CW approximation

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#### Homotopy theory
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[[!include homotopy - contents]]
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#### Topology
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[[!include topology - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

For every [[topological space]] $X$ there is a [[CW complex]] $Z$ and a [[weak homotopy equivalence]] $f  \colon Z\to X$. Such a map $f  \colon Z\to X$ is called a **CW approximation** to $X$.

Such CW-approximation may be constructed case-by-case by iteratively [[cell attachment|attaching cells] (starting from the [[empty space]]]) for each representative of a [[homotopy group]] of $X$ and further cells to kill off spurious homotopy groups introduced this way (e.g. [Hatcher, p. 352-353](#Hatcher)).

In the [[classical model structure on topological spaces]] $Top_{Quillen}$, the [[cofibrant objects]] are the [[retracts]] of [[cell complexes]], and hence CW approximations are in particular cofibrant replacements in this model structure.

The [[Quillen equivalence]] $Top_{Quillen} \stackrel{\overset{{\vert - \vert}}{\longleftarrow}}{\underset{Sing}{\longrightarrow}} sSet_{Quillen}$ to the [[classical model structure on simplicial sets]] ("[[homotopy hypothesis]]") yields a _functorial_ CW approximation (by [this proposition](geometric+realization#mono)) via

$$
  X \mapsto {\vert Sing X\vert}
$$

([[geometric realization]] of the [[singular simplicial complex]] of $X$) with the [[adjunction counit]]

$$
  {\vert Sing X\vert}
  \overset{\in W}{\longrightarrow}
  X
$$

a weak homotopy equivalence.

## Statement

### For topological spaces

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

Let $f \;\colon\; A \longrightarrow X$ be a [[continuous function]] between [[topological spaces]]. Then there exists for each $n \in \mathbb{N}$ a [[relative CW-complex]] $\hat f \colon A \hookrightarrow \hat X$ together with an [[extension]] $\phi \colon \hat X \to X$, i.e.

$$
  \array{
    A &\overset{f}{\longrightarrow}& X
    \\
    {}^{\mathllap{\hat f}}\downarrow & \nearrow_{\mathrlap{\phi}}
    \\
    \hat X
  }
$$

such that $\phi$ is [[n-connected continuous function|n-connected]].

Moreover:

* if $f$ itself is [[n-connected continuous function|k-connected]], then the relative CW-complex $\hat f$ may be chosen to have cells only of [[dimension]] $k + 1 \leq dim \leq n$.

* if $A$ is already a [[CW-complex]], then $\hat f \colon A \to X$ may be chosen to be a subcomplex inclusion.

=--

([tomDieck 08, theorem 8.6.1](#tomDieck08))

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

For every [[continuous function]] $f \colon A \longrightarrow X$ out of a [[CW-complex]] $A$, there exists a [[relative CW-complex]] $\hat f \colon A \longrightarrow \hat X$ that factors $f$ followed by a [[weak homotopy equivalence]]

$$
  \array{
     A && \overset{f}{\longrightarrow} && X
     \\
     & {}_{\mathllap{\hat f}}\searrow && \nearrow_{\mathrlap{{\phi} \atop {\in WHE}}}
     \\
     && \hat X
  }
  \,.
$$

=--

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

Apply lemma \ref{nConnectedCWApproximationOfContinuousFunction} iteratively for $n \in \mathbb{N}$ to produce a sequence with [[cocone]] of the form

$$
  \array{
    A &\overset{f_0}{\longrightarrow}& X_0 &\overset{f_2}{\longrightarrow}& X_1 &\longrightarrow & \cdots
    \\
    &{}_{\mathllap{f}}\searrow & \downarrow^{\mathrlap{\phi_0}} & \swarrow_{\mathrlap{\phi_1}} & \cdots
    \\
    && X
  }
  \,,
$$

where each $f_n$ is a [[relative CW-complex]] adding cells exactly of dimension $n$, and where $\phi_n$ in [[n-connected continuous function|n-connected]].

Let then $\hat X$ be the [[colimit]] over the sequence (its [[transfinite composition]]) and $\hat f \colon A \to X$ the induced component map. By definition of relative CW-complexes, this $\hat f$ is itself a relative CW-complex.

By the [[universal property]] of the colimit this factors $f$ as

$$
  \array{
    A &\overset{f_0}{\longrightarrow}& X_0 &\overset{f_1}{\longrightarrow}& X_1 &\longrightarrow & \cdots
    \\
    &{}_{\mathllap{}}\searrow & \downarrow^{\mathrlap{}} & \swarrow_{\mathrlap{}} & \cdots
    \\
    && \hat X
    \\
    && \downarrow^{\mathrlap{\phi}}
    \\
    && X
  }
  \,.
$$

Finally to see that $\phi$ is a weak homotopy equivalence: since [[n-spheres]] are [[compact topological spaces]], then every map $\alpha \colon S^n \to \hat X$ factors through a finite stage $i \in \mathbb{N}$ as $S^n \to X_i \to  \hat X$ (by [this lemma](Introduction+to+Stable+homotopy+theory+--+P#CompactSubsetsAreSmallInCellComplexes)). By possibly including further into higher stages, we may choose $i \gt n$.  But then the above says that further mapping along $\hat X \to X$ is the same as mapping along $\phi_i$, which is $(i \gt n)$-connected and hence an isomorphism on the homotopy class of $\alpha$.

=--

### For sequential topological spectra

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

For $X$ any [[sequential spectrum]] in [[Top]], then there exists a [[CW-spectrum]] $\hat X$ and a homomorphism $\phi \colon  \hat X \to X$ which is degreewise a [[weak homotopy equivalence]], hence in particular a [[stable weak homotopy equivalence]].

=--

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

First let $\hat X_0 \longrightarrow X_0$ be a CW-approximation of the component space in degree 0, via prop. \ref{CWApproximationForContinuousFunctions}. Then proceed by [[induction]]: suppose that for $n \in \mathbb{N}$ a [[CW-approximation]] $\phi_{k \leq n} \colon  \hat X_{k \leq n} \to X_{k \leq n}$ has been found such that all the structure maps are respected. Consider then the continuous function

$$
  \Sigma \hat X_n \overset{\Sigma \phi_n}{\longrightarrow} \Sigma X_n \overset{\sigma_n}{\longrightarrow} X_{n+1}
  \,.
$$

Applying prop. \ref{CWApproximationForContinuousFunctions} to this function factors it as

$$
  \Sigma X_n \hookrightarrow \hat X_{n+1} \overset{\phi_{n+1}}{\longrightarrow} X_{n+1}
  \,.
$$

Hence we have obtained the next stage of the CW-approximation. The respect for the structure maps is just this factorization property:

$$
  \array{
    \Sigma \hat X_n
      &\overset{\Sigma \phi_n}{\longrightarrow}&
    \Sigma X_n
    \\
    {}^{incl}\big\downarrow && \big\downarrow^{\mathrlap{\sigma_n}}
    \\
    \hat X_{n+1} &\underset{\phi_{n+1}}{\longrightarrow}& X_{n+1}
  }
  \,.
$$

=--

## Related concepts

* in [[stable homotopy theory]]: [CW-approximation for sequential spectra](CW-spectrum#CWApproximation)

* in [[equivariant homotopy theory]]: [[G-CW approximation]]

* [[Whitehead's theorem]]

* [[cellular approximation theorem]]

## References

* {#tomDieck08} [[Tammo tom Dieck]], section 8.6 of: _Algebraic topology_, EMS (2008)

* {#Hatcher} [[Allen Hatcher]], _Algebraic topology_, Cambridge Univ. Press 2002; Chapter 4, [Section 4.1 pdf](http://www.math.cornell.edu/~hatcher/AT/AT-CWapprox.pdf)

[[!redirects CW-approximation]]

[[!redirects CW approximations]]
[[!redirects CW-approximations]]

[[!redirects CW approximation theorem]]
[[!redirects CW approximation theorems]]

[[!redirects CW-approximation theorem]]
[[!redirects CW-approximation theorems]]


CW complex

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###Context###
#### Topology
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[[!include topology - contents]]
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#### Homotopy theory
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[[!include homotopy - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

A **CW-complex** is a [[nice topological space|nice]] [[topological space]] which is, or can be, built up inductively, by a process of [[attaching space|attaching]] [[n-disks]] $D^n$ along their [[boundary]] [[n-spheres|(n-1)-spheres]] $S^{n-1}$ for all $n \in \mathbb{N}$: a [[cell complex]] built from the basic topological cells $S^{n-1} \hookrightarrow D^n$.

Being, therefore,  essentially combinatorial objects, CW complexes are the principal objects of interest in [[algebraic topology]]; in fact, most spaces of interest to algebraic topologists are [[homotopy equivalence|homotopy equivalent]] to CW-complexes. Notably the [[geometric realization]] of every [[simplicial set]], hence also of every [[groupoid]], [[2-groupoid]], etc., is a CW complex.

Milnor has argued that the category of spaces which are homotopy equivalent to CW-complexes, also called [[m-cofibrant spaces]], is a [[nice category of spaces|convenient category of spaces]] for [[algebraic topology]].

Also, CW complexes are among the [[cofibrant objects]] in the [[classical model structure on topological spaces]].
In fact, _every_  topological space is _[[weak homotopy equivalence|weakly homotopy equivalent]]_ to a CW-complex (but need not be [[homotopy equivalence|strongly homotopy equivalent]] to one). See also at _[[CW-approximation]]_.  Since every topological space is a [[fibrant object]] in this [[model category]] structure, this means that the [[full subcategory]] of [[Top]] on the CW-complexes is a category of "homotopically very good representatives" of [[homotopy types]]. See at _[[homotopy theory]]_ and _[[homotopy hypothesis]]_ for more on this.

+-- {: .num_remark}
###### Remark
**(origin of the "CW" terminology)**

The terminology "CW-complex" goes back to [[John Henry Constantine Whitehead]] (and see the discussion in [Hatcher, "Topology of cell complexes", p. 520](#HatcherTopologyOfCellComplexes)).

To quote from the original paper, which was "an address delivered before the Princeton Meeting of the (American Mathematical) Society on November
2, 1946", Whitehead states:

>In this presentation we abandon [[simplicial complexes]] in favor of [[cell complexes]]. This first part consists of geometrical preliminaries, including some elementary propositions concerning what we call closure finite complexes with weak topology, abbreviated to CW-complexes, ...

Thus the "CW" stands for the following two properties shared by any CW-complex:

* **C** = "closure finiteness": a [[compact subset]] of a CW-complex intersects the [[interior]] of only finitely many cells ([prop.](classical+model+structure+on+topological+spaces#CompactSubsetsAreSmallInCellComplexes)), hence in particular so does the closure of any cell.

* **W** = "weak topology": Since a CW-complex is a [[colimit]] in [[Top]] over its cells, and as such equipped with the [[final topology]] of the cell inclusion maps, a subset of a CW-complex is open or closed precisely if its restriction to (the closure of) each cell is open or closed, respectively.

(Whitehead called the [[interior]] of the [[n-disks]] the "cells", so that their closure of each cell is the corresponding $n$-disk.)

=--

\linebreak

## Definition
  {#Definition}

In the following let [[Top]] be the [[category]] of [[topological spaces]], or any of its variants, [[convenient category of topological spaces]].

+-- {: .num_defn #SpheresAndDisks}
###### Definition
**(spheres and disks)**

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

* $D^n \in Top$ for the [[n-disk]], for instance realized (up to [[homeomorphism]]) as the [[closed ball|closed unit ball]] in the $n$-dimensional [[Euclidean space]] $\mathbb{R}^n$ and equipped with the induced [[subspace topology]];

* $S^{n-1} \in Top$ for the [[n-sphere|(n-1)-sphere]], for instance realized as the [[boundary]] of the [[n-disk]], also equipped with the corresponding [[subspace topology]];

* $i_n \;\colon\; S^{n-1} \hookrightarrow D^n$ for the [[continuous function]] that exhibits this [[boundary]] inclusion.

  We also call these functions the _generating cofibrations_ (of the [[classical model structure on topological spaces]]).

Notice that

* $S^{-1} = \emptyset$;

* $S^0 = \ast \sqcup \ast$.

