([arxiv:0905.2987](https://arxiv.org/abs/0905.2987))

On the generalization of the Cayley-Dickson construction:

* [[Abraham Adrian Albert]], *Quadratic forms permitting composition,* Ann. of Math. (2) **43** (1942), 161–177. ([doi](https://doi.org/10.2307/1968887))

On the Cayley-Dickson construction in terms of [[quasi-Hopf algebra|quasi-Hopf algebras]]:

* {#AM99quasi} Helena Albuquerque, [[Shahn Majid]]. *Quasialgebra Structure of the Octonions*. Journal of Algebra
Volume 220, Issue 1, 1 October 1999, Pages 188-224. ([doi](https://doi.org/10.1006/jabr.1998.7850))

[[!redirects Cayley-Dickson construction]]
[[!redirects Cayley–Dickson construction]]
[[!redirects Cayley--Dickson construction]]

[[!redirects Cayley-Dickson constructions]]
[[!redirects Cayley–Dickson constructions]]
[[!redirects Cayley--Dickson constructions]]

[[!redirects Cayley-Dickson double]]
[[!redirects Cayley–Dickson double]]
[[!redirects Cayley--Dickson double]]

[[!redirects Cayley-Dickson doubles]]
[[!redirects Cayley–Dickson doubles]]
[[!redirects Cayley--Dickson doubles]]

[[!redirects Cayley-Dickson double algebra]]
[[!redirects Cayley–Dickson double algebra]]
[[!redirects Cayley--Dickson double algebra]]

[[!redirects Cayley-Dickson double algebras]]
[[!redirects Cayley–Dickson double algebras]]
[[!redirects Cayley--Dickson double algebras]]

[[!redirects double of an algebra with involution]]
[[!redirects double of algebra with involution]]
[[!redirects doubles of algebras with involution]]

[[!redirects double of a star-algebra]]
[[!redirects doubles of star-algebras]]
[[!redirects double of a *-algebra]]
[[!redirects doubles of *-algebras]]

[[!redirects Cayley-Dickson algebra]]
[[!redirects Cayley-Dickson algebras]]
[[!redirects Cayley–Dickson algebra]]
[[!redirects Cayley–Dickson algebras]]
[[!redirects Cayley--Dickson algebra]]
[[!redirects Cayley--Dickson algebras]]

[[!redirects Dickson double]]
[[!redirects Dickson doubles]]
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[[!redirects Dickson double constructions]]


cd-structure

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

#Contents#
* table of contents
{:toc}

## Idea

Certain [[Grothendieck topologies]] on a category with an [[initial object]] can be presented by a very simple structure called a _cd-structure_.

A [[Grothendieck topology]] is "completely decomposable" if it is generated by [[covering families]] consisting of [[pairs]] of maps which are part of a distinguished [[commuting square]]. The collection of these distinguished squares is then called a "cd-structure".

The [[descent]]-condition for [[simplicial presheaves]] (their [[infinity-stack|$\infty$-stack]]-condition) over completely decomposable topologies may be expressed by a [[Brown-Gersten property]], hence a generalized [Mayer-Vietoris property](Whitehead-generalized+cohomology#MayerVietorisSequenceInGeneralizedCohomology), which essentially says that a simplicial presheaf takes distinguished squares to [[homotopy pullback]]-squares. Typically this is much easier to check than the generic (homotopy-)descent condition.

## Definition

+-- {: .num_defn}
###### Definition
A *cd-structure* $\chi$ on a [[category]] $C$ with an [[initial object]] is a class of [[commutative squares]] which is stable by isomorphism.
We will call its elements **$\chi$-distinguished squares**.
=--

## Properties

Any cd-structure gives rise in a canonical way to a [[Grothendieck topology]] on $C$.

+-- {: .num_defn}
###### Definition
Let $\chi$ be a cd-structure on $C$.  The associated [[Grothendieck topology]] $\tau_\chi$ on $C$ is the coarsest topology such that:

* the empty sieve covers $\emptyset_C$, the initial object of $C$;
* $\{ b \to d, c \to d \}$ generates a covering sieve for all $\chi$-distinguished squares
$$
\begin{matrix}
  a& \to & b \\
  \downarrow& &\, \downarrow p\\
  c &\underset{j}{\to} & d
\end{matrix}
$$
=--

+-- {: .num_defn}
###### Definition
A cd-structure $\chi$ is called **complete** if every [[morphism]] whose target is an [[initial object]] of $C$ is an isomorphism, and if it is stable by [[base change]] along arbitrary morphisms of $C$.
=--

+-- {: .num_defn}
###### Definition
A cd-structure $\chi$ is called **regular** if (i) each $\chi$-distinguished square $Q$ is [[cartesian square|cartesian]], (ii) the lower horizontal morphism $j : c \to d$ is a [[monomorphism]], and (iii) for each $Q \in \chi$, the induced commutative square
$$
\begin{matrix}
  a& \to & b \\
  \downarrow& &\, \downarrow\\
  a \times_c a &\to & b \times_d b
\end{matrix}
$$
belongs to $\chi$.
=--

+-- {: .num_prop}
###### Proposition
Let $\chi$ be a complete cd-structure.  If $F$ is an [[(∞,1)-presheaf]] on $C$ that maps initial objects of $C$ to terminal objects, and sends $\chi$-distinguished squares to [[cartesian squares]], then $F$ is a $\tau_\chi$-[[(∞,1)-sheaf]].

If $\chi$ is further regular, then the converse is also true.
=--

For presheaves of *sets*, this is [Voevodsky 2010, Lem. 2.9 and Prop. 2.15](#Voevodsky10). For general (∞,1)-presheaves, this is a rephrasing of [Voevodsky 2010, Cor. 5.10](#Voevodsky10).

## Examples

There are various interesting cd-structures on the category of [[schemes]] over a base $S$, which give rise to [[Grothendieck topologies]] like

* the [[Zariski topology]]. Here a square is distinguished iff the cospan part formed by [[open embeddings]], which together form an [[open cover]], and the square is a [[pullback square]].

* the [[Nisnevich topology]];

* the [[cdh topology]].

## See also

* [[Grothendieck topology]]
* [[Nisnevich topology]]
* [[cdh topology]]

## References

* Brad Drew, *Descente: Nisnevich et cdh*, Groupe de travail at Université Paris 13 (Spring 2010) [[pdf](http://www.math.univ-paris13.fr/~kelly/GdT/exposeVII.pdf)]

* {#Voevodsky10} [[Vladimir Voevodsky]], Section 2 of: *Homotopy theory of simplicial presheaves in completely decomposable topologies*, Journal of Pure and Applied Algebra **214** 8 (2010) 1384-1398  [[arXiv:0805.4578](https://arxiv.org/abs/0805.4578), [doi:10.1016/j.jpaa.2009.11.004](https://doi.org/10.1016/j.jpaa.2009.11.004)]

* [[Vladimir Voevodsky]], *Unstable motivic homotopy categories in Nisnevich and cdh-topologies*, Journal of Pure and Applied Algebra **214** 8 (2010) 1399-1406 [[arXiv:0805.4576](https://arxiv.org/abs/0805.4576)]

[[!redirects cd-structures]]
[[!redirects cd structure]]
[[!redirects cd structures]]
[[!redirects cd-topology]]
[[!redirects cd-topologies]]
[[!redirects cd topology]]
[[!redirects cd topologies]]

[[!redirects completely decomposable Grothendieck topology]]
[[!redirects completely decomposable Grothendieck topologies]]

[[!redirects completely decomposable topology]]
[[!redirects completely decomposable topologies]]


ceasing

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### Context
#### Philosophy
+-- {: .hide}
[[!include philosophy - contents]]
=--
=--
=--

## Idea

According to _[[Science of Logic]]_ §178, ceasing is [[becoming]] the other way around, hence is the [[unity of opposites]]

[[nothing]] $\dashv$ [[being]] : ceasing

| |  | [[Dasein]] | |  |
|--|--|--|--|--|
| [[becoming]] : | [[nothing]] | $\;\;\;\dashv$ | [[being]] | : [[ceasing]] |

[[!redirects Vergehen]]
[[!redirects vergehen]]


Cecilia Flori

* [webpage](https://www.perimeterinstitute.ca/people/cecilia-flori-0)

## Selected writings

On [[homotopy type theory]]:

* [[Cecilia Flori]], [[Tobias Fritz]], *Homotopy Type Theory: Univalent Foundations of Mathematics*, lecture notes (2014) [[pdf](http://tobiasfritz.science/2014/HoTT_lecturenotes.pdf), [webpage](http://tobiasfritz.science/2014/hott.html)]

## Related entries

* [[Bohr topos]]

category: people


celestial sphere

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#### Riemannian geometry
+--{: .hide}
[[!include Riemannian geometry - contents]]
=--
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
#### Gravity
+-- {: .hide}
[[!include gravity contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

Given a point in 4-dimensional [[Minkowski spacetime]], its _celestial_ (or _heavenly_) _sphere_ is the  space of lines in its [[light cone]], hence the [[projective space]] of its [[light cone]].

We can equivalently speak of rays in the past light cone (or rays in the future light cone); then your celestial sphere (the one around the point where your head is) is the sphere of which you directly perceive a portion when you look. (Since our eyes face forward, we actually see only a small portion of this sphere, but some birds see the entire sphere.) If you take the point to be Earth, then this celestial sphere is the sphere of the heavens as it appeared to the ancients.

