Dwyer-Wilkerson H-space

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## Idea

The _Dwyer-Wilkerson space_ $G_3$ ([Dwyer-Wilkerson 93](#DwyerWilkerson93)) (also denoted $D I(4)$) is a [[p-adic completion|2-complete]] [[H-space]], in fact a finite [[loop space]]/[[∞-group]], such that the mod 2 [[cohomology ring]] of its [[classifying space]]/[[delooping]] is the mod 2 [[Dickson invariants]] of rank 4. As such, it is the fifth and last space (see [below](#Cohomology)) in a series of [[∞-groups]] that starts with 4 [[compact Lie groups]], namely with the [automorphism groups of real normed division algebras](normed+division+algebra#Automorphisms):

| $n=$ | 0 | 1 | 2 | 3 | 4 |
|--|--|--|--|--|--|
| $DI(n)=$ | [[trivial group|1]] |  [[Z/2]] | [[SO(3)]] | [[G2]] | [[G3]] |
|          | = Aut(R) | [= Aut(C)](complex+number#AutomorphismsOfComplexNumbersIsZ2)        | [= Aut(H)](quaternion#AutomorphismsOfQUatrnionsAlgebraIsSO3)       |  [= Aut(O)](octonion#AutomorphismsOfOctonionAlgebraIsG2)  |   |

whence the notation "$G_3$" (suggested in [Møller 95, p. 5](#Moller95)).

While $G_3$ is not a [[compact Lie group]], it is a [[p-compact group|2-compact group]], hence a "[[homotopy Lie group]]" (see [below](#AsA2CompactGroup)).

The above progression starting with the [[automorphism groups]] of [[real numbers|real]] [[normed division algebras]] suggests that $G_3$ has a geometric or algebraic relevance in a context of [[division algebra and supersymmetry]]. This remains open, but there are speculations, see [below](#RelationToPctomionic3Times3HermitianMatrices).

## Properties

### Cohomology
 {#Cohomology}

The [[ordinary cohomology]] of the [[classifying space]]/[[delooping]] $B G_3$ with [[coefficients]] in the [[prime field]] $\mathbb{F}_2$ is, as an [[associative algebra]] over the [[Steenrod algebra]], the ring of mod 2 [[Dickson invariants]] of rank 4. This is the ring of invariants of the natural action of $GL(4, \mathbf{F}_2)$ on the rank 4 polynomial algebra $H^{\ast}((B \mathbf{Z}/2)^4, \mathbf{F}_2)$, a polynomial algebra on classes $c_8$, $c_12$, $c_14$, and $c_15$ with $Sq^4 c_8 = c_{12}$, $Sq^2 c_{12} = c_{14}$, and $Sq^1 c_{14} = c_{15}$.

([Dwyer-Wilkerson 93, Theorem 1.1](#DwyerWilkerson93))

As such, $G_3$ is the last in a series of [[∞-groups]] whose [[classifying spaces]]/[[deloopings]] have as mod 2 [[cohomology ring]] the mod 2 [[Dickson invariants]] for rank $n$, which starts with three ordinary [[compact Lie groups]]:

| $n=$ | 1 | 2 | 3 | 4 |
|--|--|--|--|--|
| $DI(n)=$ | [[Z/2]] | [[SO(3)]] | [[G2]] | [[G3]] |

([Dwyer-Wilkerson 93, top of p. 38 (2 of 28)](#DwyerWilkerson93))

This means in particular that the cohomology is an [[exterior algebra]] on generators of degree 7, 11, 13, 14 so it's (2-locally) a [[Poincaré duality space]] of [[dimension]] 45.

(...)

### Construction as a homotopy colimit

The space $B G_3$ is the 2-completion of the homotopy colimit of a diagram ([Notbohm 03, Sec. 2](#Notbohm), [Ziemianski, 0.2.3](#Ziemianski)).

### As a 2-compact group
 {#AsA2CompactGroup}

$G_3$ is the only exotic 2-group, or, in other words, the only simple
[[p-compact group|2-compact group]] not arising as the 2-completion of a compact connected Lie group ([Andersen-Grodal 06](#AndersenGrodal06)).

### Weyl group

The analog of the [[Weyl group]] for $G_3$ is $\mathbb{Z}/2 \times GL(3,\mathbb{F}_2)$.

([Dwyer-Wilkerson 93, middle of p. 38 (2 of 28)](#DwyerWilkerson93))

### Homotopy coset space $G_3/Spin(7)$
 {#HomotopyCosetSpace}

$G_3$ receives a homomorphism from [[Spin(7)]]. The [[homotopy fiber]] of the corresponding [[delooping]] map is a homotopy-[[coset space]]

$$
  G_3/Spin(7)
$$

The [[ordinary cohomology]] with [[coefficients]] in the [[prime field]] $\mathbb{F}_2$ of this space has [[Euler characteristic]] 7 ([Notbohm 03, Remark 2.3](#Notbohm), [Aguadé 10, p. 4133](#Aguade10)), equal to the index of the respective [[Weyl groups]]. (Note this corrects an error in ([Dwyer-Wilkerson 93, Theorem 1.8](#DwyerWilkerson93)).)

### Relation to the Conway group, $Co_3$
{#RelationtoCo3}

$B G_3$ receives a map from $B Co_3$, the [[delooping]]/[[classifying space]] of the [[Conway group]], $Co_3$. This map has the property that it injects the mod two cohomology of $B G_3$ as a subring over which the mod two cohomology of $B Co_3$ is finitely generated as a module (see [Benson 94](#Benson94)). This continues a pattern from $B A_5 \to B SO(3)$ and $B M_{12} \to B G_2$, where $M_{12}$ is a [[Mathieu group]]. For further developments see ([Aschbacher-Chermak 10]({#AschbacherChermak10})).

$G_3$ and $Co_3$ both contain as 2-local subgroups the non-split extension, $(\mathbb{Z}/2)^4.G L(4, \mathbb{F}_2)$.

### Relation to octonionic $3 \times 3$ matrix algebra?
 {#RelationToPctomionic3Times3HermitianMatrices}

Since, by the [above](#Cohomology), $G_3$ is (2-locally) a [[Poincaré duality space]] of [[dimension]] 45, there has been speculation that it might be related to the $8 + 2 \cdot 8 + 3 \cdot 7 = 45$-dimensional algebra

$$
  Mat^{skher}_{3 \times 3}(\mathbb{O})
$$

of skew-hermitian [[matrices]] over the [[octonions]] ([Solomon-Stancu 08, p. 175](#SolomonStancu08), [Wilson 09a, slide 94](#Wilson09a), [Benson 98, p. 19](#Benson98)). (Wilson's suggestion appears to arise from his construction of a 3-dimensional octonionic [[Leech lattice]], his representation of its automorphism group, the [[Conway group]] $Co_0$, as right multiplications by $3 \times 3$ matrices over the octonions ([Wilson 09b](#Wilson09b)), and the [relationship](#RelationtoCo3) between the latter's subgroup $Co_3$ and $G_3$.)

Incidentally, the algebra of $3\times 3$ [[hermitian matrices]] (as opposed to skew-hermitian) over the octonions

$$
  Mat^{her}_{3 \times 3}(\mathbb{O})
  \;
  \simeq_{\mathbb{R}}
  \;
  \underset{
    dim_{\mathbb{R}} = 26
  }{
  \underbrace{
    \mathbb{R}^{9,1}
      \oplus
    \mathbf{16}
  }}
  \oplus
  \mathbb{R}
  \,.
$$

is the [[exceptional Jordan algebra]]  called the _[[Albert algebra]]_ (see [there](Albert+algebra#RelationTo10dSuperSpacetime)).

### Homotopy representation

The possibility of there being a faithful 15-dimensional real homotopy representation of $G_3$ is raised in ([Baker-Bauer 19, p. 8](#BakerBauer)).

## Related concepts

[[!include coset space structure on n-spheres -- table]]

## References

Due to

*  {#DwyerWilkerson93} [[William Dwyer]], [[Clarence Wilkerson]], _A new finite loop space at the prime two_, J. Amer. Math. Soc. 6 (1993), 37-64  ([doi:10.1090/S0894-0347-1993-1161306-9](https://doi.org/10.1090/S0894-0347-1993-1161306-9))

Review:

* {#Moller95} [[Jesper Møller]], _Homotopy Lie groups_, Bull. Amer. Math. Soc. (N.S.) 32 (1995) 413-428 ([arXiv:math/9510218](https://arxiv.org/abs/math/9510218))

* {#Grodal10} [[Jesper Grodal]], _The Classification of $p$–Compact Groups and Homotopical Group Theory_, Proceedings of the International Congress of Mathematicians, Hyderabad 2010 ([arXiv:1003.4010](https://arxiv.org/abs/1003.4010), [pdf](http://web.math.ku.dk/~jg/papers/icm.pdf), [[GrodalpCompactGroups2010.pdf:file]])

* {#Notbohm} Dietrich Notbohm, _On the compact 2-group $D I(4)$_,  Journal für die reine und angewandte Mathematik (Crelles Journal), Volume 2003, Issue 555, Pages 163–185, ([pdf](https://pdfs.semanticscholar.org/76ee/dfae66955e452fc20e7ab4a19bd1f6284af4.pdf))

See also

* {#Benson94} David Benson, _Conway’s group $Co_3$ and the Dickson invariants_, Manuscripta Math (1994) 85: 177 ([dml:156016](https://eudml.org/doc/156016))

* {#Ziemianski} Krzysztof Ziemiański, _A faithful complex representation of the 2-compact group DI(4)_, 2005 ([thesis](https://www.mimuw.edu.pl/~ziemians/pap/Thesis.pdf))

* {#AndersenGrodal06} Kasper Andersen, [[Jesper Grodal]], _The classification of 2-compact groups_, J. Amer. Math. Soc. 22 (2009), 387-436 ([arXiv:math/0611437](https://arxiv.org/abs/math/0611437))

* {#BenderskyDavis07} Martin Bendersky, Donald M. Davis, _$v_1$-periodic homotopy groups of the Dwyer-Wilkerson space_ ([arXiv:0706.0993](https://arxiv.org/abs/0706.0993))

* {#AschbacherChermak10} Michael Aschbacher, Andrew Chermak, _A group-theoretic approach to a family of 2-local finite groups constructed by Levi and Oliver_, Annals of Mathematics,  Volume 171 (2010), Issue 2  ([doi:10.4007/annals.2010.171.881](http://doi.org/10.4007/annals.2010.171.881),[pdf](http://annals.math.princeton.edu/2010/171-2/p06))

* {#Aguade10} Jaume Aguadé, _The torsion index of a $p$-compact group_, Proceedings of the AMS, Vol. 138, No. 11, 2010 ([jstor:25748300](https://www.jstor.org/stable/25748300))

* {#BakerBauer19} [[Andrew Baker]], [[Tilman Bauer]], _The realizability of some finite-length modules over the Steenrod algebra by spaces_ ([arXiv:1903.10288](https://arxiv.org/abs/1903.10288))

Speculation on possible geometric roles of $G_3$:

* {#SolomonStancu08} Eon Solomon, Radu Stancu, p. 175 of: _Conjectures on finite and p-local groups_, L'Enseignement Mathématique (2) 54 (2008) 171-176 ([[SolomonStancuConjectures.pdf:file]], [doi:10.5169/seals-109929](http://doi.org/10.5169/seals-109929))

* {#Benson98} David Benson, _Cohomology of Sporadic Groups, Finite Loop Spaces, and the Dickson Invariants_, in P. Kropholler, G. Niblo, & R. Stöhr (Eds.), Geometry and Cohomology in Group Theory (London Mathematical Society Lecture Note Series, pp. 10-23), 1998. Cambridge University Press.