=--

+-- {: .num_defn #SingleCellAttachment}
###### Definition
**(single cell attachment)**

For $X$ any [[topological space]] and for $n \in \mathbb{N}$, then an **$n$-cell [[attaching space|attachment]]** to $X$ is the result of gluing an [[n-disk]] to $X$, along a prescribed image of its bounding [[n-sphere|(n-1)-sphere]] (def. \ref{SpheresAndDisks}):

Let

$$
  \phi \;\colon\; S^{n-1} \longrightarrow X
$$

be a [[continuous function]], then the _[[attaching space]]_

$$
  X \cup_\phi D^n \,\in Top
$$

is the topological space which is the [[pushout]] of the boundary inclusion of the $n$-sphere along $\phi$, hence the universal space that makes the following [[commuting diagram|diagram commute]]

$$
  \array{
    S^{n-1} &\stackrel{\phi}{\longrightarrow}& X
    \\
    {}^{\mathllap{\iota_n}}\downarrow &(po)& \downarrow
    \\
    D^n &\longrightarrow& X \cup_\phi D^n
  }
  \,.
$$

=--

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

If we take the defining boundary inclusion $\iota_n \colon S^{n-1} \to D^n$ itself as an attaching map, then we are gluing two $n$-disks to each other along their common boundary $S^{n-1}$. The result is the $n$-sphere:

$$
  \array{
    S^{n-1} &\overset{i_n}{\longrightarrow}& D^n
    \\
    {}^{\mathllap{i_n}}\downarrow &(po)& \downarrow
    \\
    D^n &\longrightarrow& S^n
  }
  \,.
$$

(graphics from Ueno-Shiga-Morita 95)

=--

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

A single cell [[attaching space|attachment]] of a 0-cell, according to def. \ref{SingleCellAttachment} is the same as forming the [[disjoint union space]] $X \sqcup \ast$ with the [[point]] space $\ast$:

$$
  \array{
     (S^{-1} = \emptyset) &\overset{\exists !}{\longrightarrow}&  X
     \\
     \downarrow &(po)& \downarrow
     \\
     (D^0 = \ast) &\longrightarrow& X \sqcup \ast
  }
  \,.
$$

In particular if we start with the [[empty topological space]] $X = \emptyset$ itself, then by [[attaching space|attaching]] 0-cells we obtain a [[discrete topological space]]. To this then we may attach higher dimensional cells.

=--

+-- {: .num_defn #CellAttachments}
###### Definition
**(attaching many cells at once)**

If we have a [[set]] of attaching maps $\{S^{n_i-1} \overset{\phi_i}{\longrightarrow} X\}_{i \in I}$ (as in def. \ref{SingleCellAttachment}), all to the same space $X$, we may think of these as one single continuous function out of the [[disjoint union space]] of their [[domain]] spheres

$$
  (\phi_i)_{i \in I}
    \;\colon\;
  \underset{i \in I}{\sqcup}
  S^{n_i-1}
   \longrightarrow
  X
  \,.
$$

Then the result of attaching _all_ the corresponding $n$-cells to $X$ is the pushout of the corresponding [[disjoint union]] of boundary inclusions:

$$
  \array{
    \underset{i \in I}{\sqcup} S^{n_i - 1}
      &\overset{(\phi_i)_{i \in I}}{\longrightarrow}&
    X
    \\
    \downarrow
      &(po)&
    \downarrow
    \\
    \underset{i \in I}{\sqcup} D^{n_i}
      &\longrightarrow&
    X \cup_{(\phi_i)_{i \in I}} \left(\underset{i \in I}{\sqcup} D^{n_i}\right)
  }
  \,.
$$

=--

Apart from attaching a set of cells all at once to a fixed base space, we may "attach cells to cells" in that after forming a given cell attachment, then we further attach cells to the resulting attaching space, and ever so on:

+-- {: .num_defn #RelativeCellComplexes}
###### Definition
**([[relative cell complexes]])

Let $X$ be a topological space, then
a _topological [[relative cell complex]]_ of countable height based on $X$ is a [[continuous function]]

$$
  f \colon X \longrightarrow Y
$$

and a [[sequential diagram]] of [[topological space]] of the form

$$
  X = X_{-1} \hookrightarrow X_0 \hookrightarrow X_1 \hookrightarrow X_2 \hookrightarrow \cdots
$$

such that

1. each $X_k \hookrightarrow X_{k+1}$ is exhibited as a cell attachment according to def. \ref{CellAttachments}, hence presented by a [[pushout]] diagram of the form

   $$
     \array{
       \underset{i \in I}{\sqcup} S^{n_i - 1}
         &\overset{(\phi_i)_{i \in I}}{\longrightarrow}&
       X_k
       \\
       \downarrow
         &(po)&
       \downarrow
       \\
       \underset{i \in I}{\sqcup} D^{n_i}
         &\longrightarrow&
      X_{k+1}
     }
     \,.
   $$

1. $Y = \underset{k\in \mathbb{N}}{\cup} X_k$ is the [[union]] of all these cell attachments, and $f \colon X \to Y$ is the canonical inclusion; or stated more abstractly: the map $f \colon X \to Y$ is the inclusion of the first component of the diagram into its [[colimit|colimiting cocone]] $\underset{\longrightarrow}{\lim}_k X_k$:

   $$
     \array{
       X = X_0 &\longrightarrow& X_1 &\longrightarrow& X_2 &\longrightarrow& \cdots
       \\
       & {}_{\mathllap{f}}\searrow & \downarrow & \swarrow && \cdots
       \\
       && Y = \underset{\longrightarrow}{\lim} X_\bullet
     }
   $$

If here $X = \emptyset$ is the [[empty space]] then the result is a map $\emptyset \hookrightarrow Y$, which is equivalently just a space $Y$ built form "attaching cells to nothing". This is then called just a _topological [[cell complex]]_ of countable hight.

Finally, a topological (relative) cell complex of countable hight is called a **CW-complex** if the $(k+1)$-st cell attachment $X_k \to X_{k+1}$ is entirely by $(k+1)$-cells, hence exhibited specifically by a pushout of the following form:

$$
  \array{
    \underset{i \in I}{\sqcup} S^{k}
      &\overset{(\phi_i)_{i \in I}}{\longrightarrow}&
    X_k
    \\
    \downarrow
      &(po)&
    \downarrow
    \\
    \underset{i \in I}{\sqcup} D^{k+1}
      &\longrightarrow&
    X_{k+1}
  }
  \,.
$$

A _[[finite CW-complex]]_ is one which admits a presentation in which there are only finitely many attaching maps, and similarly a _countable CW-complex_ is one which admits a presentation with countably many attaching maps.

Given a CW-complex, then $X_n$ is also called its $n$-[[skeleton]].

=--

A **cellular map** between CW-complexes $X$ and $Y$ is a [[continuous function]] $f\colon X \to Y$ such that $f(X_n) \subset Y_n$.

## Properties

### Topological properties
 {#TopologicalProperties}

\begin{prop}\label{CWComplexesAreLocallyContractible}
Every  CW-complex is a [[locally contractible topological space]].
\end{prop}
(e.g. [Hatcher 2002, prop. A.4](#Hatcher02)).

\begin{prop}
**([[CW-complexes are paracompact Hausdorff spaces]])**
\linebreak
Every CW-complex is a

1. a [[normal topological space]], in particular a [[Hausdorff space]],

1. a [[paracompact topological space]].

\end{prop}

\begin{prop}\label{CWComplexIsCompactlyGenerated}
Every CW-complex is a [[compactly generated topological space]].
\end{prop}

\begin{proof}
Since a CW-complex $X$ is a [[colimit]] in [[Top]] over attachments of standard [[n-disks]] $D^{n_i}$ (its cells), by the characterization of colimits in $Top$ ([prop.](Top#DescriptionOfLimitsAndColimitsInTop)) a subset of $X$ is open or closed precisely if its restriction to each cell is open or closed, respectively. Since the $n$-disks are compact, this implies one direction: if a subset $A$ of $X$ intersected with all compact subsets is closed, then $A$ is closed.

For the converse direction, since [[a CW-complex is a Hausdorff space]] and since [[compact subspaces of Hausdorff spaces are closed]], the intersection of a closed subset with a compact subset is closed.
\end{proof}

In fact:

\begin{prop}
  Every CW-complex is a [[Delta-generated topological space]].
\end{prop}
See there, [this Prop.](Delta-generated+topological+space#CWComplexesAreDTopologicalSpaces)

### Products of CW-complexes

If $A \hookrightarrow X$ is an inclusion of CW-complexes, then the quotient $X/A$ is naturally itself a CW-complex, such that the quotient map $X \to X/A$ is cellular.

If $X$ is a CW-complex and $K$ is a [[finite CW-complex]], then the [[product topological space]] $X \times K$ is naturally itself a CW-complex (see [Brooke-Taylor 2017](#BrookeTaylor17) for more and more generality, and see Prop. \ref{ClosureOfCWComplexesUnderCartesianProduct} below).

For example the [[suspension]] of a CW-complex itself carries the structure of a CW-complex.

Similarly for [[pointed topological space|pointed]] CW-complexes: the [[smash product]] of a pointed CW-complex with a finite pointed CW-complex is a pointed CW-complex. For example the [[reduced suspension]] $S^1 \wedge X$ of a  pointed CW-complex $X$ is itself a CW-complex.

+-- {: .num_prop #ClosureOfCWComplexesUnderCartesianProduct}
###### Proposition
**(product preserves CW-complexes in [[compactly generated topological spaces]])**
\linebreak
For $X$ and $Y$ [[CW-complexes]] with attaching maps $\{\phi_\alpha\}$ and $\{\Psi_\beta\}$, then the [[k-ification]] $(X \times Y)_c$ of their [[product topological space]] $X \times Y$ (hence their Cartesian product in the category of [[compactly generated topological spaces]]) is again a CW-complex with attaching maps $\{\Phi_\alpha \times \Psi_\beta\}$.

If either of the two CW-complexes is a [[locally compact topological space]] or if both are countable CW-complexes (have a [[countable set]] of cells) then

$$
  (X\times Y)_c \simeq X \times Y
$$

and so then the [[product topological space]] $X \times Y$ itself has CW-complex structure.

=--

(e.g. [Hatcher 2002, theorem A.6](#Hatcher02))

### Up to homotopy equivalence

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

Every CW complex is homotopy equivalent to (the [realization](simplicial+complex#geometric_realisations_and_polyhedra) of) a [[simplicial complex]].

=--

See [Gray](#Gray), Corollary 16.44 (p. 149) and Corollary 21.15 (p. 206).  For more see at _[[CW approximation]]_.

+-- {: .num_cor}
###### Corollary
Every CW complex is homotopy equivalent to a space that admits a [[good open cover]].
=--

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

If $Y$ has the [[homotopy type]] of a CW complex and $X$ is a [[finite CW complex]], then the [[mapping space]] $Y^X$ with the [[compact-open topology]] has the homotopy type of a CW complex.

=--

([Milnor 59](#Milnor59))

### Subcomplexes
 {#Subcomplexes}

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

For $X$ a CW complex, the inclusion $X' \hookrightarrow X$ of any subcomplex has an [[open neighbourhood]] in $X$ which is a [[deformation retract]] of $X'$. In particular such an inclusion is a _[good pair](relative+homology#GoodPair)_ in the sense of [[relative homology]].

=--

For instance ([Hatcher 2002, prop. A.5](#Hatcher02)).

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

For $A \hookrightarrow X$ the inclusion of a subcomplex into a CW complex, then the pair $(X,A)$ is often called a _[[CW-pair]]_. This appears notably in the [[axioms]] for [[generalized (Eilenberg-Steenrod) cohomology]].

=--

e.g. ([AGP 02, def. 5.1.11](#AGP02))

### Cellular approximation theorem
 {#CellularApproximationTheorem}

The _[[cellular approximation theorem]]_ states that every [[continuous map]] between [[CW complexes]] (with chosen CW presentations) is [[homotopy|homotopic]] to a [[cellular map]] (a map induced by a morphism of [[cell complexes]]).

This is the analogue for [[CW-complexes]] of the [[simplicial approximation theorem]] (sometimes also called lemma): 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).

For more see at _[[cellular approximation theorem]]_.

### Fibrations

Fibrations between CW-complexes also behave particularly well: [[a Serre fibration between CW-complexes is a Hurewicz fibration]].

### Singular homology
 {#SingularHomology}

We discuss aspects of the [[singular homology]] $H_n(-) \colon $ [[Top]] $\to$ [[Ab]] of CW-complexes. See also at _[[cellular homology]] of CW-complexes_.

Let $X$ be a CW-complex and write

$$
  X_0 \hookrightarrow X_1 \hookrightarrow X_2 \hookrightarrow \cdots \hookrightarrow X
$$

for its [[filtered topological space]]-structure with $X_{n+1}$ the topological space obtained from $X_n$ by gluing on $(n+1)$-cells. For $n \in \mathbb{N}$ write $nCells \in Set$ for the set of $n$-cells of $X$.

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

The [[relative singular homology]] of the filtering degrees is

$$
  H_n(X_k , X_{k-1})
  \simeq
  \left\{
    \array{
       \mathbb{Z}[nCells] & if\; k = n
       \\
       0 & otherwise
    }
  \right.
  \,,
$$

where $\mathbb{Z}[nCells]$ denotes the [[free abelian group]] on the set of $n$-cells.

=--

The proof is spelled out at _[Relative singular homology - Of CW complexes](relative+homology#RelativeHomologyOfCWComplexes)_.