## Properties

### Spinorial coordinates

By the [exceptional spin isomorphism](spin+group#ExceptionalIsomorphisms)

$$
  Spin(3,1) \simeq SL(2,\mathbb{C})
$$

one may identify points $(x^i) = (x^0, x^1, x^2, x^3)$ in [[Minkowski spacetime]] with [[Hermitean matrices]]

$$
  \left(x^{\alpha \beta}\right)
  \coloneqq
  (x^i \gamma_i^{\alpha \beta})
  =
  \tfrac{1}{\sqrt{2}}
  \left(
    \array{
       x^0 + x^3 & x^1 + i x^2
       \\
       x^1 - i x^2 & x^0 - x^3
    }
  \right)
$$

(where $\gamma_i$ denote the generators of the [[Clifford algebra]] given by the [[Pauli matrices]]). This is such that the [[Lorentz metric]] [[norm]] is just the [[determinant]] of this [[matrix]]

$$
  \Vert \left(x^i\right) \Vert
   =
   2 det\left(\left(x^{\alpha \beta} \right)\right)
  \,.
$$

From this one finds that $\left(x^i\right)$ is [[lightlike]] precisely if there is a [[spinor]] $\kappa$, hence a pair of [[complex numbers]] $\xi, \eta \in \mathbb{C}$

$$
  \left(\kappa^a\right) =
  \left(
    \array{
      \xi
      \\
      \eta
    }
  \right)
  \,,
$$

such that

$$
  x^{\alpha \beta} = \kappa^\alpha \overline{\kappa}^{\beta}
  \,.
$$

Therefore the celestial sphere is equivalently the space of such pairs of complex numbers, modulo rescaling $\kappa \mapsto c \kappa$ for $0 \neq c \in \mathbb{C}$. This identifies the celestial sphere with the [[complex projective space]]

$$
  CelestialSphere \simeq \mathbb{C}P^1
  \,,
$$

the [[Riemann sphere]].

### As a coset space of the Lorentz group

The celestial sphere may be given as a [[coset space]] of the [[Lorentz group]]. For the moment see [here](sphere#AsQuotientsOfLorentzGroups) at *[[n-sphere]]*.

## Related concepts

* [[twistor space]]

* [[Moebius transformation]]

* [[celestial amplitude]]

## References

* Blagoje Oblak, _From the Lorentz Group to the Celestial Sphere_ ([arXiv:1508.00920](http://arxiv.org/abs/1508.00920))

See also:

* Wikipedia, *[Celestial sphere](https://en.wikipedia.org/wiki/Celestial_sphere)*

[[!redirects celestial sphere]]
[[!redirects celestial spheres]]
[[!redirects heavenly sphere]]
[[!redirects heavenly spheres]]


cell complex

> For more details see also at _[[CW-complex]]_.

+-- {: .rightHandSide}
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#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _cell complex_ is an [[object]] in a [[category]] which is obtained by successively "gluing cells" via [[pushouts]].

## Definition

Let $C$ be a [[category]] with [[colimits]] and equipped with a [[set]] $\mathcal{I} \subset Mor(C)$ of [[morphisms]].

In practice $C$ is usually a [[cofibrantly generated model category]] with set $\mathcal{I}$ of generating cofibrations and set $\mathcal{J}$ of acyclic generating cofibrations.

An _$\mathcal{I}$-cell complex_ in $C$ is an [[object]] $X$ which is connected to the [[initial object]] $\emptyset \to X$  by a [[transfinite composition]] of [[pushouts]] of the generating cofibrations in $\mathcal{I}$.

A _relative $\mathcal{I}$-cell complex_ (relative to any object $A$) is any morphism $A \to X$ obtained like this starting from $A$.

A _finite cell complex_ or _countable cell complex_ is a cell complex with a [[finite set]] or a [[countable set]] of cells, respectively.

## Examples

* A [[CW-complex]] is a cell complex in [[Top]] with respect to the generating cofibrations in the standard [[model structure on topological spaces]].

* Every [[simplicial set]] is a cell complex with respect to the generating cofibrations in the standard [[model structure on simplicial sets]].

* A [[Sullivan model]] is a cell complex with respect to the generating cofibrations in the standard [[model structure on dg-algebras]].

* A [[cell spectrum]] is a cell complex in the category of topological [[sequential spectra]].

## Related concepts

* [[cellular category]]

* [[cellular map]]

* [[CW complex]]

* [[vertex]], [[edge]], [[face]]

* [[dimension of a cell complex]]

* [[locally finite cell complex]]

[[!include universal constructions of topological spaces -- table]]

## References

Textbook account:

* {#FritschPiccinini90} [[Rudolf Fritsch]], [[Renzo Piccinini]], _Cellular structures in topology_, 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))

A discussion in the context of [[algebraic model categories]] is in

* [[Emily Riehl]], _Cellularity in algebraic model structures_ ([blog post](http://golem.ph.utexas.edu/category/2011/12/cellularity_in_algebraic_model.html#c040417))

[[!redirects cell complexes]]

[[!redirects relative cell complex]]
[[!redirects relative cell complexes]]

[[!redirects cellular complex]]
[[!redirects cellular complexes]]

[[!redirects finite cell complex]]
[[!redirects finite cell complexes]]

[[!redirects countable cell complex]]
[[!redirects countable cell complexes]]

[[!redirects cell]]
[[!redirects cells]]

[[!redirects n-cell]]
[[!redirects n-cells]]


cell object

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{:toc}

## Definition

For $(L \dashv R) : C \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\to}} D$ a pair of [[adjoint functor]]s between  [[model categories]] $C$ and $D$, an [[object]] $X \in C$ is an $L$-**cell object** if it is connected to the [[initial object]] by a (possibly [[transfinite composition|transfinite]]) composition of [[pushout]]s of morphisms of the form $L(f)$.

## References

Definition 3.2 in

* [[Stefan Schwede]], [[Brooke Shipley]], _Equivalences of monoidal model categories_ , Algebr. Geom. Topol. 3 (2003), 287--334 ([arXiv](http://arxiv.org/abs/math.AT/0209342))
{#SchwedeShipley}

[[!redirects cell objects]]


cell spectrum

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

#Contents#
* table of contents
{:toc}

## Definition

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

A **cell spectrum** is a topological [[sequential spectrum]] $X$ realized as the [[colimit]] over a sequence of spectra $\ast = X_0 \to X_1 \to X_2 \to X_3 \to \cdots$ such that there are morphisms

$$
  j_n
   \;\colon\;
   \left(
      \underset{i \in I_n}{\sqcup} \Sigma^\infty S^{q_n}
   \right)
   \longrightarrow
   X_n
$$

with $X_{n+1}= Cone(j_n)$ (the [[mapping cone]]).

A cell spectrum is a **[[CW-spectrum]]** if each attaching map $\Sigma^\infty S^{q_n}\to X_n$ factors through a $X_k \to X_n$ with $k \lt q$.

=--

(e.g. [Lewis-May-Steinberger 86, def. 5.1, def. 52](#LewisMaySteinberger86))

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

A CW-spectrum, def. \ref{CellSpectrum}, is called a _[[finite spectrum]]_ (or countable spectrum, etc.) if it has [[finite number|finitely many]] cells (countably many cells) according to def. \ref{CellOfACWSpectrum}.

=--

## Related concepts

* [[cell complex]]

## References

Discussion in the generality of [[equivariant spectra]] is in

* {#LewisMaySteinberger86} [[L. Gaunce Lewis]], [[Peter May]], and M. Steinberger (with contributions by J.E. McClure), section I.5 of _Equivariant stable homotopy theory_ Springer Lecture Notes in Mathematics Vol.1213. 1986 ([pdf](http://www.math.uchicago.edu/~may/BOOKS/equi.pdf))

[[!redirects cell spectra]]


cell structure of projective spaces

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#### Complex geometry
+--{: .hide}
[[!include complex geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The [[projective spaces]] over [[ground ring]] $\mathbb{K}$ being the [[real numbers]] $\mathbb{R}$, [[complex numbers]] $\mathbb{C}$, [[quaternions]] $\mathbb{H}$ or [[octonions]] $\mathbb{O}$ admit, respectively, [[CW-complex|CW-]]-[[cell complex]] [[mathematical structure|structure]] with a single cell in each dimension $k$, $2 k$, $4 k$ or $8k$, respectively.

We discuss the cases $\mathbb{K} \,\in\, \{\mathbb{R}, \mathbb{C}, \mathbb{H}\}$. For the case $\mathbb{K} \,=\, \mathbb{O}$ see [Lackman 19, Lemma 3.4](#Lackman19)

## Preliminaries

Let $\mathbb{K} \,\in\, \big\{ \mathbb{R}, \mathbb{C}, \mathbb{H} \big\}$
be one of the [[associative algebra|associative]] [[real normed division algebras]], hence either the [[real numbers]], or the [[complex numbers]] or the [[quaternions]]. We regard this as a [[topological ring]] with respect to the canonical underlying [[topological space|topology]] of the [[Euclidean space]] $\mathbb{R}^{ dim_{_{\mathbb{R}}}(\mathbb{K}) }$.

For $n \in \mathbb{N}$, consider

$$
  \mathbb{K}P^n
  \;\;
  \coloneqq
  \;\;
  \big(
    \mathbb{K}^{n+1} \setminus \{0\}
  \big) / \mathbb{K}^\times
$$

the $\mathbb{K}$-[[projective space]], i.e. the [[topological space|topological]] [[quotient space]] of the [[complement]] of zero in the $n+1$-fold [[Cartesian product]] of $\mathbb{K}$ by the right (say) [[multiplication]] [[action]] by the [[group of units]] $\mathbb{K}^\times \,=\, \mathbb{K} \setminus \{0\}$.

