cyclic loop space

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## Idea
 {#Idea}

Any [[free loop space]] $\mathcal{L}X \,=\, Map(S^1,X)$ carries a canonical [[group action]] ([[infinity-action]]) of the [[circle group]] $S^1$ acting by rigid [[rotation]] on itself. The [[homotopy quotient]] $Cyc(X) \;\coloneqq\; \mathcal{L}(X) \sslash S^1$ of this action might be called the _cyclic loop space_ of $X$:

If $X$ is a [[simply connected topological space]], then the [[ordinary cohomology]] of $Cyc(X)$ is the [[cyclic cohomology]] of $X$ (Jones' theorem [[Jones (1987)](#Jones87); [Loday (1992), §7.2](#Loday92); [Loday (2015)](#Loday15)], see [below](#RelationToCyclicCohomology)).

Analogously, if $X = Spec(A)$ is an [[affine variety]] regarded in [[derived algebraic geometry]], then $\mathcal{O}(\mathcal{L}Spec(A))$ is the [[Hochschild homology]] of $A$ and $\mathcal{O}((\mathcal{L}Spec(A))/S^1)$ the corresponding [[cyclic homology]], see also the discussion at _[[Hochschild cohomology]]_.

But $Cyc(X)$ is also known as the *string space* of $X$ ([Chataur 2005. 4.8.1](#Chataur05), [Bökstedt & Ottosen 05, p. 1](#BokstedtOttosen05)).

{#StapletonTerminology} If $X = Y\sslash G$ is the [[homotopy quotient]] of a [[topological space]] by a [[topological group]] action, regarded as a locally constant $\infty$-stack, then (a [[point-set topology|point set]]-model for) the [[free loop space]] $\mathcal{L}(Y \sslash G)$ has been called the *twisted loop space* in [Witten 88](#Witten88); and this terminology is essentially carried over to the cyclification of its restriction to the constant loops
[Stapleton (2013), p. 2](#Stapleton13) in the context of the [[transchromatic character]].

A candidate lift of this construction from plain [[homotopy types]] to [[smooth homotopy types]], namely to [[orbifold]] [[stacks]], is [[Huan's inertia orbifold]]-construction.

## Properties

### As right base change along $\ast \to \mathbf{B} S^1$
 {#AsRightBaseChange}

The cyclic loop space $\mathcal{L}X  \sslash S^1$ is equivalently the right [[base change]]/[[dependent product]] along the canonical point inclusion $\ast \to B S^1$ ([this prop.](base+change#CyclicLoopSpace)) into the [[delooping]] of $S^1$ (the [[classifying space]] of the [[circle group]] when realized in the [[classical model structure on topological spaces|homotopy theory of]] [[topological spaces]]). See also at _[[double dimensional reduction]]_ ([BMSS 19, Sec. 2.2](#BMSS19), following [FSS 18, Sec. 3](#FSS18)).

### Ordinary cohomology of $\mathcal{L}X \sslash S^1$ as cyclic cohomology of $X$
 {#RelationToCyclicCohomology}

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 $\mathcal{L}X \sslash S^1$ is the [[cyclic homology]] $HC_\bullet$ of the singular chains:

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

([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

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

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

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)_.

### Stable splitting
 {#StableSplitting}

\begin{proposition}
For $X \in TopSpaces^{\ast/}$ a [[pointed topological space]] with [[reduced suspension]] denoted $\Sigma X$, the [[stabilization]] of its cyclic loop spaces splits as a [[direct sum]] of [[suspension spectra]], hence the underlying [[infinite loop space]] of that as a [[product space]], of the following form:

$$
  \Omega^\infty \Sigma^\infty
  \big(
    Maps(S^1, \Sigma X) \sslash S^1
  \big)
  \;\;
  \simeq
  \;\;
  \big(
    \Omega^\infty \Sigma^\infty B S^1
  \big)
  \times
  \underset{
    k \in \mathbb{N}_+
  }{\prod}
  \Omega^\infty \Sigma^\infty
  \big(
    (E (\mathbb{Z}/k))_+
    \wedge_{\mathbb{Z}/k}
    X^{\wedge^n}
  \big)
$$

In terms of the [[suspension spectra]] themselves:

$$
  \Sigma^\infty
  \Big(
    \underset{
      \frac{
        E S^1 \times_{S^1} Maps(S^1, \Sigma X)
      }
      {
        E S^1 \times_{S^1} \ast \,=\, B S^1
      }
    }{
    \underbrace{
      Maps
      \big(
        S^1, \, \Sigma X
      \big)
        \wedge_{S^1}
      (E S^1)_+
    }
    }
  \Big)
  \;\;
  \simeq
  \;\;
  \underset{ k \in \mathbb{N}_{\geq 1} }{\vee}
  \Sigma^\infty
  \big(
    (E \mathbb{Z}/k)_+  \wedge_{\mathbb{Z}/k} X^{\wedge^k}
  \big)
$$

\end{proposition}

([Carlsson & Cohen 1987, Cor. C](#CarlssonCohen1987), announced in [Cohen 1985, p. 194](#Cohen1985)) See also at *[[stable splitting of mapping spaces]]*.

Here

* $\wedge$ denotes the [[smash product]],

* $\mathbb{Z}/k$ a [[cyclic group]],

* $B G$ the [[classifying space]] of a [[Lie group]] $G$
  and $E G$ its [[universal principal bundle]],

* $(-)_+$ the [adjoining of a base point](pointed+topological+space#ForgettingAndAdjoiningBasepoints)

* and $(-)\wedge_{G} (-) \;\coloneqq\; \big((-) \wedge (-)\big)/G $.

\begin{example}
  For $X = S^n$ the [[n-sphere]], we have $\Sigma X \,=\, S^{n+1}$ and hence
$$
  \Omega^\infty \Sigma^\infty
  \big(
    Maps(S^1, S^{n+1}) \sslash S^1
  \big)
  \;\;
  \simeq
  \;\;
  \big(
    \Omega^\infty \Sigma^\infty B S^1
  \big)
  \times
  \underset{
    n \in \mathbb{N}_{\geq 1}
  }{\prod}
  \Omega^\infty \Sigma^\infty
  \big(
    (E (\mathbb{Z}/n))_+
    \wedge_{\mathbb{Z}/n}
    (S^{n})^{\wedge^n}
  \big)
$$
\end{example}
More on this case in [Hingston 1992](#Hingston92).

### Rational homotopy

For the [[rational homotopy type]] of cyclic loop spaces see at _[[Sullivan model for free loop space]]_.

## Related concepts

* [[double dimensional reduction]]

* [[string topology]]

* [[cyclic loop stack]]

* [[free loop space]], [[free loop stack]]

## References

The notion of the cyclic loop space of a topological space appears as:

* {#Cohen1985} [[Ralph Cohen]], p. 194 of: *A Model for the Free Loop Space of a Suspension*, in: [[Haynes Miller]], [[Douglas Ravenel]] (eds.), *Algebraic Topology*, Lecture Notes in Mathematics **1286**, Springer 1985 ([doi:10.1007/BFb0078743](https://link.springer.com/chapter/10.1007/BFb0078743))

* {#CarlssonCohen1987} [[Gunnar Carlsson]], [[Ralph Cohen]], *The cyclic groups and the free loop space*, Commentarii Mathematici Helvetici **62** (1987) 423–449  ([doi:10.1007/BF02564455](https://doi.org/10.1007/BF02564455), [dml:140092](https://eudml.org/doc/140092))

* {#Witten88} [[Edward Witten]], _The index of the Dirac operator in loop space_. In Elliptic curves and modular forms in algebraic topology (Princeton, NJ, 1986), volume 1326 of Lecture Notes in Math., pages 161–181. Springer, Berlin, 1988 ([doi:10.1007/BFb0078045](https://doi.org/10.1007/BFb0078045))

* {#Chataur05} [[David Chataur]], *A bordism approach to string topology*, International Mathematics Research Notices **2005** 46, (2005)  ([arXiv:math/0306080](https://arxiv.org/abs/math/0306080), [doi:10.1155/IMRN.2005.2829](https://doi.org/10.1155/IMRN.2005.2829))

* {#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 in their relation to [[Hochschild homology]]/[[cyclic homology]]:

* {#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))

* {#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))

* {#Loday15} [[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/))

and specifically for [[complex projective spaces]] in [Loday (2015), Sec 4](#Loday15).