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

With $k,n \in \mathbb{N}$ we have

$$
  (k \gt n) \Rightarrow (H_k(X_n) \simeq 0)
  \,.
$$

In particular if $X$ is a CW-complex of [[finite number|finite]] [[dimension of a CW-complex]] $dim X$ (the maximum degree of cells), then

$$
  (k \gt dim X) \Rightarrow (H_k(X) \simeq 0).
$$

Moreover, for $k \lt n$ the inclusion

$$
  H_k(X_n) \stackrel{\simeq}{\to} H_k(X)
$$

is an [[isomorphism]] and for $k = n$ we have an isomorphism

$$
  image(H_n(X_n) \to H_n(X)) \simeq H_n(X)
  \,.
$$

=--

This is mostly for instance in ([Hatcher 2002, lemma 2.34 b),c)](#Hatcher02)).

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

By the [[long exact sequence]] in [[relative homology]], discussed at _[Relative homology -- long exact sequences](relative+homology#LongExactSequences)_, we have an [[exact sequence]]

$$
  H_{k+1}(X_n , X_{n-1})
  \to
  H_k(X_{n-1})
  \to
  H_k(X_n)
  \to
  H_k(X_n, X_{n-1})
  \,.
$$

Now by prop. \ref{SkeletalRelativeSingularHomologyOfCW} the leftmost and rightmost homology groups here vanish when $k \neq n$ and $k \neq n-1$ and hence exactness implies that

$$
  H_k(X_{n-1})
  \stackrel{\simeq}{\to}
  H_k(X_n)
$$

is an [[isomorphism]] for $k \neq n,n-1$. This implies the first claims by [[induction]] on $n$.

Finally for the last claim use that the above exact sequence gives

$$
  H_{n-1+1}(X_n , X_{n-1})
  \to
  H_{n-1}(X_{n-1})
  \to
  H_{n-1}(X_n)
  \to
  0
$$

and hence that with the above the map $H_{n-1}(X_{n-1}) \to H_{n-1}(X)$ is surjective.

=--

## Examples
 {#Examples}

\begin{example}
Any undirected [[graph]] (loops and/or multiple edges allowed) has a geometric realization as a 1-dimensional CW complex.
\end{example}

\begin{example}
The [[geometric realization]] of any [[simplicial set]] is a CW-complex ([Milnor 57](#Milnor57)).

In particular, in the context of the [[homotopy hypothesis]] the [[Quillen equivalence]]  between [[∞-groupoids]] and [[nice topological spaces]] maps each [[∞-groupoid]] to a CW-complex.
\end{example}

\begin{example}
The [[n-spheres]] have a standard CW-complex structure, with exactly 2-cells in each dimension, obtained [[induction|inductively]] by [[space attachment|attaching]] two $n$-dimensional [[hemispheres]] to the $(n-1)$-sphere regarded as the [[equator]] in the $n$-sphere.
\end{example}

The [[infinite-dimensional sphere]] may be realized as the CW-complex which is the [[colimit]] over the resulting [[relative cell complex]]-inclusions $S^n \hookrightarrow S^{n + 1} \hookrightarrow S^{n + 2} \hookrightarrow \cdots$.
\end{example}

\begin{example}
Every [[projective space]] over the [[real numbers]], [[complex numbers]] or [[quaternions]] has the structure of a [[CW-complex]] with a single cell i in each dimension $k$, $2k$ or $4k$, respectively. See at _[[cell structure of K-projective space]]_.
\end{example}

\begin{example}
\label{SmoothManifoldsAdmitCWComplexStructure}
**(smooth manifolds)**
Every [[compact topological space|compact]] [[smooth manifold]] admits a smooth [[triangulation]] (by the [[triangulation theorem]]) and hence a CW-complex structure.

In the generality of manifolds with group actions see at  _[G-CW complex -- G-manifolds](G-CW+complex#GManifolds)_.

Every noncompact [[smooth manifold]] of [[dimension]] $n$ is [[homotopy equivalence|homotopy equivalent]] to an $(n-1)$-dimensional CW-complex. ([Napier & Ramachandran](#NapierRamachandran)).
\end{example}

## Related concepts

* [[dimension of a CW-complex]]

* [[cell complex]]

* [[cellular map]]

* [[CW approximation]]

* [[quasi-finite CW-complex]]

* [[cellular homology]]

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

* [[CW-spectrum]]

* [[G-CW complex]]

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

## References

The introduction of the term is contained in

* [[J. H. C. Whitehead]], _Combinatorial homotopy I_ , Bull. Amer. Math. Soc, 55, (1949), 213–245.

Basic textbook accounts:

* Brayton Gray, _Homotopy Theory: An Introduction to Algebraic Topology_, Academic Press, New York (1975).

* [[George Whitehead]], chapter II of _Elements of homotopy theory_, 1978

* {#May} [[Peter May]], _[[A Concise Course in Algebraic Topology]]_, U. Chicago Press (1999)

* {#Hatcher02} [[Allen Hatcher]], *Algebraic Topology*, Cambridge University Press 2002 ([ISBN:9780521795401](https://www.cambridge.org/gb/academic/subjects/mathematics/geometry-and-topology/algebraic-topology-1?format=PB&isbn=9780521795401), [webpage](https://pi.math.cornell.edu/~hatcher/AT/ATpage.html))

* {#HatcherTopologyOfCellComplexes} [[Allen Hatcher]], _Topology of cell complexes_ ([pdf](https://www.math.cornell.edu/~hatcher/AT/ATapp.pdf)) in _Algebraic Topology_

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

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

* {#AGP02} [[Marcelo Aguilar]], [[Samuel Gitler]], [[Carlos Prieto]], section 5.1 of _Algebraic topology from a homotopical viewpoint_, Springer (2002) ([toc pdf](http://tocs.ulb.tu-darmstadt.de/106999419.pdf))

Original articles include

* {#Milnor59} [[John Milnor]], _On spaces having the homotopy type of a CW-complex_, Trans. Amer. Math. Soc. 90 (2) (1959), 272-280.

* {#Milnor57} [[John Milnor]], _The geometric realization of a semi-simplicial complex_, Annals of Mathematics, 2nd Ser., __65__, n. 2. (Mar., 1957), pp. 357-362; doi:[10.2307/1969967](https://doi.org/10.2307/1969967), [Semantic scholar pdf](https://pdfs.semanticscholar.org/7cbe/0482ce422d3adcc84be80b5ab3f68520a247.pdf)

On [[products]] of CW-complexes:

* {#BrookeTaylor17} [[Andrew Brooke-Taylor]], *Products of CW complexes -- the full story*, 2017 ([pdf](http://www.math.helsinki.fi/logic/arctic/2017/Slides/BrookeTaylor_arctic2017.pdf), [[Brooke-Taylor_ProductsOfCWComplexes.pdf:file]])

See also:

 * {#NapierRamachandran} Terrance Napier,  Mohan Ramachandran,  _[Elementary Construction of Exhausting Subsolutions of Elliptic Operators](http://www.unige.ch/math/EnsMath/EM_en/)_ L'Enseignement Mathématique, t. 50 (2004), p. 367 - 389.

An inconclusive discussion [here](http://nforum.mathforge.org/discussion/4135/simplicial-homology/?Focus=33785#Comment_33785) about what parts of the definition of a CW complex should be [[stuff, structure, property|properties]] and what parts should be structure.

[[!redirects CW complex]]
[[!redirects CW-complex]]
[[!redirects CW complexes]]
[[!redirects CW-complexes]]

[[!redirects relative CW-complex]]
[[!redirects relative CW-complexes]]

[[!redirects relative CW complex]]
[[!redirects relative CW complexes]]


CW-complexes are paracompact Hausdorff spaces

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

#Contents#
* table of contents
{:toc}

## Statement and Proof

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

Every [[CW-complex]] is a *paracompactum*, i.e., a [[paracompact Hausdorff space]].

=--

For one textbook explanation, see e.g. [Fritsch-Piccinini 90, theorem 1.3.5](#FritschPiccinini90). Below we give a somewhat more [[category theory|categorically]] minded proof, linking to relevant results elsewhere in the [[nLab]].

+-- {: .proof}
###### Proof
Let $X_n$ denote the $n^{th}$ [[skeleton]] of $X$. We argue by [[induction]] that each skeleton is a paracompactum. Vacuously $X_{-1} = \emptyset$ is a paracompactum. Now suppose $X_{n-1}$ is a paracompactum, and suppose $X_n$ is formed as an [[attachment space]] with attaching map $f: \sum_{i \in I} S_i^{n-1} \to X_{n-1}$, so that

$$\array{
\sum_{i \in I} S_i^{n-1} & \stackrel{\;h\;}{\hookrightarrow} & \sum_{i \in I} D_i^{n} \\
\mathllap{f} \big\downarrow & {}^{{}_{(po)}} & \big\downarrow \mathrlap{g} \\
X_{n-1} & \underset{\;k\;}{\hookrightarrow} & X_n
}$$

is a pushout square. The embedding $h$ is a closed embedding, and so its pushout along $f$ [is a closed embedding](/nlab/show/subspace+topology#pushout) $k$. Furthermore, the spheres $S_i^{n-1}$ and disks $D_i^n$ are paracompacta since they are compact Hausdorff, and [coproducts of paracompacta are again paracompacta](https://ncatlab.org/nlab/show/paracompact+topological+space#ParacompactnessPreservedByDisjointUnion), making $h$ a closed embedding of paracompact Hausdorff spaces. By the result on [pushouts of closed embeddings of paracompacta](/nlab/show/colimits+of+paracompact+Hausdorff+spaces#ClosedEmbeddingsOfParacompactaClosedUnderPushout), it now follows that $X_n$ is a paracompactum.

Thus the CW-complex $X$ is a colimit of a sequence of closed embeddings

$$X_{-1} \hookrightarrow X_0 \hookrightarrow X_1 \hookrightarrow \ldots$$

between paracompacta. [It follows that this colimit is a paracompactum](/nlab/show/colimits+of+paracompact+Hausdorff+spaces#ColimitsOfSequencesOfParacompacta).
=--

## Related statements

* [[Hausdorff implies sober]]

* [[maps from compact spaces to Hausdorff spaces are closed and proper]]

* [[closed subsets of compact spaces are compact]]

* [[compact subspaces of Hausdorff spaces are closed]]

* [[closed subspaces of compact Hausdorff spaces are equivalently compact subspaces]]

## References
 {#References}

That every CW complexes is [[Hausdorff space|Hausdorff]] (in fact [[normal topological space|normal]]) may be [[folklore]], a proof is spelled out in:

* {#Hatcher02} [[Allen Hatcher]], Prop. A.3 of *Algebraic Topology*, Cambridge University Press 2002 ([ISBN:9780521795401](https://www.cambridge.org/gb/academic/subjects/mathematics/geometry-and-topology/algebraic-topology-1?format=PB&isbn=9780521795401), [webpage](https://pi.math.cornell.edu/~hatcher/AT/ATpage.html))

An early original article with the paracompactness statement is:

* {#Miyazaki52} Hiroshi Miyazaki, _The paracompactness of CW-complexes_, Tohoku Math. J. (2) Volume 4, Number 3 (1952), 309-313. 1952 [Euclid](https://projecteuclid.org/euclid.tmj/1178245380)

Textbook account:

* {#FritschPiccinini90} [[Rudolf Fritsch]], [[Renzo Piccinini]], Theorem 1.3.5 (p. 29 and following) of: _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))

[[!redirects every CW-complex is a Hausdorff space]]
[[!redirects every CW-complex is a Hausdorff topological space]]

[[!redirects CW-complexes are Hausdorff spaces]]
[[!redirects CW-complexes are Hausdorff]]
[[!redirects a CW-complex is a Hausdorff space]]

[[!redirects CW-complexes are paracompact]]
[[!redirects CW-complexes are paracompact spaces]]
[[!redirects CW-complexes are paracompact topological spaces]]


CW-pair

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

#Contents#
* table of contents
{:toc}

## Idea

In _[[algebraic topology]]_ by a _CW-pair_ $(X,A)$ is meant a [[CW-complex]] $X$ equipped with a sub-complex inclusion $A \hookrightarrow X$.

The concept appears prominently in the discussion of [[ordinary homology|ordinary]] [[relative homology]] and generally in the [[Eilenberg-Steenrod axioms]] for [[generalized homology]]/[[generalized cohomology]].

## Properties

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

For $X$ a [[CW complex]], the inclusion $A \hookrightarrow X$ of any subcomplex has an [[open neighbourhood]] in $X$ which is a [[deformation retract]] of $A$. In particular such an inclusion is a _[good pair](relative+homology#GoodPair)_ in the sense of [[relative homology]].

=--

For instance ([Hatcher, prop. A.5](#Hatcher)).

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

For $(X,A)$ a CW-pair, then the $A$-[[relative singular homology]] of $X$ coincides with the [[reduced singular homology]] of the [[quotient space]] $X/A$:

$$
  H_n(X , A)
  \simeq
  \tilde H_n(X/A)
 \,.
$$

=--

For instance ([Hatcher, prop. 2.22](#Hatcher)).