Notice the canonical [[topological subspace|subspace]] inclusion

$$
  \array{
    \mathbb{K}P^n
    &\hookrightarrow&
    \mathbb{K}P^{n+1}
    \\
    \big[
      z_0 \,:\, \cdots \,:\, z_n
    \big]
    &\mapsto&
    \big[
      z_0 \,:\, \cdots \,:\, z_n \,:\, 0
    \big]
    \,.
  }
$$

with induced [[complement]] $\mathbb{K}P^{n+1} \setminus \mathbb{K}P$ with [[topological boundary]]

$$
  \partial
  \big(
    \mathbb{K}P^{n+1} \setminus \mathbb{K}P
  \big)
  \;\;
    \coloneqq
  \;\;
  \overline{
    \big(
      \mathbb{K}P^{n+1} \setminus \mathbb{K}P
    \big)
  }
  \setminus
  \big(
    \mathbb{K}P^{n+1} \setminus \mathbb{K}P
  \big)
  \,.
$$

\linebreak

## Statement

+-- {: .num_prop #CellStructureOfKProjectiveSpace}
###### Proposition
**(cell structure of $\mathbb{K}$-projective space)**

We have a [[pushout diagram]] in [in topological spaces](Top#UniversalConstructions) of the form

$$
  \array{
    D^{ (n+1)\cdot dim_{{}_{\mathbb{R}}}(\mathbb{K}) }
    &\longrightarrow&
    \mathbb{K}P^{n+1}
    \\
    \big\uparrow
    &{}_{^{(po)}}&
    \big\uparrow
    \\
    S^{ (n+1)\cdot dim_{{}_{\mathbb{R}}}(\mathbb{K}) - 1 }
    &\longrightarrow&
    \mathbb{K}P^n
  }
$$

exhibiting $\mathbb{K}P^{n+1}$ as the result of an $(n+1)\cdot dim_{{}_{\mathbb{R}}}(\mathbb{K})$-[[cell attachment]] to $\mathbb{K}P^{n}$ with [[attaching map]] the canonical projection

$$
  \array{
     S^{(n+1)\cdot dim_{{}_{\mathbb{R}}}(\mathbb{K})-1}
     \;\simeq\;
     \big(
       \mathbb{K}^{n+1} \setminus \{0\}
     \big) / \mathbb{R}_+^\times
     &\longrightarrow&
     \big(
       \mathbb{K}^{n+1} \setminus \{0\}
     \big) / \mathbb{K}^\times
     \;\simeq\;
     \mathbb{K}P^n
     \,.
  }
$$

=--

\begin{proof}
If we coordinatize the inclusion of consecutive projective spaces as
\[
  \label{CanonicalInclusionsOfProjectiveSpacesByInsertingZeroOnTheRight}
  \array{
    \mathbb{K}P^n
    &\overset{\;\;\;}{\hookrightarrow}&
    \mathbb{K}P^{n+1}
    \\
    [z_0 \colon \cdots \colon z_n ]
    &\mapsto&
    [z_0 \colon \cdots \colon z_n \colon 0]
  }
\]
then the [[complement]] of this inclusion inherits coordinatization as
$$
  \mathbb{K}P^{n+1}
  \setminus
  \mathbb{K}P^{n}
  \;=\;
  \big\{
    \left.
    [z_0 \colon \cdots \colon z_n \colon z_{n + 1}]
    \,\right\vert\,
    z_{n + 1} \neq 0
  \big\}
  \;\subset\;
  \mathbb{K}P^{n+1}
  \,.
$$

In terms of these coordinates, observe the following [[homeomorphism]]:

\[
  \label{TheHomeomorphism}
  \array{
    \mathbb{K}P^{n+1} \setminus \mathbb{K}P^n
    &\overset{\;\;\;\simeq\;\;\;}{\longrightarrow}&
    Int
    \big(
      D^{(n+1) dim_{{}_{\mathbb{R}}}(\mathbb{K})}
    \big)
    \,\simeq\,
    \left\{
      \big(
        y_0, \cdots, y_n, (1-r)
      \big)
      \,\left\vert\,
      \array{
         r \in [0,1) \subset \mathbb{R}\,,
         \\
         \left\vert \vec y \right\vert^2 + (1-r)^2  = 1
      }
      \right.
    \right\}
    &
    \subset
    \mathbb{K}^{n+2}
    \\
    \big[
      z_0
      \,\colon\,
      \cdots
      \,\colon\,
      z_n
      \,\colon\,
      z_{n+1}
    \big]
    &\mapsto&
    \tfrac{1}{\left\vert \vec z\right\vert}
    \Big(
      z_0
      \cdot
      \tfrac{ z^\ast_{n+1} }{ \left\vert z_{n+1}\right\vert }
      \,,\,
      \cdots
      \,,\,
      z_n
      \cdot
      \tfrac{ z^\ast_{n+1} }{ \left\vert z_{n+1}\right\vert }
      \,,\,
      \left\vert z_{n+1}\right\vert
    \Big)
    \,.
  }
\]

In words this says: Given the class of a set of homogeneous coordinates with the last one non-zero, form the unique representative vector subject to the condition that:

1. its last coordinate is [[real number|real]];

1. its [[norm]] is unity.

Noticing that (eq:TheHomeomorphism) takes the [[topological boundary]] on the left to the boundary sphere as $r \to 1$ on the right, we see that the [[inverse morphism|inverse]] of this homeomorphism gives horizontal [[isomorphisms]] in a [[commuting square]] of the following form:

$$
  \array{
    D^{ (n+1)\cdot dim_{{}_{\mathbb{R}}}(\mathbb{K}) }
    &\overset{\;\;\simeq\;\;}{\longrightarrow}&
    \overline{
      \mathbb{K}P^{n+1}
      \setminus
      \mathbb{K}P^n
    }
    \\
    \big\uparrow
    &&
    \big\uparrow
    \\
    S^{ (n+1)\cdot dim_{{}_{\mathbb{R}}}(\mathbb{K}) - 1 }
    &\overset{\;\;\simeq\;\;}{\longrightarrow}&
    \partial
    \big(
      \mathbb{K}P^{n+1}
      \setminus
      \mathbb{K}P^n
    \big)
    \,.
  }
$$

But since $\mathbb{K}P^{n+1}$ is manifestly the union of its subspace $\mathbb{K}P^n$ with the [[topological closure]] of the [[complement]] of this subspace,
we have a pushout square as on the right of the following [[pasting diagram]]:

$$
  \array{
    D^{ (n+1)\cdot dim_{{}_{\mathbb{R}}}(\mathbb{K}) }
    &\overset{\;\;\simeq\;\;}{\longrightarrow}&
    \overline{
      \mathbb{K}P^{n+1}
      \setminus
      \mathbb{K}P^n
    }
    &\longrightarrow&
    \mathbb{K}P^{n+1}
    \\
    \big\uparrow
    &&
    \big\uparrow
    &{}_{^{(po)}}&
    \big\uparrow
    \\
    S^{ (n+1)\cdot dim_{{}_{\mathbb{R}}}(\mathbb{K}) - 1 }
    &\overset{\;\;\simeq\;\;}{\longrightarrow}&
    \partial
    \big(
      \mathbb{K}P^{n+1}
      \setminus
      \mathbb{K}P^n
    \big)
    &\longrightarrow&
    \mathbb{K}P^n
  }
$$

It follows that the total rectangle is a pushout (if you wish: by the [[pasting law]], using that all commuting squares with parallel isomorphisms are pushouts).
\end{proof}

+-- {: .num_prop #CellStructureOfKProjectiveSpace}
###### Corollary
**([[CW-complex]]-[[mathematical structure|structure]] on $\mathbb{K}$-[[projective spaces]])**

The $\mathbb{K}$-[[projective spaces]] appear in [[cotowers]]

$$
  \ast \,\simeq\,
  \mathbb{K}P^0
  \overset{\phantom{AAA}}{\hookrightarrow}
  \mathbb{K}P^1
  \overset{\phantom{AAA}}{\hookrightarrow}
  \mathbb{K}P^2
  \overset{\phantom{AAA}}{\hookrightarrow}
  \cdots
  \overset{\phantom{AAA}}{\hookrightarrow}
  \mathbb{K}P^\infty
$$

which at each stage $n$ exhibit [[CW-complex]]-[[mathematical structure|structure]] on $\mathbb{K}P^n$ with single cells in degree $k \cdot dim_{\mathbb{R}}(\mathbb{K})$.