Discussion in the context of the [[transchromatic character]]:

* {#Stapleton13} [[Nathaniel Stapleton]], _Transchromatic twisted character maps_,  J. Homotopy Relat. Struct. 10, 29–61 (2015). ([arXiv:1304.5194](https://arxiv.org/abs/1304.5194), [doi:10.1007/s40062-013-0040-9](https://doi.org/10.1007/s40062-013-0040-9))

Specifically on cyclic loop spaces of [[n-spheres]]:

* {#Hingston92} [[Nancy Hingston]], *An Equivariant Model for the Free Loop Space of $S^N$*, American Journal of Mathematics **114** 1 (1992) 139-155 ([doi:10.2307/2374740](https://doi.org/10.2307/2374740), [jstor:2374740](https://www.jstor.org/stable/2374740))

See also:

* [[Urs Frauenfelder]], *Dihedral homology and the moon*, J. Fixed Point Theory Appl. **14** (2013) 55–69 ([arXiv:1204.4549](https://arxiv.org/abs/1204.4549), [doi:10.1007/s11784-013-0146-z](https://doi.org/10.1007/s11784-013-0146-z))

A version of the [[cyclic loop stack]] of [[orbifolds]], or at least its restriction to constant loops, namely [[Huan's inertia orbifold]], is discussed in the context of [[equivariant elliptic cohomology]] via [[Tate K-theory]] in:

* {#Huan18} [[Zhen Huan]], Def. 2.14 of: _Quasi-Elliptic Cohomology I_, Advances in Mathematics, Volume 337, 15 October 2018, Pages 107-138 ([arXiv:1805.06305](https://arxiv.org/abs/1805.06305), [doi:10.1016/j.aim.2018.08.007](https://doi.org/10.1016/j.aim.2018.08.007))

following

* [[Zhen Huan]], Section 2.1.2 of: _Quasi-elliptic cohomology_, 2017 ([hdl](http://hdl.handle.net/2142/97268))

and recalled/expanded on in several followup articles, such as in

* [[Zhen Huan]], Section 2 of *Quasi-theories* ([arXiv:1809.06651](https://arxiv.org/abs/1809.06651))

The above formulation of cyclic loop spaces, in the generality of [[∞-stacks]], as right [[base change]] to the [[delooping]] of the [[circle group]], and its relation to [[double dimensional reduction]] in [[brane]]-physics, is due to:

* {#BMSS19} [[Vincent Braunack-Mayer]], [[Hisham Sati]], [[Urs Schreiber]]: Section 2.2 of _[[schreiber:Gauge enhancement of Super M-Branes|Gauge enhancement of Super M-Branes via rational parameterized stable homotopy theory]]_, Communications in Mathematical Physics, **371** 197 (2019) ([doi:10.1007/s00220-019-03441-4](https://doi.org/10.1007/s00220-019-03441-4), [arXiv:1806.01115](https://arxiv.org/abs/1806.01115))

following the analogous discussion in [[rational homotopy theory]] in

* {#FSS18} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], Section 3 of: *[[schreiber:T-Duality from super Lie n-algebra cocycles for super p-branes|T-Duality from super Lie $n$-algebra cocycles for super $p$-branes]]*, [ATMP Volume 22 (2018) Number 5](http://www.intlpress.com/site/pub/pages/journals/items/atmp/content/vols/0022/0005/), [doi:10.4310/ATMP.2018.v22.n5.a3](http://dx.doi.org/10.4310/ATMP.2018.v22.n5.a3), [arXiv:1611.06536](https://arxiv.org/abs/1611.06536))

with exposition in

* [[Urs Schreiber]], [Section 4](https://ncatlab.org/schreiber/show/Super+Lie+n-algebra+of+Super+p-branes#DoubleDimensionalReduction) of:  *[[schreiber:Super Lie n-algebra of Super p-branes]]* (2016)

[[!redirects cyclic loop spaces]]

[[!redirects string loop space]]
[[!redirects string loop spaces]]

[[!redirects twisted loop space]]
[[!redirects twisted loop spaces]]


cyclic loop stack

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## Idea

Any [[free loop stack]] $[S^1, \mathcal{X}]$ carries a canonical [[group action]] of the [[circle group]] $S^1$, given by rigid rotation of loops (explicit models are discussed in [Lupercio & Uribe 01, Sec. 3.6](free+loop+orbifold#LupercioUribe01)). This allows to form the further [[homotopy quotient]]/[[quotient stack]] by this action:

\[
  \label{CyclicLoopStack}
  Cyc(\mathcal{X})
  \;\coloneqq\;
  [S^1, \mathcal{X}] \!\sslash\! S^1
  \,.
\]

In analogy with [[cyclic loop spaces]], this may be called the *cyclic loop stack* of $\mathcal{X}$.

## Properties

### Groupoid presentations

While this is a basic and canonical construction, there seems to be little to no explicit discussion of it in the literature (as of summer 2021). However, the concrete groupoid-presentations considered in [Huan 18, Def. 2.5, 2.9](#Huan18), following [Ganter 07, Sec. 2.1](#Ganter07), may be understood as plausible proposals for models of (eq:CyclicLoopStack).

Motivated from these definitions is, in turn, that of [[Huan's inertia orbifolds]], whose [[orbifold K-theory]] defines a form of [[equivariant elliptic cohomology]] of $\mathcal{X}$.

### General abstract characterization
 {#GeneralAbstractPresentations}

Regarding

* the stack $\mathcal{X} \in \mathbf{H}$ as an object in the [[(∞,1)-topos]] of [[Smooth∞Groupoids]] (or [[DTopological∞Groupoids]] or whatever is appropriate),

* the [[circle group]] $S^1 \in Groups(SmoothManifolds) \xhookrightarrow{ Cdfflg } Groups(\mathbf{H})$ as a ([[smooth ∞-group|smooth]]) [[∞-group]] with [[delooping]]/[[moduli ∞-stack]] $\mathbf{B}S^1$,

then ([BMSS 19, Sec. 2.2](cyclic+loop+space#BMSS19)) the cyclic loop stack is equivalently the right [[base change]] ([[dependent product]]) of $\mathcal{X}$ along the point inclusion $pt_{\mathbf{B}S^1} \colon \ast \to \mathbf{B}S^1$ followed (when desired) by left [[base change]] ([[dependent sum]]) back to the absolute context:

\[
  \label{AbstracyCyclificationConstruction}
  Cyc
  \big(
    \mathcal{X}
  \big)
  \;\;
  \simeq
  \;\;
  \underset{\mathbf{B}S^1 \to \ast}{\sum}
  \underset{\ast \to \mathbf{B}S^1}{\prod}
  (\mathcal{X})
\]

From this one may extract component models by invoking the right [[Quillen adjunction]] which encodes the derived co-free $\infty$-action, discussed [here](Borel+model+structure#RelationToModelStructureOnPlainSimplicialSets).

## Related concepts

* [[Huan's inertia orbifold]]

* [[double dimensional reduction]]

## References

Groupoid models that plausibly represent cyclic loop stacks are discussed (under the notation "$Loops^{ext}$") in

* {#Huan18} [[Zhen Huan]], Def. 2.5, 2.9 in: _Quasi-Elliptic Cohomology I_, Advances in Mathematics, Volume 337, 15 October 2018, Pages 107-138 ([arXiv:1805.06305](https://arxiv.org/abs/1805.06305), [doi:10.1016/j.aim.2018.08.007](https://doi.org/10.1016/j.aim.2018.08.007))

following

* {#Ganter07} [[Nora Ganter]], Section 2.1 in: *Stringy power operations in Tate K-theory* ([arXiv:math/0701565](https://arxiv.org/abs/math/0701565))

[[!redirects cyclic loop stacks]]


cyclic object

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## Idea

The concept of _cyclic object_ is the generalization of that of _[[cyclic sets]]_ where [[Set]] may be replaced with any other [[category]].

Cyclic objects are used in the description of the cyclic structure on [[Hochschild homology]]/[[Hochschild cohomology]] and hence for the discussion of [[cyclic homology]]/[[cyclic cohomology]].

## Definition

Let $\Lambda$ denote the [[cyclic category]] of [[Alain Connes]].  A __cyclic object__  in a [[category]] $C$ is a $C$-valued [[presheaf]] on $\Lambda$.  ([[formal duality|Dually]], a *cocyclic object* is a [[copresheaf]] on the cyclic category.)