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

By assumption we can find a [[neighbourhood]] $A \stackrel{j}{\to} U \hookrightarrow X$ such that $A \hookrightarrow U$ has a [[deformation retract]] and hence in particular is a [[homotopy equivalence]] and so induces also isomorphisms on all [[singular homology]] groups.

It follows in particular that for all $n \in \mathbb{N}$ the canonical morphism $H_n(X,A) \stackrel{H_n(id,j)}{\to} H_n(X,U)$ is an [[isomorphism]], by [homotopy invariance](relative+homology#HomotopyInvariance) of [[relative singular homology]].

Given such $U$ we have an evident [[commuting diagram]] of pairs of [[topological spaces]]

\begin{center}
    \begin{tikzcd}
        {(X,A)} \arrow[r, "{(\mathrm{id}, j)}"] \arrow[d] & {(X,U)} \arrow[d] & {(X - A, U - A)} \arrow[l] \arrow[d, "\simeq"] \\
        {(X/A, A/A)} \arrow[r, "{(\mathrm{id}, j/A)}"]    & {(X/A, U/A)}      &{(X/A-A/A, U/A- A/A)} \arrow[l]
        \end{tikzcd}
\end{center}

Here the right vertical morphism is in fact a [[homeomorphism]].

Applying relative singular homology to this diagram yields for each $n \in \mathbb{N}$ the [[commuting diagram]] of abelian groups

\begin{center}
    \begin{tikzcd}
        {H_n(X,A)} \arrow[r, "{H_n(\mathrm{id}, j)}", "\simeq"'] \arrow[d] & {H_n(X,U)} \arrow[d] & {H_n(X - A, U - A)} \arrow[l, "\simeq"'] \arrow[d, "\simeq"] \\
        {H_n(X/A, A/A)} \arrow[r, "{H_n(\mathrm{id}, j/A)}", "\simeq"']    & {H_n(X/A, U/A)}      &{H_n(X/A-A/A, U/A- A/A)} \arrow[l, "\simeq"']
        \end{tikzcd}
\end{center}

Here the left horizontal morphisms are the above isomorphims induced from the deformation retract. The right horizontal morphisms are isomorphisms by [excision](relative%20homology#Excision) and the right vertical morphism is an isomorphism since it is induced by a homeomorphism. Hence the left vertical morphism is an isomorphism ([[2-out-of-3]] for isomorphisms).

=--

## References

* {#Hatcher} [[Allen Hatcher]], _[Algebraic Topology](http://www.math.cornell.edu/~hatcher/AT/ATpage.html)_, 2002

* {#AGP02} Marcelo Aguilar, [[Samuel Gitler]], Carlos Prieto, section 5.1 of _Algebraic topology from a homotopical viewpoint_, Springer (2002) ([toc
pdf](http://pi.math.cornell.edu/~hatcher/AT/AT.pdf))

[[!redirects CW-pairs]]

[[!redirects CW pair]]
[[!redirects CW pairs]]


CW-spectrum

+-- {: .rightHandSide}
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###Context###
#### Stable Homotopy theory
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[[!include stable homotopy theory - contents]]
=--
#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The notion of _CW-spectrum_ is the analogue for [[sequential spectra]] in [[Top]] of the concept of [[CW-complex]] for [[topological spaces]].

Just like [[CW-complexes]] are cofibrant objects in the [[classical model structure on topological spaces]], so CW-spectra are cofibrant objects in the [[stable model category|stable]] [[model structure on topological sequential spectra]], see [below](#Cofibrancy).

## Definition

### In components

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

A **CW-spectrum** $X_\bullet\in SeqSpec(Top)$ is a [[sequential spectrum]] in [[Top]] such that

1. all component spaces $X_n$ are [[CW-complexes]],

1. all structure maps $\Sigma X_n \longrightarrow X_{n+1}$ are inclusions of subcomplexes

=--

e.g. ([Adams 74, p. 139](#Adams74)) Beware that for instance ([Switzer 75, def. 8.1](#Switzer75)) says just "spectrum" for "CW-spectrum".

For a CW-spectrum $X$ there is a concept of "cell of a spectrum":

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

A **cell** of a CW-spectrum, def. \ref{CWSpectrum} is a cell of one of the components CW-complexes $X_n$, together with all its suspensions in all the higher component spaces $X_{\gt n}$, subject to the condition that the first cell itself is not itself the suspension of a cell in $X_{n-1}$.

=--

This way every CW-spectrum is the union of all its cells in the sense of def. \ref{CellOfACWSpectrum}. (e.g. [Switzer 75, 8.4](Switzer75)).

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

A CW-spectrum, def. \ref{CWSpectrum}, 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}.

=--

### Via spectrum attaching maps

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

For $n \in \mathbb{Z}$ (possibly negative) define $\mathbb{S}^n$ to be the [[sequential prespectrum]] with component spaces

$$
 (\mathbb{S}^{n})_k
   \coloneqq
 \left\{
   \array{
     S^{k+n} & if \; k + n \geq 0
     \\
     \ast & otherwise
   }
 \right.
$$

and with structure maps the canonical isomorphisms.

=--

([Lewis-May-Steinberger 86, def. 4.3](#LewisMaySteinberger86))

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

In def. \ref{SphereSpectrumOfIntegerDimension}

* for $n = 0$ then $\mathbb{S}^0 = \mathbb{S} = \Sigma^\infty S^0$ is standard sequential incarnation of the [[sphere spectrum]];

* for $n \geq n$ then $\mathbb{S}^n \simeq \Sigma^\infty S^n$ is the [[suspension spectrum]] on the [[n-sphere]];

* for general $n$ then $\mathbb{S}^n \simeq F_{-n} S^0$ is also known as the $(-n)$th [[free spectrum]] on $S^0$.

=--

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

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

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

([rmk.](Introduction+to+Stable+homotopy+theory+--+1-1#StrictModelStructureCellAttachmentToSpectra))

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. 5.2](#LewisMaySteinberger86), [Weiss](#Weiss))

### Category of CW-spectra
There is an obvious category of CW-spectra given as the full subcategory $CWSpec'$ of $SeqSpec(Top)$ consisting of the CW-spectra, ie. a morphism of CW-spectra $f: X \to Y$ is given by levelwise maps $f_n: X_n \to F_n$ that are compatible with the structure maps. In accordance with [Switzer 75, 8.9](Switzer75), we call morphisms in this sense **functions**.

However, often there is a more useful notion of morphisms between CW-spectra.

+-- {: .num_defn}
###### Definition
Let $E$ be a CW-spectrum. A **subspectrum** is a CW spectrum $F$ such that each $F_n$ is a subcomplex of $E_n$. A subspectrum is **cofinal** if for each cell $e_n \in E_n$, there is some $m$ such that $\Sigma^m e_n \in F_{n + m}$.
=--

We can then construct the category $CWSpec$ as the [[localization]] of $CWSpec'$ at the cofinal inclusions. Since the class of cofinal inclusions admit a [[calculus of right fractions]], the cateogry $CWSpec$ has the following concrete description - a morphism $X \to Y$ in $CWSpec$ is given by a function $\tilde{X} \to Y$, where $\tilde{X}$ is some cofinal subspectrum of $X$, quotiented by the relation that two such morphisms $f: \tilde{X} \to Y$ and $f': \tilde{X}' \to Y$ are considered equivalent if there is some further cofinal subspectrum $\tilde{X}'' \subseteq \tilde{X} \cap \tilde{X'}$ such that $f|_{\tilde{X}'} = f'|_{\tilde{X}''}$.

## Properties

### Cofibrancy
 {#Cofibrancy}

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

A [[sequential spectrum]] $X\in SeqSpec(Top)_{stable}$ is cofibrant in the stable [[model structure on topological sequential spectra]] in particular if all component spaces are [[cell complexes]] and all its structure morphisms $S^1 \wedge X_n \to X_{n+1}$ are [[relative cell complexes]]. In particular [[CW-spectra]], def. \ref{CellOfACWSpectrum}, are cofibrant in $SeqSpec(Top)_{stable}$.

=--

For the proof see [there](model+structure+on+topological+sequential+spectra#CellSpectraAreCofibrantInModelStructureOnTopologicalSequentialSpectra).

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

For $X\in SeqSpec(Top)_{stable}$ a [[CW-spectrum]], then its standard [[cylinder spectrum]] $X \wedge (I_+)$ is a _good_ [[cylinder object]], in that the inclusion

$$
  X \vee X \longrightarrow X \wedge (I_+)
$$

is a cofibration in $SeqSpec(Top)_{stable}$.

=--

See [this prop.](model+structure+on+topological+sequential+spectra#CylinderSpectrumOverCWSpectrumIsGood).

### CW-approximation
 {#CWApproximation}

The analog of [[CW-approximation]] for [[topological spaces]] holds true for topological [[sequential spectra]]:

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

For $X \in SeqSpec(Top)$ a topological [[sequential spectrum]], there exists a CW-spectrum $\hat X$ and a [[stable weak homotopy equivalence]]

$$
 \hat X \longrightarrow X
 \,.
$$

=--

(e.g. [Elmendorf-Kriz-May 95, theorem 1.5](#ElmendorfKrizMay95))

+-- {: .proof}
###### Proof 1

First let $\hat X_0 \longrightarrow X_0$ be a CW-approximation of the component space in degree 0, via [this prop.](CW+approximation#CWApproximationForContinuousFunctions). Then proceed by [[induction]]: suppose that for $n \in \mathbb{N}$ a [[CW-approximation]] $\phi_{k \leq n} \colon  \hat X_{k \leq n} \to X_{k \leq n}$ has been found such that all the structure maps are respected. Consider then the continuous function

$$
  \Sigma \hat X_n \overset{\Sigma \phi_n}{\longrightarrow} \Sigma X_n \overset{\sigma_n}{\longrightarrow} X_{n+1}
  \,.
$$

Applying [that prop.](CW+approximation#CWApproximationForContinuousFunctions) to this function factors it as

$$
  \Sigma X_n \hookrightarrow \hat X_{n+1} \overset{\phi_{n+1}}{\longrightarrow} X_{n+1}
  \,.
$$

Hence we have obtained the next stage of the CW-approximation. The respect for the structure maps is just this factorization property:

$$
  \array{
    \Sigma \hat X_n
      &\overset{\Sigma \phi_n}{\longrightarrow}&
    \Sigma X_n
    \\
    {}^{incl}\downarrow && \downarrow^{\mathrlap{\sigma_n}}
    \\
    \hat X_{n+1} &\underset{\phi_{n+1}}{\longrightarrow}& X_{n+1}
  }
  \,.
$$

=--

+-- {: .proof}
###### Proof 2

A high-powered way to see this is to use the [[Quillen equivalence]] between the stable [[model structure on topological sequential spectra]] and the stable [[Bousfield-Friedlander model structure]] (see there) on sequential spectra in [[simplicial sets]]. This implies that a CW-approximation is given by

$$
  {\vert Q Sing X\vert}
  \overset{\in W_{stable}}{\longrightarrow}
  X
  \,,
$$

where ${\vert - \vert} \dashv Sing$ is degreewise the adjunction between [[geometric realization]] and forming [[singular simplicial complex]], and $Q$ denotes any cofibrant replacement in the BF-model structure.

=--

## References

* {#Adams74} [[Frank Adams]], section III.2 and III.3 of _[[Stable homotopy and generalised homology]]_, 1974

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

* {#ElmendorfKrizMay95} [[Anthony Elmendorf]], [[Igor Kriz]], [[Peter May]], section 1 of _[[Modern foundations for stable homotopy theory]]_, in [[Ioan Mackenzie James]] (ed.), _[[Handbook of Algebraic Topology]]_,  1995 Amsterdam: North-Holland, pp. 213–253,   ([pdf](http://hopf.math.purdue.edu/Elmendorf-Kriz-May/modern_foundations.pdf))

* {#Weiss} [[Michael Weiss]], around pages 43-46 of _Homotopy theory and bordism theory_ ([pdf](http://www.maths.ed.ac.uk/~aar/papers/abdnbordism.pdf))

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 CW spectra]]

[[!redirects CW-spectrum]]
[[!redirects CW-spectra]]

[[!redirects CW spectrum]]


cycle

* in [[group theory]]: [[cycle of a permutation]]

* in [[homological algebra]]: cycle in a [[chain complex]]

  in [[algebraic topology]] a representative for a class in a [[homology theory]]

  [[!include chains and cochains - table]]

  e.g.

  * [[Baum-Douglas geometric cycle]]

category: disambiguation

[[!redirects cycles]]


cycle of a permutation

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#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

For $n \in \mathbb{N}$, with $Sym(n)$ denoting the [[symmetric group]] on $n$ elements, consider a [[permutation]] $\sigma \;\in\; Sym(n)$ of $n$ elements.

