twisted spin structure

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## Idea

An ordinary _[[spin structure]]_ on a [[special orthogonal group]]-[[principal bundle]] is a lift of the corresponding [[cocycle]] $g : X \to \mathbf{B} SO$ through the [[spin group]] [[fibration]] $\mathbf{B} Spin \to \mathbf{B} SO$. The [[obstruction]] for this to exist is a [[cohomology class]] $w_2 \in H^2(X, \mathbb{Z}_2)$ -- the second [[Stiefel-Whitney class]]: it exists precisely if this class is trivial, $[w_2(g)] = 0$.

Conversely, one can ask for an $SO$-cocycle $g$ with prescribed non-trivial obstruction $[w_2(g)] = \alpha \in H^2(X, \mathbb{Z}_2)$. These may usefully be understood as  _$\alpha$-twisted_ $spin$-structures, following the general logic of [[twisted cohomology]].

## Definition

Let

$$
  \mathbb{Z}_2 \to Spin \to SO \to \mathbf{B} \mathbb{Z}_2
$$

be the [[fiber sequence]] in $\mathbf{H} = $[[ETop∞Grpd]] or $\mathbf{H} = $[[Smooth∞Grpd]] given by the [[spin group]] extension of the [[special orthogonal group]] (regarded as a [[topological group]] or as a [[Lie group]], respectibely). Its [[delooping]] defines the second [[Stiefel-Whitney class]]

$$
  w_2 : \mathbf{B }SO \to \mathbf{B}^2 \mathbb{Z}_2
$$

so that for any $X \in \mathbf{X}$ we have a [[characteristic class]]

$$
  w_2 : \mathbf{H}(X,\mathbf{B}SO) \to \mathbf{H}(X, \mathbf{B}^2 \mathbb{Z}_2)
$$

$$
  [w_2] : SO Bund(X) \to H^2(X,\mathbb{Z}_2)
  \,.
$$

For $X$ a [[manifold]] define the **groupoid of twisted spin-structures** $SpinStruc_{tw}(X)$ to be the [[(∞,1)-pullback]]

$$
  \array{
    SpinStruc_{tw}(X) &\to& H^2(X, \mathbb{Z}_2)
    \\
    \downarrow && \downarrow
    \\
    \mathbf{H}(X, \mathbf{B}SO)
    &\stackrel{w_2}{\to}&
    \mathbf{H}(X, \mathbf{B}^2 \mathbb{Z}_2)
  }
  \,,
$$

where the right vertical morphism picks one [[cocycle]] representative in each [[cohomology]] class.

The [[cocycle]]s in $SpinStruc_{tw}(X)$ are [[twisted bundles|twisted Spin-bundles]].

The obstruction cocycles in $\mathbf{H}(X, \mathbf{B}^2 \mathbb{Z}_2)$ are $\mathbf{B}\mathbb{Z}_2$-[[principal 2-bundle]]s. These may be modeled by $\mathbb{Z}_2$-[[bundle gerbe]]s. In this incarnation the obstruction cocycles $w_2(g)$ above have been discussed as _spin gerbes_ in ([MurraySinger](#MurraySinger)).

## Related concepts

* [[cohomology]], [[twisted cohomology]], [[Whitehead tower]], [[twisted smooth cohomology in string theory]]

* [[orientation]]

* [[twisted differential c-structure]]

  * [[spin structure]], **twisted spin structure**

  * [[string structure]], [[differential string structure]]

  * [[fivebrane structure]], [[differential fivebrane structure]]

## References

A model of twisted spin structures by [[bundle gerbes]] is discussed in

* [[Michael Murray]], Michael Singer, _Gerbes, Clifford modules and the index theorem_  Annals of Global Analysis and Geometry
Volume 26, Number 4, 355-367, DOI: 10.1023/B:AGAG.0000047514.71785.96 ([arXiv:math/0302096](http://arxiv.org/abs/math/0302096))
 {#MurraySinger}

* Atsushi Tomoda, _A relation of spin-bundle gerbes and Mayer's Dirac operator_ ([pdf](http://ton.prosou.nu/official/procNCG20050413.pdf))

The general abstract discussion given above appears as an example in

* [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:Twisted Differential Structures]]_

[[!redirects twisted spin structures]]

[[!redirects twisted differential spin structures]]


twisted spin^c structure

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## Definition

### Topological

For $n \in \mathbb{N}$ the [[Lie group]] $Spin^c(n)$ is a [[central extension]]

$$
  U(1) \to Spin^c(n) \to SO(n)
$$

of the [[special orthogonal group]] by the [[circle group]]. This comes with a long [[fiber sequence]]

$$
  \cdots \to B U(1) \to B Spin^c(n)  \to B SO(n) \stackrel{W_3}{\to} B^2 U(1)
  \,,
$$

where $W_3$ is the _third integral [[Stiefel-Whitney class]]_ .

By the definition at [[twisted cohomology]], for a given class $[c] \in H^3(X, \mathbb{Z})$, a **$c$-twisted $spin^c$-structure** is a choice of [[homotopy]]

$$
  \eta : c \stackrel{\simeq}{\to} W_3(T X)
  \,.
$$

The [[space]]/[[∞-groupoid]] of all twisted $Spin^c$-structures on $X$ is the [[homotopy fiber]] $W_3 Struc_{tw}(T X)$ in the [[pasting diagram]] of [[homotopy pullback]]s

$$
  \array{
    W_3 Struc_{tw}(T X) &\to& W_3 Struc_{tw}(X) &\stackrel{tw}{\to}& H^3(X, \mathbb{Z})
    \\
    \downarrow && \downarrow && \downarrow
    \\
    * &\stackrel{T X}{\to}& Top(X, B SO(n))
    &\stackrel{W_3}{\to}&
    Top(X, B^2 U(1))
  }
  \,,
$$

where the right vertical morphism is the canonical [[effective epimorphism]] that picks one point in each connected component.

### Smooth

Since $U(1) \to Spin^c \to SO$ is a sequence of [[Lie group]]s, the above may be lifted from the [[(∞,1)-topos]] [[Top]] $\simeq$ [[∞Grpd]] to [[Smooth∞Grpd]].

More precisely, by the discussion at [[Lie group cohomology]] (and [[smooth ∞-groupoid -- structures]]) the characteristic map $W_3 : B SO \to B^2 U(1)$ in $\infty Grpd$ has, up to equivalence, a unique lift

$$
  \mathbf{W}_3 : \mathbf{B} SO \to \mathbf{B}^2 U(1)
$$

to [[Smooth∞Grpd]], where on the right we have the [[delooping]] of the smooth [[circle n-group|circle 2-group]].

By the general definition at _[[twisted differential c-structure]]_ , the [[2-groupoid]] of _smooth twisted $spin^c$-structures_ $\mathbf{W}_3 Struc_{tw}(X)$ is the joint [[(∞,1)-pullback]]

$$
  \array{
    \mathbf{W}_3 Struc_{tw}(T X) &\to& \mathbf{W}_3 Struc_{tw}(X) &\stackrel{tw}{\to}& H_{smooth}^2(X, U(1))
    \\
    \downarrow && \downarrow && \downarrow
    \\
    * &\stackrel{T X}{\to}& Smooth \infty Grpd(X, \mathbf{B} SO(n))
    &\stackrel{\mathbf{W}_3}{\to}&
    Smooth \infty Grpd(X, \mathbf{B}^2 U(1)
  }
  \,.
$$

## Applications

### Anomaly cancellation in physics

The existence of an ordinary [[spin structure]] on a space $X$ is, as discussed there, the condition for $X$ to serve as the [[target space]] for the [[spinning particle]] [[sigma-model]], in that the existence of this structure is precisely the condition that the corresponding fermionic [[quantum anomaly]] on the [[worldline]] vanishes.

Twisted $spin^c$-structures appear similarly as the conditions for the analogous quantum anomaly cancellation, but now of the open [[type II superstring]] ending on a [[D-brane]]. This is also called the **[[Freed-Witten anomaly cancellation]]**.

More precisely, in these applications the class of $W_3(TX) - H$ need not vanish, it only needs to be $n$-[[torsion]] if there is moreover a [[twisted bundle]] of rank $n$ on the $D$-brane.

See the [references below](ReferencesAnomalyCancellation) for details.

## Related concepts

* [[twisted differential c-structure|(twisted differential) c-structures]]

  * [[orientation]]

  * [[spin structure]], [[twisted spin structure]]

    [[spin^c structure]], **twisted $spin^c$ structure**

  * [[string structure]], [[differential string structure]]

  * [[fivebrane structure]], [[differential fivebrane structure]]

## References

### General

The notion of twisted $Spin^c$-structures as such were apparently first discussed in section 5 of

* [[Christopher Douglas]], _On the Twisted K-Homology of Simple Lie Groups_ Topology, Volume 45, Issue 6, November 2006, Pages 955-988 ([arXiv:0402082](http://arxiv.org/abs/math/0402082))

More discussion appears in section 3 of

* [[Bai-Ling Wang]], _Geometric cycles, index theory and twisted K-homology_ J. Noncommut. Geom. 2 (2008), no. 4, 497--552 ([arXiv:0710.1625](http://arxiv.org/abs/0710.1625))

The refinement to smooth twisted structures is discussed in section 4.1 of

* [[Urs Schreiber]], _[[schreiber:differential cohomology in a cohesive topos]]_

### In physics
 {#ReferencesAnomalyCancellation}

The need for twisted $Spin^c$-structures as [[Freed-Witten anomaly cancellation]] condition on the [[worldvolume]] of [[D-branes]] in [[string theory]] was first discussed in

* [[Daniel Freed]], [[Edward Witten]], _Anomalies in String Theory with D-Branes_ ([arXiv:hep-th/9907189](http://arxiv.org/abs/hep-th/9907189))

More details are in

* [[Anton Kapustin]], _D-branes in a topologically nontrivial B-field_ , Adv. Theor. Math. Phys.
4, no. 1, pp. 127–154 (2000), ([arXiv:hep-th/9909089](http://arxiv.org/abs/hep-th/9909089))

A clean formulation and review is provided in

* Loriano Bonora, Fabio Ferrari Ruffino, Raffaele Savelli, _Classifying A-field and B-field configurations in the presence of D-branes_ ([arXiv:0810.4291](http://arxiv.org/abs/0810.4291))

* Fabio Ferrari Ruffino, _Classifying A-field and B-field configurations in the presence of D-branes - Part II: Stacks of D-branes_ ([arXiv:1104.2798](http://arxiv.org/abs/1104.2798))

* Fabio Ferrari Ruffino, _Topics on topology and superstring theory_ ([arXiv:0910.4524](http://arxiv.org/abs/0910.4524))

and

* Kim Laine, _Geometric and topological aspects of Type IIB D-branes_ ([arXiv:0912.0460](http://arxiv.org/abs/0912.0460))
 {#Laine}

In ([Laine](#Laine)) the discussion of FW-anomaly cancellation with finite-rank gauge bundles is towards the very end, culminating in equation (3.41).

[[!redirects twisted spin^c structures]]

[[!redirects twisted spin^c-structure]]
[[!redirects twisted spin^c-structures]]


twisted tensor product

This article is about the twisted tensor product of a dga-algebra and a dg-coalgebra (and generalizations). For another notion of [[twisted tensor product of algebras]] see there.

##Idea ##

In 1959, Edgar Brown introduced a _twisted tensor product_ to give an algebraic description of a [[fibration]]. The chain complex of a total space of a principal fibration is obtained as a small perturbation (at the level of a differential) of the [[chain complex]] of the trivial fibration (hence a [[tensor product]] of chain complexes of the base and of the fiber). It is the analogue for differential algebra of the [[twisted cartesian product]] construction in the theory of simplicial fibre bundles.

##Definition##

Let $C$ be a dg-[[coalgebra]], $A$ a [[dg-algebra]], $\tau:C\to A$ the [[twisting cochain]], $L$ a right $C$-dg-co[[module]] with co[[action]] $\delta_L:L \to L\otimes C$ and $M$ a left $A$-dg-module with action $m_M:M\otimes A\to A$. The __twisted tensor product__ $L\otimes_\tau M$ is the chain complex that coincides with the ordinary tensor product $L\otimes M$ as a [[graded vector space|graded module]] over the [[ground ring]], and whose differential $d_\tau$ is given by
$$
d_\tau = d_L\otimes 1 + 1\otimes d_M + (1\otimes m_M)\circ(1\otimes\tau\otimes 1)\circ(\delta_L\otimes 1).
$$

##Literature##

* Edgar H. Brown Jr. _Twisted tensor products I_,
Annals of Math. (2) 69 1959 223--246.

* V. A. Smirnov, _Simplicial and operadic methods in algebraic topology_, Translations of mathematical monographs 198, AMS, Providence, Rhode Island 2001.

* [[Kenji Lefèvre-Hasegawa]], _Sur les $A_\infty$-catégories_, [thesis](https://arxiv.org/abs/math/0310337), (Université Denis Diderot – Paris 7, Paris, November 2003).  Corrections, by B. Keller, available [here]( http://people.math.jussieu.fr/~keller/lefevre/publ.html).