Equivalently, this is a [[simplicial object]] $F_\bullet$ together with a sequence of isomorphisms $t_n : F_n \rightarrow F_n$, $n\geq 1$, such that
$$\array{
\partial_i t_n = t_{n-1} \partial_{i-1},\,\, i \gt 0, &
\sigma_i t_n = t_{n+1} \sigma_{i-1},\,\, i \gt0, \\
\partial_0 t_n = \partial_n, & \sigma_0 t_n = t_{n+1}^2 \sigma_n,\\
t^n_{n+1} = \mathrm{id}
}$$
where $\partial_i$ are boundaries, $\sigma_i$ are degeneracies.

## Related concepts

* [[cyclic space]]

* [[cyclic set]]

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

## References

See the references at *[[cyclic category]]* and at *[[cyclic set]]* and *[[cyclic space]]*.

[[!redirects cyclic objects]]

[[!redirects cocyclic object]]
[[!redirects cocyclic objects]]


cyclic operad

# Cyclic operads
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{: toc}

## Motivation in cyclic homology

[[Hochschild homology]] and cohomology have a natural meaning via [[Tor]] and [[Ext]] groups; and they also have an [[higher category theory|infinity-categorical]] interpretation. The [[Hochschild complex]] for associative algebras has a remarkable quotient, the [[cyclic complex]]; this construction is not as general as the mentioned construction, and it can not be generalized to algebras over an arbitrary [[operad]]. Instead there is an additional structure on an operad which enables one to produce an analogue of [[cyclic homology]]. However the long exact sequence of Connes which in the classical case involves cyclic homology and Hochschild homology, here involves the cyclic homology for the original cyclic operad but also the one for the [[Koszul dual operad]] and the Hochschild. In the classical, associative case of course the operad and its Koszul dual coincide.

## Other motivations

Cyclic operads also appear in TQFT-related constructions, often with more structure. See [[modular operad]]s.

## Literature

* [[Ezra Getzler]], [[Mikhail Kapranov|M. M. Kapranov]], _Cyclic operads and cyclic homology, in "Geometry, topology and physics_," International Press, Cambridge, MA, 1995, pp. 167-201 ([pdf](http://sites.northwestern.edu/getzler/files/2019/08/cyclic.pdf))

*  Jovana Obradović, _Monoid-like definitions of cyclic operad_, [tac](http://tac.mta.ca/tac/volumes/32/12/32-12abs.html)
* [[Pierre-Louis Curien]], Jovana Obradović, _A formal language for cyclic operads_, [arXiv:1602.07502](https://arxiv.org/abs/1602.07502); _Categorified cyclic operads_, Appl. Categ. Struct. __28__, 59--112 (2020) [doi](https://doi.org/10.1007/s10485-019-09569-7)

* Benjamin C. Ward, _Maurer-Cartan elements and cyclic operads_, J. Noncommut. Geom. __10__:4 (2016) 1403--1464
[doi](https://doi.org/10.4171/JNCG/263)

[[!redirects cyclic operad]]
[[!redirects cyclic operads]]


cyclic order

# Cyclic orders
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## Idea #

If a [[linear order]] is a way of arranging the elements of a set 'on a line' (with a direction), then a *cyclic order* is a way of arranging them 'on a circle' (with a chosen direction, say clockwise or counterclockwise).

Unlike a linear order (and most other things called 'order') which are defined as a sort of binary relation, a cyclic order is defined as a sort of *ternary* [[relation]], since on a circle we can't say "$x$ comes before (or after) $y$" only "$x$, $y$, and $z$ come in clockwise (or counterclockwise) order."

## Definition #

A **cyclic relation** on a [[set]] $S$ is a ternary [[relation]] $R$ on $S$ such that, if $R(x,y,z)$ for any elements $x, y, z$ of $S$, then $R(y,z,x)$ (and hence also $R(z,x,y)$).

Then a **cyclic order** on $S$ is a cyclic relation $R$ that satisfies ternary versions of the properties of a [[linear order]]:

*  Cyclic [[irreflexive relation|irreflexivity]]:  If $R(x,y,z)$, then $y \ne z$.
*  Cyclic [[asymmetric relation|asymmetry]]:  If $R(x,y,z)$ and $R(x,z,w)$, then $y \ne w$.
*  Cyclic [[transitive relation|transitivity]]:  If $R(x,y,z)$ and $R(x,z,w)$, then $R(x,y,w)$.
*  Cyclic [[comparison]]:  If $R(x,y,z)$, then for any $w$, $R(x,y,w)$ or $R(x,w,z)$ or $R(w,y,z)$.
*  Cyclic [[connected relation|connectedness]]:  If $x \ne y$, $x \ne z$, and $y \ne z$, then $R(x,y,z)$ or $R(x,z,y)$.

Actually, none of these order axioms (except for comparison) is really complete as stated; any cyclic permutation of the axiom should also be assumed.  However, these permutations all follow automatically for a cyclic relation.

Note that for a [[constructive mathematics|constructive]] version, the set $S$ needs to be already equipped with a (tight) [[apartness relation]] for the connectedness axiom to make sense; unlike with a linear order, we can\'t recover the apartness relation from the cyclic order.  For a similar reason, it\'s difficult to state [[antisymmetric relation|antisymmetry]] correctly for a nonstrict (like a [[total order]]) version of a cyclic order.

As with a linear order, not all of these axioms are needed.  In particular, using [[excluded middle]], it\'s enough if a cyclic relation is transitive and satisfies

*  Cyclic trichotomy:  For all $x, y, z$, ($x = y$ or $x = z$ or $y = z$) xor $R(x,y,z)$ xor $R(x,z,y)$.

## Monotone functions

An [[injection|injective]] function $f:S\to T$ between cyclically ordered sets is **strictly monotone** if it preserves the ternary relation $R$, i.e. if $R_S(x,y,z)$ implies $R_T(f(x),f(y),f(z))$.  This is the cyclic counterpart of a strictly monotone function between linear orders (one which preserves the relation $\lt$).  As in that case, irreflexivity then implies that $f$ is injective as long as $S$ has at least three distinct elements; but in general, this should be stated explicitly (and interpreted in the strongest sense for constructive mathematics).

We say that $f$ is merely **monotone** if it satisfies (either of) the following _equivalent_ conditions:

* $f$ reflects the ternary relation $R$, i.e. if $R_T(f(x),f(y),f(z))$ then $R_S(x,y,z)$.
* $f$ preserves the ternary relation $R$ for elements with pairwise distinct images, i.e. if $R_S(x,y,z)$ holds and $f(x)$, $f(y)$, and $f(z)$ are pairwise distinct, then $R_T(f(x),f(y),f(z))$.

This is the cyclic counterpart of a monotone function between linear orders (one which reflects the relation $\lt$).

## Remarks

* One would like the [[cycle category]] $\Lambda$ to be the category of [[finite set|finite]] [[inhabited set|nonempty]] cyclically ordered sets and monotone functions, just as the [[simplex category]] $\Delta$ is the category of finite nonempty linearly ordered sets and monotone functions.  However, this seems to fail for the $0$-cycle, which has a nontrivial loop and is therefore not [[terminal object|terminal]] (unlike the cyclically ordered [[singleton]], which is terminal).

## Related concepts

* a [[round chord diagram]] is a cyclic order equipped with a pairing of all its elements

## References

* [[Tim Campion]], [[Jinhe Ye]], *Non-trivial higher homotopy of first-order theories* ([arXiv:2306.12011](https://arxiv.org/abs/2306.12011))

[[!redirects cyclic orders]]
[[!redirects cyclic ordering]]
[[!redirects cyclic orderings]]
[[!redirects cyclically ordered]]
[[!redirects cyclically ordered set]]
[[!redirects cyclically ordered sets]]


cyclic permutation

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## Definition

For a [[finite set]] $X$, a _cyclic permutation_ on $X$ is a [[permutation]] $\sigma \colon X \to X$ such that the induced [[group homomorphism]] $\mathbb{Z} \to Aut(X)$ from the [[integers]] to the [[automorphism group]]  (i.e. the [[symmetric group]]) of $X$, sending $n \in \mathbb{Z}$ to $\sigma^n$, defines a [[transitive action]].

One may visualize the elements of $X$ as points arranged on a [[circle]] spaced equally apart, with $\sigma(x)$ the next-door neighbor of $x$ in the counterclockwise direction, hence the name.

See also [[rotation permutation]].