Write

$$
  \langle \sigma \rangle
  \;\coloneqq\;
  \big\{
    e,
    \sigma,
    \sigma^2,
    \cdots
    ,
    \sigma^{order(\sigma)-1}
  \big\}
  \;\subset\;
  Sym(n)
$$

for the [[cyclic group]] [[finitely generated group|generated]] by $\sigma$. This cyclic group (whose [[order of a group|order]] is the [[order of a group element|order]] of $\sigma$) canonically [[action|acts]] on the [[set]] $\{1, \cdots, n\}$ through the defining [[group action]] of $Sym(n)$ on this set.

Then a *cycle* of the [[permutation]] $\sigma$ is an [[orbit]] of $\langle \sigma \rangle$ with respect to this canonical [[group action]] on $\{1, \cdots, n\}$.

More explicitly, this means that a cycle of $\sigma$ is a [[subset]] of the form

$$
  \big\{
    k, \sigma(k), \sigma(\sigma(k)), \cdots
  \big\}
  \;\subset\;
  \{1,\cdots, n\}
$$

for any $1 \leq k \leq n$.

Beware that this is really meant as a [[subset]], not as a [[tuple]]. So for instance

$$
  \big\{
    \sigma(k), \sigma(\sigma(k)), \sigma^3(k)), \cdots
  \big\}
  \;\subset\;
  \{1,\cdots, n\}
$$

is the same cycle.

The permutation action restricted to the cycle is a [[cyclic permutation]].

## Related concepts

* [[Cayley distance]], [[Kendall distance]]

* [[cyclic permutation]]

* [[orbit]]

## References

See also

* Wikipedia, *[Cyclic permutation](https://en.wikipedia.org/wiki/Cyclic_permutation)*

* MathWorld, *[Permutation Cycle](https://mathworld.wolfram.com/PermutationCycle.html)*

[[!redirects cycles of a permutation]]

[[!redirects permutation cycle]]
[[!redirects permutation cycles]]


cyclic category

# The cyclic category
* table of contents
{: toc}

## Idea

The __cyclic category__ ([Connes 83](#Connes83), see [Cartier 85](#Cartier85)) typically denoted $\Lambda$  (or sometimes $\mathcal{C}$) is a [[small category]] whose [[presheaves]] -- called _[[cyclic sets]]_ or more generally _[[cyclic objects]]_  -- are somewhere intermediate between [[simplicial sets]] and [[symmetric sets]]. It strictly contains the [[simplex category]], and has [[cyclic groups]] for [[automorphism groups]]. Among its virtues, it is a self-dual category.

The cycle category is used for the description of the cyclic structure on [[Hochschild homology]]/[[Hochschild cohomology]] and accordingly for the description of [[cyclic homology]]/[[cyclic cohomology]].

## Definitions

Multiple descriptions of the cyclic category $\Lambda$ are possible, but a convenient starting point is to consider first a category $L$ whose [[objects]] are [[natural numbers]] $n \geq 0$, and where the [[hom-set]] $L(m, n)$ consists of increasing functions $f: \mathbb{Z} \to \mathbb{Z}$ satisfying the "spiraling property", that $f(i + m + 1) = f(i) + n + 1$, with [[composition]] given by ordinary function composition. The category $L$ is ([[equivalence of categories|equivalent]] to) the category $\Lambda_\infty$ called the [[paracyclic category]] by [Nikolaus and Scholze](#NS).

Then, define $\Lambda$ to be a quotient category of $L$ having the same objects, with $\Lambda(m, n) = L(m, n)/\sim$ where $\sim$ is the equivalence relation for which $f \sim g$ means $f - g$ is a constant multiple of $n+1$. Let $q: L \to \Lambda$ be the quotient.

+-- {: .num_remark #simplex}
###### Remark
Notice that $f \in L(m, n)$ is completely determined by the values $f(0), \ldots, f(m)$. There is a [[faithful functor|faithful]] [[subcategory|embedding]] $i \colon \Delta \to L$ which on objects is the identity, where $f \in L(m, n)$ belongs to the image of $i$ iff $0 \leq f(0)$ and $f(m) \leq n$. The composite

$$\Delta \stackrel{i}{\hookrightarrow} L \stackrel{q}{\to} \Lambda$$

is again [[faithful functor|faithful]], so that the [[simplex category]] sits inside $\Lambda$.
=--

+-- {: .num_remark}
###### Remark
Of course the [[successor]] function $\tau \colon \mathbb{Z} \to \mathbb{Z}$ gives a function $\tau_n \in L(n, n)$ defined by $\tau_n(i) = i+1$, which in turn induces a function $q(\tau) \in \Lambda(n, n)$ such that $q(\tau)^{n+1} = 1_n$. In this way, we have inclusions $\mathbb{Z}/(n+1) \hookrightarrow \Lambda(n, n)$ of [[cyclic groups]] inside $\Lambda$.
=--

__[[cyclic object|Cyclic objects]]__ in a category $C$ are the [[contravariant functors]] $\Lambda^{\mathrm{op}}\to C$, [[cocyclic objects]] are the [[covariant functors]] $\Lambda\to C$.  Note that $\Lambda$ itself is, via its inclusion into $Cat$, an example of a [[cocyclic object]] in $Cat$.  (Thus, the common term "the cyclic category" to refer to $\Lambda$ is misleading, just like using "the [[simplicial category]]" to refer to the [[simplex category]] $\Delta$.)

If $A$ is an [[abelian category]] then the category of $A$-presheaves on $\Lambda$ is usually called (Connes\'s) category of __cyclic modules__ in $A$.

## Structure of the cyclic category {#structure}

To analyze the structure of $\Lambda$ further, we make a series of easy observations. These are largely based on [Elmendorf 93](#Elm).

+-- {: .num_prop}
###### Proposition
Every [[morphism]] $f$ of $L$, regarded as a functor $\mathbb{Z} \to \mathbb{Z}$, has a [[left adjoint]] $f^\ast: \mathbb{Z} \to \mathbb{Z}$ that is also a morphism of $L$. Similarly, every morphism $f$ of $L$ has a [[right adjoint]] $f_\ast$ belonging to $L$.
=--

+-- {: .proof}
###### Proof
By the spiraling property of $f$, for any $j \in \mathbb{Z}$ the comma category $(j \downarrow f)$ as a subset of $\mathbb{Z}$ has a lower bound in $\mathbb{Z}$ and hence is well-ordered. It is also nonempty, and we define $f^\ast(j)$ to be the least element of $(j \downarrow f)$. In other words $f^\ast(j)$ is the least $i$ such that $j \leq f(i)$. It is easy to check that $f^\ast$ obeys the spiraling property $f^\ast(j+n+1) = f^\ast(j)+m+1$, since

$$\array{
f^\ast(j+n+1) \leq f^\ast(j)+m+1 & iff & j+n+1 \leq f(f^\ast(j)+m+1) \\
 & iff & j+n+1 \leq f(f^\ast(j))+n+1 \\
 & iff & j \leq f(f^\ast(j)) \\
 & iff & f^\ast(j) \leq f^\ast(j)
}$$

and

$$\array{
f^\ast(j)+m+1 \leq f^\ast(j+n+1) & iff & f^\ast(j) \leq f^\ast(j+n+1)-m-1 \\
 & iff & j \leq f(f^\ast(j+n+1)-m-1) \\
 & iff & j \leq f(f^\ast(j+n+1))-n-1 \\
 & iff & j + n + 1\leq f(f^\ast(j) + n + 1) \\
 & iff & f^\ast(j+n+1) \leq f^\ast(j+n+1).
}$$

Also, since $(\mathbb{Z}, \leq)$ as a category is self-dual, every morphism $f$ of $L$ dually has a right adjoint that is a morphism of $L$.
=--

+-- {: .num_cor}
###### Corollary
$L$ is a self-dual category.
=--

+-- {: .proof}
###### Proof
The [[opposite category|duality]] functor $L^{op} \to L$ is the identity on objects and takes $f: m \to n$ to $f^\ast: n \to m$. It is contravariant since the left adjoint of a composite $f g$ is $g^\ast f^\ast = (f g)^\ast$. It is an equivalence because its inverse is the right-adjoint mapping, $f \mapsto f_\ast$.
=--

+-- {: .num_prop}
###### Proposition
$\Lambda$ is a self-dual category.
=--

+-- {: .proof}
###### Proof
If $f \sim g$ in $L(m, n)$, then $f = \tau^{k (n+1)} \circ g$ for some $k \in \mathbb{Z}$. Observe that $\tau^\ast = \tau^{-1}$, so $f^\ast = g^\ast \circ \tau^{-k(n+1)} = \tau^{-k(m+1)} \circ g^\ast$ where the last equation holds because $g^\ast: n \to m$ is spiraling. This shows $f^\ast \sim g^\ast$, i.e., the self-duality of $L$ descends to $\Lambda$.
=--

+-- {: .num_prop}
###### Proposition
For a morphism $f \in L(m, n)$, we have $f^\ast(0) \leq 0$ iff $0 \leq f(0)$, and $0 \leq f^\ast(0)$ iff $f(m) \leq f(n)$. Hence $f^\ast(0) = 0$ iff ($0 \leq f(0)$ and $f(m) \leq n$).
=--

+-- {: .proof}
###### Proof
The first assertion is immediate from the adjunction $f^\ast \dashv f$. The second follows from the deduction

$$\array{
0 \leq f^\ast(0) & iff & -1 \lt f^\ast(0) \\
 & iff & \neg (f^\ast(0) \leq -1) \\
 & iff & \neg (0 \leq f(-1)) \\
 & iff & f(-1) \lt 0 \\
 & iff & f(m) \lt n+1 \\
 & iff & f(m) \leq n
}$$

where the step to the penultimate line used the spiraling property.
=--

The previous proposition, in conjunction with the self-duality of $L$ and Remark \ref{simplex}, shows that $\Delta^{op}$ faithfully maps to $L$ by $\Delta^{op}(m, n) \cong \{f \in L(m, n): f(0) = 0\}$. Passing to the quotient $q: L \to \Lambda$, this description also realizes $\Delta^{op}$ as sitting inside $\Lambda$, and the next result is immediate.

+-- {: .num_prop #unique1}
###### Proposition
Every morphism $f: m \to n$ in $\Lambda$ may be uniquely decomposed as $f = \tau_n^{f(0)} g$ where $g$ belongs to $\Delta^{op}(m, n) \subset L(m, n)$, and the exponent $f(0)$ is considered modulo $n+1$.
=--

+-- {: .num_prop #action}
###### Proposition
The [[cyclic group]] $\mathbb{Z}/(m+1)$ [[group action|acts]] on $\Delta^{op}(m, n)$ via the following formula for $f \in L(m, n), f(0) = 0$:

$$k \cdot f = \tau^{-f(k)} \circ f \circ \tau^k$$

or in other words, via $(k \cdot f)(i) \coloneqq f(k+i) - f(k)$.
=--

+-- {: .proof}
###### Proof
Clearly $k \cdot f \in \{g \in L(m, n): g(0) = 0\}$. We calculate

$$\array{
j \cdot (k \cdot f) & = & \tau^{-(k \cdot f)(j)} \circ (k \cdot f) \circ \tau^j \\
 & = & \tau^{-(f(j+k) - f(k))} \circ \tau^{-f(k)} \circ f \circ \tau^k \circ \tau^j \\
 & = & \tau^{-f(j+k)} \circ f \circ \tau^{j+k} \\
 & = & (j + k) \cdot f.
}$$

Moreover, $((m+1)\cdot f)(i) = f(i+m+1)-f(0+m+1) = f(i)+n+1 - (f(0)+n+1) = f(i) - f(0) = f(i),$ so that the $\mathbb{Z}$-action $(k, f) \mapsto k \cdot f$ factors through a $\mathbb{Z}/(m+1)$-action.
=--

+-- {: .num_prop}
###### Proposition
Every morphism $f: m \to n$ in $\Lambda$ may be uniquely decomposed as $f = h \tau_m^{-k}$ where $h$ belongs to $\Delta$ and $k$ is unique modulo $m+1$. The cyclic group $\mathbb{Z}/(n+1)$ acts on $\Delta(m, n) \cong \{f \in L(m, n): 0 \f(0)\; and\; f(m) \leq n$ by the formula $k \cdot f = \tau^{-k} \circ f \circ \tau^{f^\ast(k)}$.
=--