* Manuel Rivera, [[Mahmoud Zeinalian]], _The colimit of an ∞-local system as a twisted tensor product_, Higher Structures 4(1), (2020) [higher-structures4.1](https://higher-structures.math.cas.cz/api/files/issues/Vol4Iss1/RiveraZeinalian) [arXiv:1805.01264](https://arxiv.org/abs/1805.01264) mpg:[pdf](https://pure.mpg.de/rest/items/item_3366473/component/file_3366474/content)


twisted tensor product of algebras

This article is about the twisted tensor product of associative algebras yielding a new algebra containing the original algebras as subalgebras. For another notion of [[twisted tensor product]] of a dg-algebra with a dg-coalgebra yielding a chain complex see there. While twisting tensor products of dg-algebra with a dg-coalgebra involves a [[twisting cochain]], the twist for algebras here is rather of different nature, boiling down to some variant of [[distributive law]]s.

## Definition

This is the definition for the relative case over a possibly noncommutative ring $R$.

$R$-ring $D$ is a twisted tensor product of an $R$-ring $A$ and $R$-ring $B$ if there are $R$-ring morphisms
$i_A:A\to D$ and $i_B:B\to D$ such that the canonical map
$A\otimes_R B\to D$ given by $a\otimes b\mapsto i(a)i(b)$ is an isomorphism of $R$-bimodules.

## Constructions

(Suppose $R$ is in the center and bimodules are central.) If $\tau:B\otimes A\to A\otimes B$ is an $R$-linear map such that $\tau(b\otimes 1) = 1\otimes b$ and $\tau(1\otimes a) = a\otimes 1$. This implies that the mixed triangle $\tau\circ(B\otimes\eta_A) = A\otimes\eta_B$ holds; clearly also the distributive laws triangles
$\mu_B\circ(B\otimes\eta_A) = B$ and $\mu_A\circ (\eta_B\otimes A)=A$. A hexagon formula
$$
\mu_{A\otimes B} = (\mu_A\otimes \mu_B)\circ(A\otimes_R\tau\otimes_R B) : (A\otimes_R B)\otimes (A\otimes_R B)\to
(A\otimes_R B)
$$
defines an associative multiplication iff
$$
\tau\circ(\mu_B\otimes_R\mu_A)= (\mu_A\otimes_R\mu_B)
\circ(A\otimes\tau\otimes B)\circ(\tau\otimes\tau)\circ(B\otimes\tau\otimes A)
$$

The following [[distributive law]] pentagons are needed
and are equivalent to the above condition (we skip adorning the relative tensor products):
$$
(A\otimes\mu_B)\circ (\tau\otimes B)\circ(B\otimes \tau)=  \tau\circ(\mu_B\otimes A): B\otimes B\otimes A\to A\otimes B
$$
$$
(\mu_A\otimes B)\circ (A\otimes\tau)\circ(\tau\otimes A)= \tau\circ(B\otimes\mu_A): B\otimes A\otimes A\to A\otimes B
$$

## Literature

The present notion is related to factorizations of algebras, hence also to distributive laws in categorical setup.

Original reference with one hexagon identity instead of two pentagons for associative algebras

* A. Čap, H. Schichl, J. Vanžura, _On twisted tensor products of algebras_, Commun. Alg. __23__ (1995) 4701 [doi](https://doi.org/10.1080/00927879508825496)

The construction for (concrete) twisted algebras with the explicit two pentagon identities for twist appear e.g. (see Eqs. (4.4), (4.5)) in

* [[Piotr Podleś]], [[S. L. Woronowicz]], _Quantum deformation of Lorentz group_, Comm. Math. Phys. __130__(2) (1990) 381--431 [doi](https://doi.org/10.1007/bf02473358)

and the dual version (C) for coalgebras at page 1004 of

* [[Georges Skandalis]], _Operator algebras and duality_, in: Proceedings of the ICM Kyoto (1990) 997--1009 [pdf](https://www.imj-prg.fr/wp-content/uploads/2020/prix/skandalis1990.pdf)

More elaborate versions for matched products and bicrossproducts in Hopf algebra theory were earlier in

* W. M. Singer, _Extension theory for connected Hopf algebras_, J. Algebra 21 (1972) 1-16
* [[Shahn Majid|S. H. Majid]], _Physics for algebraists: non-commutative and non-cocommutative Hopf algebras by a bicrossproduct construction_, J. Algebra __130__:1 (1990) 17-64 doi

In some abstract setups (factorization algebras, factorization monads etc.) more general construction was known beforehands, using distributive laws and further generalizations appear later.

For dg-algebras including a relative version, $R$-dg-[[ring over a ring|rings]],

* [[Dmitri Orlov]], _Twisted tensor products of DG algebras_, Russ. Math. Surv. 76 (2021) 1146; _Smooth DG algebras and twisted tensor product_, [arXiv:2305.19799](https://arxiv.org/abs/2305.19799)

category: algebra


twisted Umkehr map

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## Idea

For $E$ a [[cohomology theory]], and $f \colon X \to Y$ a map of suitable [[spaces]], an ordinary [[Umkehr map]] for the induced map $E^\bullet(f) \colon E^\bullet(Y) \to E^{\bullet}(X)$ is a [[dual morphism]] together with self-[[dual objects|duality]] [[equivalences]] for $E^\bullet(X)$ and $E^\bullet(Y)$ ([[orientation in generalized cohomology|orientation]]/[[Atiyah duality]]+[[Thom isomorphism]]).

More generally, $E^\bullet(X)$ may not be self-dual, but its [[dual object]] may be [[twisted cohomology]] $E^{\bullet+ \chi}(X)$ for some twist $\chi$. In this case the [[Umkehr map]] goes not between the original spaces and their cohomology, but between [[twisted cohomology]] variants of these.

## Definition

### Abstract duality and Atiyah-Milnor-Spanier duality + Pontryagin-Thom collapse

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

Write

$$
  D \coloneqq (-)^\vee\circ \Sigma^\infty_+ \coloneqq L_{whe} Top \to \mathbb{S}Mod
$$

for the [[Spanier-Whitehead duality]] map which sends a [[topological space]] first to its [[suspension spectrum]] and then that to its [[dual object]] in the [[(∞,1)-category of spectra]].

=--

([ABG 11, def 10.3](#ABG11)).

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

For $X$ a [[compact manifold]], let $X \to \mathbb{R}^n$ be an [[embedding]] and write $S^n \to X^{\nu_n}$ for the classical [[Pontryagin-Thom collapse map]] for this situation, and write

$$
  \mathbb{S} \to X^{-T X}
$$

for the corresponding [[looping]] map from the [[sphere spectrum]] to the [[Thom spectrum]] of the negative [[tangent bundle]] of $X$. Then [[Atiyah duality]] produces an [[equivalence]]

$$
  X^{- T X} \simeq D X
$$

which identifies the [[Thom spectrum]] with the [[dual object]] of $\Sigma^\infty_+ X$ in $\mathbb{S} Mod$ and this constitutes a [[commuting diagram]]

$$
  \array{
    && X^{- T X}
    \\
    & \nearrow & \downarrow^{\mathrlap{\simeq}}
    \\
    \mathbb{S}
    &\underset{D(X \to \ast)}{\to}&
    D X
  }
$$

identifying the classical [[Pontryagin-Thom collapse map]] with the abstract [[dual morphism]] construction of prop. \ref{SpanierDualityOperation}.

More generally, for $W \hookrightarrow X$ an [[embedding]] of [[manifolds]], then [[Atiyah duality]] identifies the [[Pontryagin-Thom collapse maps]]

$$
  \mathbb{S} \to X^{-T X} \to W^{- T W}
$$

with the abstract [[dual morphisms]]

$$
  \mathbb{S} \to D X \to D W
  \,.
$$

=--

([ABG 11, prop. 10.5](#ABG11)).

### Umkehr map

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

Given now $E \in CRing_\infty$ an [[E-∞ ring]], then the [[dual morphism]] $\mathbb{S} \to D X$ induces under [[smash product]] a similar Pontryagin-Thom collapse map, but now not in [[sphere spectrum]]-[[(∞,1)-modules]] but in $E$-[[(∞,1)-modules]].

$$
  E \to D X \otimes_{\mathbb{S}} E
  \,.
$$

The image of this under the $E$-[[generalized cohomology theory|cohomology]] functor produces

$$
  [D X \otimes_{\mathbb{S}} E, E] \to E
  \,.
$$

If now one has a [[Thom isomorphism]] ($E$-[[orientation in generalized cohomology|orientation]]) $ [D X \otimes_{\mathbb{S}} E, E] \simeq [X,E]$ that identifies the cohomology of the dual object with the original cohomology, then together with the above this produces the [[Umkehr map]]

$$
  [X,E] \simeq [D X \otimes_{\mathbb{S}} E, E] \to E
$$

that pushes the $E$-cohomology of $X$ to the $E$-cohomology of the point. Analogously if instead of the terminal map $X \to \ast$ we start with a more general map $X \to Y$.

More generally a [[Thom isomorphism]] may not exists, but $[D X \otimes_{\mathbb{S}} E, E]$ may still be equivalent to a [[twisted cohomology]]-variant $[X,E]_{\chi}$ of $[X,E]$, namely to $[\Gamma_X(\chi),E]$, where $\chi \colon \Pi(X) \to E Line \hookrightarrow E Mod$ is an ([[flat (∞,1)-bundle|flat]]) $E$-[[(∞,1)-module bundle]] on $X$ and and $\Gamma \simeq \underset{\to}{\lim}$ is the [[(∞,1)-colimit]] (the [[generalized Thom spectrum]] construction). In this case the above yields a **twisted Umkehr map**.

=--

([ABG 10, 9.1](#ABG10))

## Examples

* [[fiber integration in ordinary cohomology]]

* [[fiber integration in ordinary differential cohomology]]

* [[fiber integration in K-theory]]

  For a detailed discussion of an example in [[K-theory]] see also at _[[Poincaré duality algebra]]_ and at _[[Freed-Witten-Kapustin anomaly]]_.

* [[fiber integration in differential K-theory]]

## Related concepts

[[!include generalized fiber integration synonyms - table]]

## References

Twisted Umkehr maps in [[topological K-theory]] are discussed (somewhat implicitly sometimes) in the literature on [[KK-theory]]. See the references at _[[Poincaré duality algebra]]_.

The general abstract formulation in [[stable homotopy theory]] is sketched in section 9 of

* {#ABG10} [[Matthew Ando]], [[Andrew Blumberg]], [[David Gepner]], _Twists of K-theory and TMF_, in Robert S. Doran, Greg Friedman, [[Jonathan Rosenberg]], _Superstrings, Geometry, Topology, and $C^*$-algebras_, Proceedings of Symposia in Pure Mathematics [vol 81](http://www.ams.org/bookstore-getitem/item=PSPUM-81), American Mathematical Society ([arXiv:1002.3004](http://arxiv.org/abs/1002.3004))
 {#ABG10}

and in section 10 of

* {#ABG11} [[Matthew Ando]], [[Andrew Blumberg]], [[David Gepner]], _Parametrized spectra, multiplicative Thom spectra, and the twisted Umkehr map_, Geom. Topol. 22(7): 3761-3825 (2018).  ([arXiv:1112.2203](http://arxiv.org/abs/1112.2203), [doi:10.2140/gt.2018.22.3761](https://doi.org/10.2140/gt.2018.22.3761))

A review and applications to [[quantization]] of [[local prequantum field theory]] is in

* {#Nuiten13} [[Joost Nuiten]], _[[schreiber:master thesis Nuiten|Cohomological quantization of local prequantum boundary field theory]]_, master thesis, August 2013

Formalization in [[dependent linear type theory]] is discussed

* [[Urs Schreiber]], _[[schreiber:Quantization via Linear homotopy types]]_

[[!redirects twisted Umkehr maps]]


twisted Wu structure

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## Idea

The notion of [[twisted differential c-structures]] applied to [[Wu classes]].

## Applications

See at [[M5-brane]] the section _[Conformal blocks and 7d Chern-Simons dual](http://ncatlab.org/nlab/show/M5-brane#7dCSDual)_.

## Related concepts

* [[Spin Chern-Simons theory]]

## References

Topological twisted Wu structures are considered in

* [[Hisham Sati]], _Twisted topological structures related to M-branes II: Twisted $Wu$ and $Wu^c$ structures_ ([arXiv:1109.4461](http://arxiv.org/abs/1109.4461))

A smooth and differential refinement is discussed in

* [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:The moduli 3-stack of the C-field]]_ .

A [[truncated object|truncation]] of this reproduces the _differential integra Wu structures_ considered in

* [[Mike Hopkins]], [[Isadore Singer]], _[[Quadratic Functions in Geometry, Topology, and M-Theory]]_

[[!redirects twisted Wu structures]]

[[!redirects Wu structure]]
[[!redirects Wu structures]]


twisted ∞-bundles

#Contents#
* table of contents
{:toc}

## Idea

The notion of _twisted $\infty$-bundle_ is the generalization to [[nLab:higher geometry]] of the notion of [[nLab:twisted bundle]].

## Definition.

Let $\mathbf{H}$ be an [[(∞,1)-topos]], $G \in Grp(\mathbf{H})$ an [[∞-group]],  $V \in \mathbf{H}$ an object, and $\rho$ an [[∞-representation]] of $G$ on $V$, exhibited by a [[fiber sequence]]

$$
  \array{
    V &\to& V//G
    \\
    && \downarrow^{\mathrlap{\rho}}
    \\
    && \mathbf{B}G
  }
  \,,
$$

which is identified with the universal $V$-[[associated ∞-bundle]].