* {#Wilson09a} [[Robert A. Wilson]], Slide 94 of: _A new approach to the Leech lattice_, talk at University of Cambridge, 21st October 2009 ([slides pdf](http://www.maths.qmul.ac.uk/~raw/talks_files/Cambridge09.pdf))

  (on an [[octonions|octonionic]] construction of the [[Leech lattice]])

* {#Wilson09b} [[Robert A. Wilson]], _Conway’s group and octonions_, ([pdf](http://www.maths.qmul.ac.uk/~raw/pubs_files/octoConway.pdf))

[[!redirects Dwyer-Wilkerson H-spaces]]

[[!redirects Dwyer-Wilkerson space]]
[[!redirects Dwyer-Wilkerson spaces]]

[[!redirects G3]]


dyadic interval coalgebra

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## Definition ##

A **dyadic interval coalgebra** is a [[set]] $I$ with a [[linear order]] $\lt$, elements $0 \in I$ and $1 \in I$ and functions $z_0:I \to I$ and $z_1:I \to I$, such that $z_0(0) = 0$, $z_1(0) = 0$, $z_0(1) = 1$, $z_1(1) = 1$, * for all elements $a \in I$, $0 \lt a$ or $a \lt 1$, and for all elements $a \in I$, it is false that both $0 \lt z_0(a)$ and $z_1(a) \lt 1$.

This is called simply an interval coalgebra by Peter Freyd, however there exist similarly defined interval coalgebras with $n+1$ terms and $n$ zooming operations, such as the [[decimal interval coalgebra]].

## Examples ##

* The initial dyadic interval coalgebra is the [[unit interval]] on the [[dyadic rational numbers]]

* The terminal dyadic interval coalgebra is the [[unit interval]] on the [[Dedekind real numbers]]

## See also ##

* [[dyadic rational number]]
* [[rational interval coalgebra]]
* [[decimal interval coalgebra]]
* [[unit interval]]
* [[coalgebra]]

## References ##

* Peter Freyd, Algebraic real analysis, Theory and Applications of Categories, Vol. 20, 2008, No. 10, pp 215-306 ([tac:20-10](http://www.tac.mta.ca/tac/volumes/20/10/20-10abs.html))

[[!redirects dyadic interval coalgebras]]
[[!redirects interval coalgebra]]
[[!redirects interval coalgebras]]


dyadic rational algebra

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## Idea

A __dyadic rational algebra__ or __dy-algebra__ is an [[associative unital algebra]] over the [[dyadic rational numbers]]

## Examples

The [[decimal rational numbers]] $\mathbb{Z}[1/10]$, the [[rational numbers]] $\mathbb{Q}$, and the [[real numbers]] $\mathbb{R}$ are dyadic rational algebras.

Any $\mathbb{R}$-algebra, such as the [[complex numbers]], [[quaternions]], and finite dimensional [[Clifford algebras]] over the real numbers, is also a dyadic rational algebra.

## Properties

Every dyadic rational algebra is a [[symmetric midpoint algebra]] with the midpoint operation given by

$$
  a \vert b \coloneqq \frac{1}{2} (a + b)
$$

### Symmetric and commutator products

In any dyadic rational algebra, one can define the __symmetric product__ to be

$$
  a \cdot b \coloneqq \frac{1}{2} (a b + b a)
$$

and the __commutator product__ to be

$$
  a \times b \coloneqq \frac{1}{2} (a b - b a)
$$

The symmetric product in a dyadic rational algebra is [[commutative]]:

$$
  a \cdot b = b \cdot a
$$

while the commutator product is anticommutative

$$
  a \times b = -(b \times a)
$$

and satisfiea the [[Jacobi identity]]

$$
  a \times (b \times c) + b \times (c \times a) + c \times (a \times b) = 0
$$

The bivector subalgebra of a [[Clifford algebra]] over the real numbers with the commutator product provides a model for [[spin geometry]] and [[Lie groups]], and the vector submodule of a Clifford algebra over the real numbers with the symmetric product is an [[inner product space]].

## Related concepts

* [[dyadic rational number]]

* [[dyadic rational module]]

* [[symmetric midpoint algebra]]

* [[Clifford algebra]]

* [[geometric algebra]]

* [[spin geometry]]

## References

The relation to symmetric midpoint algebras could be found in

* [[Peter Freyd]], *Algebraic real analysis*, Theory and Applications of Categories, Vol. 20, 2008, No. 10, pp 215-306 ([tac:20-10](http://www.tac.mta.ca/tac/volumes/20/10/20-10abs.html))

The symmetric and commutator product is defined in this reference for Clifford algebras over the real numbers, but the definition is valid in any dyadic rational algebra:

* [[Chris Doran]], Anthony Lasenby, _Geometric algebra for physicists_, Cambridge University Press (2003) ([pdf](http://catdir.loc.gov/catdir/samples/cam033/2002035182.pdf))


dyadic rational module

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## Idea

A __dyadic rational module__ or __dy-module__ is an [[module]] over the [[dyadic rational numbers]].

## Examples

The [[decimal rational numbers]] $\mathbb{Z}[1/10]$, the [[rational numbers]] $\mathbb{Q}$, and the [[real numbers]] $\mathbb{R}$ are dyadic rational modules.

## Related concepts

* [[dyadic rational number]]

* [[dyadic rational algebra]]

* [[symmetric closed midpoint algebra]]

## References

The definition and relation to symmetric closed midpoint algebras could be found in

* [[Peter Freyd]], *Algebraic real analysis*, Theory and Applications of Categories, Vol. 20, 2008, No. 10, pp 215-306 ([tac:20-10](http://www.tac.mta.ca/tac/volumes/20/10/20-10abs.html))


dyadic rational number

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## Idea

A __dyadic rational number__ is a [[rational number]] $r \in \mathbb{Q}$ such that the [[binary number|binary]] expansion of $r$ has finitely many digits.

## Definition

### As a subset of the rational numbers

A __dyadic rational number__ is a [[rational number]] $r \in \mathbb{Q}$ such that there exists $n \in \mathbb{N}$ and $a \in \mathbb{Z}$ such that $r = \frac{a}{2^n}$.

### As the localisation of the integers at 2

The [[commutative ring]] of dyadic rational numbers $\mathbb{Z}[1/2]$ is the [[localization of a commutative ring|localization]] of the [[integers]] $\mathbb{Z}$ [[localisation of a commutative ring away from an element|away from]] $2$.

### As an initial object of a category

For lack of a better name, let us define a *set with dyadic rational numbers* to be a set $A$ with a function $\iota \in \mathbb{Z} \times \mathbb{N} \to A$, such that

$$\forall a \in \mathbb{Z}. \forall b \in \mathbb{N}. \forall c \in \mathbb{Z}. \forall d \in \mathbb{N}. (a \cdot 2^d = c \cdot 2^b) \implies (\iota(a, b) = \iota(c, d))$$

The integer $a:\mathbb{Z}$ represents the integer if one ignores the separator in the [[binary number|binary numeral representation]] of the dyadic rational number, and the natural number $b:\mathbb{N}$ represents the number of digits to the left of the final digit where the separator ought to be placed after in the binary numeral representation of the dyadic rational number.  The axiom above is used to state that equivalent numeral representations are equal: i.e. 1.00 = 1.000 with binary numeral representations.

A *homomorphism of sets with dyadic rational numbers* between two sets with dyadic rational numbers $A$ and $B$ is a function $f:A \to B$ such that

$$\forall a \in \mathbb{Z}. \forall b \in \mathbb{N}. f(\iota_A(a, b)) = \iota_B(a, b)$$

The *category of sets with dyadic rational numbers* is the category $SwDRN$ whose objects $Ob(SwDF)$ are sets with dyadic rational numbers and whose morphisms $Mor(A,B)$ for sets with dyadic rational numbers $A \in Ob(SwDRN)$, $B \in Ob(SwDRN)$ are homomorphisms of sets with dyadic rational numbers. The set of **dyadic rational number**, denoted $\mathbb{Z}[1/2]$, is defined as the initial object in the category of sets with dyadic rational number.

## Properties

### Algebraic closure

The [[algebraic closure]] $\overline{\mathbb{Z}[1/2]}$ of the dyadic rational numbers is called the [[field]] of _[[algebraic numbers]]_, and is thus isomorphic to $\overline{\mathbb{Q}}$, the algebraic closure of the [[rational numbers]].

### Topologies

There are several interesting [[topological space|topologies]] on $\mathbb{Z}[1/2]$ that make $\mathbb{Z}[1/2]$ into a [[topological group]] under addition, allowing us to define interesting fields by taking the [[complete space|completion]] with respect to this topology:

1.  The _[[discrete topology]]_ is the most obvious, which is already complete.

2.  The _absolute-value topology_ is defined by the [[metric space|metric]] $d(x,y) \coloneqq {|x - y|}$; the completion is the field of [[real numbers]].

    (This topology is [[totally disconnected space|totally disconnected]].)

3.  The _$2$-adic topology_ is defined by the [[ultrametric]] $d(x,y) \coloneqq 1/n$ where $n$ is the highest exponent on $2$ in the [[prime factorization]] of ${|x - y|}$; the completions of each metric are the fields of $2$-[[adic number|adic numbers]].

## Related entries

* [[decimal rational number]]

* [[Urysohn's lemma]]

* [[halving group]]

* [[symmetric midpoint algebra]]

* [[dyadic rational module]]

* [[dyadic rational algebra]]

* [[dyadic interval coalgebra]]

## References

* Wikipedia, _[Dyadic rational](https://en.wikipedia.org/wiki/Dyadic_rational)_

[[!redirects dyadic rational numbers]]

[[!redirects dyadic rational]]
[[!redirects dyadic rationals]]

[[!redirects dyadic number]]
[[!redirects dyadic numbers]]


dyadic space

A __dyadic compactum__ is an image under a continuous map of some cartesian (that is [[product topology|Tihonov]]) product $2^K$ of discrete 2-point topological spaces. A __dyadic space__ is a topological space which is homeomorphic to a dense subspace of a dyadic compactum. The basic example is the [[Cantor set]] $2^\omega$.