=--

## Consequences
 {#Consequences}

Fixing an [[orthonormal basis]] $\mathrm{i}, \mathrm{j}, \mathrm{k} \,\in\, \mathbb{H}$ of [[imaginary number|imaginary]] [[quaternions]] with $\mathrm{i} \cdot \mathrm{j} \,=\, \mathrm{k}$ induces, in particular, a [[star-algebra]] inclusion of the [[complex numbers]] into the [[quaternions]]

\begin{xymatrix@C=20pt}
  \mathbb{C}
  \;
  \ar@{^{(}->}[rr]
  &&
  \mathbb{H}
\end{xymatrix}

and a [[direct sum]] decomposition of $\mathbb{H}$ as a $\mathbb{C}$-[[bimodule]], with right action by direct right multiplication but left action by [[complex conjugation|complex conjugate]] left multiplication in the second variable:

\begin{xymatrix@C=20pt@R=6pt}
  \mathbb{C} \oplus \mathbb{C}^\ast
  \ar[rr]^-{ \simeq }
  &&
  \mathbb{H}
  \\
  ( z, z' )
  \ar@{|->}[rr]
  &&
  z + \mathrm{j} \cdot z'
\end{xymatrix}

\begin{proposition}
  \label{CellStructureCompatibleUnderPassingFromCToH}
  Under a decomposition $\mathbb{H} \,\simeq\, \mathbb{C} \oplus \mathbb{C}^\ast$ as above, the cell structures on [[complex projective spaces]] and [[quaternionic projective spaces]] from Prop. \ref{CellStructureOfKProjectiveSpace} are compatible, in that for all $k \in \mathbb{N}$ we have a [[pasting diagram]] of the form

\begin{xymatrix@C=20pt}
  S^{4k+3}
  \;
  \ar@{^{(}->}[r]
  \ar[d]
  \ar@{}[dr]
    |-{
      \mbox{\tiny\rm(po)}
    }
  &
  D^{4k+4}
  \ar[d]
  \\
  \mathbb{C}P^{2k+1}
  \;
  \ar@{^{(}->}[r]
  \ar[d]
  \ar@{}[dr]
    |-{
      \mbox{\tiny\rm(po)}
    }
  &
  \mathbb{C}P^{2k+2}
  \ar[d]
  \\
  \mathbb{H}P^k
  \;
  \ar@{^{(}->}[r]
  &
  \mathbb{K}P^{k+1}
  \,,
\end{xymatrix}

where the top square and the total rectangle are the [[pushout]]-squares from Prop. \ref{CellStructureOfKProjectiveSpace}, while in the bottom square

1. the bottom left vertical morphism is the canonical projection (the "[[twistor fibration]]" for $k = 1$);

1. the bottom right vertical morphism is the canonical inclusion $\mathbb{C}P^{2k + 2} \hookrightarrow \mathbb{C}P^{2k+3}$ (eq:CanonicalInclusionsOfProjectiveSpacesByInsertingZeroOnTheRight) followed by that canonical projection.

In particular, it follows by the [[pasting law]] that this bottom square is also a [[pushout]].

\end{proposition}

\begin{proof}
  First, the bottom left morphism clearly has to be the claimed projection for the total left morphism to be the assumed projection.

  Next, by the [[universal property]] of the top [[pushout]], the bottom right morphism is unique once it is such as to induce the given total rectangle. So we just have to check that the total right vertical morphism factors as claimed. This is a straightforward unwinding of the construction of these morphisms in the proof of Prop. \ref{CellStructureOfKProjectiveSpace}.

The following diagram means to make this evident for the case that $k = 1$:

\begin{xymatrix@R=6pt@C=4pt}
    &&
    &&
    \left\{
    \!
    \frac{
      (z_0 + \mathrm{j} \cdot z_1,
      z'_0 + \mathrm{j} \cdot z'_1, 0)
    }{
      \left\vert
        (z_0 + \mathrm{j} \cdot z_1,
         z'_0 + \mathrm{j} \cdot z'_1)
      \right\vert
    }
    \!
    \right\}
    \ar@{=}[d]
    \;
    \ar@{^{(}->}[rr]
    &&
    \left\{
    \!
    \frac{
      (z_0 + \mathrm{j} \cdot z_1,
      z'_0 + \mathrm{j} \cdot z'_1, 1-r)
    }{
      \left\vert
        (z_0 + \mathrm{j} \cdot z_1,
         z'_0 + \mathrm{j} \cdot z'_1, 1-r)
      \right\vert
    }
    \!
    \right\}
    \ar@{=}[d]
    \\
    S^7
    \;
    \ar@{^{(}->}[rr]
    \ar[dd]
    \ar@{}[ddrr]
    &&
    D^8
    \ar[dd]
    &&
    \left\{
    \!
    \frac{
      (z_0, z_1, z'_0, z'_1, 0)
    }{
      \left\vert
        (z_0, z_1, z'_0, z'_1)
      \right\vert
    }
    \!
    \right\}
    \;
    \ar@{^{(}->}[rr]
    \ar[dd]
    &&
    \left\{
    \!
    \frac{
      (z_0, z_1, z'_0, z'_1, 1-r)
    }{
      \left\vert
        (z_0, z_1, z'_0, z'_1, 1-r)
      \right\vert
    }
    \!
    \right\}
    \ar[dd]
    \\
    \\
    \mathbb{C}P^3
    \;
    \ar@{^{(}->}[rr]
    \ar[dd]
    &&
    \mathbb{C}P^4
    \ar[dd]
    &&
    \big\{
      \!
      \scalebox{.7}{$
      \big[
        z_0 : z_1 :
        z'_0. : z'_1
      \big]
      \!
      $}
    \big\}
    \;
    \ar@{^{(}->}[rr]
    \ar[dd]
    &&
    \big\{
      \!
      \scalebox{.7}{$
      \big[
        z_0 : z_1 :
        z'_0. : z'_1 :
        1 - r
      \big]
      $}
      \!
    \big\}
    \ar[dd]
    \\
    \\
    \mathbb{H}P^1
    \;
    \ar@{^{(}->}[rr]
    &&
    \mathbb{H}P^2
    &&
    \big\{
    \!
    \scalebox{.7}{$
      \big[
        z_0 + \mathrm{j} \cdot z_1
        :
        z'_0 + \mathrm{j} \cdot z'_1
      \big]
    $}
    \!
    \big\}
    \;
    \ar@{^{(}->}[rr]
    &&
    \big\{
    \!
    \scalebox{.7}{$
      \big[
        z_0 + \mathrm{j} \cdot z_1
        :
        z'_0 + \mathrm{j} \cdot z'_1
        :
        1 - r
      \big]
    $}
    \!
    \big\}
\end{xymatrix}

Here the notation closely alludes to the construction inside the proof of Prop. \ref{CellStructureOfKProjectiveSpace}: In particular $1-r$ denotes a _real_ number (regarded inside the complex numbers or quaternions under the chosen embedding above), using that there is always a homogeneous coordinate representative with the last entry of this form. With this understood, the maps are given by sending coordinate labels "to themselves", and, if necessary, by including a last coordinate $1 - r = 0$.

\end{proof}

## Related statements

* [[zero-section into Thom space of universal line bundle is weak equivalence]]

## References

The case of [[octonionic projective space]]:

* {#Lackman19} [[Malte Lackmann]], Lemma 3.4 in: _The octonionic projective plane_ ([arXiv:1909.07047](https://arxiv.org/abs/1909.07047))

[[!redirects cell structure of projective space]]

[[!redirects cell structures of projective spaces]]
[[!redirects cell structures of projective spaces]]

[[!redirects cell structure on projective space]]
[[!redirects cell structure on projective spaces]]
[[!redirects cell structures on projective space]]
[[!redirects cell structures on projective spaces]]

[[!redirects cell structure of K-projective spaces]]
[[!redirects cell structure of K-projective space]]
[[!redirects cell structures of K-projective spaces]]
[[!redirects cell structures of K-projective space]]


cellular approximation theorem

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

# Contents #
* table of contents
{: toc}

## Idea

The **cellular approximation theorem** states that every [[continuous map]] between [[CW complexes]] (with chosen CW presentations) is [[homotopy|homotopic]] to a [[cellular map]], hence a map that respects the [[cell complex]]-structure, mapping [[n-skeleta]] to $n$-skeleta for all $n$.

This is the analogue for [[CW-complexes]] of the [[simplicial approximation theorem]]: that every continuous map between the [[geometric realizations]] of [[simplicial complexes]] is [[homotopy|homotopic]] to a map induced by a map of [[simplicial complexes]] (after subdivision).

## Statement

+-- {: .num_theorem #CellularApproximationTheorem}
###### Theorem

Given a [[continuous map]] $f \colon (X, A) \to (X', A')$ between [[relative CW-complexes]] that is cellular on a subcomplex $(Y, B)$ of $(X, A)$, there is a cellular map $g \colon (X, A) \to (X', A')$ that is [[homotopy|homotopic]] to $f$ relative to $Y$.

=--

It follows that if two cellular maps between CW-complexes are homotopic, then they are so by a cellular homotopy.

([Spanier 66, p. 404](#Spanier66), review in [May 99, Section 10.4](#May99), [Hatcher 02, Thm. 4.8](#Hatcher02))

## Applications

### Finite-dimensional universal bundles
 {#FiniteDimensionalUniversalBundles}

For $G$ a suitable [[topological group]], consider the [[universal principal bundle]] $E G \to B G$ over the [[classifying space]] equipped with some [[CW-complex]]-structure (as typically comes with its construction as a [[sequential colimit]] of [[Grassmannians]]).

Then the cellular approximation theorem (Thm. \ref{CellularApproximationTheorem}) implies at once that the [[pullback bundle|pullback]] $E_d G$ of the [[universal principal bundle]] $E G$ from $B G$ to its $d+1$-[[cellular skeleton|skeleton]]

\begin{xymatrix@=28pt}
  E_{d} G
  \ar[r]
  \ar[d]
  \ar@{}[dr]|-{\mbox{\tiny\rm(pb)}}
  & E G
  \ar[d]
  \\
  \mathrm{sk}_{d+1} B G
  \;
  \ar@{^{(}->}[r]
  &
  B G
\end{xymatrix}

is universal for $G$-[[principal bundles]] over $d$-dimensional [[cell complexes]] $X^d$ (in particular: over $d$-dimensional [[smooth manifolds]], via any of their [[smooth triangulations]]) -- in that forming [[pullback bundles|pullback]] of $E_d G$ identifies [[homotopy classes]] of maps  $X^d \to sk_{d+1} B G$ with [[isomorphism classes]] of $G$-principal bundles over $X^d$.