Then for $P \to X$ a $G$-[[principal ∞-bundle]], correspoding to a [[cocycle]] $g : X \to \mathbf{B}G$ with coefficients in the [[moduli ∞-stack]] of $G$-principal $\infty$-bundles, the [[∞-groupoid]] of [[sections]] of the [[associated ∞-bundle]] $P times_G V$ is equivalently the cocycle $\infty$-groupoid of $g$-[[twisted cohomology]] relative to $\rho$:

$$
  \Gamma_X(P\times_G V) \simeq \mathbf{H}_{/\mathbf{B}G}(g,\rho)
  \,.
$$

Each such cocycle may be understood as locally having coefficients in  $\Omega V$, but globally being _twisted_ by $P$.

The geometric structure canonically associated to this is a (twisted $G$-equivariant) $\Omega V$-[[principal ∞-bundle]] $Q \to P$ on the total space $P$ of the twisting $\infty$-bundle, seen by applying the [[pasting law]] of [[(∞,1)-pullbacks]] to the [[pasting diagram]]

$$
  \array{
    P &\to& *
    \\
    \downarrow && \downarrow
    \\
    P &\to& V &\to& *
    \\
    \downarrow && \downarrow && \downarrow
    \\
    X &\stackrel{\sigma}{\to}& V//G &\to& \mathbf{B}G
  }
  \,.
$$

## Examples

(...)

## References

* [[Thomas Nikolaus]], [[Urs Schreiber]], [[Danny Stevenson]], _[[schreiber:Principal ∞-bundles -- Theory, presentations and applications]]_

[[!redirects twisted infinity-bundles]]


twisting cochain

# Twisting cochains
* table of contents
{: toc}

## Definition

Let $(C,d_C)$ be a dg-[[coalgebra]] with comultiplication $\Delta$ and $(A,d_A)$ a [[dg-algebra]] with multiplication $\mu$. A __twisting cochain__ is a morphism $\tau:C\to A[1]$ such that the following [[Maurer-Cartan equation]] in the [[convolution algebra]] holds:
$$d_A\circ\tau+\tau\circ d_C+\mu\circ(\tau\otimes\tau)\circ\Delta = 0.$$
Notice that the last, perturbation term describes the square $\tau\star\tau$ in the [[convolution]] algebra of homogeneous maps in $\mathrm{Hom}(C,A)$.

## Relation to the adjunction bar-cobar

Let $\mathrm{Cogc}$ be the category of cocomplete dg-co(al)gebras and $\mathrm{Alg}$ the category of dg-algebras. There is a [[bar and cobar construction|bar-construction]] functor $B :\mathrm{Alg}\to\mathrm{Cogc}$ which is a [[adjoint functor|right adjoint]] to the [[bar and cobar construction|cobar-construction]] functor $\Omega:\mathrm{Cogc}\to\mathrm{Alg}$. Starting from a map $f\in\mathrm{Cogc}(C,B A)$, one constructs a twisting cochain $\tau_f$ by postcomposing $f: C\to B A$ by the natural projection $B A\to A[1]$; the [[Maurer-Cartan equation]] for $\tau_f$ translates to saying that $f$ is a chain map, $d_{B A}\circ f = f\circ d_C$. One then replaces $\tau_f$ by the composition of the evident canonical map $\tau_0:\Omega C\to C[-1]$ (called the _canonical twisting cochain_) and $\tau_f[-1]:C[-1]\to A$ to obtain a morphism $f':\Omega C\to A$. The Maurer--Cartan equation for $\tau$ is equivalent also to saying that $f'$ is a chain map, i.e. $d_A\circ f'=f'\circ d_{\Omega C}$.

## Some usages of twisting cochains

A twisting cochain is a datum used to define the [[twisted tensor product]] $L\otimes_\tau M$ for any right $C$-co[[module]] $L$ and any left $A$-module $M$,
as well as the [[twisted module of homomorphisms]] $\mathrm{Hom}_\tau(N,P)$ where $N$ is a left $C$-dg-comodule and $P$ a left $A$-dg-module.

B. Keller and his student Kenji Lefèvre-Hasegawa have shown that [[Koszul duality]] is closely related to twisting cochains. Given a twisting cochain $\tau$, one always has a pair of [[adjoint functor]]s $\otimes_\tau A$ and $\otimes_\tau C$ between the [[derived category]] of modules over $A$ and the coderived category of comodules over $C$ (where $C$ is in $\mathrm{Cogc}$ and the coderived category is just the localization of the category of complexes of comodules at the class of [[weak equivalence]]s, which are by definition those morphisms which became [[quasi-isomorphism]]s after applying $\otimes_{\tau_0}\Omega C$ where $\tau_0:\Omega C\to C[-1]$ is the canonical twisting cochain). This pair of adjoint functors is an [[adjoint equivalence]] iff the composition $\Omega C\to C[-1]$ by $\tau[-1]:C[-1]\to A$ (compare reasoning above) is a quasi-isomorphism. This can also be expressed by saying that the canonical map
$$
A\otimes_\tau C\otimes_\tau A\to A
$$
is a quasiisomorphism. In that case, Keller calls the triple $(C,A,\tau)$ the _Koszul--Moore triple_. Lefevre-Hasegawa's thesis (pdf) asserts that in that case $A$ determines $C$ up to a weak equivalence (defined above) and $C$ determines $A$ up to a quasi-isomorphism. Moreover,
$$H_* C = \mathrm{Tor}^A_*(k,k)\quad \text{and}\quad H^* A = \mathrm{Ext}_C(k,k)$$
where $k$ is the [[ground field]]. Notice that such a formulation of Koszul duality using coalgebras and coderived categories avoids various finiteness conditions present when Koszul duality is phrased as relating algebras to algebras.

## Remarks

John C. Moore was one of the people who studied the subject of 'differential coalgebra', including twisting cochains, in the 1960s and 1970s and gave a survey of the area during his ICM address.

There are variants of the notion of twisting cochain in a variety of other contexts.

A [[twisting function]] is an analogue of a twisting cochain in the context of [[simplicial set]]s.

Apart from original usage for the algebraic models for fibrations, twisting cochains and variants are used in [[homological perturbation theory]] (sometimes abbreviated HPT), rational homotopy theory, deformation theory, study of $A_\infty$-categories, Grothendieck duality on complex manifolds (Toledo-Tong) and so on.

An old query archived [here](http://www.math.ntnu.no/~stacey/Mathforge/nForum/comments.php?DiscussionID=3399&Focus=27888#Comment_27888).

## References

* Edgar H. Brown Jr. Twisted tensor products. I.  Ann. of Math. (2)  69  (1959) 223--246.

* Kenji Lefèvre-Hasegawa, Sur les A-infini catégories, ([pdf](http://people.math.jussieu.fr/~keller/lefevre/TheseFinale/tel-00007761.pdf))

* V. A. Smirnov, Simplicial and operad methods in algebraic topology, Transl. Math. Monogr. 198, AMS 2001 (transl. by G. L. Rybnikov)

* D. Barnes and L. A. Lambe, A Fixed Point Approach to Homological Perturbation Theory, with Don Barnes, Proc. AMS, 112 (1991), 881--892.

* K. Hess, The cobar construction: a modern perspective ([pdf](http://sma.epfl.ch/~hessbell/Minicourse_Louvain_Notes.pdf)) (lecture notes from a minicourse at Louvain)

* Alexander I. Efimov, Valery A. Lunts, Dmitri O. Orlov, Deformation theory of objects in homotopy and derived categories, [part 1](http://arxiv.org/abs/math/0702838), [part 2](http://arxiv.org/abs/math/0702839), [part 3](http://arxiv.org/abs/math/0702840))

* N. R. O'Brian, Geometry of twisting cochains.  Compositio Math.  63  (1987),  no. 1, 41--62.

* D. Toledo, Yue Lin L.  Tong, _Duality and intersection theory in complex manifolds. I._  Math. Ann.  __237__  (1978), no. 1, 41--77; II. The holomorphic Lefschetz formula.  Ann. of Math. (2) __108__ (1978), no. 3, 519--538.

* [[Henri Gillet]], The $K$-theory of twisted complexes, in "Applications of algebraic $K$-theory to algebraic geometry and number theory", Part I, II (Boulder, Colo., 1983), 159--191, Contemp. Math., 55, AMS 1986.

* Richard M. Hain, _Twisted cochains and duality between minimal algebras and minimal Lie algebras_, Trans. AMS __277__, 1 (1983) 397--411.

[[!redirects twisting cochain]]
[[!redirects twisting cochains]]


twisting function

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=--
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#Contents#
* table of contents
{:toc}

##Idea

The notion of _twisting functions_ is an explicit [[simplicial set|simplicial]] formula for a [[cocycle]] with values in a [[simplicial group]]. The _twisted product_ induced from the twisting function is an explicit simplicial formula for a [[simplicial principal bundle]], a model for a [[discrete ∞-groupoid|discrete]] [[principal ∞-bundle]] classified by this cocycle.

In a fibre [[bundle]] or more generally in a fibration, the fibre 'twists' as one goes around a loop in the base space, (the standard simple example is the [[Möbius band]]). Such fibre bundles are usually restricted to being 'locally trivial', that is locally a product of an open set in the base with the fibre.

In the  setting of simplicial [[homotopy theory]], one can attempt to construct analogues of fibre bundles by starting with a base [[simplicial set]] $X_\bullet$ and a fibre $F_\bullet$ and trying to 'deform' the simplicial product $X_\bullet \times F_\bullet$ to get some non-trivial fibred object. In the Cartan seminars of the period 1956--57 ([numdam](http://www.numdam.org/numdam-bin/fitem?id=SHC_1956-1957__9__A1_0)), (about pages 1--10), a neat solution was described: by leaving all but one of the face maps of the product alone, and deforming the last. The result is a "twisted [[cartesian product]]" (see below). The deformation required a simplicial [[automorphism]] of the fibre of course, and the resulting twisting function went from the base $X_\bullet$ to the automorphisms of $F_\bullet$, completely mirroring the topological example. The simplicial identities force the twisting function to obey certain equations.

##Definition

### Twisting function

Let $X_\bullet$ be a [[simplicial set]] and $G_\bullet$ a [[simplicial group]]. Then a __twisting function__ $\phi :X_\bullet\to G_\bullet$ is a family of maps $\phi=\{\phi_n : X_n\to G_{n-1}\}_{n\gt 1}$ such that

$$\array{
d_0 \phi(x) = (\phi(d_0 x))^{-1} \phi(d_1 x)\\
d_i \phi(x) = \phi(d_{i+1}x), i\gt 0,\\
s_i\phi(x) = \phi(s_{i+1}x), i\geq 0,\\
\phi(s_0 x) = 1_G.
}$$

### Twisted Cartesian products

Given a [[simplicial set]] $F_\bullet$ with left $G_\bullet$-[[action]], one then defines a __twisted Cartesian product__, (TCP),  $X_\bullet \times_\phi F_\bullet$ with
$(X_\bullet \times_\phi F_\bullet)_n = X_n\times F_n$ and
$$\array{
d_i(x,f) = (d_i x, d_i f), i\gt 0\\
d_0 (x,f) = (d_0 x, \phi(x)(d_0 f)),\\
s_i(x,y) = (s_i x,s_i y).
}$$

Thus the only difference from the usual Cartesian product of simplicial sets is in $d_0$.

## Properties

* A twisting function $\phi :X_\bullet\to G_\bullet$ corresponds exactly to a simplicial map from $X$ to $\overline{W}(G_\bullet)$ [delooping of the simplicial group](http://ncatlab.org/nlab/show/simplicial+group#Delooping).  There is a universal twisting function $\overline{W}(G_\bullet)_\bullet\to G_\bullet$.  See [[simplicial principal bundle]] for more.