## Related concepts

* [[permutation cycle]]

* [[cyclic group]]

## References

See also:

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

[[!redirects cyclic permutations]]


cyclic set

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## Definition

A **cyclic set** is a [[presheaf]] on the _[[cyclic category]]_ (which is often called [[Connes' cyclic category]] though it is cocyclic, with the usual contravariant confusion), which is intermediate between a [[symmetric set]] and a [[simplicial set]]. With [[Set]] replaced by a general [[category]] one speaks of a _[[cyclic object]]_.

The concept of cyclic sets/objects is used in the description of the cyclic structure on [[Hochschild homology]]/[[Hochschild cohomology]] and hence for the discussion on [[cyclic homology]]/[[cyclic cohomology]].

Cyclic sets and more generally [[cyclic objects]] can be described in terms of standard generators:

A $\mathbf{Z}$-cyclic object (synonym: [[paracyclic object]])in a [[category]] $C$ is a [[simplicial object]] $F_\bullet$ in $C$ together with a sequence of [[isomorphisms]] $t_n : F_n \rightarrow F_n$, $n\geq 1$, such that
$$\array{
\partial_i t_n = t_{n-1} \partial_{i-1},\,\, i \gt 0, &
\sigma_i t_n = t_{n+1} \sigma_{i-1},\,\, i \gt0, \\
\partial_0 t_n = \partial_n, & \sigma_0 t_n = t_{n+1}^2 \sigma_n,
}$$
where $\partial_i$ are boundaries, $\sigma_i$ are degeneracies. A $\mathbf Z$-cocyclic ([[paracocyclic object|paracocyclic]]) object in $C$ is a $\mathbf{Z}$-cyclic object in $C^{\mathrm{op}}$.
$\mathbf Z$-(co)cyclic object is (co)cyclic if,
in addition, $t_n^{n+1} = 1$

## Properties

### As a classifying topos

The [[category]] of cyclic sets, being a [[presheaf category]] is a [[topos]], and hence is the [[classifying topos]] for some [[geometric theory]]. This turns out to be the theory of [[abstract circles]] ([Moerdijk 96](#Moerdijk96)).
A further analysis can be found in ([Caramello Wentzlaff 14](#CaramelloWentzlaff14)).
Accordingly there is an [[infinity-action]] of the [[circle group]] on the [[geometric realization]] of a cyclic set (see also [Drinfeld 03](#Drinfeld03)).

### Model category structure and $S^1$-equivariant homotopy theory
 {#ModelCategoryStructure}

There is a [[model category]]-structure on the category of
cyclic sets, which makes it a presentation for
$S^1$-[[equivariant homotopy theory]] ([Spalinski 95](#Spalinski95), [Blumberg 04](#Blumberg04)).

## Related concepts

* [[cyclic category]]

* [[cyclic cohomology]]

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

* [[skew-simplicial set]], [[symmetric set]], [[simplicial set]]

## References

The original definition:

* {#Connes83} [[Alain Connes]], _Cohomologie cyclique et foncteurs $Ext^n$_, C.R.A.S. **296** (1983), Série I, 953-958 [MR777584](https://mathscinet.ams.org/mathscinet-getitem?mr=777584) [Zbl0534.18009](https://zbmath.org/?q=an:0534.18009) ([pdf](https://alainconnes.org/wp-content/uploads/n83.pdf), [[Connes_CohomologieCyclique.pdf:file]]).

* [[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))

Textbook account (in the generality of [[cyclic spaces]]):

* [[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 3 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))

Exposition:

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

Connections to [[simplicial sets]] are in:

* [[Alain Connes]], Caterina Consani, _Cyclic structures and the topos of simplicial sets_, [1309.0394](http://arxiv.org/abs/1309.0394)

The identification of the category of cyclic sets as the [[classifying topos]] for [[abstract circles]] is due to

* {#Moerdijk96} [[Ieke Moerdijk]], _Cyclic sets as a classifying topos_, 1996 ([[MoerdijkCyclic.pdf:file]])

* {#CaramelloWentzlaff14} Olivia Caramello, Nicholas Wentzlaff, _Cyclic theories_, 2014 ([arXiv:1406.5479](http://arxiv.org/abs/1406.5479))

The resulting circle-action on the ([[geometric realization]] of) cyclic sets is also discussed in

* {#Drinfeld03} [[Vladimir Drinfeld]], _On the notion of geometric realization_ ([arXiv:0304064](http://arxiv.org/abs/math/0304064))

The [[homotopy theory]] of cyclic sets and its relation to $S^1$-[[equivariant homotopy theory]] is discussed in

* {#Spalinski95} J. Spalinski, _Strong homotopy theory of cyclic sets_, J. of Pure and Appl. Alg. 99 (1995), 35–52.

* {#Blumberg04} [[Andrew Blumberg]], _A discrete model of $S^1$-homotopy theory_ ([arXiv:math/0411183](http://arxiv.org/abs/math/0411183))

An old query is archived in $n$Forum [here](http://nforum.mathforge.org/discussion/5822/cyclic-set/?Focus=46240#Comment_46240).

There are fairly recent slides by Spalinski on the subject [here](http://www.mimuw.edu.pl/~cat09/slides/Spalinski.pdf), which also discuss relationships with [[dihedral sets]] and [[quaternionic set]]s, as studied by [[Loday]].

[[!redirects cyclic sets]]


cyclic space

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## Idea

A *cyclic space* is a [[cyclic object]] in a [[category]] of [[spaces]], usually in one of the standard presentations of [[homotopy types]], hence in [[TopologicalSpaces]] or [[SimplicialSets]].

## Related concepts

* [[cyclic homology]]

* [[cyclic category]]

* [[cyclic object]]

* [[cyclic set]]

* [[cyclotomic space]]

* [[cyclotomic spectrum]]

## References

* D. Burghelea Z. Fiedorowicz, Section 1 of: *Cyclic homology and algebraic K-theory of spaces—II*, Topology Volume 25, Issue 3, 1986, Pages 303-317 (doi:10.1016/0040-9383(86)90046-7)

* [[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 3 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/))

See also:

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

In the context of [[geometric realization of topological stacks|geometric realization]] of [[loop stacks]] and [[inertia stacks]]:

* [[Ernesto Lupercio]], [[Bernardo Uribe]], Section 7.2 of: _Inertia orbifolds, configuration spaces and the ghost loop space_, Quarterly Journal of Mathematics __55__ 2 (2004) 185-201 ([arxiv:math.AT/0210222](http://arxiv.org/abs/math/0210222), [doi:10.1093/qmath/hag053](https://doi.org/10.1093/qmath/hag053))

[[!redirects cyclic spaces]]


cyclic vector

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## Idea

A [[vector]] --- hence an [[element]] of some [[vector space]] $\mathcal{H}$ --- is called _cyclic_ with respect to the [[action]]/[[representation]] of some [[algebra]] $\mathcal{A}$ on $\mathcal{H}$ if every element of $\mathcal{H}$ may be obtained by acting on $v$ with some algebra element $A \in \mathcal{A}$, or rather if one may always find a [[sequence]] of elements $A_n$ whose action on $v$ [[convergence of a sequence|converges]] to the given element.

## Definition

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

Let

1. $\mathcal{A}$ be a [[C*-algebra]];

1. $\mathcal{H}$ a [[Hilbert space]];

1. $\pi \;\colon\; \mathcal{A} \longrightarrow \mathcal{B}(\mathcal{H})$ a [[C*-representation]] of $\mathcal{A}$ on $\mathcal{H}$.

Then a [[vector]] $v \in \mathcal{H}$ is called a **cyclic vector** if the [[image]] of $v$ under $\mathcal{A}$ acting via $\rho$ is a [[dense subspace]] of $\mathcal{H}$:

$$
  im\left( \rho(-)(v) \right)
   \overset{\text{dense}}{\hookrightarrow}
  \mathcal{H}
  \,.
$$

=--

## Properties

The notions of cyclic vector is dual to that of [[separating vector]] with respect to the [[commutant]] $\mathcal{M}'$, that is a vector is cyclic for $\mathcal{M}$ iff it is separating for $\mathcal{M}'$.

## Applications

* In [[algebraic quantum field theory]] the [[state on a star-algebra|states]] corresponding to cyclic vectors appear as _[[vacuum states]]_. See [[Reeh-Schlieder theorem]].

* [[GNS construction]]

## References

* [[eom]], _[Cyclic vector](https://www.encyclopediaofmath.org/index.php/Cyclic_vector)_

[[!redirects cyclic vectors]]


cyclotomic field

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## Idea

A [[field extension]] $\mathbb{F}(\zeta_n)$ by a primitive [[root of unity]] $\zeta_n$.