Beware here the required skeletal degree: On the one hand, the cellular approximation theorem gives that every isomorphism class of a $G$-principal bundle on $X^d$ is hit already by pullback from just the $d$-skeleton $sk_d B G$. But in order for the isomorphism relation of bundles to be reflected in the homotopy relation of their classifying maps one needs the $(d+1)$-skeleton: Because the [[cylinder]] $X^d \times [0,1]$ on which [[left homotopy]] of maps is defined, is $d+1$-skeletal of $X^d$ is $d$-skeletal.

Such finite-dimensional $G$-principal bundles, universal for base spaces of fixed bounded dimension, have the advantage that they carry an ordinary [[smooth manifold]]-[[mathematical structure|structure]] (instead of just a [[generalized smooth space]]-structure) and as such an ordinary [[principal connection]], which is a [[universal connection]] for bundles over fixed bounded-dimensional base spaces. In this way these finite-dimensional universal bundles serve as a foundation for [[Chern-Weil theory]] ([Chern 51 -- p. 45 and 67](Chern-Weil+theory#Chern51), [Narasimhan-Ramanan 61](Chern-Weil+theory#NarasimhanRamanan61), [Narasimhan-Ramanan 63](Chern-Weil+theory#NarasimhanRamanan63), [Sclafly 80](Chern-Weil+theory#Schlafly80)).

## Related concepts

* [[CW-approximation theorem]]

* [[Whitehead's theorem]]

## References

### General

* {#Spanier66} [[Edwin Spanier]], ch. 7. sec. 6, cor. 18 (p. 404)  of: _Algebraic topology_, Springer 1966 ([doi:10.1007/978-1-4684-9322-1](https://link.springer.com/book/10.1007/978-1-4684-9322-1))

* {#May99} [[Peter May]],  sec. 10.4 in: _[[A Concise Course in Algebraic Topology]]_, University of Chicago Press 1999 ([ISBN: 9780226511832](https://www.press.uchicago.edu/ucp/books/book/chicago/C/bo3777031.html), [pdf](http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf))

* {#Hatcher02} [[Allen Hatcher]], Theorem 4.8, [p. 349](http://pi.math.cornell.edu/~hatcher/AT/AT.pdf#page=358) in: _Algebraic topology_, Cambridge Univ. Press 2002 ([web](http://pi.math.cornell.edu/~hatcher/AT/ATpage.html))

See also

* Wikipedia, _[Cellular approximation](http://en.wikipedia.org/wiki/Cellular_approximation)_

### In equivariant homotopy theory
 {#ReferencesInEquivariantHomotopyTheory}

Cellular approximation for [[G-CW complexes]] in [[equivariant homotopy theory]] is due to:

* {#Matumoto71} [[Takao Matumoto]], Theorem 4.4 in: _On $G$-CW complexes and a theorem of JHC Whitehead_, J. Fac. Sci. Univ. Tokyo Sect. IA 18, 363-374, 1971 ([[matumoto.pdf:file]])

and, independently, due to:

* {#Illman72} [[Sören Illman]], Prop. 2.4 of: *Equivariant singular homology and cohomology for actions of compact lie groups* ([doi:10.1007/BFb0070055](https://doi.org/10.1007/BFb0070055)) In: H. T. Ku, L. N. Mann, J. L. Sicks, J. C. Su (eds.), *Proceedings of the Second Conference on Compact Transformation Groups* Lecture Notes in Mathematics, vol 298. Springer 1972 ([doi:10.1007/BFb0070029](https://link.springer.com/book/10.1007/BFb0070029))

See also

* {#Illman73} [[Sören Illman]], p. 90 (5 of 6) in: *Equivariant algebraic topology*, Annales de l'Institut Fourier, Tome 23 (1973) no. 2, pp. 87-91 ([doi:10.5802/aif.458](https://doi.org/10.5802/aif.458))

Textbook account:

* {#tomDieck87} [[Tammo tom Dieck]], Theorem II.2.1 in: _[[Transformation Groups]]_, de Gruyter 1987  ([doi:10.1515/9783110858372]( https://doi.org/10.1515/9783110858372))

[[!redirects equivariant cellular approximation theorem]]


cellular category

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
#### Limits and colimits
+--{: .hide}
[[!include infinity-limits - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A *cellular category* in the sense of [Makkai & Rosický 2014](#MakkaiRosický14) is a ([[cocomplete category|cocomplete]]) [[category]] equipped with a minimum of extra information indicating that some of its [[morphisms]] behave like [[relative cell complexes]].

## Definition

\begin{definition}
\label{CellularCategory}
([Makkai & Rosický 2014, Def. 2.1](#MakkaiRosický14))
\linebreak
A *cellular category* is:

1. a [[cocomplete category]] $\mathcal{C}$,

1. a [[class]] of morphisms $Cell(\mathcal{C}) \,\subset\, Mor(\mathcal{C})$,

such that

1. $Cell(\mathcal{C})$ contains all [[isomorphisms]],

1. $Cell(\mathcal{C})$ is closed under [[pushouts]] and [[transfinite composition]].

\end{definition}

## Examples

\begin{example}
**(model categories with their cofibrations)**
\linebreak
  Given a [[model category]] $\big(\mathcal{C}, (\mathrm{W},\, Fib, Cof) \big)$, then -- understood as equipped (just) with its [[cofibrations]] $Cof$ or (just) with its [[acyclic cofibrations]] $Cof \cap \mathrm{W}$ -- it is a cellular category in the sense of Def. \ref{CellularCategory}.

More to the point, if such a [[model category]] is [[cofibrantly generated]] by generating sets $I \,\subset\, Cof$ (and $J \,\subset\, Cof \cap \mathrm{W}$), then $\mathcal{C}$ equipped with the closure of $I$ under [[pushout]] and [[transfinite composition]] is cellular (by construction) and in this case the class $Cell(-)$ is that which one ordinarily addresses as [[relative cell complexes]] with [[attaching maps]] in $I$. (The full class of cofibrations is the re-obtained by further closing the class of relative cell complexes under forming [[retracts]]).
\end{example}

## Related concepts

* [[cell complex]], [[relative cell complex]]

* [[cofibrantly generated model category]]

* [[model structure on functors]]

* [[transferred model structure]]

* the notions of *cellular categories* and of *[[cellular model categories]]* are vaguely related, but not as systematically as the (independently chosen) terminology might suggest

## References

The notion is due to:

* {#MakkaiRosický14} [[M. Makkai]], [[J. Rosický]], _Cellular categories_, J. Pure Appl. Alg. **218** (2014) 1652-1664 [[arXiv:1304.7572](https://arxiv.org/abs/1304.7572), [doi:10.1016/j.jpaa.2014.01.005](https://doi.org/10.1016/j.jpaa.2014.01.005)]

[[!redirects cellular categories]]


* [[Ulrik Buchholtz]], [[Favonia|Kuen-Bang Hou (Favonia)]]:

  **Cellular Cohomology in Homotopy Type Theory**

  Logical Methods in Computer Science, **16** 2 (2020)

  [arXiv:1802.02191](https://arxiv.org/abs/1802.02191), [lmcs:6518](https://lmcs.episciences.org/6518)

on [[cellular cohomology]].

> **Abstract.** We present a development of [[cellular]] [[ordinary cohomology|cohomology]] in [[homotopy type theory]]. [[ordinary cohomology|Cohomology]] associates to each space a sequence of [[abelian groups]] capturing part of its structure, and has the advantage over [[homotopy groups]] in that these abelian groups of many common spaces are easier to compute in many cases. Cellular cohomology is a special kind of [[ordinary cohomology|cohomology]] designed for [[cell complexes]]: these are built in stages by [[attaching space|attaching]] [[spheres]] of progressively higher [[dimension]], and cellular cohomology defines the groups out of the combinatorial description of how spheres are attached. Our main result is that for finite cell complexes, a wide class of [[cohomology theories]] (including the ones defined through [[Eilenberg-MacLane spaces]]) can be calculated via cellular cohomology. This result was formalized in the Agda proof assistant.

category: reference


cellular homology

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

# Contents
* Table of contents
{: toc}

##Idea

Cellular homology is a very efficient tool for computing the [[ordinary homology|ordinary]] [[homology groups]] of [[topological spaces]] which are [[CW complexes]], based on the [[relative singular homology]] of their [[cell complex]]-decomposition and using [[degree of a continuous function|degree]]-computations.

Hence cellular homology uses the combinatorial structure of a CW-complex to define, first a [[chain complex]] of _celluar chains_ and then the corresponding [[chain homology]]. The resulting _cellular homology_ of a CW-complex is [[isomorphism|isomorphic]] to its [[singular homology]], hence to its [[ordinary homology]] as a topological space, and hence provides an efficient method for computing the latter.

## Definition

### CW-Complex

For definiteness and to fix notation which we need in the following, we recall the definition of _[[CW-complex]]_. The actual definition of cellular homology is [below](#CellularHomology).

For $n \in \mathbb{N}$ write

* $S^n \in $ [[Top]] for the standard $n$-[[sphere]];

* $D^n \in $ [[Top]] for the standard $n$-[[disk]];

* $S^n \hookrightarrow D^{n+1}$ for the [[continuous function]] that includes the $n$-sphere as the [[boundary]] of the $(n+1)$-disk.

Write furthermore $S^{-1} \coloneqq \emptyset$ for the [[empty set|empty]] topological space and think of $S^{-1} \to D^0 \simeq *$ as the boundary inclusion of the (-1)-sphere into the 0-disk, which is the [[point]].

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

A **[[CW complex]] of [[dimension]] $(-1)$** is the [[empty set|empty]] [[topological space]].