+-- {: .proof}
###### Proof
This follows from previous propositions by dualizing. For $f \in L(m, n)$ we write $f^\ast: n \to m$ uniquely in the form $\tau_m^k g$ with $g \in \Delta^{op}(n, m)$, by Proposition \ref{unique1}. Taking right adjoints, $f = g_\ast \tau_m^{-k}$ where $g_\ast \in \Delta(m, n)$. We define the action on $\Delta(m, n)$ by conjugating the action on $\Delta^{op}(n, m)$ provided by Proposition \ref{action}, i.e., for $f \in \Delta(m, n)$ we define

$$k \cdot f = (k \cdot f^\ast)_\ast = [\tau^{-f^\ast(k)} \circ f^\ast \circ \tau^k]_\ast = (\tau^k)_\ast \circ f^\ast_\ast \circ (\tau^{-f^\ast(k)})_\ast = \tau^{-k} \circ f \circ \tau^{f^\ast(k)}$$

and this conjugation preserves the action axioms.
=--

Denoting the generator $q(\tau_n)$ of $\Aut_\Lambda([n])$ also by $\tau_n$, we saw $\tau_n^{n+1} = \mathrm{id}_{[n]}$. One may read off from the development above a (perhaps more standard, and equivalent) presentation of $\Lambda$ by [[presentation of a category by generators and relations|generators and relations]]. In addition to the cosimplicial identities between the coboundaries $\delta_i$ and codegeneracies $\sigma_j$ and $\tau^{n+1}_n = \mathrm{id}$ there are the following identities:
$$\array{
\tau_n\delta_i = \delta_{i-1}\tau_{n-1},\,\, 1\leq i \leq n\\
\tau_n\delta_0 = \delta_n\\
\tau_n\sigma_i = \sigma_{i-1}\tau_{n-1},\,\, 1\leq i \leq n\\
\tau_m\sigma_0 = \sigma_n\tau_{n+1}^2
}$$

## Properties

### General

We reiterate the development in the [section on structure](#structure) in summary form:

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

1. $\Aut_\Lambda([n]) = \mathbf{Z}/(n+1)\mathbf{Z}$

2. $\Lambda([n],[m]) = \Delta([n],[m])\times \mathbf{Z}/(n+1)\mathbf{Z}$ (as a set)

3. Any morphism $f$ in $\Lambda([n],[m])$ can be uniquely written as a composition $f = \phi\circ g$ where $\phi\in\Delta([n],[m])$ and $g\in\Aut_\Lambda([n])$.
=--

The generalizations of this theorem are the starting point of the theory of [[skew-simplicial set]]s of Krasauskas or equivalently crossed simplicial groups of Loday and Fiedorowicz.

The cyclic category is a [[generalized Reedy category]], as explained [here](http://arxiv.org/abs/0809.3341).

### Generalized Reedy model structure

The cycle category is a [[generalized Reedy category]] (see [Berger-Moerdijk 08, example 2.7](#BM08)). Hence "cyclic spaces" carry a [[generalized Reedy model structure]].

## Related concepts

* [[minimal simplicial circle]]

* [[cyclic object]]

* [[cyclic set]], [[cyclic space]],

* [[cyclotomic space]], [[cyclotomic spectrum]]

* [[cyclic homology]]

## References

The original definition:

* {#Connes83} [[Alain Connes]], _Cohomologie cyclique et foncteurs $Ext^n$_, C.R.A.S. **296** (1983), Série I, 953-958 ([pdf](https://alainconnes.org/wp-content/uploads/n83.pdf), [[Connes_CohomologieCyclique.pdf:file]]).

* {#Cartier85} [[Pierre Cartier]], Section 1.6 of: *Homologie cyclique : rapport sur des travaux récents de Connes, Karoubi, Loday, Quillen...*, Séminaire Bourbaki: volume 1983/84, exposés 615-632, Astérisque, no. 121-122 (1985), Exposé no. 621 ([numdam:SB_1983-1984__26__123_0](http://www.numdam.org/item/?id=SB_1983-1984__26__123_0))

Exposition:

* {#Malkiewich15} [[Cary Malkiewich]], _A visual introduction to cyclic sets and cyclotomic spectra_, 2015 ([pdf](http://math.uiuc.edu/~cmalkiew/ytm_2015.pdf))

Textbook account:

* [[Jean-Louis Loday]], *The Cyclic Category, Tor and Ext Interpretation* ([doi:10.1007/978-3-662-21739-9_6](https://doi.org/10.1007/978-3-662-21739-9_6)) and *Cyclic Spaces and $S^1$-Equivariant Homology* ([doi:10.1007/978-3-662-21739-9_7](https://link.springer.com/chapter/10.1007/978-3-662-21739-9_7))

  Chapters 6 and 7 in: *Cyclic Homology*, Grundlehren **301**, Springer 1992 ([doi:10.1007/978-3-662-21739-9](https://link.springer.com/book/10.1007/978-3-662-21739-9))

See also:

* [[V. Drinfeld]], _On the notion of geometric realization_, [arXiv:math.CT/0304064](http://front.math.ucdavis.edu/0304.5064)

* [[Alain Connes]], _Noncommutative geometry_, Academic Press 1994
(also at <http://www.alainconnes.org>)

* R. Krasauskas, _Skew-simplicial groups_, (Russian)  Litovsk. Mat. Sb.  __27__ (1987),  no. 1, 89--99, [MR88m:18022](http://www.ams.org/mathscinet-getitem?mr=88m:18022) (English transl. [[krasauskas.pdf:file]])

* [[William Dwyer]], [[Daniel Kan]], _Normalizing the cyclic modules of Connes_, Comment. Math. Helv. 60 (1985), no. 4, 582--600.

* [[William Dwyer]], [[Mike Hopkins]], [[Daniel Kan]],  _The homotopy theory of cyclic sets_, Trans. Amer. Math. Soc. 291 (1985), no. 1, 281--289.

* Z. Fiedorowicz, [[Jean-Louis Loday]], _Crossed simplicial groups and their associated homology_,  Trans. Amer. Math. Soc. __326__ (1991),  no. 1, 57--87, [MR91j:18018](http://www.ams.org/mathscinet-getitem?mr=91j:18018), [doi](http://dx.doi.org/10.2307/2001855)

* {#Elm} [[Anthony Elmendorf]], *A simple formula for cyclic duality*, Proc. Amer. Math. Soc. Volume 118, Number 3 (July 1993), 709-711. ([pdf](http://www.ams.org/journals/proc/1993-118-03/S0002-9939-1993-1143017-0/S0002-9939-1993-1143017-0.pdf))

As a [[generalized Reedy category]]:

* {#BM08} [[Clemens Berger]], [[Ieke Moerdijk]], _On an extension of the notion of Reedy category_ ([arXiv:0809.3341](https://arxiv.org/abs/0809.3341))

Relation to the paracyclic category:

* {#NS} [[Thomas Nikolaus]], [[Peter Scholze]], *On topological cyclic homology* ([arxiv:1707.01799](https://arxiv.org/abs/1707.01799))

See also

* Wikipedia, _[Cyclic category](https://en.wikipedia.org/wiki/Cyclic_category)_

* Blog [discussion](http://golem.ph.utexas.edu/category/2007/06/the_curious_incident_of_the_do.html)

[[!redirects cyclic categories]]

[[!redirects Connes' cyclic category]]
[[!redirects category of cycles]]
[[!redirects cycle category]]

[[!redirects paracyclic category]]

category: category


cyclic derivation

Given a field $F$, a cyclic derivation on an $F$-algebra $A$ (example: algebra of formal noncommutative power series in $n$-variables) is an $F$-linear map $D:A\to End_F(A)$ satisfying
$$
D(f_1 f_2)(f) = D(f_1)(f_2 f)+D(f_2)(f f_1)
$$
Given a cyclic derivation $D$ a corresponding cyclic derivative $\delta:A\to A$ is defined by $\delta f = (D f)(1)$.

They are appearing in the definition of [[Jacobian algebra]](also called Jacobi algebra) of a quiver with potential, see there.

* [[Gian-Carlo Rota]], Bruce Sagan, Paul R.Stein, _A cyclic derivative in noncommutative algebra_, J. Algebra __64__:1 (1980) 54-75 doi [MR575782](https://mathscinet.ams.org/mathscinet-getitem?mr=575782)
* [[Christophe Reutenauer]], _Cyclic derivation of noncommutative algebraic power series_, J. Alg. __85__, 32-39 (1983)
* Daniel Lopez-Aguayo, _Cyclic derivations, species realizations and potentials_, [pdf](https://revistas.unal.edu.co/index.php/recolma/article/view/84083/73220)

A class of identities involving multiple zeta functions is described using cyclic derivations in

* Michael E.Hoffman, Yasuo Ohno, _Relations of multiple zeta values and their algebraic expression_, J. Algebra __262__:2 (2003) 332-347 doi

category: algebra

[[!redirects cyclic derivative]]


cyclic group

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

# Cyclic groups
* table of contents
{: toc}

## Definition

A __cyclic group__ is a [[quotient group]] of the [[free group]] on the [[singleton]].

Generally, we consider a cyclic group as a [[group]], that is without specifying which element comprises the generating singleton.  But see [Ring structure](#Ring) below.

## Examples

There is (up to [[isomorphism]]) one cyclic group for every [[natural number]] $n$, denoted

$$
  \mathbb{Z}_n \coloneqq \mathbb{Z}/n\mathbb{Z}
  \,.
$$

For $n \gt 0$, the [[order of a group|order]] ([[cardinality]]) of $\mathbb{Z}_n$ is $n$ (so [[finite set|finite]]); for $n = 0$, which is the group of [[integers]]

$$
  \mathbb{Z}_0 \coloneqq \mathbb{Z}
$$

the [[order of a group|order]] is [[countable set|countable]] but [[infinite set|infinite]].

If we identify the [[free group]] on the singleton with the additive group $\mathbb{Z}$ of [[integers]], then the infinite cyclic group is $\mathbb{Z}$ itself, while the finite cyclic group of order $n$ is $\mathbb{Z}/n$, that is $\mathbb{Z}$ modulo (the [[normal subgroup]] generated by) the integer $n$.  Of course, $\mathbb{Z}$ itself is also $\mathbb{Z}/0$.  One could also consider $\mathbb{Z}/n$ for negative $n$, but this is the same as $\mathbb{Z}/{|n|}$.

The cyclic group of order $n$ may also be identified with a [[subgroup]] of the multiplicative group of [[complex numbers]] (or [[algebraic numbers]]): the group of $n$th roots of $1$.  For $n = 0$, we may pick any non-zero complex number (or even something else) that is *not* a root of $1$ (but there is no standard choice) and take the subgroup generated by it.

Dedicated entries exist for:

* [[cyclic group of order 2]]

* [[cyclic group of order 3]]

* [[cyclic group of order 4]]

## Notation

Besides '$\mathbb{Z}/n$', the cyclic group of order $n$ may also be denoted in other ways: some more complicated variation of '$\mathbb{Z}/n$' (to put 'the \[normal\] subgroup generated by' explicitly in the notation), or else the simplified form '$\mathbb{Z}_n$' (which however conflicts with notation for the $n$-[[adic integers]]).  When written multiplicatively, it may be denoted '$Z_n$' (note the font change) or '$C_n$'; either letter here stands for 'cyclic' in one language or another.  (It is a coincidence that the German words 'Zahl', which gives us '$\mathbb{Z}$', and 'zyklisch', which gives us '$Z$', begin with the same letter.)

Besides '$\mathbb{Z}$', the infinite cyclic group may also be denoted in other ways: some variation of '$(\mathbb{Z},+)$' to indicate that we are using *addition* of integers, or any of the above notations with either '$0$' or '$\infty$' in place of '$n$' (depending on whether we think of it as $\mathbb{Z}$ modulo $0$ or the cyclic group with order $\infty$).

When written additively, the notation for the elements of a cyclic group are usually just the notation for integers; for the finite cyclic group of order $n$, we use the natural number less than $n$.  In the finite case, we may also use brackets or some other notation to indicate [[equivalence classes]].  When written multiplicatively, any letter ('$e$', '$x$', '$a$', '$\xi$', etc) may be taken to stand for the generating element; then any other element is a power of this generator.  When thought of as a multiplicative group of complex numbers, one generator is $\mathrm{e}^{2\mathrm{i}\pi/n}$, and the notation may reflect that.

## Properties

### Ring structure
   {#Ring}

Let $A$ be a cyclic group, and let $x$ be a generator of $A$.  Then there is a unique [[ring]] structure on $A$ (making the original group the additive group of the ring) such that $x$ is the multiplicative identity $1$.

If we identify $A$ with the additive group $\mathbb{Z}/n$ and pick (the equivalence class of) the integer $1$ for $x$, then the resulting ring is precisely the [[quotient ring]] $\mathbb{Z}/n$.

In this way, a cyclic group equipped with the [[extra structure]] of a generator is the same thing (in the sense that their [[groupoids]] are [[equivalence of categories|equivalent]]) as a ring with the [[extra property]] that the underlying additive group is cyclic.