* wikipedia:[dyadic space](https://en.wikipedia.org/wiki/Dyadic_space)

Related notions include [[Cantor cube]], [[Cantor set]]

category: topology

[[!redirects dyadic compactum]]


Dylan Allegretti

* [webpage](http://users.math.yale.edu/~dga4/)

category: people


Dylan McDermott

* [personal page](https://dylanm.org/)

## Selected writings

On [[strong monad|strong]] [[monads in computer science]]:

* {#McDermottUustalu22} [[Dylan McDermott]], [[Tarmo Uustalu]], *What Makes a Strong Monad?*, EPTCS **360** (2022) 113-133 [[arXiv:2207.00851](https://arxiv.org/abs/2207.00851), [doi:10.4204/EPTCS.360.6](https://doi.org/10.4204/EPTCS.360.6)]

On [[graded monads]] [[monad in computer science|in computer science]] as [[enriched monad|enriched]] [[relative monads]] (and on [[locally graded categories]]):

* [[Dylan McDermott]], [[Tarmo Uustalu]]. _Flexibly graded monads and graded algebras_, in: *Mathematics of Program Construction MPC 2022*, Lecture Notes in Computer Science **13544**, Springer (2022) [[doi:10.1007/978-3-031-16912-0_4](https://doi.org/10.1007/978-3-031-16912-0_4)]

On [[formal category theory|formal theory]] of [[relative monads]] via [[virtual equipments]]:

* [[Nathanael Arkor]], [[Dylan McDermott]], *The formal theory of relative monads* [[arXiv:2302.14014](https://arxiv.org/abs/2302.14014)]

* {#AM23b} [[Nathanael Arkor]], [[Dylan McDermott]], _Relative monadicity_ (2023) [[arXiv:2305.10405](https://arxiv.org/abs/2305.10405)]

category: people


Dylan Thurston

* [webpage](http://www.math.indiana.edu/people/profile.phtml?id=dpthurst)

## Selected writings

Computation of the perturbative [[Wilson loop observable]] ([[universal Vassiliev invariant]] of the [[unknot]] ("[[Wheels theorem]]"):

* [[Dror Bar-Natan]], [[Le Tu Quoc Thang]], [[Dylan Thurston]], _Two applications of elementary knot theory to Lie algebras and Vassiliev invariants_, Geom. Topol. Volume 7, Number 1 (2003), 1-31 ([euclid.gt/1513883092](https://projecteuclid.org/euclid.gt/1513883092))

following

* [[Dror Bar-Natan]], [[Stavros Garoufalidis]], [[Lev Rozansky]], [[Dylan Thurston]], _Wheels, wheeling, and the Kontsevich integral of  the unknot_ ([q-alg/9703025](http://arxiv.org/abs/q-alg/9703025))

## Related $n$Lab entries

* [[complex volume]]

category: people


Dylan Wilson

* [webpage](https://sites.google.com/a/uw.edu/dwilson/)

category: people


dynamic lifting

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## Idea

In [[quantum computation]] the term "dynamic lifting" (mostly used in the *[[Quipper]]*-community, starting around [Green et al (2013), p. 5](#GLRSV13)) refers to the outcome of [[quantum measurement]]-results --- which can be known only during runtime (ie. "dynamically"), due to the intrinsic stochastic nature of [[quantum physics]]  --- being turned into (ie. "lifted" back to) [[variables]] on which subsequent classical [[computation]] may be conditioned, which may then in turn [[controlled quantum gate|control]] further [[quantum circuits]], etc.:

\begin{imagefromfile}
    "file_name": "SQRAMschema-WithDynLifting.jpg",
    "width": 500,
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 10
    },
    "caption": "SQRAM model adapted from [Nagarajan, Papanikolaou & Williams 2007, Fig 1](#NagarajanPapanikolaouWilliams07)"
\end{imagefromfile}

Dynamic lifting is thought (eg. [Rand (2018),p. 52](#Rand18)) to be optional in cases like the [[quantum teleportation protocol]] but necessary for applications like

* [[quantum error correction]] (e.g. [Paler, Herr & Devitt (2019), §4.1](quantum+error+correction#PalerHerrDevitt19)), where error correction steps are conditioned on [[quantum measurement]] of syndrome [[qbits]];

* [[repeat-until-success computing]], where a quantum circuit is re-run on its input data until a failure syndrome measurement indicates success.

There have been several (partial) proposals for the precise [[categorical semantics]] of this idea, see the references [below](#ReferencesCategoricalSemantics).
Another formalization, via [[monadic effects]] of [potentiality modalities](necessity+and+possibility#RelationToPotentiality) in [[dependent linear type theory]], is discussed at *[[quantum circuits via dependent linear types]]* ([CQTS 2022](#CQTS22), see [there](quantum+circuits+via+dependent+linear+types#QuantumMeasurement)).

## References

### General

The general but informal idea of quantum computation with classical control that may itself be conditioned on quantum measurement results (see also at *[Quantum Computation -- Classical control and quantum data](quantum+computation#ClassicalControlQuantumData)*) may be argued to be implicit already in [Knill (1996)](quantum+computation#Knill96).

Explicit discussion includes:

* {#NagarajanPapanikolaouWilliams07} Rajagopal Nagarajan, Nikolaos Papanikolaou, David Williams, *Simulating and Compiling Code for the Sequential Quantum Random Access Machine*, Electronic Notes in Theoretical Computer Science Volume 170, 6 March 2007, Pages 101-124 [[doi:10.1016/j.entcs.2006.12.014](https://doi.org/10.1016/j.entcs.2006.12.014)]

The term "dynamic lifting" is mainly used in discussion of the [[quantum programming language]] *[[Quipper]]*.  Its first (brief) appearance might be that in:

* {#GLRSV13} [[Alexander Green]], [[Peter LeFanu Lumsdaine]], [[Neil Ross]], [[Peter Selinger]], [[Benoît Valiron]], p.5 of: _Quipper: A Scalable Quantum Programming Language_, ACM SIGPLAN Notices **48** 6 (2013) 333-342 [[arXiv:1304.3390](https://arxiv.org/abs/1304.3390), [doi:10.1145/2499370.2462177](https://doi.org/10.1145/2499370.2462177)]

More in-depth discussion of "dynamic lifting" is in:

* {#Rand18} [[Robert Rand]], pp. 40 of: *Formally Verified Quantum Programming*, UPenn (2018) [[ediss:3175](https://repository.upenn.edu/edissertations/3175)]

### Categorical semantics
 {#ReferencesCategoricalSemantics}

Proposals for [[categorical semantics]] of "dynamic lifting", mostly following along the lines of some [[dependent linear type theory]], such as found in [[Quipper]]:

* [[Mathys Rennela]], [[Sam Staton]], *Classical Control, Quantum Circuits and Linear Logic in Enriched Category Theory*, Logical Methods in Computer Science **16** (2020) lmcs:6192 [[arXiv:1711.05159](https://arxiv.org/abs/1711.05159), doi:10.23638/LMCS-16(1:30)2020]

  > (cf. *[[EWIRE]]*)

* [[Dongho Lee]], Sebastien Bardin, [[Valentin Perrelle]], [[Benoît Valiron]], *Formalization of a Programming Language for Quantum Circuits with Measurement and Classical Control*, talk at *[Journées Informatique Quantique 2019](https://quantcert.github.io/GT-IQ'19/)* (Nov 2019) [[pdf](https://quantcert.github.io/GT-IQ%2719/slides/Lee.pdf), [[LBPV_QCWithClassicalControl.pdf:file]]]

* [[Dongho Lee]], [[Valentin Perrelle]], [[Benoît Valiron]], Zhaowei Xu, *Concrete Categorical Model of a Quantum Circuit Description Language with Measurement*, Leibniz International Proceedings in Informatics **213** (2021) 51:1-51:20 [[arXiv:2110.02691](https://arxiv.org/abs/2110.02691), [doi:10.4230/LIPIcs.FSTTCS.2021.51](https://doi.org/10.4230/LIPIcs.FSTTCS.2021.51)]

* [[Andrea Colledan]], [[Ugo Dal Lago]], *On Dynamic Lifting and Effect Typing in Circuit Description Languages*, talk at *TYPES Workshop*, Nantes (2022)  [[arXiv:2202.07636](https://arxiv.org/abs/2202.07636), [pdf](https://types22.inria.fr/files/2022/06/TYPES_2022_slides_13.pdf), [[ColledanLago-DynamicsLifting.pdf:file]]]

* [[Peng Fu]], [[Kohei Kishida]], [[Neil J. Ross]], [[Peter Selinger]], *A biset-enriched categorical model for Proto-Quipper with dynamic lifting*, in proceedings of *[Quantum Physics and Logic 2022](https://www.qplconference.org/)* [[arXiv:2204.13039](https://arxiv.org/abs/2204.13039)]

* [[Peng Fu]], [[Kohei Kishida]], [[Neil J. Ross]], [[Peter Selinger]], *Proto-Quipper with dynamic lifting*, in proceedings of *[Quantum Physics and Logic 2022](https://www.qplconference.org/)* [[arXiv:2204.13041](https://arxiv.org/abs/2204.13041)]

* {#CQTS22} [[Urs Schreiber]], *[Quantum Data Types via Linear HoTT](https://ncatlab.org/schreiber/show/QDataInLHoTT#QTML2022)* (Nov 2022)

* [[Dongho Lee]], *Formal Methods for Quantum Programming Languages*, Paris Saclay (Dec 2022) [[hal:tel-03895847](https://hal.science/LMF/tel-03895847)]

* [[Hisham Sati]], [[Urs Schreiber]], *[[schreiber:The Quantum Monadology|The Quantum Monadology]]* [[arXiv:2310.15735](https://arxiv.org/abs/2310.15735)]


dynamical extension of a monoidal category

## Idea

Given a monoid $L$ in a monoidal category, there is an induced monad $T^L$ obtained by tensoring with $L$ from the right. Consider the [[Kleisli category]] of the monad $T^L$. It is not necessarily a monoidal category; there is a need for an additional structure ([[distributive law]] of a sort) in order to make such a lift. If the monoidal category $C$ in question is the monoidal category of left modules over a Hopf algebra $H$, then any commutative algebra $L$ in the center $Z(C)$ provides a canonical lift to the Kleisli category

#### Physical motivation

Dynamical extension or a dynamization of a monoidal category is an abstract framework to aid the construction of so called dynamical quantum groups which are in turn related to [[dynamical quantum Yang-Baxter equation]].

## Definition

...

## Properties and remarks

In Donin-Mudrov (2006) a relation to [[bialgebroid]]s has been exhibited.

## Literature
* J. Donin, A. Mudrov, _Dynamical Yang-Baxter equation and quantum vector bundles_, Comm. Math. Phys. __254__ (2005), no. 3, 719--760,  [math.QA/0306028](http://arxiv.org/abs/math/0306028), [MR2006i:81099](http://www.ams.org/mathscinet-getitem?mr=2126489), [doi](http://dx.doi.org/10.1007/s00220-004-1247-8)
* J. Donin, A. Mudrov, _Quantum groupoids and dynamical categories_,
J. Algebra __296__ (2006), no. 2, 348–384, [math.QA/0311316](http://arxiv.org/abs/math/0311316), [MR2007b:17022](http://www.ams.org/mathscinet-getitem?mr=2201046), [doi](http://dx.doi.org/10.1016/j.jalgebra.2006.01.001); MPIM-2004-21, [dvi](http://www.mpim-bonn.mpg.de/preblob/1921) with hyperlinks, [ps](http://www.mpim-bonn.mpg.de/preblob/1922)


dynamical system

#Contents#
* table of contents
{:toc}

## Idea

A __dynamical system__ is a [[space]] $X$ (often a bare [[set]] or a [[manifold]]) together with a "law of motion" expressed by the [[action]] of a [[monoid]] $A$:

$$
  A \times X \longrightarrow X
  \,.
$$

To model continuous [[time]]-evolution one may take $A = (\mathbb{R}, +)$ to be the additive [[group]] of [[real numbers]].