* By the adjunction between $W$-bar and the [[Dwyer-Kan loop groupoid]] functor, a twisting function also corresponds to a morphism of simplicial groupoids $G(X_\bullet)\to G_\bullet$.

\begin{remark}\label{RelationToTwistingCochains}
Twisting functions are the analogue of [[twisting cochains]] in the context of [[simplicial sets]]; but twisting cochains were introduced by [Brown 1959](#Brown59), whilst twisting functions were discussed already in [Cartan 1956](#Cartan56).
\end{remark}

## Literature

* {#Cartan56} [[Henri Cartan]], *Sur la théorie de Kan*, Séminaire Henri Cartan **9** (1956-1957) talk no. 1 [[numdam:SHC_1956-1957__9__A1_0](http://www.numdam.org/item?id=SHC_1956-1957__9__A1_0)]

* {#Brown59} [[Edgar H. Brown]] Jr. _Twisted tensor products. I._  Ann. of Math. **69** 2  (1959) 223-246 [[doi:10.2307/1970101](https://doi.org/10.2307/1970101), [jstor:1970101](https://www.jstor.org/stable/1970101)]

* [[Michael G. Barratt]], [[Victor K.A.M. Gugenheim]], [[John C. Moore]], _On semisimplicial fibre-bundles_, Amer. J. Math. __81__ (1959) 639-657 [[doi:10.2307/2372920](https://doi.org/10.2307/2372920), [jstor:2372920](https://www.jstor.org/stable/2372920), MR0111028]

The link between Kan fibrations and simplicial fibre bundles, and thus TCPs is neatly summarised in:

*  [[Edward Curtis]], _Simplicial Homotopy Theory_, Advances in Math. __6__ (1971) 107-209 [[MR279808](http://www.ams.org/mathscinet-getitem?mr=279808) doi]

Review:

* [[Peter May]], §IV in: _Simplicial objects in algebraic topology_, University of Chicago Press (1967) [[ISBN:9780226511818](https://press.uchicago.edu/ucp/books/book/chicago/S/bo5956688.html), [djvu](http://www.math.uchicago.edu/~may/BOOKS/Simp.djvu), [[May_SimplicialObjectsInAlgebraicTopology.pdf:file]]]

[[!redirects twisting functions]]

[[!redirects twisted Cartesian product]]
[[!redirects twisted Cartesian products]]

[[!redirects twisted cartesian product]]
[[!redirects twisted cartesian products]]


twistor fibration

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#### Complex geometry
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#Contents#
* table of contents
{:toc}

## Idea

The _twistor fibration_ $\mathbb{C}P^3 \to S^4$ ([Atyiah-Hitchin-Singer 78, Ex. 2 on p. 438](#AtyiahHitchinSinger78), [Atiyah 79, Sec. III.1](#Atiyah79), see also [Bryant 82](#Bryant82), [ArmstrongSalamon14](#ArmstrongSalamon14), [ABS 19](#ABS19)), also called, in its [[coset space]]-version $SO(5)/U(2) \to SO(5)/SO(4)$, the _Calabi-Penrose fibration_ (apparently starting with [Lawson 85, Sec. 3](#Lawson85), see also, e.g., [Loo 89](#Loo89), [Seade-Verjovsky 03, 3](#SeadeVerjovsky03) for this usage, see [Nordstrom 08, Lemma 2.31](#Nordstrom08) for review of Calabi's construction [Calabi 67](#Calabi67), [Calabi 68](#Calabi68)) is a [[fiber bundle]]-structure on [[complex projective 3-space]] over the [[4-sphere]] with [[2-sphere]] ([[Riemann sphere]]) [[fibers]]:

$$
  \array{
    \mathbb{C}P^1
      &\longrightarrow&
    \mathbb{C}P^3
    \\
    &&
    \big\downarrow^{\mathrlap{p}}
    \\
    && S^4
  }
$$

If one identifies the 4-sphere as the [[quaternions|quaternionic]] [[projective space|projective line]]  $S^4 \simeq \mathbb{H}P^1$, then the fibration $p$ here is given by sending [[complex number|complex]] lines to the [[quaternions|quaternionic]] lines which they span ([Atiyah 79, III (1.1)](#Atiyah79), see also [Seade-Verjovsky 03, p. 198](https://www.e-periodica.ch/digbib/view?pid=ens-001:2003:49::488#513)):

$$
  \array{
    \mathbb{C}P^3
    &\overset{p}{\longrightarrow}&
    \mathbb{H}P^1
    \\
    \{x \cdot z \vert z \in \mathbb{C}\}
    &\mapsto&
    \{x \cdot q \vert q \in \mathbb{H}\}
  }
  \,,
$$

for any $x \in \mathbb{C}^4 \simeq_{\mathbb{R}} \mathbb{H}^2$.

## Generalizations

It is possible to define a twistor fibration over each $S^{2n}$, where the resulting manifold is a [[complex manifold]] endowed with a [[holomorphic]] $n$-plane field transverse to the fibers of the mapping. Namely, writing $S^{2n}=SO(2n+1)/SO(2n)$, then, using the inclusion $U(n) \subset SO(2n)$, one has the coset fibration
$$Z_n=SO(2n+1)/U(n) \to SO(2n+1)/SO(2n).$$
The manifold $Z_n$ canonically has the structure of a complex manifold and is known as the _twistor space_ of $S^{2n}$.

There are generalizations of this picture for each of the so-called 'inner' [[symmetric spaces]] $G/K$ where $K$ is the fixed subgroup of an [[involution]] that is an [[inner automorphism]] of $G$. The twistor fibration is of the form $G/U \to G/K$ where $U \subset K$ is a subgroup such that $K/U$ (the typical fiber of the fibration) is an Hermitian symmetric space. There are also other kinds of twistor spaces over $G/K$ that are [[flag manifolds]] of the form $G/T$ where $T \subset K$ is a [[maximal torus]]. (For these generalizations see [Bryant 85](#Bryant85).)

## Related concepts

* [[quaternionic Hopf fibration]]

* [[twistor space]], [[complex projective 3-space]]

* [[metric cone over complex projective 3-space]]

## References

* {#Calabi67} [[Eugenio Calabi]], _Minimal immersions of surfaces in euclidean spheres_, J. Differential Geometry (1967), 111-125 ([euclid:jdg/1214427884](https://projecteuclid.org/euclid.jdg/1214427884))

* {#Calabi68} [[Eugenio Calabi]], _Quelques applications de l’analyse complexe aux surfaces d’aire minima_, Topics in Complex Manifolds (Ed. H. Rossi), Les Presses de l’Universit ́e de Montr ́eal (1968), 59-81 ([naid:10006413960](https://ci.nii.ac.jp/naid/10006413960))

* {#AtyiahHitchinSinger78} [[Michael Atiyah]], [[Nigel Hitchin]], [[Isadore Singer]], Example 2 on p. 438 in: _Self-Duality in Four-Dimensional Riemannian Geometry_, Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences Vol. 362, No. 1711 (Sep. 12, 1978), pp. 425-461 (37 pages) ([jstor:79638](https://www.jstor.org/stable/79638), [doi:10.1098/rspa.1978.0143](https://doi.org/10.1098/rspa.1978.0143))

* {#Atiyah79} [[Michael Atiyah]], Section III.1 of: _Geometry of Yang-Mills fields_, Pisa, Italy: Sc. Norm. Sup. (1979) 98 ([spire:150867](https://inspirehep.net/literature/150867), [pdf](https://pdfs.semanticscholar.org/b523/d376f11a531fa5a4692401902a17ee18861f.pdf))

* {#Bryant82} [[Robert Bryant]], Section 1 of: _Conformal and minimal immersions of compact surfaces into the 4-sphere_, J. Differential Geom. Volume 17, Number 3 (1982), 455-473 ([euclid:jdg/1214437137](https://projecteuclid.org/euclid.jdg/1214437137))

* {#Bryant85} [[Robert Bryant]], _Lie groups and twistor spaces_, Duke Mathematical Journal 52 (1985), pp. 223–261, ([euclid:dmj/1077304286](https://projecteuclid.org/euclid.dmj/1077304286))

* {#Lawson85} [[H. Blaine Lawson]], _Surfaces minimales et la construction de Calabi-Penrose_,  Séminaire Bourbaki : volume 1983/84, exposés 615-632, Astérisque no. 121-122  (1985), Talk no. 624, p. 197-211 ([numdam:SB_1983-1984__26__197_0](http://www.numdam.org/item/SB_1983-1984__26__197_0))

* [[Simon Salamon]], p. 218-220 of: _Harmonic and holomorphic maps_,  In: E. Vesentini  (eds.) _Geometry Seminar "Luigi Bianchi" II - 1984_, Lecture Notes in Mathematics, vol 1164. Springer 1985 ([doi:10.1007/BFb0081912](https://doi.org/10.1007/BFb0081912))

* [[James Eells]], [[Simon Salamon]], Section 8 of: _Twistorial construction of harmonic maps of surfaces into four-manifolds_,  Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 4, Volume 12 (1985) no. 4, p. 589-640 ([numdam:ASNSP_1985_4_12_4_589_0](http://www.numdam.org/item/ASNSP_1985_4_12_4_589_0))

* {#Loo89} [[Bonaventure Loo]], _The space of harmonic maps of $S^2$ into $S^4$_, Transactions of the American Mathematical Society Vol. 313, No. 1 (1989) ([jstor:2001066](https://www.jstor.org/stable/2001066))

* {#SeadeVerjovsky02} [[José Seade]], [[Alberto Verjovsky]], Section 2 of: _Higher dimensional complex Kleinian groups_,  Math Ann 322, 279–300 (2002) ([doi:10.1007/s002080100247](https://doi.org/10.1007/s002080100247))

* {#SeadeVerjovsky03} Le, [[José Seade]], [[Alberto Verjovsky]], Section 4 of: _Quadrics, orthogonal actions and involutions in complex projective space_, L'Enseignement Mathématique, t.  49 (2003) ([e-periodica:001:2003:49::488](https://www.e-periodica.ch/digbib/view?pid=ens-001:2003:49::488#488))

* {#Nordstrom08} Jonas Nordstrom, _Calabi's construction of Harmonic maps from $S^2$ to $S^n$_, Lund University 2008 ([pdf](http://www.matematik.lu.se/matematiklu/personal/sigma/students/Jonas-Nordstrom-BSc.pdf), [[NordstromCalabiConstruction.pdf:file]])

* Angel Cano, Juan Pablo Navarrete, [[José Seade]], Section 10.1 in: _Kleinian Groups and Twistor Theory_, In: _Complex Kleinian Groups_, Progress in Mathematics, vol 303. Birkhäuser 2013 ([doi:10.1007/978-3-0348-0481-3_10](https://doi.org/10.1007/978-3-0348-0481-3_10))

* {#ArmstrongSalamon14} John Armstrong, [[Simon Salamon]], _Twistor Topology of the Fermat Cubic_, SIGMA 10 (2014), 061, 12 pages ([arXiv:1310.7150](https://arxiv.org/abs/1310.7150))

* [[Bobby Acharya]], [[Robert Bryant]], [[Simon Salamon]], _A circle quotient of a $G_2$ cone_, Differential Geometry and its Applications Volume 73, December 2020, 101681  ([arXiv:1910.09518](https://arxiv.org/abs/1910.09518), [doi:10.1016/j.difgeo.2020.101681](https://doi.org/10.1016/j.difgeo.2020.101681))

In [[noncommutative geometry]]:

* Simon Brain, Giovanni Landi, _Differential and Twistor Geometry of the Quantum Hopf Fibration_, Commun. Math. Phys. 315 (2012):489-530 ([arXiv:1103.0419](https://arxiv.org/abs/1103.0419))

In higher dimensions:

Over $\mathbb{H}P^3$:

* {#LooVerjovsky94} [[Bonaventure Loo]], [[Alberto Verjovsky]], _On quotients of Hopf fibrations_, in: Rendiconti dell’Istituto di Matematica dell’Università di Trieste. An International Journal of Mathematics, 26 (1994), pp. 103-108 ([hdl:10077/4637](	http://hdl.handle.net/10077/4637), [pdf](https://www.openstarts.units.it/bitstream/10077/4637/1/LooVerjovskyRendMat26.pdf))

[[!redirects twistor fibrations]]

[[!redirects Calabi-Penrose fibration]]
[[!redirects Calabi-Penrose fibrations]]


twistor space

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#Contents#
* table of contents
{:toc}

## Idea

_Twistor space_ in the original sense of ([Penrose 67](#Penrose67)) is a [[complex manifold]] whose [[complex geometry]] is usefully related to the [[conformal geometry]] of ([[conformal compactification|conformally compactified]] [[complexification|complexified]]) 4-dimensional [[Minkowski spacetime]]. The _Penrose transform_ from twistor space to this [[spacetime]] yields powerful computational tools for studying certain [[quantum field theories]], in particular 4d [[Yang-Mills theory]]. Notably it sends [[ordinary cohomology]] classes on twistor space to [[self-dual Yang-Mills theory|self-dual Yang-Mills fields]] on spacetime ([Ward 77](#Ward77), [Ward-Wells 90](#WardWells90)). Morever, when twistor space is taken as a [[target space]] for [[twistor string theory]], then it serves to compute the [[MHV amplitudes]] in [[super Yang-Mills theory]] ([Witten 03](#Witten03)).

This _Penrose transform_ is exhibited by a [[correspondence]] of [[coset spaces]]/[[flag varieties]]/[[Grassmannians]] which is a special case of general such [[correspondences]] as they are studied in [[Schubert calculus]], [[geometric representation theory]], [[parabolic geometry]]. Therefore one can consider "generalized twistors" to be elements of certain [[flag varieties]] and "generalized Penrose transforms" to be those induced by the relevant [[correspondences]]. ([Baston-Eastwood 89](#BastonEastwood89), [Cap 01](#Cap01)).

In this generality given a [[semisimple Lie group]] $G$ and two [[parabolic subgroups]] $P_1$ and $P_2$ with [[intersection]] $P_1 \cap P_2$, then the _twistor correspondence_ is the [[correspondence]] (see also at _[Schubert calculus -- Correspondences](http://ncatlab.org/nlab/show/Schubert+calculus#Correspondences)_ and at _[[horocycle correspondence]]_) of the form

$$
  \array{
      && G/(P_1 \cap P_2)
      \\
      & {}^{\mathllap{p_1}}\swarrow && \searrow^{\mathrlap{p_2}}
     \\
     G/P_1 && && G/P_2
  }
$$

given by the [[projections]] onto the two [[coset spaces]]/[[flag varieties]] and a _Penrose transform_ is an [[integral transform]]/pull-[[fiber integration|push]] $(p_2)_! \circ (p_1)^\ast$ through this correspondence.

In particular there are higher twistor spaces corresponding to higher dimensional [[Minkowski spacetimes]] which are useful for describing higher dimensional [[quantum field theory]], notably there are twistor spaces for 6-dimensional spacetime useful for the study of the [[6d (2,0)-superconformal QFT]] on the [[worldvolume]] of the [[M5-brane]] ([MREIC 11](#MREIC11)) with its [[self-dual higher gauge field]], the [[B-field]] .