The default meaning where $\mathbb{F}$ is left unspecified is $\mathbb{F} = \mathbb{Q}$, the field of [[rational numbers]], where $\zeta_n$ may be taken to be $\exp(2\pi i/n)$ (hence the smallest subfield of the [[complex numbers]] $\mathbb{C}$ that contains this [[exponential function|exponential]] value).

## Related concepts

* [[cyclotomic integer]]

* [[field extension]], [[Galois group]]

* [[algebraic numbers]]

* [[anti-cyclotomic field]]

* [[cyclotomic spectrum]]

## References

* [[Keith Conrad]], *Cyclotomic extensions* [[pdf](https://kconrad.math.uconn.edu/blurbs/galoistheory/cyclotomic.pdf), [[Conrad-CyclotomicExtensions.pdf:file]]]

* Wikipedia, _[cyclotomic field](http://en.wikipedia.org/wiki/Cyclotomic_field)_

[[!redirects cyclotomic fields]]

[[!redirects cyclotomic extension]]
[[!redirects cyclotomic extensions]]

[[!redirects cyclotomic field extension]]
[[!redirects cyclotomic field extensions]]


cyclotomic integer

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## Idea

For $\zeta_n$ an $n$th [[root of unity]] and $k$ a [[field]], consider the [[cyclotomic field]] $k(\zeta_n)$. The [[ring of integers]] of this field is called the ring of _cyclotomic integers._

## Examples

* For $\mathbb{Q}$ the [[rational numbers]], consider the [[cyclotomic field]] $\mathbb{Q}(\zeta_n)$. The [[ring of integers]] of this field is $\mathbb{Z}[\zeta_n]$, i.e. [[polynomials]] in $\zeta_n$ with [[integer]] [[coefficients]].

* [[characters are cyclotomic integers]]

## References

See also

* Wikipedia, _[Cyclotomic field -- Properties](https://en.wikipedia.org/wiki/Cyclotomic_field#Properties)_

[[!redirects cyclotomic integers]]


cyclotomic space

#Contents#
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## Definition

Let $\rho_r: S^1 \to S^1/C_r$ be the homeomorphism $z \mapsto \sqrt[r] z$, $C_r \coloneqq \mathbb{Z}/r\mathbb{Z}$.

Given an $S^1$-space $X$, denote by $\rho^{\ast}_r X^{C_r}$ the $S^1$–space obtained by pulling back the $S^1/C_r$ action on $X^{C_r}$ via $\rho_r$, where $X^{C_r}$ is the [[homotopy fixed point]] space of the induced [[cyclic group]] action. Then $\rho^{\ast}_r(\rho^{\ast}_s X^{C_s})^{C_r} =\rho^{\ast}_{r s} X^{C_{r s}}$.

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

A _cyclotomic space_ is

1. a [[topological space]] $X$

1. equipped with a [[continuous function|continuous]] [[circle group]] [[action]]

   (hence a [[topological G-space]] for $G = S^1$)

1. and equipped with a family of $S^1$-equivariant [[weak homotopy equivalences]] $R_r: \rho^{\ast}_r X^{C_r} \to X$, for $r \geq 1$, such that $R_1 = id$, and $R_{r s} = R_r \circ \rho^{\ast}_r R_s^{C_r}$.

=--

([Schl 09, def. 4.3](#Schl09), [AGHL 12, def. 3.6](#AGHL12))

## Properties

A [[free loop space]], $\mathcal{L} X$, is a cyclotomic space, where $S^1$ acts by rotation of loops.

Let $R X = \underset{R_r}{holim} \rho^{\ast}_r X^{C_r}$. Then $R$ is a [[comonad]] on the category of cyclotomic spaces.

The $S^1$-equivariant suspension spectrum of a cyclotomic space is a [[cyclotomic spectrum]].

## Related concepts

* [[cyclotomic spectrum]]

* [[cyclic space]]

## References

* {#Schl09} [[Christian Schlichtkrull]], _The cyclotomic trace for symmetric ring spectra_, Geometry & Topology Monographs 16 (2009), 545–592, ([pdf](/http://msp.org/gtm/2009/16/gtm-2009-16-014s.pdf)), ([arXiv:0903.3495](https://arxiv.org/abs/0903.3495))

* {#AGHL12} [[Vigleik Angeltveit]], [[Teena Gerhardt]], [[Michael Hill]], Ayelet Lindenstrauss, _On the algebraic K-theory of truncated polynomial algebras in several variables_ ([arXiv:1206.0247](https://arxiv.org/abs/1206.0247))

[[!redirects cyclotomic spaces]]


cyclotomic spectrum

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## Idea

A _cyclotomic spectrum_ is an $S^1$-[[equivariant spectrum]] $E$ with [[fixed points]] for all the [[finite group|finite]] [[cyclic groups]] $C_p = \mathbb{Z}/p\mathbb{Z} \hookrightarrow S^1$ inside the [[circle group]], and equipped with $S^1$-equivariant identifications $E^{C_p} \stackrel{\simeq}{\to} E$ of the $C_P$-[[fixed points]] with the full object.

The [[topological Hochschild homology]]-spectra $E = THH(A)$ are naturally cyclotomic spectra, and this is where the concept originates: by the discussion at _[[Hochschild cohomology]]_ $THH(A)$ is the [[E-infinity ring]] of functions on the [[free loop space]] of $Spec(A)$, and cyclotomic structure reflects the structure of free loop spaces: loops that repeat with period $p$ are equivalent to plain loops.

Cyclotomic structure is the origin of the [[cyclotomic trace]] map $THH \longrightarrow TC$ from [[topological Hochschild homology]] to [[topological cyclic homology]].

## Definition
 {#Definition}

Throughout, for $p$ a [[prime number]] write $C_p \subset S^1$ for the [[cyclic group]] $\mathbb{Z}/p\mathbb{Z}$ of order $p$, regarded as a subgroup of the [[circle group]].

A definition says that a cyclotomic spectrum  is an [[circle group]]-[[genuine equivariant spectrum]] $X$ (modeled on [[orthogonal spectra]]) equipped with equivalences to its naive point-set fixed point spectra $\Phi^{C_p} X$ for all the [[cyclic group|cyclic]] [[subgroups]] $C_p \subset S^1$.

A more abstract definition was given in [Nikolaus-Scholze 17](#NikolausScholze17):[^HarmonizeLiterature]

[^HarmonizeLiterature]: Maybe this definition does not exactly agree with other definitions in the literature; see $n$Forum discussion [here](https://nforum.ncatlab.org/discussion/6851/cyclotomic-spectrum/?Focus=89589#Comment_89589).

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

A _cyclotomic spectrum_ is

1. a [[spectrum]] $X$

1. a [[circle group]] [[∞-action]] on $X$, i.e. an [[(∞,1)-functor]] $B S^1 \to Spectra$ which takes [[generalized the|the]] unique point of $B S^1$ to $X$;

1. for each prime number $p$ a homomorphism of spectra with such circle group action

   $$
     F_p \;\colon\; X \longrightarrow X^{t C_p}
   $$

   to the [[Tate spectrum]] (the [[homotopy cofiber]] $X^{t C_p} \coloneqq cofib( X_{C_p} \overset{norm_p}{\to} X^{C_p} )$ of the [[norm map]]),

   {#CircleQuotient} where the circle action on the Tate spectrum comes from the canonical identification $S^1/C_p \simeq S^1$.

(These morphisms $F_p$ are called the _[[Frobenius morphisms]]_ of the cyclotomic structure, due to [this def. ](Frobenius+morphism#EInfinityFrobenius), [this example](Frobenius+morphism#EInfinityFrobeniusReducesToOrdinaryFrobenius)).

=--

([Nikolaus-Scholze 17, def. 1.3, def. II.1.1](#NikolausScholze17)).

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

For $X$ a spectrum with [[stable homotopy groups]] bounded below, then
def. \ref{CyclotomicSpectrumViaTateSpectra} is equivaent to the traditional:

There is an [[(∞,1)-functor]]

$$
  CycSp_-^{gen} \overset{\simeq}{\longrightarrow} CycSp_-
$$

from traditional ("genuine") cyclotomic spectra bounded below to bounded below cyclotomic spectra in the sense of def. \ref{CyclotomicSpectrumViaTateSpectra}, and this is an [[equivalence of (∞,1)-categories]].

=--

([Nikolaus-Scholze 17, prop. II.3.4, theorem II.6.9](#NikolausScholze17)).