For discrete time evolution $A = (\mathbb{Z}, +)$ is the additive group of [[integers]] (or just the monoid $(\mathbb{N}, +)$ of [[natural numbers]]). In this case the law of motion is often given by an [[initial value problem]] for a [[differential equation]] "of evolution type".

In this case, the dynamical system is equivalently a space $X$ equipped with an [[automorphism]] (this being the action of the [[unit element]] in $\mathbb{Z}$, sometimes called the "shift").

Sometimes the evolution is only partially defined; this is most often in dynamical systems induced by evolution differential equations which do not necessarily have existence of solutions for arbitrary large time, or the dynamical system is defined only for nonnegative time.

The definition evidently makes sense quite generally [[internalization|internal to]] various ambient [[categories]]: For instance one considers complex dynamics, [[algebraic dynamics]], arithmetic dynamics and so on.

Dynamical systems are used to describe not only physical motions but also the behaviour of parameters of various systems, e.g. in sociological, financial, weather and other models.

## Literature

## Related concepts

* [[categorical systems theory]]

* [[zeta function of a dynamical system]]

* [[arithmetic topology]]

* [[cohomology of dynamical systems]]

* [[chaos]]

* [[Furstenberg-Zimmer structure theorem]]

## References

### General

* Boris Hasselblatt, Anatole Katok, _Handbook of dynamical systems_, [contents](http://www.math.psu.edu/katok_a/handbook.html)

See also

* Wikipedia [dynamical system](http://en.wikipedia.org/wiki/Dynamical_system), [autonomous system](http://en.wikipedia.org/wiki/Autonomous_system_%28mathematics%29)

### Category-theoretic formulations

Discussion of dynamical systems in terms of [[category theory]] (see also at *[[categorical systems theory]]*):

* [[William Lawvere]], [_Functorial Remarks on the General Concept of Chaos_](http://www.ima.umn.edu/preprints/Functional-Remarks-General-Concept-Chaos), IMA reprint 87, 1984 ([pdf](http://www.ima.umn.edu/sites/default/files/87s.pdf))

* George Dimitrov, Fabian Haiden, [[Ludmil Katzarkov]], [[Maxim Kontsevich]], _Dynamical systems and categories_, [[arxiv/1307.8418](http://arxiv.org/abs/1307.8418)]

* [[Michael Barr]], [[John Kennison]], [[Robert Raphael]], *Flows: cocyclic and almost cocyclic*, Theory Appl. Categories **25** 18 (2010) 490–507 [[tac:25-18](http://www.tac.mta.ca/tac/volumes/25/18/25-18abs.html)]

* [[Patrick Schultz]], [[David I. Spivak]], [[Christina Vasilakopoulou]], *Dynamical Systems and Sheaves*, Appl Categor Struct **28** (2020) 1–57 [[arXiv:1609.08086](https://arxiv.org/abs/1609.08086), [doi:10.1007/s10485-019-09565-x](https://doi.org/10.1007/s10485-019-09565-x)]

* [[David Jaz Myers]], *Double Categories of Open Dynamical Systems*, EPTCS **333** (2021) 154-167 [[arXiv:2005.05956](https://arxiv.org/abs/2005.05956), [doi:10.4204/EPTCS.333.11](https://doi.org/10.4204/EPTCS.333.11)]

Monograph:

* {#Myers} [[David Jaz Myers]], *Categorical systems theory*, book project [[github](https://github.com/DavidJaz/DynamicalSystemsBook/tree/master/book), [pdf](http://davidjaz.com/Papers/DynamicalBook.pdf)]

Exposition:

* [[David Jaz Myers]], *[Categorical systems theory](https://topos.site/blog/2021/11/categorical-systems-theory/)*, [Topos Institute Blog](https://topos.site/blog/) (Nov 2021)

* [[David Jaz Myers]], *Double Categories of Dynamical Systems* (2020) [[pdf](http://davidjaz.com/Talks/DJM_Dyn2020.pdf)]

* [[David Jaz Myers]], *A general definition of open dynamical systems*, talk at *MIT Category Theory Seminar* (2020) [video:[YT](https://youtu.be/8T-Km3taNko)]

[[!redirects dynamical systems]]


dynamical Yang-Baxter equation

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## Idea

(...)

## Related concepts

* [[dynamical extension of a monoidal category]].

[[!include Yang-Baxter equations -- contents]]

## References

Classical and quantum dynamical Yang-Baxter equation

* J.-L. Gervais, A. Neveu, _Novel triangle relation and absence of tachyons in Liouville string field theory_, Nuclear Phys. B 238 (1984), no. 1, 125--141, [MR86f:81088](http://www.ams.org/mathscinet-getitem?mr=747093), doi

* G. Felder, _Conformal field theory and integrable systems associated to elliptic curves_, Proc. ICM, Vol. 1, 2 (Zürich, 1994), 1247--1255, Birkhäuser 1995, [MR97m:81015](http://www.ams.org/mathscinet-getitem?mr=1404026)

[[!redirects dynamical quantum Yang-Baxter equation]]


Dynkin diagram

> under construction

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## Idea

A [[Dynkin diagram]] is a labeled graph that possesses a one-to-one correspondence with a finite indecomposable reduced [[root system]], thus with a simple complex finite dimensional [[Lie algebra]] or [[Cartan matrix]].

The construction of a [[Dynkin diagram]] from the [[Cartan Matrix]], $A = (a_{i,j})_{i,j=1}^n$ is obtained from the following procedure:

1. Number of vertices = Number of simple roots = size of $A$ = n

2. If $a_{i,j} = a_{i,j} = -1$ then $i,j$ are connected by a nonlabeled edge.

3. If $a_{i,j} = -1$ and  $a_{j,i} = -l$ then $i,j$ are connected by $l$ arrows labeled by <.

## Properties

### Classification of simple Lie groups

[[classification of simple Lie groups]]:

> graphics grabbed from [Schwichtenberg](http://jakobschwichtenberg.com/classification-of-simple-lie-groups/)

### ADE-Classification

Those Dynkin diagrams in the [[ADE classification]] are the following

[[!include ADE -- table]]

### Dynkin Index

Let $\mathcal{g}$ be a finite simple complex [[Lie algebra]] with a [[Killing form]] $k$ on $\mathcal{g}$ given by the trace in the adjoint representation,
$k(x,y) = Tr R_{ad}(x)R_{ad}(y)$ for $x,y \in \mathcal{g}$.

For any irreducible finite representation $R_{\lambda}$ of $\mathcal{g}$,
$TrR_\lambda(x)R_\lambda(y) = I_\lambda k(x,y)$
Where $I_\lambda$ is the _Dynkin Index_ of $R_\lambda$.

The Dynkin index can also be defined in terms of the eigenvalue $C_\lambda$ of the quadratic [[Casimir operator]]:
$ I_\lambda = \frac{dim(R_\lambda)}{dim(\mathcal{g}}C_\lambda $.

\begin{remark}\label{DynkinIndexInInstantonNumber}
In [[mathematical physics]],in the context of [[embeddings]] of [[gauge fields]], the Dynkin index is used in the calculation of topological charges ([[instanton number]]).
To demonstrate this, let $A_\mu$ be a gauge potential on the Euclidean space $E^4$ given by
$$
A_\mu(x) = A_{\mu}^{\alpha}X_\alpha
$$
where $X_\alpha$ are the generators of a compact [[gauge group]] $G$. If the gauge field $A_\mu$ is an [[embedding]] of $\tilde{G}$ in $G$ then the Dynkin index of the embedding is denoted as $j_{\tilde{G} \over G}$ . The topological charge is
$$
q = j_{\tilde{G} \over G} q_1
$$
where $q_1$ is the charge of $A_\mu$ treated as a $\tilde{G}$ [[gauge field]].
The proof was carried out for $\tilde{G} \simeq SU(2)$ by Bitar and Sorba see [Myers, de Roo & Sorba 79, Sec. 2](#MyersdeRooSorba79), which can be extended to arbitrary simple [[compact groups]].
\end{remark}

## Related concepts

* [[Dynkin quiver]]

* [[ADE classification]]

## References

### General

See also:

* Wikipedia, _[Dynkin diagram](http://en.wikipedia.org/wiki/Dynkin_diagram)_

### Dynkin index

* Bianchi M. et al. (2004) Dynkin Index. In: Duplij S., Siegel W., Bagger J. (eds) Concise Encyclopedia of Supersymmetry. Springer, Dordrecht. _[publisher](https://doi.org/10.1007/1-4020-4522-0_169)_

* {#MyersdeRooSorba79} C. Meyers, [[Mees de Roo]], P. Sorba,  _Group-theoretical aspects of instantons_. Nuov Cim A 52, 519–530 (1979) ([doi:10.1007/BF02770858](https://doi.org/10.1007/BF02770858))

[[!redirects Dynkin diagrams]]

[[!redirects simply laced Dynkin diagram]]
[[!redirects simply laced Dynkin diagrams]]


Dynkin quiver

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## Definition

A _Dynkin quiver_ is a [[quiver]] whose underlying [[undirected graph]] is a one of the [[Dynkin diagrams]] in the [[ADE classification|ADE series]].

>  graphics grabbed from [Qiu 15](#Qiu15)

See e.g. [Qiu 15, Def. 2.1](#Qiu15)

## Properties

### Characterization as finite-type quivers

[[Gabriel's theorem]] ([Gabriel 72](#Gabriel72)) says that connected [[quivers]] with a [[finite number]] of [[indecomposable object|indecomposable]] [[quiver representations]] over an [[algebraically closed field]] are precisely the _[[Dynkin quivers]]_: those whose underlying [[undirected graph]] is a [[Dynkin diagram]] in the [[ADE classification|ADE series]]

Moreover, the [[indecomposable object|indecomposable]] [[quiver representations]] in this case are [[bijection]] with the positive [[root (in representation theory)|roots]] in the [[root system]] of the [[Dynkin diagram]].