By [[induction]], for $n \in \mathbb{N}$ a **[[CW complex]] of [[dimension]] $n$** is a [[topological space]] $X_{n}$ obtained from

1. a $CW$-complex $X_{n-1}$ of dimension $n-1$;

1. an index set $Cell(X)_n \in Set$;

1. a set of [[continuous maps]] (the **attaching maps**) $\{ f_i \colon S^{n-1} \to X_{n-1}\}_{i \in Cell(X)_n}$

as the [[pushout]] $X_n \simeq \coprod_{j \in Cell(X)_n} D^n \coprod_{j \in Cell(X)_n S^{n-1}} X_n$

$$
  \array{
    \coprod_{j \in Cell(X)_{n}} S^{n-1} &\stackrel{(f_j)}{\to}& X_{n-1}
    \\
    \downarrow && \downarrow
    \\
    \coprod_{j \in Cell(X)_{n}} D^{n} &\to& X_{n}
  }
  \,.
$$

By this construction an $n$-dimensional CW-complex is canonical a [[filtered topological space]] with filter inclusion maps

$$
  \emptyset \hookrightarrow X_0 \hookrightarrow X_1 \hookrightarrow \cdots \hookrightarrow X_{n-1} \hookrightarrow X_n
$$

the right vertical morphisms in these pushout diagrams.

A general **[[CW complex]]** $X$ is a [[topological space]] given as the [[sequential colimit]] over a [[tower]] [[diagram]] each of whose morphisms is such a filter inclusion

$$
  \emptyset \hookrightarrow X_0 \hookrightarrow X_1 \hookrightarrow \cdots \hookrightarrow X
  \,.
$$

=--

For the following a CW-complex is all this data: the chosen [[filtered topological space|filtering]] with the chosen attaching maps.

### Cellular homology
 {#CellularHomology}

We define "[[ordinary homology|ordinary]]" cellular homology with [[coefficients]] in the [[group]] $\mathbb{Z}$ of [[integers]]. The analogous definition for other [[coefficients]] is immediate.

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

For $X$ a [[CW-complex]], def. \ref{CWComplex}, its
**cellular chain complex** $H_\bullet^{CW}(X) \in Ch_\bullet$ is the [[chain complex]] such that for $n \in \mathbb{N}$

* the [[abelian group]] of [[chains]] is the [[relative singular homology]] group of $X_n \hookrightarrow X$ relative to $X_{n-1} \hookrightarrow X$:

  $$
    H_n^{CW}(X) \coloneqq H_n(X_n, X_{n-1})
    \,,
  $$

* the [[differential]] $\partial^{CW}_{n+1} \colon H_{n+1}^{CW}(X) \to H_n^{CW}(X)$ is the [[composition]]

  $$
    \partial^{CW}_n
    \colon
    H_{n+1}(X_{n+1}, X_n)
     \stackrel{\partial_n}{\to}
    H_n(X_n)
     \stackrel{i_n}{\to}
    H_n(X_n, X_{n-1})
    \,,
  $$

  where $\partial_n$ is the [[boundary]] map of the [[singular chain complex]] and where $i_n$ is the morphism on [[relative homology]] induced from the canonical inclusion of pairs $(X_n, \emptyset) \to (X_n, X_{n-1})$.

=--

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

The composition $\partial^{CW}_{n} \circ \partial^{CW}_{n+1}$ of two differentials in def. \ref{CellularChainComplex} is indeed zero, hence $H^{CW}_\bullet(X)$ is indeed a [[chain complex]].

=--

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

On representative singular [[chains]] the morphism $i_n$ acts as the identity and hence $\partial^{CW}_{n} \circ \partial^{CW}_{n+1}$ acts as the double singular boundary, $\partial_{n} \circ \partial_{n+1} = 0$.

=--

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

By the discussion at _[Relative homology - Relation to reduced homology of quotient spaces](relative%20homology#RelationToQuotientTopologicalSpaces)_ the relative homology group $H_n(X_n, X_{n-1})$ is isomorphic to the the [[reduced homology]] $\tilde H_n(X_n/X_{n-1})$ of $X_n/X_{n-1}$.

This implies in particular that

* a **cellular $n$-chain** is a singular $n$-chain required to sit in filtering degree $n$, hence in $X_n \hookrightarrow X$;

* a **cellular $n$-cycle** is a singular $n$-chain whose singular boundary is not necessarily 0, but is contained in filtering degree $(n-2)$, hence in $X_{n-2} \hookrightarrow X$.

=--

## Properties

### Cellular chains

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

For every $n \in \mathbb{N}$ we have an [[isomorphism]]

$$
  H^{CW}_n(X)
  \coloneqq
  H_n(X_n, X_{n-1})
  \simeq
  \mathbb{Z}(Cell(X)_n)
$$

that the group of cellular $n$-chains with the [[free abelian group]] whose set of [[basis]] elements is the set of $n$-[[disks]] attached to $X_{n-1}$ to yield $X_n$.

=--

This is discussed at _[Relative homology - Homology of CW-complexes](relative%20homology#RelativeHomologyOfCWComplexes)_.

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

Thus, each cellular differential $\partial^{CW}_n$ can be described as a [[matrix]] $M$ with [[integer]] entries $M_{i j}$. Here an index $j$ refers to the attaching map $f_j \colon S^n \to X_n$ for the $j^{th}$ disk $D^{n+1}$. The integer entry $M_{i j}$ corresponds to a map

$$\mathbb{Z} \cong H_{n+1}(D^{n+1}, S^n) \to H_n(S^n) \to H_n(D^n, S^{n-1}) \cong H_n(S^n) \cong \mathbb{Z}$$

and is computed as the [[degree of a continuous function]]

$$S^n \stackrel{f_j}{\to} X_n \to X_n/(X_n - D^n) \cong D^n/S^{n-1} \cong S^n$$

where the inclusion $X_n - D^n \hookrightarrow X_n$ corresponds to the attaching map for the $i^{th}$ disk $D^n$.

=--

### Relation to singular homology

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

For $X$ a [[CW-complex]], its cellular homology $H^{CW}_\bullet(X)$ agrees with its [[singular homology]] $H_\bullet(X)$:

$$
  H^{CW}_\bullet(X)
  \simeq
  H_\bullet(X)
  \,.
$$

=--

This appears for instance as ([Hatcher, theorem 2.35](#Hatcher)). A proof is below as the proof of cor. \ref{CelluarEquivalentToSingularFromSpectralSequence}.

### Relation to the spectral sequence of the filtered singular complex
 {#RelationToSpectralSequenceOfFilteredSingularComplex}

The structure of a [[CW-complex]] on a [[topological space]] $X$, def. \ref{CWComplex} naturally induces on its [[singular simplicial complex]] $C_\bullet(X)$ the structure of a [[filtered chain complex]]:

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

For $X_0 \hookrightarrow X_1 \hookrightarrow \cdots \hookrightarrow X$ a [[CW complex]], and $p \in \mathbb{N}$, write

$$
  F_p C_\bullet(X) \coloneqq C_\bullet(X_p)
$$

for the [[singular chain complex]] of $X_p \hookrightarrow X$. The given  [[topological subspace]] inclusions $X_p \hookrightarrow X_{p+1}$ induce [[chain map]] inclusions $F_p C_\bullet(X) \hookrightarrow F_{p+1} C_\bullet(X)$ and these equip the singular chain complex $C_\bullet(X)$ of $X$ with the structure of a bounded [[filtered chain complex]]

$$
  0
  \hookrightarrow
  F_0 C_\bullet(X)
  \hookrightarrow
  F_1 C_\bullet(X)
  \hookrightarrow
  F_2 C_\bullet(X)
  \hookrightarrow
  \cdots
  \hookrightarrow
  F_\infty C_\bullet(X)
  \coloneqq
  C_\bullet(X)
  \,.
$$

(If $X$ is of finite [[dimension]] $dim X$ then this is a bounded filtration.)

Write $\{E^r_{p,q}(X)\}$ for the [[spectral sequence of a filtered complex]] corresponding to this filtering.

=--

We identify various of the pages of this spectral sequences with structures in singular homology theory.

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

* $r = 0$ -- $E^0_{p,q}(X) \simeq C_{p+q}(X_p)/C_{p+q}(X_{p-1})$ is the group of $X_{p-1}$-[[relative homology|relative (p+q)-chains]] in $X_p$;

* $r = 1$ -- $E^1_{p,q}(X) \simeq H_{p+q}(X_p, X_{p-1})$ is the $X_{p-1}$-[[relative singular homology]] of $X_p$;

* $r = 2$ -- $E^2_{p,q}(X) \simeq \left\{ \array{ H_p^{CW}(X) & for\; q = 0 \\ 0 & otherwise }  \right.$

* $r = \infty$ -- $E^\infty_{p,q}(X) \simeq F_p H_{p+q}(X) / F_{p-1} H_{p+q}(X) $.

=--

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

(...)

=--

This now directly implies the isomorphism between the cellular homology and the singular homology of a CW-complex $X$:

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

$$
  H^{CW}_\bullet(X) \simeq H_\bullet(X)
$$

=--

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

By the third item of prop. \ref{PagesInTheSpectralSequenceOfTheFilteredSingularComplex} the $(r = 2)$-page of the spectral sequence $\{E^r_{p,q}(X)\}$ is concentrated in the $(q = 0)$-row. This implies that all [[differentials]] for $r \gt 2$ vanish, since their domain and codomain groups necessarily have different values of $q$. Accordingly we have

$$
  E^\infty_{p,q}(X) \simeq E^2_{p,q}(X)
$$

for all $p,q$. By the third and fourth item of prop. \ref{PagesInTheSpectralSequenceOfTheFilteredSingularComplex} this is equivalently

$$
  G_p H_{p}(X) \simeq H^{CW}_p(X)
  \,.
$$

Finally observe that $G_p H_p(X) \simeq H_p(X)$ by the definition of the filtering on the homology as $F_p H_p(X) \coloneq image(H_p(X_p) \to H_p(X))$ and by standard properties of singular homology of [[CW complexes]] discusses at _[CW complex -- singular homology](CW+complex#SingularHomology)_.