## Examples
 {#Examples}

+-- {: .num_example #THHCarriesCyclotomicStructure}
###### Example
**([[topological Hochschild homology]])**

For every [[A-∞ ring]] $A$, the [[topological Hochschild homology]] spectrum $THH(A)$ naturally carries the structure of a cyclotomic spectrum (def. \ref{CyclotomicSpectrumViaTateSpectra}).

=--

([Nikolaus-Scholze 17, section II.2, def. III.2.3](#NikolausScholze17))

+-- {: .num_example #TrivialCyclotomicSpectra}
###### Example
**(trivial cyclotomic spectra)**

Every [[spectrum]] $X$ becomes a cyclotomic spectrum $X^{triv}$ in the sense of def. \ref{CyclotomicSpectrumViaTateSpectra} by equipping it

1. with the trivial [[circle group]] [[∞-action]]

1. for each prime $p$ with the composite morphism

   $$
     f_p
      \;\colon\;
     \mathbb{S}
      \longrightarrow
     \mathbb{S}^{C_p}
       \longrightarrow
     \mathbb{S}^{t C_p}
   $$

   (the first being the $( B C_p \times (-) \dashv (-)^{C_p} )$-[[unit of an adjunction|unit]] into the [[homotopy fixed points]], the second the defining morphism into the [[Tate spectrum]]  )

1. the $S^1/C_p$-equivariant structure on these morphisms given under the [[adjunction]] between trivial action and homotopy fixed points by the [[adjunct]] morphisms

   $$
     X \longrightarrow \left(X^{t C_p}\right)^{S^1/C_p}
   $$

   as the composite

   $$
     X \to X^{S^1}
       \simeq
    \left( X^{C_p} \right)^{S^1/C_p}
     \longrightarrow
    \left( X^{t C_p} \right)^{S^1/C_p}
    \,.
   $$

This construction constitutes a left [[adjoint (infinity,1)-functor]]
to taking [[topological cyclic homology]]

$$
  CycSpectra
    \underoverset{\underset{TC}{\longrightarrow}}{\overset{(-)^{triv}}{\longleftarrow}}{\bot}
  Spectra
  \,.
$$

=--

([Nikolaus-Scholze 17, example II.1.2 (ii) and middle of p. 126](#NikolausScholze17))

+-- {: .num_example #CyclotomicSphereSpectrum}
###### Example
**(cyclotomic sphere spectrum)**

The [[sphere spectrum]] regarded as a cyclotomic spectrum via example \ref{TrivialCyclotomicSpectra} is called the _cyclotomic sphere spectrum_.

As such it is equivalently its [[topological Hochschild homology]] according to example \ref{THHCarriesCyclotomicStructure}:

$$
  \mathbb{S}^{triv} \simeq THH(\mathbb{S})
  \,.
$$

=--

([Nikolaus-Scholze 17, example II.1.2 (ii)](#NikolausScholze17))

## Properties

### Monoidal structure
 {#MonoidalStructure}

The [[tensor unit]] in the [[symmetric monoidal (infinity,1)-category]] of cyclotomic spectra is the cyclotomic sphere spectrum from example \ref{CyclotomicSphereSpectrum} ([Blumberg-Mandell 13, example 4.9](#BlumbergMandell13))

## Related concepts

* [[cyclotomic space]]

* [[cycle category]], [[cyclic set]], [[cyclic object]]

* [[Frobenius homomorphisms]]

* [[Tate spectrum]]

## References

* {#BlumbergMandell13} [[Andrew Blumberg]], [[Michael Mandell]], _The homotopy theory of cyclotomic spectra_, Geom. Topol. **19** (2015) 3105-3147 [[arXiv:1303.1694](http://arxiv.org/abs/1303.1694), [doi:10.2140/gt.2015.19.3105](https://doi.org/10.2140/gt.2015.19.3105)]

* {#BarwickGlasman16} [[Clark Barwick]], [[Saul Glasman]], _Cyclonic spectra, cyclotomic spectra, and a conjecture of Kaledin_ ([arXiv:1602.02163](http://arxiv.org/abs/1602.02163))

* [[Clark Barwick]], [[Saul Glasman]], _Noncommutative syntomic realization_ ([pdf](http://dl.dropbox.com/u/1741495/papers/tornado2.pdf))

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

* {#NikolausScholze17} [[Thomas Nikolaus]], [[Peter Scholze]], _On topological cyclic homology_,  Acta Math. **221** 2 (2018) 203-409 [[arXiv:1707.01799](https://arxiv.org/abs/1707.01799), [doi:10.4310/ACTA.2018.v221.n2.a1](https://dx.doi.org/10.4310/ACTA.2018.v221.n2.a1)]

[[!redirects cyclotomic spectra]]


cyclotomic trace

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{:toc}

## Idea

Given a ring $R$, then there is a natural morphism of [[spectra]]

$$
  \array{
     && \mathbf{TC}(R)
     \\
      & \nearrow & \downarrow
     \\
     \mathbf{K}(R) &\longrightarrow& \mathbf{THH}(R)
  }
$$

from the [[algebraic K-theory]] spectrum to the [[topological Hochschild homology]] spectrum and factoring through the [[topological cyclic homology]] spectrum (a [[cyclotomic spectrum]]) called the _cyclotomic trace_ which much like a [[Chern character]] map for [[algebraic K-theory]].

This is a refinement to [[spectra]] of the [[Dennis trace]].

## Related concepts

* [[Beilinson regulator]]

## References

* [[Marcel Bökstedt]], W.C. Hsiang, [[Ib Madsen]], _The cyclotomic trace and algebraic K-theory of spaces_, Invent. Math. __111__ (1993), 463-539, [MR94g:55011](http://www.ams.org/mathscinet-getitem?mr=1202133), [doi](http://dx.doi.org/10.1007/BF01231296)
* [[Ib Madsen]], _The cyclotomic trace in algebraic K-theory_, In: Joseph, A., Mignot, F., Murat, F., Prum, B., Rentschler, R. (eds) First European Congress of Mathematics Paris, July 6–10, 1992. Progress in Mathematics __120__, Birkhäuser 1994 [doi](https://doi.org/10.1007/978-3-0348-9112-7_9)

* [[Andrew Blumberg]], [[David Gepner]], [[Goncalo Tabuada]], _Uniqueness of the multiplicative cyclotomic trace_ ([arXiv:1103.3923](http://arxiv.org/abs/1103.3923))

* [[David Ayala]], [[Aaron Mazel-Gee]], [[Nick Rozenblyum]], _The geometry of the cyclotomic trace_, ([arXiv:1710.06409](https://arxiv.org/abs/1710.06409))

For a very brief, very elementary introduction, see

* [[Aaron Mazel-Gee]], _The geometry of the cyclotomic trace_, ([pdf](https://etale.site/writing/cyclo-summary.pdf))

For more see the references at _[[topological Hochschild homology]]_.


cylinder factorisation system

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###Context###
#### Factorization systems
+--{: .hide}
[[!include factorization systems - contents]]
=--
#### Category theory
+--{: .hide}
[[!include category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A **cylinder factorisation system** on a category is a generalisation of a [[factorisation system]] in which we have a pair $(E, M)$, where $E$ is a class of small cocones, and $M$ is a class of small cones, such that every cylinder factorises uniquely as a cocone in $E$ followed by a cone in $M$.

Cylinder factorisation systems are algebras for a pseudomonad structure given by sending each category to its [[Isbell envelope]]: see [Garner 2015](#Garner).

## Related concepts

* [[factorisation system]], [[orthogonal factorisation system]]

* [[Isbell envelope]], [[Isbell adjunction]]

## References

Cylinder factorisation systems were introduced in:

* {#Garner} [[Richard Garner]], _The Isbell monad_, Advances in Mathematics **274** (2015) pp.516-537. ([draft](http://comp.mq.edu.au/~rgarner/Papers/Isbell.pdf))

The special case in which $E$ and $M$ comprise classes of discrete cones was introduced in:

* [[Horst Herrlich]], and Walter Meyer, _Factorization of flows and completeness of categories_, Quaestiones Mathematicae 17.1 (1994): 1-11.

See also:

* George E. Strecker, _Flows with respect to a functor_, Applied Categorical Structures 8 (2000): 559-578.