### McKay correspondence

[[!include ADE -- table]]

## References

[[Gabriel's theorem]] is due to

* {#Gabriel72} [[Peter Gabriel]], _Unzerlegbare Darstellungen. I_, Manuscripta Mathematica 6: 71–103, (1972)

Discussion of [[Bridgeland stability conditions]] on [[quiver representations]] over Dynkin Quivers includes

* {#Qiu15} [[Yu Qiu]], _Stability conditions and quantum dilogarithm identities for Dynkin quivers_, Adv. Math., 269 (2015), pp 220-264 ([arXiv:1111.1010](https://arxiv.org/abs/1111.1010))

* [[Tom Bridgeland]], [[Yu Qiu]], Tom Sutherland, _Stability conditions and the $A_2$ quiver_ ([arXiv:1406.2566](https://arxiv.org/abs/1406.2566))

[[!redirects Dynkin quivers]]


dyon

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## Idea

In [[physics]] a _dyon_ is an object that carries both [[electric charge]] as well as [[magnetic charge]]. For [[fundamental particles]] this would correspond to [[magnetic monopoles]] which are also electrically charged. More generally, higher dimensional [[branes]] may (or may not) be dyons in that they are electrically as well as magnetically charged under the appropriate [[higher gauge field]].

## References

See also

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

### Dyonic branes

[[dyon|dyonic]]$\,$[[black brane|black]] [[M2-branes]] ([[M5-branes]] [[wrapped brane|wrapped]] on a [[3-manifold]], see also at _[NS5-branes -- D2/D4/NS5-bound states](NS5-brane#ReferencesNS5D4D2BoundStates)_):

* {#ILPT95} J.M. Izquierdo, [[Neil Lambert]], [[George Papadopoulos]], [[Paul Townsend]], _Dyonic Membranes_, Nucl. Phys. B460:560-578, 1996 ([arXiv:hep-th/9508177](https://arxiv.org/abs/hep-th/9508177))

* [[Michael Green]], [[Neil Lambert]], [[George Papadopoulos]], [[Paul Townsend]], _Dyonic $p$-branes from self-dual $(p+1)$-branes_, Phys.Lett.B384:86-92, 1996 ([arXiv:hep-th/9605146](https://arxiv.org/abs/hep-th/9605146))

* {#Petri18} Nicolò Petri, slide 14 of _Surface defects in massive IIA_, talk at [Recent Trends in String Theory and Related Topics](http://physics.ipm.ac.ir/conferences/stringtheory3/) 2018 ([pdf](http://physics.ipm.ac.ir/conferences/stringtheory3/note/N.Petri.pdf))

[[!redirects dyons]]


dyslectic geometry

As introduced in

* Shai Haran, _An invitation to dyslectic geometry_,  Journal of Algebra __155__:2, (1993) 455-481 [doi](http://dx.doi.org/10.1006/jabr.1993.1054)

the term dyslectic geometry refers to the geometry of (braided-)commutative
objects in the braided monoidal category of
$k[G]^*\sharp k[G]$-modules ($k$ a field, $G$ a finite group) and in some generalizations where $k[G]$ is replaced by a more general Hopf algebra. If $G = \mathbb{Z}_2$ this reduces to [[supersymmetry]].


Dyson formula

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## In quantum mechanics

In [[quantum mechanics]] what is called the _Dyson formula_ is what in _[[mathematics]]_ is called the _[[iterated integral]]-expression for [[parallel transport]]_: It an expression for the solution to a [[differential equation]] of the form

$$
  \frac{d}{d t} \psi(t)
  \;=\;
  i H(t)
  \psi(t)
$$

where $t \mapsto \psi(t) \in \mathcal{H}$ is a one-parameter family of elements of some [[Hilbert space]] and $t \mapsto H(t)$ is a one-parameter family of [[linear operators]] on this Hilbert space.

This appears prominently in the expression of the [[Schrödinger equation]] in the [[interaction picture]], in which case $H(t)$ is the [[interaction]] [[Hamiltonian]] in the [[Heisenberg picture]] of the [[free field|free]] theory. In this case the Dyson series gives the _[[S-matrix]]_.
This is the context in which the term "Dyson formula" orginates. But of course also the plain Schrödinger equation ("in the [[Schrödinger picture]]") is already of this form if the Hamiltonian is time-dependent.

The idea is to think of the solution as given by the [[limit of a sequence|limit]]

$$
  \psi(t)
  \;=\;
  \underset{N \to \infty}{\lim}
  \underset{N \, \text{factors}}{
  \underbrace{
  \left(
    id + \tfrac{i}{N}H(t)
  \right)
  \left(
    id + \tfrac{i}{N}H(t(N-1)/N)
  \right)
  \cdots
  \left(
    id + \tfrac{i}{N}H(t/N)
  \right)
  }
  }
  \psi(0)
$$

and to think of this as the result of forming the [[exponential]] expression $\exp(\int_{[0,t] } H(t)\,d t)$ and then re-ordering in the resulting sum of products all the factors of $H(t)$ such that they are _time ordered_ with larger values of $t$ ordered to the left of smaller values.

Accordingly, in  the physics literature solutions to this equation are written

$$
  \psi(t)
  \;=\;
  T\left(
    \exp\left(
      \int_{[0,t]} i H(t) \,d t
    \right)
  \right)
  \psi(0)
  \,,
$$

with $T(-)$ the _time ordering operator_ producing _[[time-ordered products]]_. The corresponding [[series]] of [[iterated integrals]]

$$
  \psi(t)
  \;=\;
  \left(
    \underoverset{n = 0}{\infty}{\sum}
    (i)^n
    \underset{0 \leq t_1 \leq \cdots \leq t_n \leq t}{\int}
    H(t_n) H(t_{n-1})\cdots H(t_1)
  \right)\psi(0)
$$

is called the _Dyson series_.

## In quantum field theory

The idea generalizes to [[relativistic field theory]] [[perturbative quantum field theory]] on [[Lorentzian manifold|Lorentzian]] [[spacetimes]] if due care is exercised (including [[adiabatic switching]] and [[point-extension of distributions|point extension]] of [[operator-valued distributions]]). Here the "time ordering" is generalized to a [[causal locality|causal ordering]], this is the content of the construction of the [[S-matrix]] and the [[interacting field algebra]] in _[[causal perturbation theory]]_.

See at _[[time-ordered product]]_ and _[[S-matrix]]_ for details.

## Examples

### Kontsevich integral is Dyson formula of KZ-connection

[[!include Knizhnik-Zamolodchikov-Kontsevich construction -- definition]]

## Related concepts

* [[parameterized quantum system]]

* [[path integral]]

* [[iterated integral]]

* [[parallel transport]]

[[!redirects Dyson series]]


Döring-Harding-Hamhalter theorem

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## Idea

Theorems observed by Harding, Döring and Hamhalter say that under some conditions the [[Jordan algebra]] structure of a [[C*-algebra]] is effectively captured by its [[poset of commutative subalgebras]].

Via the [[Alfsen-Shultz theorem]], which states that two [[C*-algebra]] have the same [[states]] precisely if they are isomorphic as [[Jordan algebras]], this is related to [[Gleason's theorem]], which says that states on an [[algebra of bounded operators]] are determined by their restriction to commutative subalgebras (if the underlying Hilbert space has dimenion $\gt 2$).

## Statements

For $A$ an [[associative algebra]] write $A_J$ for its corresponding [[Jordan algebra]], where the commutative product $\circ : A_J \otimes A_J \to A_J$ is the symmetrization of the product in $A$: $a \circ b = \frac{1}{2}(a b + b a)$.

+-- {: .num_lemma}
###### Observation

There exist [[von Neumann algebras]] $A$, $B$ such that there exists a [[Jordan algebra]] isomorphism $A_J \to B_J$ but not an algebra isomorphism $A \to B$.

=--

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

By

* [[Alain Connes]], _A factor not anti-isomorphic to itself_, Annals of Mathematics, 101 (1975), no. 3, 536–554. ([JSTOR](http://www.jstor.org/stable/1970940))

there is a [[von Neumann algebra factor]] $A$ with no algebra isomorphism to its opposite algebra $A^{op}$. But clearly $A_J \simeq (A^{op})_J$.

=--

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

Let $A, B$ be [[von Neumann algebra]]s without a type $I_2$-[[von Neumann algebra factor]]-summand and let $ComSub(A)$, $ComSub(B)$ be their posets of [[commutative C-star-algebra|commutative]] sub-von Neumann algebras.

Then every [[isomorphism]] $ComSub(A) \to ComSub(B)$ of [[poset]]s comes from a unique [[Jordan algebra]] isomorphism $A_J \to B_J$.

=--

This is the theorem in ([Harding-Döring 10](#HardingDoering)).

There is a generalization of this theorem to more general [[C-star algebras]] in ([Hamhalter 11](#Hamhalter)).

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

This is related to the [[Alfsen-Shultz theorem]], which says that two $C^*$-algebras have the
same [[state on an operator algebra|states]] precisely if they are Jordan-isomorphic.

=--

## Related theorems

Other theorems about the foundations and [[interpretation of quantum mechanics]] include:

* [[order-theoretic structure in quantum mechanics]]

  * [[Kochen-Specker theorem]]

  * [[Alfsen-Shultz theorem]]

* [[Fell's theorem]]

* [[Gleason's theorem]]

* [[Wigner theorem]]

* [[Bell's theorem]]

* [[Bub-Clifton theorem]]

* [[Kadison-Singer problem]]

## References

The relation to Jordan algebras of $ComSub(A)$ is discussed in

* {#HardingDoering} [[Andreas Döring]], [[John Harding]],  *Abelian subalgebras and the Jordan structure of a von Neumann algebra*, Houston Journal of Mathematics **42** 2 (2016) [[arXiv:1009.4945](http://arxiv.org/abs/1009.4945), [issue](https://www.math.uh.edu/~hjm/Vol42-2.html), [pdf](https://wordpress.nmsu.edu/hardingj/files/2019/10/2010-Abelian-subalgebras-and-the-Jordan-structure-of-von-Neumann-algebras.pdf)]

for $A$ a [[von Neumann algebra]] and more generally for $A$ a [[C*-algebra]] in

* {#Hamhalter} [[Jan Hamhalter]], _Isomorphisms of ordered structures of abelian $C^\ast$-subalgebras of $C^\ast$-algebras_, J. Math. Anal. Appl. 383 (2011) 391–399 ([journal](http://dx.doi.org/10.1016/j.jmaa.2011.05.035))

* [[Jan Hamhalter]], E. Turilova, _Structure of associative subalgebras of Jordan operator algebras_ ([arXiv:1111.7240](http://arxiv.org/abs/1111.7240))

[[!redirects Harding-Döring-Hamhalter theorem]]


e

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Arithmetic
+--{: .hide}
[[!include arithmetic geometry - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

The [[irrational number]] conventionally denoted $e$ (a notation credited to [[Euler]], hence also called the _[[Euler number]]_) is the base of the natural [[logarithm]]; it is approximately 2.71828182845 in decimal notation.

## Definition

There are numerous ways of defining $e$. One is

$$e \coloneqq \sum_{n \geq 0} \frac1{n!} = 1 + 1 + \frac1{2!} + \frac1{3!} + \ldots.$$

This can be interpreted as the [[groupoid cardinality]] for $core(FinSet)$. Perhaps more important than the [[constant function|constant]] $e$ is the standard [[exponential function]] (defined for all [[complex numbers]] $x$)

$$\exp(x) = \sum_{n \geq 0} \frac{x^n}{n!}$$

for which $e = \exp(1)$. This exponential function is especially convenient because it is uniquely characterized as a function $f(x)$ equal to its own [[derivative]] such that $f(0) = 1$ (necessary in order that it satisfy the exponential law $f(x + y) = f(x)f(y)$).