## Software

There are convenient software implementations for large-scale computations of cellular homology: one may use [LinBox](http://www.linalg.org), [CHomP](http://chomp.rutgers.edu) or [Perseus](http://www.sas.upenn.edu/~vnanda/perseus).
=--

## Related concepts

* [[ordinary homology]]

* [[singular homology]], [[Čech homology]]

## References

[[!include cohomology -- early references]]

### General

A standard textbook account is from p. 139 on in

*  [[Allen Hatcher]], _[Algebraic topology](http://www.math.cornell.edu/~hatcher/AT/ATpage.html)_, Cambridge Univ. Press 2002,

Lecture notes include

* Lisa Jeffrey, _Homology of CW-complexes and Cellular homology_ ([pdf](http://www.math.toronto.edu/~mat1300/oldnotes/cellular-homology.pdf))

Formulation in [[homotopy type theory]]:

* [[Ulrik Buchholtz]], [[Favonia|Kuen-Bang Hou (Favonia)]], _[[Cellular Cohomology in Homotopy Type Theory]]_, Logical Methods in Computer Science, **16** 2 (2020) ([arXiv:1802.02191](https://arxiv.org/abs/1802.02191), [lmcs:6518](https://lmcs.episciences.org/6518))

[[!redirects cellular chain complex]]

[[!redirects cellular cohomology]]

[[!redirects cellular cochain complex]]

[[!redirects cellular]]


cellular map

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

#Contents#
* table of contents
{:toc}

## Idea

A [[continuous function]] between [[topological spaces]] which respects a given [[cell complex]]-[[mathematical structure|structure]] on these spaces is called a _cellular map_.

## Definition

Let $X$ and $Y$ be [[CW-complexes]] and let $X_n$ (respectively $Y_n$ ) denote the $n$-[[skeleton]] of $X$ (respectively $Y$). Then a [[continuous function]] $f:X \rightarrow Y$ is said to be _cellular_ if it takes $n$-skeletons to $n$-skeletons for all $n = 0,1,2,...,$ i.e, if

$$
  f(X_n ) \subseteq Y_n
$$

for all [[natural numbers]] $n$.

## Related concepts

* [[cell complex]]

* [[cellular homology]]

* [[simplicial set]], [[geometric realization]]

[[!redirects cellular maps]]

[[!redirects cellular function]]
[[!redirects cellular functions]]


cellular model

# Cellular models

* table of contents
{: toc}

## Definition

Let $E$ be a [[Grothendieck topos]], or more generally a [[locally presentable category]].  A **cellular model** for $E$ is a [[set]] of [[monomorphisms]] $I$ in $E$ that [[combinatorial wfs|cofibrantly generates]] a [[weak factorization system]] whose left class is precisely the monomorphisms.

## Existence

+--{: .un_theorem}
###### Theorem
A cellular model exists in any Grothendieck topos.
=--

For presheaf toposes this is [Cisinski 06, prop. 1.2.27](#Cisinski06); the result for general toposes is stated without proof as [Cisinski 03, prop. 1.2.2](#Cisinski03).  The proof below is a direct generalization of the presheaf version.

+--{: .proof}
###### Proof
Let $E=Sh(C)$ be the category of [[sheaves]] on some small [[site]] $C$, let $a : [C^{op},Set] \to Sh(C)$ be the [[associated sheaf]] functor, and for each $c\in C$ let $y_c = C(-,c)$ denote the [[representable presheaf]].  Let $I$ be a set of representatives for isomorphism classes of all monomorphisms $i:A\to B$ such that there exists an [[epimorphism]] $a y_c \to B$ in $E$ for some $c\in C$.  Since toposes are both [[well-powered category|well-powered]] and well-copowered, $I$ is a small set.

It suffices to show that every monomorphism in $E$ can be written as an $I$-[[cell complex]].  Let $f:X\to Y$ be a monomorphism and choose an surjection $w : \lambda \to  \coprod_{c\in C} Y(c)$ for some [[ordinal number]] $\lambda$.  For each $\beta\le \lambda$, let $X_\beta$ be the smallest subobject of $Y$ that contains $X$ and also $w_\alpha$ for all $\alpha\lt\beta$.  Then $X_0 = X$ and $X_\lambda = Y$, exhibiting $f$ as the transfinite composite of the inclusions $X_\beta \hookrightarrow X_{\beta+1}$, so it will suffice to show that each of these inclusions is a pushout of a morphism in $I$.

Now by the Yoneda lemma, $w_\beta\in Y(c)$ corresponds to a map of presheaves $y_c \to Y$, hence a map of sheaves $a y_c \to Y$.  Let $D_\beta$ be the [[image]] of $a y_c \to Y$ in $E$ (which may not coincide with the image in $[C^{op},Set]$, of course).  Then $X_{\beta+1} = X_\beta \cup D_\beta$ as subobjects of $Y$, so since $E$ is a [[coherent category]] we can write it as a pushout $X_{\beta+1} = X_\beta \sqcup_{(X_\beta \cap D_\beta)} D_\beta$.  However, the inclusion $X_\beta \cap D_\beta \hookrightarrow D_\beta$ is (isomorphic to) some morphism in $I$.
=--

More generally, cellular models exist in any [[locally presentable category|locally presentable]] [[coherent category]] whose monomorphisms are closed under [[transfinite composition]].  The proof is similar, using a [[strong generating set]] instead of the representables, and the fact that every locally presentable category is well-powered and well-copowered; see [Beke00, Proposition 1.12](#Beke00).

## Related pages

* [[Cisinski model structure]]

## References

* [[Denis-Charles Cisinski]], _[[joyalscatlab:Les préfaisceaux comme type d'homotopie|Les préfaisceaux comme modèles des types d'homotopie]]_, Astérisque, Volume 308, Soc. Math. France (2006), 392 pages ([pdf](http://www.math.univ-toulouse.fr/~dcisinsk/ast.pdf))
 {#Cisinski06}

* [[Denis-Charles Cisinski]], _Faisceaux localement asphériques_ (preliminary version), 2003, [pdf](http://www.mathematik.uni-regensburg.de/cisinski/mtest2.pdf)
 {#Cisinski03}

* [[Tibor Beke]], _Sheafifiable homotopy model categories_. Math. Proc. Cambridge Philos. Soc. 129 (2000), no. 3, 447--475
 {#Beke00}

[[!redirects cellular models]]


cellular model category

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

#Contents#
* table of contents
{:toc}

## Idea ##

A _cellular model category_ is a particularly convenient form of a [[model category]].

It is similar to a [[combinatorial model category]]. (For the moment, see there for more details.)

## Definition ##

A **cellular model category** is a [[cofibrantly generated model category]] such that there is a [[set]] of generating [[cofibrations]] $I$ and a set of generating [[acyclic cofibrations]] $J$, such that:

* all [[domain]] and [[codomain]] [[objects]] of elements of $I$ are [[compact objects]] relative to $I$ (in the sense of Hirschhorn);

* the domain objects of the elements of $J$ are [[small objects]] relative to $I$;

* the [[cofibrations]] are [[effective monomorphism]]s.

## Examples ##

* [[Top]] with the standard [[Quillen model structure on topological spaces]] (same for [[pointed topological spaces]])

* [[SSet]] with the standard [[Quillen model structure on simplicial sets]] (same for pointed simplicial sets).

For $C$ a cellular model category we have that

* the [[functor category]] $[D,C]$ for any [[small category]], $D$, with its projective [[global model structure on functors]] is again a cellular model category;

* for $c \in C$ any object, the [[over category]] $C/c$ is again a cellular model category.

## Applications ##

For cellular model categories $C$ that are [[proper model category|left proper model categories]] all left [[Bousfield localization]]s $L_S C$ at any set $S$ of morphisms are guaranteed to exist.

## Related concepts

* the notions of *[[cellular categories]]* and of *cellular model categories* are vaguely related, but not as systematically as the (independently chosen) terminology might suggest

* [[combinatorial model category]]

## References

Textbook account:

* {#Hirschhorn02} [[Philip Hirschhorn]], Section 12 of: _[[Model Categories and Their Localizations]]_, AMS Math. Survey and Monographs Vol 99 (2002) [[ISBN:978-0-8218-4917-0](https://bookstore.ams.org/surv-99-s/), [pdf toc](http://www.gbv.de/dms/goettingen/360115845.pdf), [pdf](http://www.maths.ed.ac.uk/~aar/papers/hirschhornloc.pdf)]

Discussion in the context of [[algebraic model categories]]:

* [[Emily Riehl]], _Cellularity in algebraic model structures_ ([blog post](http://golem.ph.utexas.edu/category/2011/12/cellularity_in_algebraic_model.html#c040417))

[[!redirects cellular model categories]]


cellular set

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

#Contents#
* table of contents
{:toc}

## Idea

A _cellular set_ is a [[presheaf]] on the [[Theta-category]] (also called the _cell category_ or _(Joyal's) disk category_), analogous to how a [[simplicial set]] is a presheaf on the [[simplex category]].

Cellular sets provide one approach to [[higher category theory|higher categories]] and abstract study of [[homotopy n-type|homotopy type]]s, for example in Joyal's approach to higher [[quasi-category|quasicategories]], which are a variety of [[(∞,n)-category|(∞,n)-categories]] (for $n=1$).