The special case in which $E$ is a class of [[epimorphisms]] and $M$ is a class of cones was studied in the following papers, including an application to [[colimits in categories of algebras]]:

* [[Rudolf-E. Hoffmann]], _Factorization of cones_, Mathematische Nachrichten **87** 1 (1979) 221-238. [[doi:10.1002/mana.19790870120](https://doi.org/10.1002/mana.19790870120)]

* [[Rudolf-E. Hoffmann]], _Factorization of cones II, with applications to weak Hausdorff spaces_, in: *Categorical Aspects of Topology and Analysis: Proceedings of an International Conference Held at Carleton University, Ottawa, August 11–15, 1981*, Springer (1981) [[doi:10.1007/BFb0092878](https://doi.org/10.1007/BFb0092878)]

Earlier still, the following considered the special case in which the right class furthermore is required to comprise discrete cones:

* [[Horst Herrlich]]. _Topological functors_, General Topology and its Applications 4.2 (1974): 125-142.

See also:

* [[Horst Herrlich]], G. Salicrup, and R. Vazquez, _Dispersed factorization structures_, Canadian Journal of Mathematics 31.5 (1979): 1059-1071.

* A. Melton, and G. E. Strecker, _On the structure of factorization structures_, Category Theory: Applications to Algebra, Logic and Topology Proceedings of the International Conference Held at Gummersbach, July 6–10, 1981. Springer Berlin Heidelberg, 1982.

* [[Abstract and Concrete Categories]]

[[!redirects cylinder factorisation systems]]
[[!redirects cylinder factorization system]]
[[!redirects cylinder factorization systems]]


cylinder functor

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

#Contents#
* table of contents
{:toc}

## Definition

For $C$ a [[category]], a **cylinder functor** on $C$ is a [[functor]] denoted

$$
  (-)\times I : C \to C
$$

equipped with three [[natural transformations]]

$$
  e_0, e_1 : Id_C \to (-)\times I
$$

$$
 \sigma : (-)\times I \to Id_C
$$

such that $\sigma e_0 = \sigma e_1 = Id_C$.

## Remarks

* A cylinder functor functorially provides _[[cylinder objects]]_ used for talking about [[homotopy]].

* The notation is supposed to be suggestive of a [[product]] with an object $I$. While this is the motivating example, the cylinder functor need not be of that form.

## Cylindrical model structures

[[Richard Williamson]] has developed a way to build a model structure from the simple point of departure of a structured interval in a monoidal category - more generally, a structured cylinder and a structured co-cylinder in a category. This is given in ([Williamson 2012](#Williamson12)).

## References

A very brief introduction to cylinder functors is given starting on page 9 of [[Abstract-Homotopy.pdf|Abstract Homotopy Theory:file]].

A fuller development of their properties is given in

* [[Klaus Heiner Kamps]], [[Timothy Porter]], _Abstract Homotopy and Simple Homotopy Theory_ ([GoogleBooks](http://books.google.de/books?id=7JYKxInRMdAC&dq=Porter+Kamps&printsec=frontcover&source=bl&ots=uuyl_tIjs4&sig=Lt8I92xQBZ4DNKVXD0x76WkcxCE&hl=de&sa=X&oi=book_result&resnum=3&ct=result#PPP1,M1))

Cylinder functors also form one of the key elements in Baues' approach to [[algebraic homotopy]]:

* [[Hans-Joachim Baues]], _Algebraic Homotopy_, Cambridge studies in advanced mathematics 15, Cambridge University Press, (1989).

* [[Hans-Joachim Baues]], _Combinatorial Homotopy and 4-Dimensional Complexes_, de Gruyter Expositions in Mathematics 2, Walter de Gruyter, (1991).

* [[Hans-Joachim Baues]], _Homotopy Types_, in I.M.James, ed., _[[Handbook of Algebraic Topology]]_, 1--72, Elsevier, (1995).

Cylindrical model structures are discussed in

* {#Williamson12} [[Richard Williamson]], _Cylindrical model structures_, PhD thesis, 2012 ([arXiv:1304.0867](https://arxiv.org/abs/1304.0867))

[[!redirects cylinder functors]]


cylinder object

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

#Contents#
* table of contents
{:toc}

## Idea

The concept of a **cylinder object** in a [[category]] is an abstraction of the construction in [[Top]] which associates to any [[topological space]] $X$ the _cylinder_ $X \times [0,1]$ over $X$, where $[0,1]$ is the standard [[topological interval]]. It is notably used to define the concept of [[left homotopy]], say in a [[model category]].

The standard topological cylinder $X \times [0,1]$ naturally comes equipped with a continuous map

$$
  X \coprod X \to X \times [0,1]
$$

that identifies $X$ as the two ends $X \times \{0\}$ and $X \times \{1\}$ of the cylinder, and with a map

$$
  X \times [0,1] \to X
$$

that collapses the cylinder back onto $X$.

The composite of these two maps is the [[codiagonal]] $(Id,Id) :  X \coprod X \to X$. Moreover, the cylinder $X \times [0,1]$ is [[homotopy equivalence|homotopy equivalent]] to $X$.

These properties are the characterizing properties of the cylinder that can be abstracted and realized in other categories.

The notion dual to _cylinder object_ is [[path space object]], which is thus sometimes alternatively called a cocylinder. Cylinder objects and path space objects are used to define [[left homotopies]] and [[right homotopies]], respectively.

There are several views on the role of cylinders /  cocylinders in homotopy theory. If there is a natural notion of weak equivalence or [[quasi-isomorphism]] then the cylinder is used to encode a notion of homotopy equivalence compatible with the weak equivalences. In some other situations, a 'cylinder' , often functorially given and well structured in some way, may be the _primitive_ notion that allows a notion of 'homotopy equivalence' to be put forward.  Below we give a definition optimised for the former situation. Some indication of the second context is given in the entry [[cylinder functor]].

## Definition

In a [[category with weak equivalences]] $C$ that has [[coproducts]] a **cylinder object** $Cyl(X)$ for an [[object]] $X$ is a factorization

$$
  X \coprod X \to Cyl(X) \stackrel{\simeq}{\to} X
$$

of the codiagonal $X \coprod X \to X$ out of the [[coproduct]] of $X$ with itself, such that $Cyl(X) \to X$ is a weak equivalence and such that the morphism $X \coprod X \to Cyl(X)$ is "nice" in some way.

In some situations the assignment of cylinder objects may exist functorially, in which case one speaks of a [[cylinder functor]].

If $C$ has the structure of a [[model category]] then "nice" means that $X \coprod X \hookrightarrow Cyl(X)$ is a [[cofibration]].  The factorization axiom of a model category ensures that for each object there is a cylinder object with this property; in fact, one with the additional property that $Cyl(X) \to X$ is an acyclic fibration.  Cylinder objects such that $X \coprod X \hookrightarrow Cyl(X)$ is a cofibration are sometimes called *good*, and those for which moreover $Cyl(X) \to X$ is an acyclic fibration are then called *very good*.

## Examples

* In [[sSet]] equipped with the standard [[model structure on simplicial sets]] the standard cylinder object for any $S \in sSet$ is $S \times \Delta[1]$.

* In [[Top]], the standard cylinder $X\times [0,1]$ is a cylinder object for both the [[classical model structure on topological spaces]] $Top_{Quillen}$ (the one with [[Serre fibrations]]) as well as for the [[Strøm model structure]] $Top_{Strom}$ (the one with [[Hurewicz fibrations]]).

  This standard cylinder is generally a "good cylinder" in the above sense only for $Top_{Stron}$  (in which case it is in fact a "very good cylinder").

  In $Top_{Quillen}$ a sufficient condition for the standard cylinder $X\times I$ to be good is that $X$ is a [[CW-complex]].

## Related concepts

* [[cylinder functor]]

* [[left homotopy]]

* [[mapping cylinder]]

* [[path space object]]

* [[reduced cylinder]]

* [[cylinder spectrum]]

* [[h-cobordism]]

## References

The general definition in [[model categories]] is due to:

* {#Quillen67} [[Daniel Quillen]], Section I.1, Def. 4, p. 9 (15 of 165) in: _Axiomatic homotopy theory_ in: _[[Homotopical Algebra]]_, Lecture Notes in Mathematics 43, Springer 1967 ([doi:10.1007/BFb0097438](https://doi.org/10.1007/BFb0097438))

Lecture notes:

* *[[Introduction to Homotopy Theory]]*, around [this Def.](Introduction+to+Homotopy+Theory#PathAndCylinderObjectsInAModelCategory)

The precise argument that for $X$ a [[cell complex]] then also the standard cyclinder $X\times I$ is a cell complex is spelled out in

* {#Ottina14} Ottina, Prop. 2.9 in: _An A-based cofibrantly generated model category_ ([arXi:1405.2086](http://arxiv.org/abs/1405.2086))

[[!redirects cylinder objects]]
[[!redirects cylinder]]
[[!redirects cylinders]]

[[!redirects good cylinder object]]
[[!redirects good cylinder objects]]


cylinder on a presheaf

#Contents#
* automatic table of contents goes here
{:toc}

## Idea

--

## Definition

###Cylinder Object

Let $A$ be a small category, and let $Psh(A)$ denote the functor category $[A^{op}, Set]$.  An object $X\in Ob(Psh(A))$ is called a [[presheaf]] (of sets).