## Lay geometric description

Construct a [[polar coordinates|polar coordinate grid]] (consisting of radial lines through a point called the origin, and concentric circles centered at the origin). Draw a curve starting at any point except the origin in such a way that at each of its points $p$, the tangent at $p$ meets the radial line at $p$ in a 45 degree angle. This curve is called a _logarithmic spiral_. Then, following the trajectory of the spiral inward (towards the origin, so to speak) through one radian from $p$ to a second point $q$, the distance from $p$ to the origin differs to the distance from $q$ to the origin by a factor of $e$.

Equivalently, imagine four ants situated at the corners of a square, and imagine that at some instant each begins crawling toward its neighbor looking clockwise from above, each at the same speed. The trajectory of each ant is a logarithmic spiral as described above, and the same description of $e$ applies.

## Irrationality

It is a simple matter to show that $e$ is [[irrational number|irrational]]. For if on the contrary we have $e = p/q$, then $e \cdot n!$ would be an integer for any $n \geq q$. However,

$$e \cdot n! = integer + \frac1{n+1} + \frac1{(n+1)(n+2)} + \ldots$$

where the nonzero tail after the integer part is bounded above by $\sum_{k=1}^\infty 1/(n+1)^k = 1/n \lt 1$ for $n \gt 1$, giving a contradiction.

It is harder to show that $e$ is transcendental. An online proof (written up by David Richeson) may be found [here](https://divisbyzero.com/2010/09/28/the-transcendence-of-e/).

## Related entries

* [[Euler number]]

* [[exponential map]]

* [[pi]]

* [[golden ratio]]

## References

* [Wikipedia](http://en.wikipedia.org/wiki/E_%28mathematical_constant%29)

* Eli Maor, _e: The Story of a Number_, Princeton University Press (1994). ISBN 0-691-05854-7.


E-category

+-- {: .rightHandSide}
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### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
=--
=--

# Contents
* table of contents
{:toc}

## Idea

An **E-category** is a category [[enriched category|enriched]] over [[setoids]].  This is mainly used in [[dependent type theory]]; from that point of view, it is related to a [[precategory]] but with specified [[equivalence relations]] rather than [[identity type]] as the "equality of morphisms".

## Definition

In [[intensional type theory]], an **E-category** or $\mathcal{C}$ consists of

* a [[type]] of [[objects]] $Ob(\mathcal{C})$,
* for each object $A:Ob(\mathcal{C})$ and $B:Ob(\mathcal{C})$, a [[setoid]] $(Hom(A, B), \sim_{Hom(A, B)})$ of [[morphisms]]
* for each object $A:Ob(\mathcal{C})$, $B:Ob(\mathcal{C})$, and $C:Ob(\mathcal{C})$, a [[binary function]]
$$(-)\circ_{A, B, C}(-) :Hom(B, C) \times Hom(A, B) \to Hom(A, C)$$
such that for each morphism $f:Hom(A, B)$, $g:Hom(A, B)$, $h:Hom(B, C)$ and $k:Hom(B, C)$, there is a witness of extensionality
$$\mathrm{ext}(A, B, C, f, g, h, k):(f \sim_{Hom(A, B)} g) \times (h \sim_{Hom(B, C)} k) \to (h \circ_{A, B, C} f \sim_{Hom(A, C)} k \circ_{A, B, C} g)$$
* for each object $A:Ob(\mathcal{C})$, a morphism $\mathrm{id}_A:Hom(A, A)$
* such that
  * the composition of morphisms is [[associative]]: for each object $A:Ob(\mathcal{C})$, $B:Ob(\mathcal{C})$, $C:Ob(\mathcal{C})$, and $D:Ob(\mathcal{C})$, and for each morphism $f:Hom(A, B)$, $g:Hom(B, C)$, and $h:Hom(C, D)$, there is a witness of associativity
$$\mathrm{assoc}(A, B, C, D, f, g, h):h \circ_{A, C, D} (g \circ_{A, B, C} f) \sim_{Hom(A, D)} h \circ_{B, C, D} (g \circ_{A, B, D} f)$$
  * the composition of morphisms satisfies the left and right [[unit laws]]: for each object $A:Ob(\mathcal{C})$ and $B :Ob(\mathcal{C})$ and morphism $f:Hom(A, B)$, there are witnesses for left and right unitality
$$\mathrm{lunital}(A, B, f):\mathrm{id}_B \circ_{A, B, B} f \sim_{Hom(A, B)} f$$
$$\mathrm{runital}(A, B, f):f \circ_{A, A, B} \mathrm{id}_A \sim_{Hom(A, B)} f$$

## Properties

An E-category is **locally univalent** or a **[[precategory]]** if for all objects $A$ and $B$ and morphisms $f:Hom(A, B)$ and $g:Hom(A, B)$ the canonical function
$$idtoequivrel(A, B, f, g):f =_{Hom(A, B)} g \to f \sim_{Hom(A, B)} g$$
is an equivalence of types.

An isomorphism in a E-category is a morphism $f:Hom(A, B)$ with a morphism $g:Hom(B, A)$ and witnesses
$$\mathrm{ret}(A, B, f, g): g \circ f \sim_{Hom(A, A)} id_A$$
$$\mathrm{sec}(A, B, f, g): f \circ g \sim_{Hom(B, B)} id_B$$
The type of all isomorphisms between $A$ and $B$ is represented by $A \cong_{\mathcal{C}} B$.

An E-category is **univalent**, or a **[[univalent category]]**, if it is locally univalent and for all objects $A$ and $B$ the canonical function
$$idtoiso(A, B):A =_{Ob(\mathcal{C})} B \to A \cong_{\mathcal{C}} B$$
is an equivalence of types.

## See also

* [[precategory]]

* [[univalent category]]

* [[prebicategory]]

* [[locally univalent bicategory]]

* [[univalent bicategory]]

[[!redirects E-categories]]
[[!redirects e-category]]
[[!redirects e-categories]]


E-infinity algebra

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###Context###
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

Given a [[stable (infinity,1)-category|stable]] [[symmetric monoidal (infinity,1)-category]] $C$, an **$E_\infty$-algebra** in $C$ is synonymous with a [[commutative monoid in a symmetric monoidal (infinity,1)-category|commutative monoid]] in $C$.
Most often $C$ is the [[(infinity,1)-category of chain complexes]] over a [[field]] or [[commutative ring]], or the [[stable (infinity,1)-category]] of [[spectra]] (see also [[E-infinity ring]]).

In terms of [[operads]], an $E_\infty$-algebra is an [[algebra over an operad]] for an [[E-∞ operad]].

## Realizations

### In chain complexes

$E_\infty$-algebras in [[chain complex|chain complexes]] are equivalent to those in abelian [[simplicial group|simplicial groups]].

For details on this statement see [[monoidal Dold-Kan correspondence]] and [[operadic Dold-Kan correspondence]].

The [[singular cohomology]] $H^*(X,\mathbb{Z})$ of a topological space
is a graded-commutative algebra over the integers, but the the [[singular cohomology|singular cochain]] complex $C^*(X,\mathbb{Z})$ is not: instead, it is an $E_\infty$-algebra.

A concrete construction of an $E_\infty$-algebra structure
on [[singular cochains]], and, more generally,
[[simplicial cochains]] of a [[simplicial set]]
can be found in [McClure and Smith](#MCO).
Their work gives a precise description of the involved __sequence operad__, which is an $E_\infty$-operad,
as well as its action on [[simplicial cochains]],
which involves a generalization of Steenrod's cup-$i$ products.
Recall that the cup-1 product controls the noncommutativity of
the ordinary [[cup product]]:
$$d(x\cup_1 y)=x\cup y-(-1)^{|x|\cdot|y|}y\cup x.$$
It can be defined using a formula similar to the one used
for the [[cup product]], and the higher operations
also have similar nature.

Remarkably, for [[simply connected]] spaces of [[finite type]] this $E_\infty$-algebra knows everything about the [[weak homotopy type]] of $X$.  In fact a stronger statement holds:

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

Finite type [[nilpotent spaces]] $X$ and $Y$ are [[weak equivalence|weakly homotopy equivalent]] if and only if the $E_\infty$-algebras $C^*(X,\mathbb{Z})$ and $C^*(Y,\mathbb{Z})$ are quasi-isomorphic.

=--

This was proved by [Mandell](#Mandell) in 2003.

### In topological spaces

+-- {: .un_theorem}
###### Theorem (May recognition theorem)

A [[connected space]] of the [[homotopy type]] of a [[CW-complex]] with a non-degenerate basepoint that has the [[homotopy type]] of a $k$-fold [[loop space]] for all $k \in \mathbb{N}$ admits the structure of an $E_\infty$-space.

=--

### In simplicial sets

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

The [[model structure on algebras over an operad]] over [[E-∞ operad]]s in [[Top]] and in [[sSet]] are [[Quillen equivalence|Quillen equivalent]].

=--

This is in [BergerMoerdijk I](#BergerMoerdijkHomotopy), [BergerMoerdijk II](#BergerMoerdijkAlgebras).

### In spectra

An $E_\infty$-algebra in [[spectra]] is an [[E-∞ ring]].

### In $\infty$-stacks

See [[Ek-Algebras]].

### In $(\infty,n)$-categories

See _[[symmetric monoidal (∞,n)-category]]_.

## Related concepts

* [[A-∞ algebra]]

* [[Hopf E-∞ algebra]]

## References

* [[Peter May]], _The geometry of iterated loop spaces_ ([pdf](http://www.math.uchicago.edu/~may/BOOKS/gils.pdf))
{#May}

* {#MCO} [[James E. McClure]], [[Jeffrey H. Smith]], _Multivariable cochain operations and little $n$-cubes__,[arXiv](https://arxiv.org/abs/math/0106024), [Journal of the AMS 16:3 (2003), 681–704](https://doi.org/10.1090/S0894-0347-03-00419-3).