The original _twistor correspondence_ ([Penrose 67](#Penrose67)) is the [[correspondence]]

$$
  \left(
  \array{
    && Gr_{1,2}(\mathbb{C}^4)
    \\
    & \swarrow && \searrow
    \\
    \mathbb{C}P^3 && && Gr_2(\mathbb{C}^4)
  }
  \right)
  \;\;\;\;\;
    \simeq
  \;\;\;\;\;
  \left(
  \array{
    && SL_\mathbb{C}(4)/SL_{\mathbb{C}}(2)
    \\
    & \swarrow && \searrow
    \\
    SL_{\mathbb{C}}(4)/SL_{\mathbb{C}}(3)
     && &&
    SL_{\mathbb{C}}(4)/(SL_{\mathbb{C}}(2)\times SL_{\mathbb{C}}(2))
  }
  \right)
  \,,
$$

where

* the [[Grassmannian]] $G_2(\mathbb{C}^4)$ of [[planes]] in complexified 4d [[Minkowski spacetime]];

* the **twistor space** is the [[complex projective 3-space]] $\mathbb{C}P^3 = Gr_1(\mathbb{C}^4)$;

* the correspondence space $Gr_{1, 2}(\mathbb{C}^4)$ is the space of lines in planes in $\mathbb{C}^4$.

(e.g. [Ward-Wells 90](#WardWells90))

## Details

### Twistors for 4d Minkowski space
 {#TwistorsFor4dMinkowskiSpacetime}

We discuss the original twistors for the description of [[physics]] in 4d
[[Minkowski spacetime]]. In summary, twistor space of 4d Minkowski space is the space of pairs consisting of a [[momentum]] vector and an [[angular momentum]] [[tensor]] subject to the constraint that the momentum is [[lightlike]] and of definite [[helicity]].

The formulation is all motivated from the form that basic quantities of [[special relativity|special relativistic physics]] take when [[vectors]] are expressed in [[spinor]] coordinates via the [exceptional spin isomorphism](spin+group#ExceptionalIsomorphisms)

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

Under this identification a chiral [[spinor]] $\kappa$ is represented just by a pair of [[complex numbers]] $\xi, \eta \in \mathbb{C}$ and one may write the spinor as

$$
  \left(
    \kappa^\alpha
  \right)
  =
  \left(
    \array{
     \xi
     \\
     \eta
    }
  \right)
  \in
  \;\; \mathbb{C}^2
  \,.
$$

Moreover, via the non-degenerate bilinear pairing

$$
  \Gamma \;\colon\; S \otimes S \longrightarrow V
$$

given by the [[Clifford algebra]] (see at [[spin representation]] for details) one can express vectors

$$
  (x^i) = (x^0, x^1, x^2, x^3)
$$

in [[Minkowski spacetime]] as "bi-spinors" given by 2x2 [[Hermitean matrices]]

$$
  \begin{aligned}
    \left(x^{\alpha \beta}\right)
    & \coloneqq
    (x^i \gamma_i^{\alpha \beta})
    \\
    &
    =
    \tfrac{1}{\sqrt{2}}
    \left(
      \array{
         x^0 + x^3 & x^1 + i x^2
         \\
         x^1 - i x^2 & x^0 - x^3
      }
    \right)
  \;\;
  \in Mat_{2}(\mathbb{C})
  \end{aligned}
  \,,
$$

where $\gamma_i$ denote the generators of the [[Clifford algebra]] given by the [[Pauli matrices]].

This is such that the [[Lorentz metric]] [[norm]] is just the [[determinant]] of this [[matrix]]

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

This spinorial re-expression of vectors turns out to yield very efficient expressions particularly for those kinds of terms that appear in the observed physics of massless chiral particles, such as they appear in the [[standard model of particle physics]].

For instance a skew rank-2 [[tensor]] (2-from) $(F_{i j})$ with $F_{i j} = - F_{j i}$ (for instance the [[field strength]] of an [[electromagnetic field]]) has spinorial expression of the simple form

$$
  \begin{aligned}
    F_{\alpha_1 \beta_1 \alpha_2 \beta_2}
    & \coloneqq
    F_{i_1 i_2} \gamma^{i_1}_{\alpha_1 \beta_1} \gamma^{i_2}_{\alpha_2 \beta_2}
    \\
    & =
    \phi_{\alpha_1 \alpha_2} \epsilon_{\beta_1 \beta_2}
    +
    \psi_{\beta_1 \beta_2} \epsilon_{\alpha_1 \alpha_2}
  \end{aligned}
$$

for complex numbers $(\phi_{\alpha_1 \alpha_2})$ and $(\psi_{\beta_1 \beta_2})$. Here if $F$ itself is real-number valued, then $\psi$ is the [[complex conjugate]] of $\phi$ and hence any real 2-form is encoded equivalently by a bispinor $\phi$ via

$$
  F_{\alpha_1 \beta_1 \alpha_2 \beta_2}
  \coloneqq
  \phi_{\alpha_1 \alpha_2} \epsilon_{\beta_1 \beta_2}
  +
  \overline{\phi}_{\beta_1 \beta_2} \epsilon_{\alpha_1 \alpha_2}
$$

Crucially the [[Hodge star|Hodge dual]] of a 2-form has then the simple expression

$$
  (\star F)_{\alpha_1 \beta_1 \alpha_2 \beta_2}
  =
  - i \phi_{\alpha_1 \alpha_2} \epsilon_{\beta_1 \beta_2}
  +
  i \psi_{\beta_1 \beta_2} \epsilon_{\alpha_1 \alpha_2}
  \,,
$$

which is the first sign that [[self-dual Yang-Mills theory]] has a simpler expression in terms of such spinorial coordinates.

Now characterizing an [[elementary particle]] is (by the discussion at _[[unitary representation of the Poincaré group]]_) the [[momentum]] 4-vector $(p^i)$ and its [[angular momentum]] [[tensor]] $(M^{i j})$. A **twistor** is effectively a pair of spinorial coordinates expression this data for _massless_ and _chiral_ particles.

Here chiral means this: from combining the [[momentum]] and [[angular momentum]] one obtains the  [[Pauli-Lubanski vector]]

$$
  S_i \coloneqq \tfrac{1}{2} (\star M)_{i j} p^j
$$

and chiral particles satisfy

$$
  S_i  = s p_i
$$

for some constant $s$.

A _twistor_ is a set of spinorial coordinates for encoding tensors $\left(\left(p^i\right), \left(M^{i j}\right)\right)$ which satisfy

1. masslessness: $p^i p_i = 0$

1. chirality $S_i = s p_i$.

By the discussion at _[[celestial sphere]]_ we have that the first condition means equivalently that there is a single [[spinor]] $(\pi_\alpha)$ such that the spinorial expression for the momentum is

$$
  p_{\alpha \beta} = \pi_\alpha \overline{\pi}_\beta
  \,.
$$

By the above discussion of 2-forms we know moreover that $M^{i j}$ is encoded by a bispinor $\mu_{\alpha \beta}$ and imposing the chirality constraint one finds, using the above formula for the Hodge dual, that its solutions are parameterized precisely by another spinor $(\omega_\alpha)$ via

$$
  \overline{\mu}_{\alpha \beta} = i \omega_{(\alpha} \overline{\pi}_{\beta)}
$$

(where the parenthesis denote symmetrization of indices).

In summary, the possible momentum and angular momentum $(p^i, M^{i j})$ of massless chiral particles in 4d Minkowski spacetime is parameterized precisely by pair of spinors

$$
  Z =
  \left(
    (\omega^\alpha), (\pi_\beta)
  \right)
  \;\;
  \in
  \mathbb{C}^4
  \,.
$$

The image of this under modding out by a global complex factor is a Penrose _twistor_

$$
  Z \in \mathbb{C}P^3
  \,,
$$

an element of the [[complex projective 3-space]], called _twistor space_.

### Twistor space
 {#TwistorSpace}

To discuss twistor space for Minkowski spacetime, it is useful to work more generally with $d$-dimensional Minkowski spacetime for $d \in \{3,4,6,10\}$. If we also use the corresponding irreducible real [[spin representation]] then by the discussion [there](spin+representation#InTermsOfNormedDivisionAlgebraInDimension3To10), these are equivalently 2-component vectors over the real [[normed division algebras]] $\mathbb{K} \in \{\mathbb{R}, \mathbb{C}, \mathbb{H}, \mathbb{O}\}$, respectively. (See also at _[[supersymmetry and division algebra]]_, we follow [Bengtsson-Cederwall 88](#BengtssonCederwall88)).

With this, the vector space underlying $d$-dimensional [[Minkowski spacetime]] $\mathbb{R}^{d-1,1}$ is identified with the space of $2 \times 2$ [[hermitian matrices]] with entries in $\mathbb{K}$

$$
  \mathbb{R}^{d-1,1} \simeq_{linear} Mat_{2 \times 2}^{hermitian}(\mathbb{K})
$$

and the [[Minkowski metric]] [[norm]]-square is then identified with the [[determinant]] of matrices:

$$
  \eta(v,v) = det(v)  \;\;\; v\in \mathbb{R}^{d-1,1} \simeq_{linear} Mat_{2 \times 2}^{hermitian}(\mathbb{K})
  \,.
$$

A choice of [[linear basis]] for this space is provided by the [[Pauli matrices]] with coefficients in $\mathbb{K}$, namely the matrices

$$
  (\Gamma_0^{\dot \alpha \alpha})
    \coloneqq
   \left[
      \array{
        1 & 0
        \\
        0 & -1
      }
   \right]
   \;\;\,,
   \;\;\;\;
  (\Gamma_1^{\dot \alpha \alpha})
    \coloneqq
   \left[
      \array{
        1 & 0
        \\
        0 & 1
      }
   \right]
   \;\;\,,
   \;\;\;\;
  (\Gamma_{n+1}^{\dot \alpha \alpha})
    \coloneqq
   \left[
      \array{
        0 & e_n
        \\
        \overline{e}_n & 0
      }
   \right]
$$

where $\{e_n\}_{n = 2}^{d}$ denotes an [[orthonormal basis]] for the elements of $\mathbb{K}$. Here we write

$$
  \overline{(-)}
  \;\colon\;
  \mathbb{K}
    \longrightarrow
  \mathbb{K}
$$

for the operation of complex conjugation in $\mathbb{K}$, i.e. the $\mathbb{R}$-linear operation which sends the unit $1$ to itself, and sends imaginary elements to their negative.

In terms of this basis then a $d$-component vector $(X^\mu) \in \mathbb{R}^{d-1,1}$ is identified with the hermitian $2 \times 2$ matrix

$$
  (X^{\dot \alpha \alpha})
  \;\coloneqq\;
  (X^\mu \Gamma_\mu^{\dot \alpha \alpha})
   \;=\;
  \left[
    \array{
      x^0 + x^1 & x^{n} e_n
      \\
      x^n \overline{e}_n & x^0 - x^1
    }
  \right]
  \,.
$$

This presentation of [[Minkowski spacetime]] by $2 \times 2$ hermitian matrices with coefficients in $\mathbb{K}$ makes it very natural to consider also its "$\mathbb{K}$-ification" (hence [[complexification]] in the case that $d = 4$ and $\mathbb{K} = \mathbb{C}$), namely the space of _all_ $2 \times 2$ [[matrices]] with entries in $\mathbb{K}$:

$$
  \mathbb{K}\mathbb{R}^{d-1,1}
    \coloneqq
   Mat_{2\times 2}(\mathbb{K})
  \,.
$$

This space carries the operation of [[Hermitian conjugation]]

$$
  X \mapsto X^\dagger
$$

as an [[involution]], and ordinary $d$-dimensional Minkowski spacetime is hence the [[fixed point]] set of this involution.

By the discussion [there](spin+representation#InTermsOfNormedDivisionAlgebraInDimension3To10), the Spin-invariant pairing of two real spinors $\psi, \phi \in \mathbb{K}^2$ to a vector $\overline{\psi} \Gamma \phi \in Mat_{2\times 2}^{hermitian}(\mathbb{K})$ is given by ([prop.](spin+representation#RealSpinorPairingsViaDivisionAlg))

$$
  (\overline{\psi}\Gamma \phi)
   \coloneqq
   \psi \phi^\dagger + \phi \psi^\dagger
  \,.
$$

In particular the pairing of a spinor $\psi = (\psi_{\dot \alpha})$ with itself is

$$
  (\psi \psi^\dagger)_{\alpha \dot\alpha}
  =
  2
  \overline{\psi}_\alpha \psi_{\dot \alpha}
  \,.
$$

Since, as hermitian matrices, the $2 \times 2$ matrices arising this way are manifestly precisely the hermitian projection operators on 1-dimensional linear subspaces of $\mathbb{K}^2$, these are precisely the $2 \times 2$ hermitian matrices with vanishing determinant. Hence the future-oriented [[lightlike]] vectors in Minkowski spacetime in $d = \{3,4,6,10\}$ are precisely those that arise as the pairing of a chiral spinor with itself.

The above formula shows that as we rescale $\psi \mapsto k \psi$ with $k \in \mathbb{K}$, then the corresponding lightlike vector is sent to itself, rescaled by the real number $k \overline{k} \in \mathbb{R}$. Hence the direction of lightlike vectors is parameterized by the [[projective space]] $\mathbb{K}P^1$. At least for $\mathbb{K} = \mathbb{C}$ (the [[complex numbers]]) this is called the [[celestial sphere]] (in this case: the [[Riemann sphere]] $\mathbb{C}P^1$).