For $n \gt 0$, the number of ring structures on the cyclic group $\mathbb{Z}/n$, which is the same as the number of generators, is $\phi(n)$, the [[totient function|Euler totient]] of $n$; the generators are those $i$ that are [[relatively prime numbers|relatively prime]] to $n$.  While $\phi(1) = 1$, otherwise $\phi(n) \gt 1$ (another way to see that we have a structure and not just a property).  For $\mathbb{Z}$ itself, there are two ring structures, since $1$ and $-1$ are the generators (and these are relatively prime to $0$).

\lineabreak

### Fundamental theorem of cyclic groups
  {#FundamentalTheoremOfCyclicGroups}

For $n \in \mathbb{N}$, there is precisely one [[subgroup]] of the cyclic group $\mathbb{Z}/n\mathbb{N}$ of [[order of a group|order]] $d \in \mathbb{N}$ for each factor of $d$ in $n$, and this is the subgroup [[generators and relations|generated]] by $n/d \in \mathbb{Z} \to \mathbb{Z}/n\mathbb{Z}$.

Moreover, the [[lattice of subgroups]] of $\mathbb{Z}/n\mathbb{Z}$ is equivalently the dual of the lattice of natural numbers $\leq n$ ordered by divisibility.

(e.g [Aluffi 09, pages 83-84](#Aluffi09))

This is a special case of the _[[fundamental theorem of finitely generated abelian groups]]_. See there for more.

### Relation to finite abelian groups

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

Every [[finite abelian group]] is a [[direct sum of abelian groups]] over cyclic groups.

=--

See at _[[finite abelian group]]_ for details.

### Group cohomology

For discussion of the [[group cohomology]] of cyclic groups see

* at _[[projective resolution]]_ the section _[Cohomology of cyclic groups](http://ncatlab.org/nlab/show/projective+resolution#CohomologyOfCyclicGroups)_

* at *[[finite rotation group]]* the section *[Group cohomology](finite+rotation+group#GroupCohomology)*

* in [Golasiński & Gonçalves 2011](#GolasińskiGonçalves11) for the cases $H^n_{Grp}(/mathbb{Z}/n;\, \mathbb{Z}/m)$

For example, relevant for [[Dijkgraaf-Witten theory]] is the fact:

$$
  H^3_{grp}\big(\mathbb{Z}/n\mathbb{Z}, U(1)\big)
  \;\simeq\;
  \mathbb{Z}/n\mathbb{Z}
  \,.
$$

### Linear representations

We discuss some of the [[representation theory]] of cyclic groups.

+-- {: .num_example #IrreducibleRealRepresentationsOfCyclicGroups}
###### Example
**([[irreducible representation|irreducible]] [[real numbers|real]] [[linear representations]] of [[cyclic groups]])**

For $n \in \mathbb{N}$, $n \geq 2$,
the [[isomorphism classes]] of [[irreducible representation|irreducible]] [[real numbers|real]] [[linear representations]] of the [[cyclic group]] $\mathbb{Z}/n$ are given by precisely the following:

1. the 1-[[dimension|dimensional]] [[trivial representation]] $\mathbf{1}$;

1. the 1-[[dimension|dimensional]] [[sign representation]] $\mathbf{1}_{sgn}$;

1. the 2-[[dimension|dimensional]] standard representations $\mathbf{2}_k$ of [[rotations]] in the [[Euclidean plane]] by [[angles]] that are [[integer]] multiples of $2 \pi k/n$, for $k \in \mathbb{N}$, $0 \lt k \lt n/2$;

   hence the [[restricted representations]] of the defining real rep of [[SO(2)]] under the [[subgroup]] inclusions $\mathbb{Z}/n \hookrightarrow SO(2)$, hence the representations generated by [[real number|real]] $2 \times 2$ [[trigonometric function|trigonometric]] [[matrices]] of the form

   $$
    \rho_{\mathbf{2}_k}(1)
    \;=\;
     \left(
       \array{
         cos(\theta) & -sin(\theta)
         \\
         sin(\theta) & \phantom{-}cos(\theta)
       }
     \right)
     \phantom{AA}
     \text{with}
     \;
     \theta = 2 \pi \tfrac{k}{n}
     \,,
   $$

  (For $k = n/2$ the corresponding 2d representation is the [[direct sum]] of two copies of the [[sign representation]]: $\mathbf{2}_{n/2} \simeq \mathbf{1}_{sgn} \oplus \mathbf{1}_{sgn}$, and hence not [[irrep|irreducible]]. Moreover, for $k \gt n/2$ we have that $\mathbf{2}_{k}$ is irreducible, but [[isomorphism|isomorphic]] to $\mathbf{2}_{n-k} \simeq \mathbf{2}_{-k}$).

In summary:

$$
  Rep^{irr}_{\mathbb{R}}
  \big(
    \mathbb{Z}/n
  \big)_{/\sim}
  \;=\;
  \big\{
    \mathbf{1},
    \mathbf{1}_{sgn},
    \mathbf{2}_k
    \;\vert\;
    0 \lt k \lt n/2
  \big\}
$$

=--

(e.g. [tom Dieck 09 (1.1.6), (1.1.8)](representation+theory#tomDieck09))

## Related concepts

* [[group of order 2]]

* [[binary cyclic group]]

* [[permutation cycle]], [[cyclic permutation]]

* [[multiplicative group of integers modulo n]]

* [[Q/Z]]

* [[integers modulo n]]

* [[p-localization]]

* [[profinite completion of the integers]]

* [[cyclotomic field]]

* [[cyclotomic spectrum]]

* [[cyclically ordered group]]

## References

Historical discussion of the cyclic group in the context of the [[classification of finite rotation groups]]:

* {#Klein1884} [[Felix Klein]], chapter I.3 of: _Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade_, 1884, translated as _Lectures on the Icosahedron and the Resolution of Equations of Degree Five_ by George Morrice 1888, [online version](https://archive.org/details/cu31924059413439)

Textbook accounts:

* {#Aluffi09} [[Paolo Aluffi]], Part 0 of: *Algebra: Chapter 0*, Graduate Studies in Mathematics **104**,  AMS (2009) [[ISBN:978-1-4704-1168-8](https://bookstore.ams.org/gsm-104/)]

* [[Joseph A. Gallian]], Section 4 of: *Contemporary Abstract Algebra*, Chapman and Hall/CRC (2020)  [[doi:10.1201/9781003142331](https://doi.org/10.1201/9781003142331), [pdf](https://ict.iitk.ac.in/wp-content/uploads/CS203-Mathematics-for-Computer-Science-III-Gallian.pdf)]

Further review:

*  Philippe B. Laval, _Cyclic groups_ [[pdf](http://math.kennesaw.edu/~plaval/math4361/groups_cyclic.pdf)]

On the [[fundamental theorem of cyclic groups]]:

*  {#Gallian10} Joseph A. Gallian, _Fundamental Theorem of Cyclic Groups_, Contemporary Abstract Algebra, p. 77, (2010)

On the [[group cohomology]] of cyclic groups with [[coefficients]] in cyclic groups:

* {#GolasińskiGonçalves11} [[Marek Golasiński]], [[Daciberg Lima Gonçalves]], *On cohomologies and extensions of cyclic groups*, Topology and its Applications **158** (2011) 1858–1865 [[doi:10.1016/j.topol.2011.06.022](https://doi.org/10.1016/j.topol.2011.06.022)]

[[!redirects cyclic group]]
[[!redirects cyclic groups]]

[[!redirects infinite cyclic group]]
[[!redirects infinite cyclic groups]]


cyclic group of order 3

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###Context###
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
=--
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#Contents#
* table of contents
{:toc}

## Idea

The [[cyclic group]] of [[order of a group|order]] 3.

### ADE-Classification

[[!include ADE -- table]]

## References

* group names, _[C3](https://people.maths.bris.ac.uk/~matyd/GroupNames/1/C3.html)_


cyclic group of order 4

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###Context###
#### Group Theory
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[[!include group theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The [[cyclic group]] of [[order of a group|order]] 4.

## Properties

### General
 {#General}

Besides the [[Klein four-group]] $\mathbb{Z}/2 \times \mathbb{Z}/2$, the cyclic group of order 4 is the only other [[group]] of [[order]] 4, up to [[isomorphism]].

### ADE-Classification

[[!include ADE -- table]]

## References

* group names, _[C4](https://people.maths.bris.ac.uk/~matyd/GroupNames/1/C4.html)_


cyclic homology

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###Context###
#### Cohomology
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[[!include cohomology - contents]]
=--
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#Contents#
* table of contents
{:toc}

## Idea

[[Hochschild homology]] may be understood as the [[cohomology]] of [[free loop space object]]s (as described there). These free loop space objects are canonically equipped with a [[circle group]]-[[action]] that rotates the loops. _Cyclic homology_ is the corresponding $S^1$-[[equivariant cohomology]] of free loop space objects.

Like [[Hochschild homology]], [[cyclic homology]] is an _additive invariant_ of [[dg-categories]] or [[stable infinity-categories]], in the sense of [[noncommutative motives]].  It also admits a [[Dennis trace map]] from [[algebraic K-theory]], and has been successful in allowing computations of the latter.

There are several definitions for the cyclic homology of an [[associative algebra]] $A$ (over a [[commutative ring]] $k$).  [[Alain Connes]] originally defined cyclic homology over [[fields]] of [[characteristic zero]], as the [[homology]] groups of a cyclic variant of the [[chain complex]] computing [[Hochschild homology]].  [[Jean-Louis Loday]] and [[Daniel Quillen]] gave a definition via a certain [[double complex]] (for arbitrary commutative rings).  [[Alain Connes|Connes]] gave another definition by associating to $A$ a [[cyclic object|cyclic]] [[vector space]] $A^\sharp$, and showing that the cyclic homology of $A$ may be computed as via [[Ext]]-groups $Ext^*(A^\sharp, k^\sharp)$.  A fourth definition was given by [[Christian Kassel]], who showed that the cyclic homology groups may be computed as the homology groups of a certain [[mixed complex]] associated to $A$.

Following [[Alexandre Grothendieck]], [[Charles Weibel]] gave a definition of [[cyclic homology]] (and [[Hochschild homology]]) for [[schemes]], using [[hypercohomology]].  On the other hand, the definition of [[Christian Kassel]] via [[mixed complexes]] was extended by [[Bernhard Keller]] to [[linear categories]] and [[dg-categories]], and he showed that the cyclic homology of the [[dg-category]] of [[perfect complexes]] on a (nice) [[scheme]] $X$ coincides with the cyclic homology of $X$ in the sense of [[Charles Weibel|Weibel]].

There are closely related variants called [[periodic cyclic homology]] and [[negative cyclic homology]].  There is a version for [[ring spectra]] called [[topological cyclic homology]].

## Definition

### The chain complex for cyclic homology

Let $A$ be an [[associative algebra]] over a [[ring]] $k$. Write $C_\bullet(A,A)$ for the [[Hochschild homology]] [[chain complex]] of $A$ with coefficients in $A$.

For each $n \in \mathbb{N}$ let $\lambda : C_n(A,A) \to C_n(A,A)$ be the $k$-linear map that cyclically permutes the elements and introduces a sign:

$$
  \lambda : (a_0, a_1, \cdots, a_{n-1}, a_n)
  \mapsto
   (-1)^n
   (a_n, a_0  , \cdots, a_{n-1})
  \,.
$$

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

The **cyclic homology complex** $C^\lambda_\bullet(A)$ of $A$ is the quotient of the Hochschild homology complex of $A$ by cyclic permutations:

$$
  C_\bullet^\lambda(A) := C_\bullet(A,A)/im(Id-\lambda)
  \,.
$$

The [[homology]] of the cyclic complex, denoted

$$
  HC_n(A) := H_n( C_\bullet^\lambda(A) )
$$

is called the **cyclic homology** of $A$.
=--

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

The **cyclic cohomology** groups of $A$ Are the [[cohomology]] groups of the dual [[cochain complex]], denoted $HC^n(A)$.
=--

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

If $I\subset A$ is an ideal, then the **relative cyclic homology** groups $HC_n(A,I)$ are the homology groups of the complex $C_\bullet(A,I) = ker(C_\bullet(A)\to C_\bullet(A/I))$.
=--

### Ordinary cohomology of $\mathcal{L}X/S^1$ and cyclic homology of $X$

Let $X$ be a [[simply connected topological space|simply connected]] [[topological space]].

The [[ordinary cohomology]] $H^\bullet$ of its free loop space is the [[Hochschild homology]] $HH_\bullet$ of its [[singular cohomology|singular chains]] $C^\bullet(X)$:

$$
  H^\bullet(\mathcal{L}X)
    \simeq
  HH_\bullet( C^\bullet(X) )
  \,.
$$

Moreover the $S^1$-equivariant cohomology of the loop space, hence the ordinary cohomology of the [[cyclic loop space]] $\mathcal{L}X/^h S^1$ is the [[cyclic homology]] $HC_\bullet$ of the singular chains:

$$
  H^\bullet(\mathcal{L}X/^h S^1)
    \simeq
  HC_\bullet( C^\bullet(X) )
$$

([Jones 87, Thm. A](#Jones87), review in [Loday 92, Cor. 7.3.14](#Loday92), [Loday 11, Sec 4](#Loday11))

If the [[coefficients]] are [[rational numbers|rational]], and $X$ is of [[finite type]] then this may be computed by the _[[Sullivan model for free loop spaces]]_, see there the section on _[Relation to Hochschild homology](Sullivan+model+of+free+loop+space#RelationToHochschildHomology)_.