* [[Mike Mandell]], _Cochains and homotopy type_, Publ. Math. IHES (2006) 103: 213–246.  ([arXiv](https://arxiv.org/abs/math/0311016))
{#Mandell}

* [[Martin Markl]], Steve Shnider, [[Jim Stasheff]], _Operads in algebra, topology and physics_, Math. Surveys and Monographs __96__, Amer. Math. Soc. 2002.
{#MarklShniderStasheff}

In the context of [[(infinity,1)-operad]]s $E_\infty$-algebras are discussed in

* [[Jacob Lurie]], _[[Ek-Algebras]]_
{#Lurie}

A systematic study of [[model category]] structures on operads and their algebras is in

* [[Clemens Berger]], [[Ieke Moerdijk]], _Axiomatic homotopy theory for operads_ Comment. Math. Helv. 78 (2003), 805–831. ([arXiv:math/0206094](http://arxiv.org/abs/math/0206094))
{#BergerMoerdijkHomotopy}

The induced model structures and their properties on [[algebras over operads]] are discussed in

* [[Clemens Berger]], [[Ieke Moerdijk]], _Resolution of coloured operads and rectification of homotopy algebras_ ([arXiv:math/0512576](http://arxiv.org/abs/math/0512576))
{#BergerMoerdijkAlgebras}

[[!redirects E-infinity-algebra]]
[[!redirects E-infinity algebras]]
[[!redirects E-infinity-algebras]]

[[!redirects E-∞ algebra]]
[[!redirects E-∞ algebras]]

[[!redirects E-∞-algebra]]
[[!redirects E-∞-algebras]]


E-infinity monoid in a symmetric monoidal model category

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

#Contents#
* table of contents
{:toc}

## Idea

An _$E_\infty$-monoid_ (or _E-infinity monoid object_) in a [[symmetric monoidal model category]] $C$ is an [[algebra over an operad]] over the [[E-infinity operad]].
Assuming that $C$ is [[cofibrantly generated model category|cofibrantly generated]], there is a model structure on $E_\infty$-monoids, given by the [[model structure on algebras over an operad]] over the [[E-infinity operad]] (which is [[cofibrant]]).
This [[model category]] presents the [[(infinity,1)-category]] of [[commutative monoids in a symmetric monoidal (infinity,1)-category]] (in the [[symmetric monoidal (infinity,1)-category]] presented by $C$).

## Rectification

In some [[symmetric monoidal model categories]], $E_\infty$-monoids can be rectified to (strictly) [[commutative monoids in a symmetric monoidal model category]].
See there for more.

## Related concepts

* [[model structure on commutative monoids in a symmetric monoidal model category]]

* [[model structure on monoids in a monoidal model category]]

* [[model structure on algebras over an operad]]

* [[model structure on algebras over a monad]]

* [[model structure on simplicial T-algebras]]

## References

* [[Jacob Lurie]], _[[Higher Algebra]]_.

[[!redirects E-infinity monoids in a symmetric monoidal model category]]

[[!redirects model structure on E-infinity monoids in a symmetric monoidal model category]]
[[!redirects model category of E-infinity monoids in a symmetric monoidal model category]]

[[!redirects E-infinity monoid in a model category]]
[[!redirects E-infinity monoids in a model category]]


E-infinity operad

[[!redirects E-∞ operad]]

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

#Contents#
* table of contents
{:toc}

## Idea

An **$E_\infty$-operad** is a [[topological operad]] that is a [[homotopy theory|homotopy theoretic]] [[resolution]] of [[Comm]], the operad for [[commutative monoid]]s: an [[algebra over an operad]] over an $E_\infty$-operad is an [[E-∞ ring|E-∞ algebra]].

## Definition

The definition of $E_\infty$-operads depends a bit on which [[presentable (∞,1)-category|presentation]] of the [[(∞,1)-category]] of [[(∞,1)-operad]]s one uses:

* abstractly, in the [[(∞,1)-category]] of [[(∞,1)-operad]]s, the operad [[Comm]] itself already _is_ an $E_\infty$-operad, in that its [[∞-algebras over an (∞,1)-operad]] are [[E-∞ ring|E-∞ algebras]];

* when presenting $(\infty,1)-Operad$ by the [[model structure on operads]] for [[topological operad]]s, forming the homotopy-algebras over any operad means forming the ordinary [[algebras over an operad]] for any of its cofibrant [[resolution]]s. Therefore one say: an $E_\infty$-operad is (any) cofibrant resolution of [[Comm]] in the standard [[model structure on operads]] over the [[model structure on topological spaces]].

## Properties

For every $E_\infty$-operad $P$, all the spaces $P_n$ are [[contractible]].

In fact,  _every_ [[topological operad]] $P$ for which $P_n \simeq *$ for all $n \in \mathbb{N}$ is weakly equivalent to [[Comm]]: because $Comm_n = *$ there is a unique morphism of operads (necessarily respecting the action of the [[symmetric group]])

$$
  P \to Comm
$$

and for each $n$ this is by assumption a [[weak homotopy equivalence]]

$$
  P_n \to Comm_n = *
$$

of topological spaces.

The only extra condition on an operad $P$ with contractible operation spaces to be $E_\infty$ is that it is in addition _cofibrant_ . This imposes the condition that the [[action]] of the [[symmetric group]] $\Sigma_n \times P_n \to P_n$ in each degree is _free_ .

## Examples

* In some sense the [[universal construction|universal]] model for an $E_\infty$-operad is the [[Barratt-Eccles operad]].

* The [[little k-cubes operad]] for $k \to \infty$ is $E_\infty$.

## Related concepts

* [[A-∞ operad]]

* [[E-∞ ring]]

[[!redirects E-infinity operad]]
[[!redirects E-∞ operad]]
[[!redirects E-infinity-operad]]
[[!redirects E-∞-operad]]


E-infinity space

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

#Contents#
* table of contents
{:toc}

## Definition

An _$E_\infty$-space_ is a [[commutative ∞-monoid]] in [[Top]].

## Properties

Grouplike $E_\infty$-spaces are equivalently [[infinite loop spaces]]. See there for more details.

## Related concepts

* [[I-space]]

* [[connective spectrum]]

* [[model structure for dendroidal left fibrations]]

* [[∞-group completion]]

* [[very commutative object]]

[[!include k-monoidal table]]

## References

* [[Peter May]] with contributions by [[Frank Quinn]], [[Nigel Ray]] and [[Jørgen Tornehave]], _$E_\infty$-Ring spaces and $E_\infty$ ring spectra_, Lecture Notes in Mathematics 577, Springer 1977 ([pdf](http://www.math.uchicago.edu/~may/BOOKS/e_infty.pdf), [cds:1690879](https://cds.cern.ch/record/1690879))

[[!redirects E-∞ space]]

[[!redirects E-∞ spaces]]
[[!redirects E-infinity spaces]]


E-infinity-ring

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### Stable Homotopy theory
+--{: .hide}
[[!include stable homotopy theory - contents]]
=--
#### Higher linear algebra
+-- {: .hide}
[[!include homotopy - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

An __$E_\infty$-ring__ is a [[commutative monoid in an (∞,1)-category|commutative monoid]] in the [[stable (∞,1)-category of spectra]], an [[E-∞ algebra]] in [[spectra]]. This is (up to equivalence) also called a _[[highly structured ring spectrum]]_.

This means that an $E_\infty$-ring is an [[A-∞ ring]] that is commutative up to [[coherence|coherent]] [[higher homotopies]]. $E_\infty$-rings are the analogue in [[higher algebra]] of the [[commutative rings]] in ordinary [[algebra]].

In terms of [[model categories]], and $E_\infty$-rings may be modeled as ordinary [[commutative monoids]] with respect to the [[symmetric monoidal smash product of spectra]] -- [[highly structured ring spectra]], a fact sometimes referred to as "[[brave new algebra]]".
For details see _[[Introduction to Stable homotopy theory]]_, _[[Introduction to Stable homotopy theory -- 1-2|Part 1-2 -- Structured spectra]]_.

## Properties

### Postnikov tower

The [[Postnikov tower in an (∞,1)-category|Postnikov tower]] of a [[connective E-infinity-ring]] is a sequence of [[square-zero extensions]] ([Lurie, section 8.4](#Lurie)).

## Examples

* The [[sphere spectrum]] $\mathbb{S}$ becomes an $E_\infty$-ring via the canonical maps $S^{n_1} \wedge S^{n_2} \stackrel{\simeq}{\longrightarrow} S^{n_1 + n_2}$. As such the sphere spectrum is [[generalized the|the]] [[initial object]] in $E_\infty$-rings.

* Given any [[∞-group]], there is its [[∞-group ∞-ring]].

## Related concepts

* [[commutative ring]], [[symmetric ring groupoid]]

* [[ring spectrum]], [[functor with smash products]]

* [[symmetric ring spectrum]]

* [[model structure for ring spectra]]

* [[Hopf E-infinity ring]]

* [[∞-group of units]]

* [[logarithmic cohomology operation]]

* [[étale morphism of E-∞ rings]]

* [[higher algebra]]

* [[E-∞ arithmetic geometry]]

[[!include k-monoidal table]]

## References

The theory is laid out in

* [[Peter May]] with contributions by [[Frank Quinn]], [[Nigel Ray]] and [[Jørgen Tornehave]], _$E_\infty$-Ring spaces and $E_\infty$ ring spectra_, Lecture Notes in Mathematics 577, Springer 1977 ([pdf](http://www.math.uchicago.edu/~may/BOOKS/e_infty.pdf), [cds:1690879](https://cds.cern.ch/record/1690879))

* {#Lurie} [[Jacob Lurie]], _[[Higher Algebra]]_

For survey see also

* [[Moritz Groth]], _A short course on infinity-categories_, [pdf](http://www.math.ru.nl/~mgroth/preprints/groth_scinfinity.pdf).

Discussion of a  [[Blakers-Massey theorem]] for $E_\infty$-rings is in

* Michael Ching, [[John Harper]], _Higher homotopy excision and Blakers-Massey theorems for structured ring spectra_ ([arXiv:1402.4775](http://arxiv.org/abs/1402.4775))

In [[K(n)-local stable homotopy theory]]:

* [[Michael Hopkins]], _$K(1)$-local $E_\infty$-Ring spectra_ ([pdf](http://www.math.rochester.edu/people/faculty/doug/otherpapers/knlocal.pdf))

See also

* [[Benoit Fresse]], _The Bar Complex of an E-infinity algebra_ ([pdf](http://math.univ-lille1.fr/~fresse/Bar-StructureUniqueness.pdf))

* [[Birgit Richter]], [[Brooke Shipley]], _An algebraic model for commutative HZ-algebras_, [arXiv:1411.7238](http://arxiv.org/abs/1411.7238).

[[!redirects E-∞ ring]]
[[!redirects E-infinity ring]]
[[!redirects E-∞-ring]]
[[!redirects E-infinity-rings]]
[[!redirects E-∞ rings]]
[[!redirects E-infinity rings]]
[[!redirects E-∞-rings]]

[[!redirects E-∞ ring spectrum]]
[[!redirects E-∞ ring spectra]]

[[!redirects E-infinity ring spectrum]]
[[!redirects E-infinity ring spectra]]


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#### Algebraic topology
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#### Cobordism theory
+--{: .hide}
[[!include cobordism theory -- contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The [[e-invariant]] is, a priori, a [[homotopy invariant]] of [[morphisms]] in the [[stable homotopy category]], hence in particular of the [[stable homotopy groups of spheres]]. Under the identification of the latter, via the [[Pontrjagin-Thom isomorphism]], with the [[framed bordism ring]], the [[e-invariant]] turns out to equal the [[Todd class]] of any [[coboundary|cobounding]] [[(U,fr)-manifold]] ([Conner-Floyd 66](#ConnerFloyd66)).