Lightlike vectors in $\mathbb{R}^{d-1,1}$ up to scale are equivalently loci of lightlike [[geodesics]] through the origin. A general lightlike geodesic in $\mathbb{R}^{d-1,1}$ is instead parameterized by

$$
  \tau \mapsto (X^{\dot \alpha \alpha} + \tau {\psi}^\alpha \overline{\psi}^{\dot \alpha})
$$

for some offset $(X^{\dot \alpha \alpha})$. To encode this offset into spinors, consider the image of $\psi$ under the Clifford action with $X$:

$$
  \omega^\alpha \coloneqq \psi_{\dot \alpha} X^{\dot \alpha \alpha}
  \,.
$$

This relation bewteen $\omega$ and $\psi$ is called the _incidence relation_ in this context.

The locus of the lightlike geodesic does not change as we rescale $\psi$ by elements $k \in \mathbb{K}$, and then $\omega$ gets rescaled accordingly. Hence the pair $(\omega^\alpha, \psi_{\dot \alpha})$ regarded as encoding lightlike geodesics in Minkowski spacetime should be thought of as an element of the [[projective space]]

$$
  \mathbb{K}P^3
   \coloneqq \{ (Z^{a}) = (\omega^\alpha,\psi_{\dot \alpha}) \in \mathbb{K}^4 \}_{( (Z^a) \sim (k Z^a) \vert k \in \mathbb{K} )}.
$$

This is the _twistor space_ corresponding to the given Minkowski spacetime.

However, by the above definition of $\omega^\alpha$, not every element of this space corresponds to a lightlike geodesic. Rather (for $d \in \{3,4,6\}$) it is only those satisfying

$$
  \omega^\alpha \overline{\psi}_\alpha
    =
  \psi_{\dot \alpha} \overline{\omega}^{\dot \alpha}
$$

reflecting the fact that $(X^{\dot \alpha \alpha})$ is a hermitian matrix, instead of a general $2\times 2$ matrix. (For $d = 10$ there is no elegant statement like this, due to the non-associativity of the [[octonions]] $\mathbb{K} = \mathbb{O}$. But there is nevertheless an analogue of the twistor transform in 10d [Witten 86](#Witten86)).

Hence the subspace inclusion

$$
  \mathbb{R}^{d-1,1} \hookrightarrow \mathbb{K}\mathbb{R}^{d-1,1}
$$

of Minkowski-spacetime into its $\mathbb{K}$-ification, corresponds to a subspace

$$
  \mathcal{N} \hookrightarrow \mathbb{K}P^3
$$

of twistor space. This is called the space of _null twistors_, or the _spin shell_.

If we consider the metric on $\mathbb{K}^4$ given in component by

$$
  \left[
    \array{
       0 & \delta^\beta_\alpha
       \\
       - \delta^{\dot \alpha}_{\dot \beta} & 0
    }
  \right]
$$

then this subspace is the [[quadric]] given by the equation

$$
  Z^a \overline{Z}_a = 0
  \,.
$$

Now conversely, given $Z^a = (\omega^\alpha, \psi_{\dot\alpha})$ not equal to zero and satisfying this condition in that there is a hermitian $X^{\dot \alpha \alpha}$ with $\omega^{\alpha} = \psi_{\dot \alpha} X^{\dot \alpha \alpha}$, then also every hermitian matrix of the form

$$
  X^{\dot \alpha \alpha} + \tau \psi^{\dot \alpha}\psi^\alpha
$$

(for $\tau \in \mathbb{R}$, hence every point on the lightlike geodesic through $X$ in the direction of $\psi \psi^\dagger$) satisfies the relation (since the spinor index is raised with $(\epsilon^{\dot \alpha \dot \beta})$, which is skew symmetric).  This way non-zero points in $\mathcal{N} \hookrightarrow \mathbb{K}P^3$ correspond to lightlike geodesics in Minkowski spacetime.

<img src="http://ncatlab.org/nlab/files/TwistorSpaceCorrespondence.jpg" width="700">

(graphics from [Adamo 13](#Adamo13))

This correspondence extends also to the 0-point in $\mathcal{N}$ if one passes to "compactified Minkowski spacetime", given by the [[Grassmannian]] $Gr_2(\mathbb{C}^4)$ of complex planes in $\mathbb{C}^4$ (review includes [Fioresi-Lledo-Varadarajan 07, section 2](#FioresiLledoVaradarajan07)).

## Related concepts

* [[twistor fibration]]

* A real analog of this is the _[[Radon transform]]_.

* [[spinor]]

* [[twistor string theory]]

* [[MHV amplitude]]

* [[Radon transform]]

* [[Schubert calculus]]

* [[celestial sphere]], [[quadric]], [[Klein quadric]]

## References

### General

The notion originates in

* {#Penrose67} [[Roger Penrose]], _Twistor algebra_,  Journal of Mathematical Physics 8 (2): 345–366, (1967) ([doi:10.1063/1.1705200](https://doi.org/10.1063/1.1705200))

* [[Roger Penrose]], _The twistor programme, Reports on Mathematical Physics 12 (1): 65–76 (1977)

Key developments concerning [[scattering amplitudes]] are due to

* [[Andrew Hodges]], _Twistor diagrams_ ([web](http://www.twistordiagrams.org.uk/papers/index.html))

The relation to [[self-dual Yang-Mills theory]] is due to

* {#Ward77} R. S. Ward, _On Selfdual gauge fields_, Phys. Lett. A61 (1977) 81-82.

Introductions and surveys:

* [[Yuri Manin]], chapters 1 and 2 of _[[Gauge Field Theory and Complex Geometry]], Grundlehren der Mathematischen Wissenschaften 289, Springer 1988

* Fedja Hadrovich, _Twistor primer_ ([html](http://users.ox.ac.uk/~tweb/00004), [pdf](http://henry.pha.jhu.edu/twistors.pdf))

* Paul Bair, _Introduction to twistors_ ([pdf](http://www.math.jussieu.fr/~helein/encyclopaedia/baird-twistors.pdf))

* S. A. Huggett, K. P. Todd, _An introduction to twistor theory_, Cambridge University Press (1985)

* Liana David, _The Penrose transform and its applications_, 2001 ([pdf](http://www.maths.ed.ac.uk/pg/thesis/david.pdf))

* {#Dunajski09} Maciej Dunajski, _Twistor theory and differential equations_, J.Phys. __A42__:404004 (2009) [arXiv:0902.0274](http://arxiv.org/abs/0902.0274) [doi](https://doi.org/10.1088/1751-8113/42/40/404004)

* [[Roger Penrose]], _Twistor theory_, talk at _[[New Spaces for Mathematics and Physics]]_, IHP Paris, 2015 ([video recording](https://www.youtube.com/watch?v=kmYfYOW0vSg))

* {#FioresiLledoVaradarajan07} R. Fioresi, M. A. Lledo, [[Veeravalli Varadarajan]], _The Minkowski and conformal superspaces_, J.Math.Phys.48:113505,2007 ([arXiv:0609813](https://arxiv.org/abs/math/0609813))

* [[Michael Atiyah]], [[Maciej Dunajski]], [[Lionel J. Mason]], _Twistor theory at fifty: from contour integrals to twistor strings_,  Proc. R. Soc. A473: 20170530 ([arXiv:1704.07464](https://arxiv.org/abs/1704.07464), [doi:10.1098/rspa.2017.0530](http://dx.doi.org/10.1098/rspa.2017.0530))

* {#Adamo17} Tim Adamo, _Lectures on twistor theory_ ([arXiv:1712.02196](https://arxiv.org/abs/1712.02196))

* [[Lionel J. Mason]], N. M. J. Woodhouse, _Integrability, self-duality and twistor theory_, OUP 1996

* R. S. Ward, R. O. Wells, Jr. _Twistor geometry and field theory_, Cambridge Univ. Press 1990

* [[Roger Penrose]], *Twistor Theory: A Geometric Perspective for Describing the Physical World* ([doi:10.1017/9781108854399.011](https://doi.org/10.1017/9781108854399.011)), chapter in: [[Gabriel Catren]], [[Mathieu Anel]] (eds.) *[[New Spaces for Mathematics and Physics]]*, Cambridge University Press (2021) ([doi:10.1017/9781108854429](https://doi.org/10.1017/9781108854429))

Review of the application in [[super Yang-Mills theory]] includes

* {#Adamo13} [[Tim Adamo]], _Twistor actions for gauge theory and gravity_ ([arXiv:1308.2820](https://arxiv.org/abs/1308.2820))

See also

* Wikipedia, _[Twistor theory](http://en.wikipedia.org/wiki/Twistor_theory)_

### Relation to geometric representation theory and parabolic geometry

A discussion in the general context of [[geometric representation theory]] is in

* {#BastonEastwood89} [[Robert J. Baston]], [[Michael G. Eastwood]], _The Penrose transform its interaction with representation theory_ Oxford Science Publications, Clarendon Press, 1989 (ISBN:9780198535652, [spire:290342](https://inspirehep.net/literature/290342), [pdf](http://www.math.utah.edu/~ptrapa/math-library/baston-eastwood/baston-eastwood.pdf))

and the further generalization to [[Cartan geometry]]/[[parabolic geometry]] is discussed in

* {#Cap01} [[Andreas Čap]], _Correspondence spaces and twistor spaces for parabolic geometries_, J. Reine Angew. Math. 582 (2005) 143-172 ([arXiv:math/0102097](http://arxiv.org/abs/math/0102097))

### Palatial twistor theory

* Roger Penrose, _Palatial twistor theory and the twistor googly problem_, Phil. Trans. Royal Soc. A373: 20140237 (2015) [doi](http://dx.doi.org/10.1098/rsta.2014.0237)
* R. Penrose, _Twistor theory as an approach to fundamental physics_, in: Foundations of mathematics and physics one century after Hilbert, 253-285 (2018), ed. Joseph Kouneiher [doi](https://doi.org/10.1007/978-3-319-64813-2_10)

### Application to quantum field theory

More on traditional applications to [[quantum field theory]] is in

* {#WardWells90} R.S. Ward, R.O. Wells, _Twistor geometry and field theory_, Cambridge Univ. Press 1990 ([doi:10.1017/CBO9780511524493](https://doi.org/10.1017/CBO9780511524493))

The relation of twistor geometry to [[MHV amplitudes]] in 4d [[Yang-Mills theory]] and [[twistor string theory]] is due to

* {#Witten03} [[Edward Witten]], _Perturbative Gauge Theory As A String Theory In Twistor Space_, Commun. Math. Phys. 252:189-258, 2004 ([arXiv:hep-th/0312171](http://arxiv.org/abs/hep-th/0312171))

Surveys of the resulting modern application of twistors in field theory include

* David Skinner, _The geometry of scattering amplitudes_, talk notes, November 2009 ([pdf](http://research.physics.unc.edu/string/transparencies/trans_20091112.pdf))

### Twistors in higher dimensions

Discussion of twistors in dimensions 3,4,6 and 10 using the [[normed division algebras]] (as in [[supersymmetry and division algebras]]) is in

* {#BengtssonCederwall88} [[Ingemar Bengtsson]], [[Martin Cederwall]], _Particles, Twistors and the Division Algebras_, Nucl.Phys. B302 (1988) 81-103 ([spire:247269](http://inspirehep.net/record/247269), <a href="https://doi.org/10.1016/0550-3213(88)90667-0">doi:10.1016/0550-3213(88)90667-0</a>)

* {#Cederwall93} [[Martin Cederwall]], _Introduction to Division Algebras, Sphere Algebras and Twistors_ ([arXiv:hep-th/9310115](https://arxiv.org/abs/hep-th/9310115))

and specifically for dimension 10

* {#Witten86} [[Edward Witten]], _Twistor-like transform in ten dimensions_, Nuclear Physics __B266__:2, 245-264. (1986) (<a href="http://dx.doi.org/10.1016/0550-3213(86)90090-8">10.1016/0550-3213(86)90090-8</a>)

and its [[anti de Sitter spacetime|AdS]] version:

* [[Alex Arvanitakis]], Alec E. Barns-Graham, [[Paul Townsend]], _Twistor description of spinning particles in AdS_, JHEP 01 (2018) 059 ([arXiv:1710.09557](https://arxiv.org/abs/1710.09557))

#### Application to the 6d self-dual 2-form field
 {#ReferencesApplicationToSelfDual2FormField}

A general discussion of Penrose-Ward-type transforms sending [[circle 2-bundles]] ([[holomorphic line 2-bundles]]) on some twistor space to [[circle 2-bundles with connection]] and self-dual [[curvature]] 3-form on spacestime (expected to play a role in the description of the [[6d (2,0)-superconformal QFT]]) is in

* [[David Chatterjee]], sections 4 and 8 of _On gerbs_, 1998 ([pdf](http://people.maths.ox.ac.uk/hitchin/hitchinstudents/chatterjee.pdf))

* [[Lionel J. Mason]], R. A. Reid-Edwards, A. Taghavi-Chabert, appendix of _Conformal field theories in six-dimensional twistor space_, J. Geom. Phys. 62 (2012), no. 12, 2353-2375 ([arXiv:1111.2585](http://arxiv.org/abs/1111.2585))
 {#MREIC11}

* [[Christian Saemann]], [[Martin Wolf]], _On twistors and conformal field theories from six dimensions_, J. Math. Phys. 54:013507, 2013  ([arXiv.1111.2539](http://arxiv.org/abs/1111.2539))

More generally, there are arguments that the [[worldvolume]] theory of several coincident [[M5-branes]] carries not just an abelian but a nonabelian [[higher gauge field]] given by a [[principal 2-bundle]] [[principal 2-connection]].