In the special case that the [[topological space]] $X$ carries the structure of a [[smooth manifold]], then the singular cochains on $X$ are equivalent to the [[dgc-algebra]] of [[differential forms]] (the [[de Rham algebra]]) and hence in this case the statement becomes that

$$
  H^\bullet(\mathcal{L}X)
    \simeq
  HH_\bullet( \Omega^\bullet(X) )
  \,.
$$

$$
  H^\bullet(\mathcal{L}X/^h S^1)
    \simeq
  HC_\bullet( \Omega^\bullet(X) )
  \,.
$$

This is known as _[[Jones' theorem]]_ ([Jones 87](#Jones87))

An [[(infinity,1)-category theory|infinity-category theoretic]] proof of this fact is indicated at _[Hochschild cohomology -- Jones' theorem](Hochschild+cohomology#JonesTheorem)_.

## Properties

### Loday-Quillen-Tsygan theorem

The _[[Loday-Quillen-Tsygan theorem]]_ ([Loday-Quillen 84](#LodayQuillen84), [Tsygan 83](#Tsygan83)) states that for any [[associative algebra]], $A$ in [[characteristic zero]], the [[Lie algebra homology]] $H_\bullet(\mathfrak{gl}(A))$  of the infinite [[general linear Lie algebra]] $\mathfrak{gl}(A)$ with [[coefficients]] in $A$ is, up to a degree shift, the [[exterior algebra]] $\wedge(HC_{\bullet - 1}(A))$ on the [[cyclic homology]] $HC_{\bullet - 1}(A)$ of $A$:

$$
  H_\bullet(\mathfrak{gl}(A))
  \;\simeq\;
  \wedge( HC_{\bullet - 1}(A) )
$$

(see e.g [Loday 07, theorem 1.1](#Loday07)).

## Related entries

* [[cyclic category]]

* [[cyclic object]]

  [[cyclic set]] [[cyclic space]]

* [[cyclic loop space]]

* [[topological cyclic homology]]

* [[Hochschild homology]], [[topological Hochschild homology]]

* [[Sullivan model of free loop space]]

* [[dihedral homology]]

* [[Loday-Quillen-Tsygan theorem]]

* [[additive K-theory]]

## References

Foundational articles:

* [[A. Connes]], _Noncommutative differential geometry, Part I, the Chern character in $K$-homology_, Preprint, Inst. Hautes Études Sci., Bures-sur-Yvette, 1982; _Part II, de Rham homology and noncommutative algebra_, Preprint, IHÉS 1983; _Cohomologie cyclique et foncteurs $Ext^n$_, C. R. Acad. Sci. Paris __296__, (1983), pp. 953–958, [MR86d:18007](http://www.ams.org/mathscinet-getitem?mr=777584)

* {#Connes83} [[Alain Connes]], _Cohomologie cyclique et foncteurs $Ext^n$_, C.R.A.S. **296** (1983), Série I, 953-958 ([pdf](https://alainconnes.org/wp-content/uploads/n83.pdf), [[Connes_CohomologieCyclique.pdf:file]]).

* [[B. L. Tsygan]], _The homology of matrix Lie algebras over rings and the Hochschild homology_, Uspekhi Mat. Nauk, 38:2(230) (1983), 217–218.

* [[Jean-Louis Loday]], [[Daniel Quillen]], _Cyclic homology and the Lie algebra homology of matrices_, Comment. Math. Helvetici 59 (1984) 565-591 ([doi:10.1007/978-3-662-21739-9](https://link.springer.com/book/10.1007/978-3-662-21739-9))

* [[Christian Kassel]], _Cyclic homology, comodules and mixed complexes_, J. Alg. 107 (1987), 195–216 (doi:10.1016/0021-8693(87)90086-X)

Monographs:

* {#Loday98} [[Jean-Louis Loday]], _Cyclic homology_, Grundlehren Math.Wiss. __301__, Springer (1998)

* [[Alain Connes]], _Noncommutative geometry_, Acad. Press 1994, 661 p. [PDF](http://www.alainconnes.org/docs/book94bigpdf.pdf)
* [[Max Karoubi]], _Homologie cyclique et K-théorie_, Astérique __149__, Société Mathématique de France (1987).
* [[Ib Madsen]], _Algebraic K-theory and traces_, Current Developments in Mathematics, 1995.

Lecture notes:

* {#Loday07} [[Jean-Louis Loday]], _Cyclic Homology Theory, Part II_, notes taken by Pawel Witkowsk (2007) ([pdf](https://www.impan.pl/swiat-matematyki/notatki-z-wyklado~/loday_cht_2.pdf))

* [[D. Kaledin]], _Tokyo lectures "Homological methods in non-commutative geometry"_, ([pdf](http://imperium.lenin.ru/~kaledin/tokyo/final.pdf), [TeX](http://imperium.lenin.ru/~kaledin/tokyo/final.tex)); and related but different [Seoul lectures](http://imperium.lenin.ru/~kaledin/seoul)

* [[Masoud Khalkhali]], _A short survey of cyclic cohomology_, [arxiv/1008.1212](http://arxiv.org/abs/1008.1212)

Some modern treatments:

* [[Bernhard Keller]], _On the cyclic homology of ringed spaces and schemes_, Doc. Math. J. DMV 3 (1998), 231-259, [pdf](http://www.math.jussieu.fr/~keller/publ/KellerHCSchemes.pdf).

* [[Bernhard Keller]], _On the cyclic homology of exact categories_, Journal of Pure and Applied Algebra 136 (1999), 1-56, [pdf](http://www.math.jussieu.fr/~keller/publ/cyex.pdf).

* [[Bernhard Keller]], _Invariance and Localization for Cyclic Homology of DG algebras_, Journal of Pure and Applied Algebra, 123 (1998), 223-273, [pdf](http://www.math.jussieu.fr/~keller/publ/ilc.pdf).

* [[Charles Weibel]], _Cyclic homology for schemes_, Proceedings of the AMS, 124 (1996), 1655-1662, [web](http://www.math.uiuc.edu/K-theory/0043/).

* [[D. Kaledin]], _Cyclic homology with coefficients_, [math.KT/0702068](http://arxiv.org/abs/math.KT/0702068), to appear in Yu. Manin's 70th anniversary volume.

* [[E. Getzler]], [[M. Kapranov]], _Cyclic operads and cyclic homology_, in: "Geometry, Topology and Physics for R. Bott",
ed. S.-T. Yau, p. 167-201, International Press, Cambridge MA, 1995, [pdf](http://www.math.northwestern.edu/~getzler/Papers/cyclic.pdf)

* [[Teimuraz Pirashvili]], [[Birgit Richter]], _Hochschild and cyclic homology via functor homology_, K-Theory __25__ (2002), no. 1, 39–49, [MR2003c:16011](http://www.ams.org/mathscinet-getitem?mr=1899698), [doi](http://dx.doi.org/10.1023/A:1015064621329)

* [[Jolanta Słomińska]], _Decompositions of the category of noncommutative sets and Hochschild and cyclic homology_, Cent. Eur. J. Math. __1__ (2003), no. 3, 327–331 ([doi:10.2478/BF02475213](http://dx.doi.org/10.2478/BF02475213), [MR2004f:16011](http://www.ams.org/mathscinet-getitem?mr=1992895))

Some influential original references from 1980s:

* [[Boris Tsygan]], [[Boris Feigin]],  _[[Additive K-theory]]_, in LNM 1289 (1987), edited by Yu. I. Manin, pp. 67--209, seminar 1984-1986 in Moscow), [MR89a:18017](http://www.ams.org/mathscinet-getitem?mr=923136);
_Аддитивная K-теория и кристальные когомологии_, Функц. анализ и его прил., 19:2 (1985), 52–-62, [pdf](http://www.mathnet.ru/php/getFT.phtml?jrnid=faa&paperid=1358&what=fullt&option_lang=rus), [MR88e:18008](http://www.ams.org/mathscinet-getitem?mr=800920); Engl. transl. in B. L. Feĭgin, B. L. Tsygan, _Additive $K$-theory and crystalline cohomology_, Functional Analysis and Its Applications, 1985, 19:2, 124–132.
* [[T. Goodwillie]], _Cyclic homology, derivations, and the free loopspace_, Topology __24__ (1985),
no. 2, 187–215, [MR87c:18009](http://www.ams.org/mathscinet-getitem?mr=793184), doi

The relation to [[cyclic loop spaces]]:

* {#Jones87} [[John D.S. Jones]], _Cyclic homology and equivariant homology_, Invent. Math. __87__, 403-423 (1987) ([pdf](https://math.berkeley.edu/~nadler/jones.pdf), [doi:10.1007/BF01389424](https://doi.org/10.1007/BF01389424))

reviewed in:

* {#Loday92} [[Jean-Louis Loday]], *Cyclic Spaces and $S^1$-Equivariant Homology* ([doi:10.1007/978-3-662-21739-9_7](https://link.springer.com/chapter/10.1007/978-3-662-21739-9_7))

  Chapter 7 in: *Cyclic Homology*, Grundlehren **301**, Springer 1992 ([doi:10.1007/978-3-662-21739-9](https://link.springer.com/book/10.1007/978-3-662-21739-9))

* {#Loday11} [[Jean-Louis Loday]], Section 4 of: _Free loop space and homology_, Chapter 4 in: Janko Latchev, Alexandru Oancea (eds.): *Free Loop Spaces in Geometry and Topology*, IRMA Lectures in Mathematics and Theoretical Physics **24**, EMS 2015 ([arXiv:1110.0405](https://arxiv.org/abs/1110.0405), [ISBN:978-3-03719-153-8](https://bookstore.ams.org/emsilmtp-24/))

On the [[ordinary cohomology]] of [[cyclic loop spaces]] ("string cohomology") computed directly (instead of via the isomorphism to cyclic homology):

* {#BokstedtOttosen05} [[Marcel Bökstedt]], [[Iver Ottosen]], *A spectral sequence for string cohomology*, Topology Volume 44, Issue 6, November 2005, Pages 1181-1212 ([arXiv:math/0411571](https://arxiv.org/abs/math/0411571), [doi:10.1016/j.top.2005.04.006](https://doi.org/10.1016/j.top.2005.04.006))

and specifically for [[complex projective spaces]]:

* [[Marcel Bökstedt]], [[Iver Ottosen]], *String cohomology groups of complex projective spaces*, Algebr. Geom. Topol. 7(4): 2165-2238 (2007). ([arXiv:math/0605754](https://arxiv.org/abs/math/0605754), [doi:10.2140/agt.2007.7.2165](https://projecteuclid.org/journals/algebraic-and-geometric-topology/volume-7/issue-4/String-cohomology-groups-of-complex-projective-spaces/10.2140/agt.2007.7.2165.full))

On the cyclic homology of the [[groupoid convolution algebra]] of [[étale groupoid|étale]] [[Lie groupoids]], hence of [[orbifolds]]:

* [[Jean-Luc Brylinski]], [[Victor Nistor]], *Cyclic cohomology of étale groupoids*,  K-theory 8.4 (1994): 341-365 ([pdf](https://www.researchgate.net/profile/Victor-Nistor/publication/226520448_Cyclic_cohomology_of_etale_groupoids/links/5509d9d30cf20ed529e227b0/Cyclic-cohomology-of-etale-groupoids.pdf), [[BrylinskiNistor_CyclicCohomology.pdf:file]])

* [[Marius Crainic]], *Cyclic Cohomology of Étale Groupoids; The General Case*, K-Theory 17: 319–362, 1999 ([arXiv:funct-an/9712001](https://arxiv.org/abs/funct-an/9712001), [published pdf](https://access.portico.org/Portico/rest/au/getAUContent?auId=ark:/27927/pgg1zfpvw2x&fileName=pgg1zfprgx2.pdf&viewType=PDF))

The [[Loday-Quillen-Tsygan theorem]] is originally due, independently, to

* {#LodayQuillen84} [[Jean-Louis Loday]], [[Daniel Quillen]], _Cyclic homology and the Lie algebra homology of matrices_ Comment. Math. Helv., 59(4):569–591, 1984 ([doi:10.1007/BF02566367](https://doi.org/10.1007/BF02566367))

and

* {#Tsygan83} [[Boris Tsygan]], _Homology of matrix algebras over rings and the Hochschild homology_, Uspeki Math. Nauk., 38:217–218, 1983.

[[!redirects cyclic (co)homology]]
[[!redirects cyclic cohomology]]

[[!redirects string cohomology]]