A *cylinder on a presheaf* $X$ is a presheaf $I X$ with the following data:

* Two jointly monomorphic morphisms $\partial^0_X, \partial^1_X\in Hom_{Psh(A)}(X,I X)$ admitting a common [[retraction]] $\sigma_X:I X\to X$.  That is, the induced map $\partial^0_X\coprod \partial^1_X:X\coprod X\to I X$ is a monomorphism, and $\sigma_X\circ \partial^j_X=id_X$ for $j\in \{0,1\}$.

That is, a cylinder is an object $I X$ with the morphisms above making the diagram below commute:

(PLACEHOLDER!)

A morphism of cylinders on presheaves $X$ and $Y$ is given by a pair of morphisms $\phi:X\to Y$ and $\psi:I X\to I Y$ making the following diagram commute:

$$\begin{matrix}
&&\partial^j_X &&\sigma_X&&\\
&X&\to&I X&\to&X&\\
\phi&\downarrow&&\psi\downarrow&&\downarrow&\phi\\
&Y&\to&I Y&\to&Y&\\
&&\partial^j_Y&&\sigma_Y&&\end{matrix}$$

In particular, $\phi$ is a [[retract]] of $\psi$.

###Cylinder Functor

A _cylinder functor_ is a cylinder object for the identity functor $1_Psh(A)$ in the endofunctor category $[Psh(A),Psh(A)]$.

####Elementary Homotopy Data

A presheaf category is said to have a _elementary homotopy data_ if it is equipped with a cylinder functor I such that

* the functor I commutes with all small colimits;
* the functor I respects monomorphisms;
* the natural transformation $\partial^j_{(-)}:1_{Psh(A)}\to I$ sends arrows of Psh(A) to commutative squares in Psh(A) in the obvious way.  We require that it sends all monomorphisms to [[pushout|cartesian squares]].

(More to come..)


cylinder spectrum

#Contents#
* table of contents
{:toc}

## Idea

For $S_\bullet$ a [[spectrum]], its _cylinder spectrum_ $Cyl(E)_\bullet$ has as component spaces the [[reduced cylinder]] spaces over the original component spaces:

$$
  Cyl(E)_n \coloneqq I_+ \wedge E_n
  \,,
$$

i.e. the [[smash product]] (of [[pointed topological spaces]]) of $E_n$ with the unit inveral with a dijoint basepoint adjoined.

Cylinder spectra serve to define [[left homotopy]] of spectra as for other [[cylinder objects]].

## References

Early lecture notes include

* [[Frank Adams]], part III, section 2 _[[Stable homotopy and generalised homology]]_, 1974

[[!redirects cylinder spectra]]


Cynthia Lester

* [Mathematics Genealogy page](https://www.mathgenealogy.org/id.php?id=264387)

## Selected writings

On the [[canonical Grothendieck topology]]:

* [[Cynthia Lester]], *Covers in the Canonical Grothendieck Topology* ([arXiv:1909.03384](https://arxiv.org/abs/1909.03384))

and an analog in [[homotopy theory]]:

* [[Cynthia Lester]], *The canonical Grothendieck topology and a homotopical analog*, PhD thesis (2019) [[uoregon:1794/24924](https://scholarsbank.uoregon.edu/xmlui/handle/1794/24924) [pdf](https://scholarsbank.uoregon.edu/xmlui/bitstream/handle/1794/24924/Lester_oregon_0171A_12522.pdf?sequence=1&isAllowed=y)]

* [[Cynthia Lester]], *The canonical Grothendieck topology and a homotopical analog* [[arXiv:1909.03188](https://arxiv.org/abs/1909.03188)]

category: people


Cyril Cohen

* [webpage](http://perso.crans.org/cohen/)

## Publications

* [[Sophie Bernard]], [[Cyril Cohen]], [[Assia Mahboubi]], [[Pierre-Yves Strub]], *Unsolvability of the Quintic Formalized in Dependent Type Theory*, ITP 2021 - 12th International Conference on Interactive Theorem Proving, Jun 2021, Rome / Virtual, France ([hal:hal-03136002](https://hal.inria.fr/hal-03136002))

## related $n$Lab entries

* [[cubical type theory]]

category: people


Cécile DeWitt-Morette

* [Wikipedia entry](https://en.wikipedia.org/wiki/C%C3%A9cile_DeWitt-Morette)

* [webpage](http://www.ph.utexas.edu/person/dewitt_cecile)

## Selected writings

Early use of the [[configuration space of points]] in discussion of [[spin-statistics theorem|particle statistics]] (see also at *[[braid group statistics]]*):

* [[Michael G. G. Laidlaw]], [[Cécile Morette DeWitt]], *Feynman Functional Integrals for Systems of Indistinguishable Particles*, Phys. Rev. D **3** (1971) 1375 $[$[doi:10.1103/PhysRevD.3.1375](https://doi.org/10.1103/PhysRevD.3.1375)$]$

On [[analysis]] and [[differential geometry]] on [[smooth manifolds]] with application to [[mathematical physics]]:

* [[Yvonne Choquet-Bruhat]], [[Cécile DeWitt-Morette]], *Analysis, manifolds and physics*, North Holland (1982, 2001) $[$[ISBN:9780444860170](https://www.elsevier.com/books/analysis-manifolds-and-physics-revised-edition/choquet-bruhat/978-0-444-86017-0)$]$

On [[path integrals]] such as in [[quantum physics]]:

* [[Pierre Cartier]], [[Cécile DeWitt-Morette]], _Functional integration: action and symmetries_, Cambridge Monographs on Mathematical Physics (2006) [[ISBN:9780521143578](https://www.cambridge.org/ae/academic/subjects/physics/theoretical-physics-and-mathematical-physics/functional-integration-action-and-symmetries?format=PB#contentsTabAnchor)]

On the early history of [[quantum gravity]] along the memoirs of [[Bryce DeWitt]]:

* [[Cécile DeWitt-Morette]], *The Pursuit of Quantum Gravity -- Memoirs of Bryce DeWitt from 1946 to 2004*, Springer (2011) [[doi:10.1007/978-3-642-14270-3_4](https://doi.org/10.1007/978-3-642-14270-3_4)]

Many more to go here...

## Related pages

* [[mathematical physics]]

Many more to go here...

category: people

[[!redirects Cecile DeWitt-Morette]]
[[!redirects Cécile Morette DeWitt]]


Cédric Ho Thanh

* [website](https://cedric.hothanh.fr/about/)

## Publications

* [[Cédric Ho Thanh]], [[Chaitanya Leena Subramaniam]] *Opetopic algebras I: Algebraic structures on opetopic sets* ([arXiv:1911.00907](https://arxiv.org/abs/1911.00907))

category: people


Cédric Villani

__Cédric Villani__ is a French mathematician working in [[partial differential equations]], especially those appearing in [[classical physics|classical]] [[mathematical physics]] and differential geometry. Villani extensively worked on the topic of optimal transport and its applications to differential geometry. Villani was awarded the Fields medal at Hyderabad in 2010 for the study of Boltzmann equation (especially its hydrodynamic limit) and nonlinear Landau damping,

Villani is a director of the Institut Henri Poincaré in Paris and holds a part time research appointment at IHÉS.

* [wikipedia](http://en.wikipedia.org/wiki/C%C3%A9dric_Villani)
* [cedricvillani.org](http://cedricvillani.org)
* IMU [Fields Medal laudatio](http://www.icm2010.org.in/imu-prizes/prize-winners-2010/fields-medal-cedric-villani)

category: people

[[!redirects Cedric Villani]]


César Gómez

* [Mathematics Genealogy Page](http://www.genealogy.ams.org/id.php?id=139573)

## related $n$Lab entries

* [[D6-brane]]

category: people

[[!redirects Cesar Gomez]]