The idea of generalizing the Penrose-Ward transform to one that takes nonabelian [[principal 2-bundles]] to [[self-dual higher gauge field|self-dual]] [[principal 2-connections]] is explored in

* [[Christian Saemann]], [[Martin Wolf]], _Six-dimensional superconformal field theories from principal 3-bundles over twistor space_ ([arXiv:1305.4870](http://arxiv.org/abs/1305.4870))

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twistor string theory

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[[!include functorial quantum field theory - contents]]
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#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

It seems ([Nair 1988](#Nair88), [Witten 2004, Sec. 3](#Witten04)) that the [[scattering amplitudes]] of $n$ [[gluons]] ([[n-point functions]]) in pure [[Yang-Mills theory]] on 4d [[Minkowski spacetime]], regarded as [[distributions]] on the [[product space]] of $n$ copies of [[twistor space]] [[complex projective 3-space|$\mathbb{C}P^3$]], have [[support of a distribution|support]] -- after [[complexification]] -- on (the [[diagonal morphism|diagonal]] [[image]] of) a [[complex curve]] in [[twistor space]], hence on the [[image]] of a [[holomorphic function]]

\[
  \label{ComplexCurveInTwistorSpace}
  \phi \;\colon\; \Sigma \longrightarrow \mathbb{C}P^3
\]

from a [[Riemann surface]] $\Sigma$ to [[complex projective 3-space]]. Here the [[genus of a surface|genus]] $g$ of $\Sigma$ is bounded by the [[loop order]] $l$ of the [[scattering amplitude]],  $g \leq l$ and the [degree](homotopy+of+rational+maps#eq:DegreeOfAMapToCPn) of the map (eq:ComplexCurveInTwistorSpace) depends on the number $q$ of [[negative number|negative]] [[helicity]] gluon states in the process, as $d \,=\, q - 1 + l$.

The idea of [[twistor string theory]] is to understand these [[complex curves]] in [[twistor space]] as [[worldsheet]] [[instantons]] (or rather [[D1-brane]]-instantons) of a [[string theory]] -- specifically of the [[B-model]] [[topological string]], whose [[target space]] is the [[twistor space]] [[complex projective 3-space|$\mathbb{C}P^3$]], or rather the [[supermanifold]] $\mathbb{C}^{3\vert 4}$ -- and to use this to explain the structure of [[MHV amplitudes]] in [[N=4 D=4 super Yang-Mills theory]].

Somehow. The original proposals ([Witten 04](#Witten04), [Berkovits 04](#Berkovits04)) were only partially succesful as [[string theories]] (e.g. [Skinner 09](#Skinner09), [Spradlin](#Spradlin)).

But the idea as such of [[scattering amplitudes]] localized on [[complex curves]] in [[twistor space]] remains fruitful in pure [[quantum field theory]], for instance it seems to generalize to [[perturbative quantum gravity]] in the form of [[D=4 N=8 supergravity]] ([Cachazo & Skinner 2012](#CachazoSkinner12)).

## Related entries

* [[string theory results applied elsewhere]]

* [[homotopy of rational maps]]

## References

Precursors:

* [[Edward Witten]], *An Interpretation of Classical Yang-Mills Theory*,     Phys. Lett. B 77 (1978) 394-398 ([spire:129655](https://inspirehep.net/literature/129655), doi:10.1016/0370-2693(78)90585-3 [pdf](https://lib-extopc.kek.jp/preprints/PDF/1978/7806/7806147.pdf))

* {#Nair88} [[Velayudhan Parameswaran Nair]], *A current algebra for some gauge theory amplitudes*, Physics Letters B Volume 214, Issue 2, 17 November 1988, Pages 215-218 (doi:10.1016/0370-2693(88)91471-2, [spire:24212](https://inspirehep.net/literature/24212))

Original articles:

* {#Witten04} [[Edward Witten]], _Perturbative Gauge Theory As A String Theory In Twistor Space_, Commun. Math. Phys. 252:189-258, 2004 ([arXiv:hep-th/0312171](http://de.arxiv.org/abs/hep-th/0312171))

* {#Berkovits04} [[Nathan Berkovits]], *An Alternative String Theory in Twistor Space for $N = 4$ Super-Yang-Mills*, Phys. Rev. Lett. 93 (2004) 011601 ([arXiv:hep-th/0402045](https://arxiv.org/abs/hep-th/0402045))

* {#RoibanVolovich04} [[Radu Roiban]], [[Anastasia Volovich]], *All Googly Amplitudes from the B-model in Twistor Space*, Phys. Rev. Lett. 93 (2004) 131602 ([arXiv:hep-th/0402121](https://arxiv.org/abs/hep-th/0402121))

* {#RoibanSpradlinVolovich04} [[Radu Roiban]], [[Marcus Spradlin]], [[Anastasia Volovich]], *On the Tree-Level S-Matrix of Yang-Mills Theory*, Phys. Rev. D70 : 026009, 2004 ([arXiv:hep-th/0403190](https://arxiv.org/abs/hep-th/0403190))

Survey:

* {#Skinner09} [[David Skinner]], _The geometry of scattering amplitudes_, talk notes, November 2009 ([pdf](http://research.physics.unc.edu/string/transparencies/trans_20091112.pdf), [[Skinner_GeometryOfScattering.pdf:file]])

* {#Spradlin} [[Marcus Spradlin]], *Progress and Prospects in Twistor String Theory* ([pdf](https://conservancy.umn.edu/bitstream/handle/11299/130081/1/spradlin.pdf))

* [[Michael Atiyah]], [[Maciej Dunajski]], [[Lionel J. Mason]], Section 7 of: _Twistor theory at fifty: from contour integrals to twistor strings_,  Proc. R. Soc. A473: 20170530 ([arXiv:1704.07464](https://arxiv.org/abs/1704.07464), [doi:10.1098/rspa.2017.0530](http://dx.doi.org/10.1098/rspa.2017.0530))

Further developments:

* Yvonne Geyer, Arthur E. Lipstein, [[Lionel Mason]], _Ambitwistor strings in 4-dimensions_, Phys. Rev. Lett. 113, 081602 (2014) ([arXiv:1404.6219](http://arxiv.org/abs/1404.6219))

Discussion for [[D=4 N=8 supergravity]]:

* {#CachazoSkinner12} Freddy Cachazo, [[David Skinner]], *Gravity from Rational Curves*, Phys. Rev. Lett. 110, 161301, 2012 ([arXiv:1207.0741](https://arxiv.org/abs/1207.0741), [doi:10.1103/PhysRevLett.110.161301](https://doi.org/10.1103/PhysRevLett.110.161301))

* {#Adamo15} Tim Adamo, *Gravity with a cosmological constant from rational curves*, JHEP **2015** 098, 2015 ([arXiv:1508.02554](https://arxiv.org/abs/1508.02554))

[[!redirects twistor string]]
[[!redirects twistor strings]]


two

[[!redirects 2]]

\tableofcontents

## Disambiguation

$2$ could refer to a number of things in mathematics:

* In [[number theory]], the successor of the number [[one]] in the [[natural numbers]], or in any other set of numbers which contain the natural numbers as a [[subobject|sub]][[rig]], such as the [[integers]], the [[rational numbers]], or the [[real numbers]].

* In [[set theory]] and [[dependent type theory]], the [[set]] with two elements; i.e. the [[boolean domain]].

* In [[category theory]], the [[interval category]] or the [[interval poset]]

## Related concepts

* [[zero]]

* [[one]]

category: disambiguation


two dimensional sheaf theory

[[Ross Street]] has written the articles

* _Two dimensional sheaf theory_, J. Pure Appl. Algebra 23 (1982) 251-270
* _Characterization of bicategories of stacks_, in: Category theory (Gummersbach 1981) Springer Lecture Notes in Mathematics __962__, 1982, pp 282-291 [[zoranskoda:Characterization of Bicategories of Stacks|transcript]]

where the stacks are considered on a 2-site. A _2-site_ is a [[2-category]] with a Grothendieck 2-topology (compare [[Grothendieck topology]]), which is in turn defined in terms of 2-sieves (compare [[sieve]]). There is a Giraud-type theorem proved in this context. In a later article there were some errata mentioned.

This should be related to the "$\infty$-dimensional sheaf theory" described at [[(infinity,1)-category of (infinity,1)-sheaves]], somehow. Compare also [[derived stack]].

+--{ .query}
_Zoran_: Could one define $(\infty,1)$-sieves somehow as subobjects (in quasi-category sense) of representables in enriched quasi-category setup ?
=--

+-- {: .query}
So if $\infty$-stacks are really $(\infty,1)$-sheaves, and stacks are really $(2,1)$-sheaves, then these are the real $2$-sheaves, that is $(2,2)$-sheaves? (with notation following that of $(n,r)$-[[(n,r)-category|category]]).  ---Toby
=--

category: sheaf theory


two-level type theory

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### Deduction and Induction
+-- {: .hide}
[[!include deduction and induction - contents]]
=--
#### Constructivism, Realizability, Computability
+-- {: .hide}
[[!include constructivism - contents]]
=--
#### Foundations
+-- {: .hide}
[[!include foundations - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

##Idea

**Two-level type theory** (2LTT) refers to versions of [[Martin-Lof type theory]] that combine two [[type theories]]: one level as a [[homotopy type theory]], which may include [[univalence axiom|univalent]] universes and [[higher inductive types]], and the second level as a traditional form of type theory validating [[uniqueness of identity proofs]]. The second layer may be understood as the internalised meta-theory of the first.

The rules are much inspired by the homotopical semantics in [[fibration categories]] and [[model categories]], according to which the types of the homotopy layer are sometimes called **[[fibrant object|fibrant]] types** and the others **non-fibrant types**.  An alternative terminology is simply **types** (since these are the objects of real interest) and **pretypes** (an auxiliary structure used to study the types).

In other words, where default [[homotopy type theory]] has [[categorical semantics]] (see at _[[relation between category theory and type theory]]_) in suitable [[type-theoretic model categories]] but in such a way that only the [[(infinity,1)-category]] [[presentable (infinity,1)-category|presented]] by that really matters, in two-level type theory one adds explicit control over the presenting [[model category]] (or other kind of [[fibration category]]), thus apparently breaking the $(\infty,1)$-categorical "[[principle of equivalence]]" but providing more tools for handling the presentation.  It is an open question to what extent the principle of equivalence is actually broken, i.e. whether results proven in two-level type theory can be transferred to any model categorical presentation.

## Type theories

* The first proposal for two-level type theory was [[Vladimir Voevodsky]]'s [[Homotopy Type System]].  This system had a reflection rule collapsing the non-fibrant "exact equality" to [[judgmental equality]], making some things easier but making type-checking undecidable.

* More recently the proposals of [ACK](#ACK17) simply assumes [[uniqueness of identity proofs]] for the exact equality; this seems to suffice for most if not all purposes.

* [[computational type theory|Computational]]  [[cubical type theory]] can also include  a non-fibrant layer: its syntax is interpreted as cubical sets, where "fibrancy" is a *defined* condition on pretypes (existence of Kan operations).

## Applications

### Semisimplicial types

Two-level type theories (including HTS and ACK) were motivated to a large extent by the technical difficulties encountered in formalizing [[semisimplicial types]] in default [[homotopy type theory]] (HoTT). This problem comes precisely from the fact that HoTT is really the [[internal language]] of some [[(infinity,1)-topos]] and hence defining [[simplicial objects]] or similar here means to speak of [[simplicial objects in an (infinity,1)-category]], which means that the [[simplicial identities]] hold only up to [[coherence|coherent]] higher [[homotopy]]. The problem of syntactically encoding the infinite amount of this [[coherence]] data remains unsolved to date. For [[semisimplicial types]] the difficulties greatly reduce (since the evident iterated [[dependent type]]-definition of a semisimplicial type is automatically interpreted by a [[Reedy model structure|Reedy]]-[[fibrant object]] in the given [[type-theoretic model category]], which takes care of all the homotopy coherence by the power of [[model category]] theory, see at _[[internal (infinity,1)-category]]_ for more on this) but technical problems remain even in formulating the plain 1-categorical [[simplicial identities]] to all degrees.

The "exact equality" of two-level type theories solves this problem because such equalities can be defined by induction.

## Variations

### Natural numbers

One important choice to be made in writing down a two-level type theory is whether there is one natural numbers type or two.  There must certainly be a fibrant natural numbers type; the question is whether the elimination rule for this "fibrant nat" allows elimination also into non-fibrant types.  If it does not, then there should also be a "non-fibrant nat" which is not fibrant, but whose elimination principle can eliminate into other non-fibrant types.

Semantically, this distinction corresponds to whether the natural numbers object of the model category presentation is already fibrant, or whether it needs to be fibrantly replaced to represent a (fibrant) type.  In some models, such as [[simplicial sets]], it is already fibrant; but in others, such as [[local model structure on simplicial presheaves]], it is not.  It is an open question whether any [[(infinity,1)-topos]] can be presented by a model category in which the natural numbers object is fibrant, and thus serve as semantics for a two-level type theory with only one natural numbers type.

Since the definition of semisimplicial types requires an inductive definition of a (non-fibrant) exact equality, if there are two nats then it must use the non-fibrant one.  This means that, without additional rules, a two-level type theory with two nats cannot define a *fibrant* type of *untruncated* semisimplicial types: for each non-fibrant $n$ it can define a fibrant type of $n$-truncated semisimplicial types, but the limit of these types over the *non-fibrant* nat is no longer fibrant.  Assuming only one nat is sufficient to solve this problem, but as mentioned above this may exclude desirable models.  Another weaker axiom, which *is* satisfied in all sufficiently nice model categories, is that fibrant types are closed under limits of towers of fibrations indexed by the non-fibrant nat.  (In two-level type theory terminology, this is a strengthening of the assumption that the non-fibrant nat is "cofibrant".)