One detailed development of a model of [[(∞,n)-category|(∞,n)-categories]] in terms of presheaves on disk categories is the notion of [[Theta space]].

## Properties

### Segal condition

A cellular set that satisfies the cellular [[Segal condition]] is an [[omega-category]].

### Model category strucuture

There is a [[model category]] structure on presheaves on $\Theta_n$ which models [[(∞,n)-categories]]. See at _[[model structure on cellular sets]]_ and at _[[n-quasicategory]]_.

## Related concepts

* [[simplicial set]], [[globular set]], [[cubical set]], [[dendroidal set]]

* [[test category]]

* [[globular operad]], [[omega-category]]

## References

* [[Andre Joyal]], _Disks, duality and $\theta$-categories_ , preprint (1997).

* [[Mihaly Makkai]], Marek Zawadowski, Duality for simple $\omega$-categories and disks, Theory and Applications of Categories, Vol. 8, 2001, No. 7, pp 114-243, [link](http://www.tac.mta.ca/tac/volumes/8/n7/8-07abs.html)

* [[Clemens Berger]], _Structures cellulaires en théorie d'homotopie_, habilitation thesis, <a href="http://math.unice.fr/%7Ecberger/habil.pdf"> pdf </a>

* [[Clemens Berger]], Opérades cellulaires et espaces de lacets itérés, Ann. Inst. Fourier 46 (1996), 1125-1157.  MR 98c:55011, [pdf](http://math.unice.fr/~cberger/opercell.pdf)

* [[Clemens Berger]], _Cellular structures for $E_n$-operads_, talk at Workshop on Operads, Bielefeld (1999), [pdf](http://math.unice.fr/~cberger/cell.pdf)

* [[Clemens Berger]], _A cellular nerve for higher categories_, Adv. Math. 169 (2002), 118-175, [pdf](http://math.unice.fr/~cberger/nerve.pdf).

A sketch of some of related combinatorics "on open boxes and prisms" is in one of the chapters in Joyal's Barcelona [course](https://mat.uab.cat/~kock/crm/hocat/advanced-course/Quadern45-2.pdf).

[[!redirects cellular sets]]
[[!redirects cellular object]]
[[!redirects cellular objects]]


center

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In general, the _center_ (or _centre_) of an [[algebra|algebraic]] [[object]] $A$ is the collection of [[elements]] of $A$ which "commute with all elements of $A$."  This has a number of specific incarnations.

## Definitions

### Of groups and monoids

The original example is the __center__ $Z(G)$ of a [[group]] $G$, which is defined to be the [[subgroup]] consisting of all elements $g\in G$ such that for all elements $h\in G$ the equality $g h=h g$ holds.  The center is an [[abelian group|abelian]] subgroup, but not every abelian subgroup is in the center.  See also [[centralizer]].

This notion of center of a group can be generalized to the center of a [[monoid]] in an obvious way.

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

Let $C$ be an object in a [[2-category]]. The center of $C$, $Z(C)$ is the [[monoid]] of [[endomorphisms]] of the identity morphism, $id_C : C \rightarrow C$.
=--

One can invoke the [[Eckmann-Hilton argument]] to prove that vertical and horizontal composition agree on $Z(C)$ and are commutative.

### Of rings

The __center__ $Z(R)$ of a [[ring]] $R$ is defined to be the [[multiplicative subset]] consisting of all elements $r \in R$ such that for all elements $s \in R$, $r \cdot s = s \cdot r$ is true. $R$ is a [[commutative ring]] if $R$ is isomorphic to $Z(R)$.

### Of Lie algebras

The __center of a [[Lie algebra]]__ $L$ is an abelian Lie subalgebra $Z(L)$, consisting of all elements $ z\in L$ such that $[l,z]=0$ for all $l\in L$. There are generalizations for some other kinds of algebras.

### Of (higher) categories
  {#OfCategories}

#### Of general categories

The notion of *[[center]] of a [[monoid]]* has a [[horizontal categorification]] to a notion of *[[center of a category]]*.

For $C$ a category, its center is defined to be the [[commutative monoid]]

$$
  Z(C)
  \;\coloneqq\;
  [C,C](Id_C,Id_C)
$$

of [[endomorphism|endo]]-[[natural transformation]] of the [[identity functor]] $Id_C \,\colon\, C \to C$, i.e. the [[endomorphism monoid]] of $Id_C$ in the [[functor category]] $[C,C]$.

It is straightforward to check that this reduces to the usual definition of the center of monoid $D$ in the case that $C = \mathbf{B}(A,\cdot)$ is the corresponding [[delooping]].

* For a [[generator]] $G$ of a category $\mathcal{C}$ there is an embedding of $Z(\mathcal{C})$ into the monoid $Hom(G,G)$ given by $\eta\mapsto\eta _G$. In particular, if $Hom(G,G)$ or $Z(Hom(G,G))$ is trivial, as happens e.g. for $Set$ with $G=\ast$, then so is $Z(\mathcal{C})$
[[Hoffmann (1975)](#Hoffmann75)]

* For [[Cauchy completion|Cauchy complete]] $\mathcal{C}$ the idempotent elements of $Z(\mathcal{C})$ correspond precisely to the _quintessential localizations_ of $\mathcal{C}$ [[Johnstone (1996)](#Johnstone96)]

#### Of abelian categories

If a category carries further [[structure]] this may be inherited by its center. Notably the center of an [[additive category]] is not just a [[commutative monoid]] but a [[commutative ring]] (the [[endomorphism ring]] of its [[identity functor]]).

For more on this see at *[[center of an abelian category]]*.

#### Of higher categories

The notion of center also has a  [[vertical categorification]]:  It is easy to categorify the notion of center of a category as defined above: if $C$ is an [[n-category]], then its _center_ is the monoidal $(n-1)$-category $[C,C](Id_C,Id_C)$ of endo-transformations of its identity functor.  One expects that in general, this center will actually admit a natural structure of *braided* monoidal $(n-1)$-category, just as the center of a category is actually a commutative monoid, not merely a monoid.

For instance if $C = \mathbf{B}_\otimes \mathcal{C}$ is the [[delooping]] of a [[monoidal category]], then this center is called the _[[Drinfeld center]]_ of $(C, \otimes)$.

Generally, we can now obtain a notion of the center of a monoidal $n$-category by regarding it as a one-object $(n+1)$-category, according to the [[delooping hypothesis]].  It follows that the center of a monoidal $n$-category should naturally be a braided monoidal $n$-category.  This is known to be true when $n=0$ (the center of a monoid is a commutative monoid) and also for $n=1$ and $n=2$.

Note that a monoidal $n$-category has two different centers: if we regard it as a one-object $(n+1)$-category, then its center is a braided monoidal $n$-category, but if we regard it merely as an $n$-category, then its center is a braided monoidal $(n-1)$-category. The latter construction makes no reference to the monoidal structure. Likewise, a braided monoidal $n$-category has three different centers, depending on whether we regard it as an $n$-category, a connected $(n+1)$-category, or a 2-connected $(n+2)$-category, and so on (a $k$-tuply monoidal $n$-category has $k+1$ different centers).

It seems that in applications, however, one is usually most interested in the sort of center of a monoidal $n$-category $C$ obtained by regarding it as a one-object $(n+1)$-category, thereby obtaining a braided monoidal $n$-category.  It is in this case, and seemingly this case only, that the center comes with a natural forgetful functor to $C$, corresponding to the classical inclusion of the center of a monoid.  (For $n\gt 0$, however, this functor will not be an inclusion; the objects of the center of $C$ are objects of $C$ equipped with additional [[stuff, structure, property|structure]].)

Moreover, one expects that if we perform this "canonical" operation on a [[k-tuply monoidal n-category]] (for $k\ge 1$), the resulting braided monoidal $n$-category will actually be $(k+1)$-tuply monoidal.  This is known to be true in the cases $n\le 4$: the center of a braided monoidal category is symmetric monoidal, the center of a braided monoidal 2-category is sylleptic, and the center of a sylleptic monoidal 2-category is symmetric.

Finally, if we decategorify further, we find that the center of a [[set]] (i.e. a [[0-category]]) is a monoidal [[(-1)-category]], i.e. the [[truth value]] "true."  This is what we ought to expect, since when $C$ is a set, there is precisely one endo-transformation of its identity endofunction (namely, the identity).

### Of $\infty$-groups

See *[[center of an ∞-group]].

## Related concepts

* [[center of a category]]

  [[center of an additive category]]

  [[Drinfeld center]]

* [[central product of groups]]

## References

See also

* Wikipedia, _Center (group theory)_

On the notion of [[center of a category]]:

* {#Hoffmann75} [[Rudolf-E. Hoffmann]], *Über das Zentrum einer Kategorie*, Math. Nachr. **68** (1975) 299-306 [[doi:10.1002/mana.19750680122](https://doi.org/10.1002/mana.19750680122)]

* {#Johnstone96} [[Peter Johnstone]], _Remarks on Quintessential and Persistent Localizations_, TAC **2** 8 (1996) 90-99 [[tac:2-08](http://www.tac.mta.ca/tac/volumes/1996/n8/2-08abs.html), [pdf](http://www.tac.mta.ca/tac/volumes/1996/n8/n8.pdf)]

and of an [[enriched category]]:

* [[Harald Lindner]]. _Center and trace_, Archiv der Mathematik 35 (1980): 476-496. ([doi:10.1007/BF01235372](https://doi.org/10.1007/BF01235372))

[[!redirects centre]]
[[!redirects centers]]
[[!redirects centres]]

[[!redirects center of a group]]
[[!redirects center of a monoid]]