## See also

* [[Homotopy Type System]]
* [[type theory with shapes]]

## References

* Thorsten Altenkirch, Paolo Capriotti, Nicolai Kraus, _Extending Homotopy Type Theory with Strict Equality_, ([ arXiv:1604.03799](https://arxiv.org/abs/1604.03799))

* Danil Annenkov, Paolo Capriotti, Nicolai Kraus, Christian Sattler, _Two-Level Type Theory and Applications_, ([arXiv:1705.03307](https://arxiv.org/abs/1705.03307))
{#ACK17}

[[!redirects 2-level type theory]]
[[!redirects 2-level type theories]]
[[!redirects 2LTT]]
[[!redirects fibrant type]]
[[!redirects fibrant types]]
[[!redirects non-fibrant type]]
[[!redirects non-fibrant types]]


two-out-of-six property

+-- {: .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

\begin{definition}
\label{TheDefinition}
In a [[category]] $C$, a [[class]] $W\subseteq Mor(C)$ of [[morphisms]] is said to satisfy **2-out-of-6** if for any sequence of three composable morphisms

$$ X\xrightarrow{u} Y \xrightarrow{v} Z \xrightarrow{w} K$$

if $w v$ and $v u$ are in $W$, then so are $u$, $v$, $w$, and $w v u$.
\end{definition}

A category equipped with a class of "[[category with weak equivalences|weak equivalences]]" satisfying 2-out-of-6 and containing all the [[identity morphisms]] is called a *[[homotopical category]]* by [Dwyer, Hirschhorn, Kan & Smith (2004)](homotopical+category#DwyerHirschhornKanSmith04) (but not all authors follow this terminology, see [[homotopical category|there]]).

## Examples

\begin{example}
\label{IsomorphismsSatisfyTwoOutOfSix}
The class of [[isomorphisms]] in any category satisfies 2-out-of-6 (Def. \ref{TheDefinition}).  This case is the archetype of most of the cases in which the property is invoked: 2-out-of-6 is characteristic of morphisms that have a notion of [[inverse]].
\end{example}

\begin{example}
\label{ModelCategoriesSatisfyTwoOutOfSix}
The weak equivalences in any [[model category]] satisfy two-out-of-six (Def. \ref{TheDefinition}).

This follows from Exp. \ref{IsomorphismsSatisfyTwoOutOfSix} and the fact ([here](Introduction+to+Homotopy+Theory#MorphismIsWeakEquivalenceIfIsoInHomotopyCategoryForQuillen)) that a morphism in a model category is a weak equivalence iff its image in the [[homotopy category of a model category|homotopy category]] is an [[isomorphism]].

This is a special case of the general discussion of *saturation* [below](#Saturation). See also a comment by [[Charles Rezk]] on [Math.SE:a/1495891](https://math.stackexchange.com/a/1495891/58526).
\end{example}

## Relation to other concepts

### 2-out-of-3

\begin{proposition}\label{When2OutOf6ImpliesTwoOutOfThree}
If all [[identity morphisms]] are among the [[weak equivalences]] (as is usually assumed, cf. *[[homotopical category]]*) then the 2-out-of-6 property implies the [[two-out-of-three property]].
\end{proposition}
\begin{proof}
Because, on the one hand, if $f$ and $g$ are in $W$, then applying 2-out-of-6 to $\xrightarrow{f} \xrightarrow{id} \xrightarrow{g}$, we find that $g f\in W$.  On the other hand, if $f$ and $g f$ are in $W$, then applying 2-out-of-6 to $\xrightarrow{id} \xrightarrow{f} \xrightarrow{g}$, we find that $g\in W$, and similarly if $g$ and $g f$ are in $W$.
\end{proof}

### Identities and isomorphisms

If $W$ satisfies 2-out-of-6 and contains the identities (i.e. $C$ is a homotopical category), then $W$ contains all isomorphisms.  For if $f$ has inverse $g$, then applying 2-out-of-6 to $\xrightarrow{g} \xrightarrow{f} \xrightarrow{g}$ we find that $g$ and $f$ are in $W$.

### Closure under retracts

The 2-out-of-6 property is closely related to the property that $W$ is closed under [[retracts]], as a [[subcategory]] of the [[arrow category]].  For instance, we have the following theorem due to [Blumberg-Mandell](#BlumbergMandell) (stated there in the context of [[Waldhausen categories]]):

+-- {: .num_theorem #CofibRetracts}
###### Theorem
Suppose a [[category with weak equivalences]] $\mathcal{C}$ has an additional class of maps called [[cofibrations]] which satisfy the following properties:

* All [[pushouts]] of cofibrations exist.

* The pushout of a cofibration that is also a weak equivalence is again a cofibration and a weak equivalence.

* Every weak equivalence factors as a weak equivalence that is a cofibration followed by a weak equivalence that is a [[retraction]].

Then if the weak equivalences in $\mathcal{C}$ are closed under retracts, they also satisfy 2-out-of-6.

=--

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

Suppose the first three assumptions on the cofibrations, and let
$$A \xrightarrow{u} B \xrightarrow{v} C \xrightarrow{w} D$$
be a sequence of composable maps, with $w v$ and $v u$ weak equivalences.  Factor $v u\colon A\to C$ as $A \xrightarrow{i} C' \xrightarrow{p} C$ where $i$ is a cofibration weak equivalence and $p$ is a weak equivalence with a [[section]]] $s\colon C\to C'$.  Let $B'$ be the pushout
$$
  \array{
    A & \overset{i}{\to} & C'\\
  ^u\downarrow && \downarrow^h\\
  B& \underset{k}{\to} & B'}
$$

Since $p i = v u$, we have a unique map $g\colon B' \to C$ such that $g h = p$ and $g k = v$.  Define $f = h s$; then $g f = g h s = p s = 1_C$.

Since $i$ is a cofibration weak equivalence, so is $k$.  And since $w g k = w v\colon B\to D$ is a weak equivalence, by two-out-of-three, $w g\colon B' \to D$ is also a weak equivalence.  But now we have a commutative diagram

$$\array{C & \overset{f}{\to} & B' & \xrightarrow{g} & C \\
  ^w\downarrow && \downarrow^{w g} && \downarrow^w\\
  D& \underset{=}{\to} & D & \underset{=}{\to} & D}
$$

exhibiting $w$ as a retract of $w g$ in the arrow category.  Thus, by assumption $w$ is a weak equivalence.  By successive applications of two-out-of-three, so are $v$, $u$, and $w v u$.
=--

Of course, there is a dual theorem for fibrations.  Note that the fibrations in a [[category of fibrant objects]] satisfy (the duals of) all the above conditions.  They are not implied by the axioms for the cofibrations in a [[Waldhausen category]] (the factorization axiom is what is missing), but many Waldhausen categories do satisfy them.

### Saturation
 {#Saturation}

The 2-out-of-6 property is also closely related to the property that $W$ is *saturated*, in the sense that any morphism which becomes an isomorphism in the [[localization]] $C[W^{-1}]$ is already a weak equivalence.  (This is unrelated to the notion of [[saturated class of maps]] used in the theory of [[weak factorization systems]].)

Clearly saturation implies 2-out-of-6, but we also have the following two converses.

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

Suppose $W$ admits a [[calculus of fractions]].  Then $W$ satisfies two-out-of-six if and only if it is saturated.

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

This is from 7.1.20 of _[[Categories and Sheaves]]_.  Suppose $f\colon X\to Y$ becomes an isomorphism in $\mathcal{C}[W^{-1}]$, and represent its inverse by $Y \xrightarrow{g} X' \overset{s}{\leftarrow} X$ with $s\in W$.  Then since $g f$ and $s$ represent the same morphism in $\mathcal{C}[W^{-1}]$, there is a morphism $t\colon X'\to X''$ in $W$ such that $t g f = t s$.  Since $t s\in W$, it follows by 2-out-of-3 that $g f\in W$.

Now applying this same argument to $g$, we obtain an $h$ such that $h g \in W$.  But then by 2-out-of-6, we have $f\in W$ as desired.
=--

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

Suppose $C$ has a class of "cofibrations" satisfying the properties in Theorem \ref{CofibRetracts}, and moreover the pushout of any weak equivalence along a cofibration is a weak equivalence.  Then $W$ satisfies two-out-of-six if and only if it is saturated (and hence, if and only if it is closed under retracts).
=--
+-- {: .proof}
###### Proof
See [Blumberg-Mandell](#BlumbergMandell) for details; an outline follows.

We first observe that $W$ admits a [[homotopy calculus of left fractions]], and in particular that every morphism in $\mathcal{C}[W^{-1}]$ can be represented by a zigzag $A \to C \overset{\sim}{\leftarrow} B$ in which $B\xrightarrow{\sim} C$ is a cofibration and a weak equivalence.  See [Blumberg-Mandell](#BlumbergMandell), section 5 for a detailed proof.  The idea is that given any zigzag $A \overset{\sim}{\leftarrow} D \to B$, we factor $D\to A$ as a cofibration weak equivalence followed by a retraction weak equivalence, then push out the cofibration along $D\to B$ and use the section to obtain a map from $A$ into the pushout.

Now suppose $a\colon A\to B$ becomes an isomorphism in $C[W^{-1}]$, and represent its inverse by $B \xrightarrow{b} C \overset{c}{\leftarrow} A$ with $c$ a cofibration weak equivalence.  Since the composite $A \xrightarrow{b a} C \overset{c}{\leftarrow} A$ represents $1_A$, we have $b a \in W$.  Consider the following diagram where the squares are pushouts:
$$\array{ && A & \overset{a}{\to} & B & \xrightarrow{b} & C \\
 &&  ^c\downarrow && \downarrow && \downarrow\\
B & \underset{b}{\to} & C& \underset{}{\to} & B' & \underset{}{\to} & C'}$$
All the vertical maps are cofibration weak equivalences, by assumption.  Moreover, the bottom map $C\to C'$ is a weak equivalence, since it is the pushout of the weak equivalence $b a$ along the cofibration $c$.  And since the zigzag
$$ B \xrightarrow{b} C \to B' \overset{\sim}{\leftarrow} B $$
represents the same morphism as
$$ B \xrightarrow{b} C \overset{c}{\leftarrow} A \xrightarrow{a} B$$
which represents $1_B$, we have that $B\xrightarrow{b} C \to B'$ is a weak equivalence.  Thus, by 2-out-of-6, $b$ is a weak equivalence, hence so is $a$ by 2-out-of-3.
=--

Of course, there is a dual theorem for fibrations.

## References

* [[William Dwyer]], [[Philip Hirschhorn]], [[Daniel Kan]], [[Jeff Smith]], _[[Homotopy Limit Functors on Model Categories and Homotopical Categories]]_

* [[Andrew Blumberg]] and [[Michael Mandell]], *Algebraic $K$-theory and abstract homotopy theory*
{#BlumbergMandell}

* [[Masaki Kashiwara]], [[Pierre Schapira]], _[[Categories and Sheaves]]_

* Andrei Radulescu-Banu, *Cofibrations in Homotopy Theory*, ArXiv 0610009, [pdf](https://arxiv.org/pdf/math/0610009.pdf)

[[!redirects two-out-of-six]]
[[!redirects two-out-of-six-property]]
[[!redirects 2-out-of-6 property]]
[[!redirects 2-out-of-6-property]]
[[!redirects 2-out-of-6]]


two-out-of-three

+-- {: .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 [[class]] $K \subset Mor(C)$ of [[morphism]]s in $C$ is said to satisfy **2-out-of-3** if for all composable $f,g \in Mor(C)$ we have that if two of the three morphisms $f$, $g$ and the composite $g \circ f$ is in $K$, then so is the third.

$$
  \array{
    \\
    {}^{\mathllap{f}}\nearrow \searrow^{\mathrlap{g}}
    \\
    \stackrel{g \circ f}{\to}
  }
  \,.
$$

So in particular this means that $K$ is closed under composition of morphisms.

This definition has immediate generalization also to [[higher category theory]]. For instance in [[(∞,1)-category theory]] its says that:

a class of 1-morphisms in an [[(∞,1)-category]] satisfies two out of 3, if for every [[2-morphism]] of the form

$$
  \array{
    & {}^{\mathllap{}f}\nearrow &\Downarrow^{\simeq}& \searrow^{\mathrlap{g}}
    \\
    &&\stackrel{h}{\to}&&
  }
$$

we have that if two of $f$, $g$ and $h$ are in $C$, then so is the third.

## Examples

* The class of [[isomorphism]]s in any category satisfies 2-out-of-3. This case is the archetype of most of the cases in which the property is invoked: 2-out-of-3 is characteristic of morphisms that have a notion of [[inverse]].

* A [[category with weak equivalences]] is defined as a category with a subcategory that contains all [[isomorphism]]s and satisfies 2-out-of-3.

## Related concepts

* [[category with weak equivalences]]
* [[weak factorization system]]
* [[2-out-of-6 property]]

[[!redirects 2-out-of-3]]
[[!redirects 2-out-of-3 property]]
[[!redirects 2 out of 3]]
[[!redirects 2 out of 3 property]]
[[!redirects two out of three property]]
[[!redirects two-out-of-three property]]