## Preliminaries

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

In generalization to how the [[U-bordism ring]] $\Omega^U_{2k}$ is [[Brown representability theorem|represented]] by [[homotopy classes]] of [[maps]] into the [[Thom spectrum]] [[MU]], so the [[(U,fr)-bordism ring]] $\Omega^{U,fr}_{2k}$ is represented by maps into the [[quotient spaces]] $MU_{2k}/S^{2k}$ (for $S^{2k} = Th(\mathbb{C}^{k}) \to Th( \mathbb{C}^k \times_{U(k)} E U(k) ) = MU_{2k}$ the canonical inclusion):

\[
  \label{InTermsOfHomotopyGroupsOfQuotientedThomSpace}
  \Omega^{(U,fr)}_\bullet
  \;=\;
  \pi_{\bullet + 2k}
  \big(
    MU_{2k}/S^{2k}
  \big)
  \,,
  \;\;\;\;\;
  \text{for any}
  \;
  2k \geq \bullet + 2
  \,.
\]

=--

([Conner-Floyd 66, p. 97](#ConnerFloyd66))

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

The [[bordism rings]] for [[MU]], [[MUFr]] and [[MFr]] sit in a [[short exact sequence]] of the form

\[
  \label{ShortExactSequenceOfUFrBordismRings}
  0
  \to
  \Omega^U_{\bullet+1}
  \overset{i}{\longrightarrow}
  \Omega^{U,fr}_{\bullet+1}
  \overset{\partial}{
    \longrightarrow
  }
  \Omega^{fr}_\bullet
  \to
  0
  \,,
\]

where $i$ is the evident inclusion, while $\partial$ is restriction to the [[boundary]].

(By [this Prop.](MUFr#AShortExactSequenceOfUFrBordismRings) at _[[MUFr]]_.)

In particular, this means that $\partial$ is [[surjective function|surjective]], hence that every $Fr$-manifold is the boundary of a [[(U,fr)-manifold]].

=--

## Statement

+-- {: .num_prop #EInvariantIsToddClassOnCoboundingUFrManifold}
###### Proposition
**([[Todd class]] on [[coboundary|cobounding]] [[(U,fr)-manifolds]] computes [[e-invariant]])**

Evaluation of the [[Todd class]] on [[(U,fr)-manifolds]] yields [[rational numbers]] which are [[integers]] on actual $U$-manifolds. It follows with the [[short exact sequence]] (eq:ShortExactSequenceOfUFrBordismRings) that assigning to $Fr$-manifolds the Todd class of any of their cobounding $(U,fr)$-manifolds yields a well-defined element in [[Q/Z]].

Under the [[Pontrjagin-Thom isomorphism]] between the [[framed bordism ring]] and the [[stable homotopy group of spheres]] $\pi^s_\bullet$, this assignment coincides with the [[Adams e-invariant]] in [its Q/Z-incarnation](Adams+e-invariant#TheEInvariantAsAnElementOfQModZ):

\[
  \label{ToddClassesOnShortExactSequenceOfUFrBordismRings}
  \array{
  0
  \to
  &
  \Omega^U_{\bullet+1}
  &
  \overset{i}{\longrightarrow}
  &
  \Omega^{U,fr}_{\bullet+1}
  &
  \overset{\partial}{
    \longrightarrow
  }
  &
  \Omega^{fr}_\bullet
  &
  \simeq
  &
  \pi^s_\bullet
  \\
  &
  \big\downarrow{}^{\mathrlap{Td}}
  &&
  \big\downarrow{}^{\mathrlap{Td}}
  &&
  \big\downarrow{}^{}
  &&
  \big\downarrow{}^{e_{\mathbb{C}}}
  \\
  0
  \to
  &
  \mathbb{Z}
  &\overset{\;\;\;\;\;}{\hookrightarrow}&
  \mathbb{Q}
  &\overset{\;\;\;\;}{\longrightarrow}&
  \mathbb{Q}/\mathbb{Z}
  &=&
  \mathbb{Q}/\mathbb{Z}
  }
  \,,
\]

=--

([Conner-Floyd 66, Theorem 16.2](#ConnerFloyd66))

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

The first step in the proof of (eq:ToddClassesOnShortExactSequenceOfUFrBordismRings) is the observation ([Conner-Floyd 66, p. 100-101](#ConnerFloyd66)) that the representing map (eq:InTermsOfHomotopyGroupsOfQuotientedThomSpace) for a [[(U,fr)-manifold]] $M^{2k}$ cobounding a $Fr$-manifold represented by a map $f$ is given by the following [[homotopy coherent diagram|homotopy]] [[pasting diagram]] (see also at _[[Hopf invariant]] -- [In generalized cohomology](Hopf+invariant#InGeneralizedCohomology)_):

\begin{imagefromfile}
        "file_name": "CoboundingUFrManifoldDiagrammatically.jpg",
        "web": "nlab",
        "width": 470,
        "unit": "px",
        "margin": {
            "top": -20,
            "right": 0,
            "bottom": 20,
            "left": 20,
            "unit": "px"
        },
        "alt": "homotopy pasting diagram exhibiting cobounding UFr-manifolds",
        "caption": "from [SS21](https://ncatlab.org/schreiber/show/Equivariant+Cohomotopy+and+Oriented+Cohomology+Theory)"
\end{imagefromfile}

From this, Conner-Floyd conclude essentially  by considering the following [[homotopy coherent diagram|homotopy]] [[pasting diagram]] (the diagrammatic perspective here follows [[schreiber:M-Theory as Mf-Theory|SS21]], for more see at _[Adams e-invariant via unit cofiber theories](Adams+e-invariant#ConstructionViaUnitCofiberTheories)_):

\begin{xymatrix@C=15pt}
   S^{2(n + d)-1}
   \ar[dd]_-{ c }
   \ar[rr]
   \ar@{}[ddrr]|-{
     \rotatebox[origin=c]{-45}{\color{orange}$\big\Downarrow$}
     \mbox{\tiny (po)}
   }
   &&
   \ast
   \ar[dr]
   \ar[dd]
   \\
   && &
   \ast
   \ar[dd]|-{
     \mathclap{\phantom{\vert^{\vert}}}
     0
     \mathclap{\phantom{\vert_{\vert}}}
   }
   \\
   S^{2n}
   \ar[dd]
   \ar[rr]
   \ar@{}[ddrr]|-{
     \rotatebox[origin=c]{-45}{\color{orange}$\big\Downarrow$}
     \mbox{\tiny (po)}
   }
   \ar@/_.7pc/[drrr]
     |<<<<<{ \;\Sigma^{2n} (1^{M\mathrm{U}})\;\; }
     |>>>>>>>>>{ {\phantom{AA}} \atop {\phantom{AA}} }
   &&
   C_c
   \ar[dd]
   \ar@{-->}[dr]^-{
   }
   %\ar@/_2pc/[dddrrr]
   %  |>>>>>>>>>>>>>>>{
   %    \mathclap{\phantom{\vert}}
   %    \;{\color{magenta} e(f)}\;
   %  }
   &&
   \\
   && &
   M\mathrm{U}^{2n}
   \ar[dd]
   \ar[dr]|-{
     \;\sigma\;
   }
   \ar@/^1.7pc/[ddrr]|>>>>>>>>>>>{
     \;
     \mathclap{\phantom{\vert^{\vert}}}
     \mathrm{Td}
     \mathclap{\phantom{\vert_{\vert}}}
     \;
   }
   \\
   \ast
   \ar@/_.7pc/[drrr]|-{ \;0\; }
   \ar[rr]
   &&
   S^{2(n+d)}
   \ar@{-->}[dr]|-{
     \;
     \mathclap{\phantom{\vert^{\vert}}}
     \color{magenta}M^{2d}
     \mathclap{\phantom{\vert_{\vert}}}
   }
   &&
   \mathrm{KU}^{2n}
   \ar[dr]|-{
     \;
     \mathrm{ch}
     \;
     \mathclap{\phantom{\vert_{\vert}}}
   }
   \ar[dd]|>>>>>>{ \phantom{\vert_{\vert}} }
   \\
   && &
   (M\mathrm{U}/\mathbb{S})^{2n}
   \ar@/^1.7pc/[ddrr]
     |>>>>>>>>>{
       \mathclap{\phantom{\vert^{\vert}}}
       \;\mathrm{Td}/\mathbb{S}\;
       \mathclap{\phantom{\vert_{\vert}}}
     }
   \ar[dr]|-{ \;\sigma/\mathbb{S}\; }
   &&
   (H^{\mathrm{ev}}\mathbb{Q})^{2n}
   \ar[dd]
   \\
   && &&
   (\mathrm{KU}/\mathbb{S})^{2n}
   \ar[dr]|-{ \;\mathrm{ch}/\mathbb{S}\; }
   \\
   && && &
   \big(
     (H^{\mathrm{ev}}\mathbb{Q})/\mathbb{S}
   \big)^{2n}
\end{xymatrix}

Here for the identification of the [[Todd class]] $Td$ in the bottom of the diagram we used that [[the rational Todd class is the Chern character of the Thom class]].

That this Todd class equals the [[Q/Z]]-valued [[e-invariant]] follows now since the latter is the top-degree component of the [[Chern character]] of any [[complex topological K-theory]]-class $V_{2n} \coloneqq \sigma \circ c$ lifting $\Sigma^{2n} 1 \in \widetilde K(S^{2n})$ to $C_f$ ([this Prop.](Adams+e-invariant#QModZValuedEInvariantIsTopDegreeCoefficientOfChernCharacterOnCofiberSpace)).

=--

## Variants
 {#Variants}

An analogous but finer construction works for [[special unitary group]]-structure instead of [[unitary group]]-structure and in dimensions $8\bullet + 4$:

Since on $(8 \bullet + 4)$-dimensional $SU$-manifolds the [[Todd class]] is divisible by 2 ([Conner-Floyd 66, Prop. 16.4](#ConnerFloyd66)) we have ([Conner-Floyd 66, p. 104](#ConnerFloyd66)) the following [[short exact sequence]] of [[MSUFr]]-[[bordism rings]], in variation of (eq:ToddClassesOnShortExactSequenceOfUFrBordismRings):

\[
  \label{HalfToddClassesOnShortExactSequenceOfSUFrBordismRings}
  \array{
  0
  \to
  &
  \Omega^{SU}_{8\bullet+4}
  &
  \overset{i}{\longrightarrow}
  &
  \Omega^{SU,fr}_{8\bullet+4}
  &
  \overset{\partial}{
    \longrightarrow
  }
  &
  \Omega^{fr}_{8\bullet + 3}
  &
  \simeq
  &
  \pi^s_{8\bullet + 3}
  \\
  &
  \big\downarrow{}^{\tfrac{1}{2}\mathrlap{Td}}
  &&
  \big\downarrow{}^{\tfrac{1}{2}\mathrlap{Td}}
  &&
  \big\downarrow{}^{}
  &&
  \big\downarrow{}^{e_{\mathbb{R}}}
  \\
  0
  \to
  &
  \mathbb{Z}
  &\overset{\;\;\;\;\;}{\hookrightarrow}&
  \mathbb{Q}
  &\overset{\;\;\;\;}{\longrightarrow}&
  \mathbb{Q}/\mathbb{Z}
  &=&
  \mathbb{Q}/\mathbb{Z}
  }
  \,.
\]

This produces $e_{\mathbb{R}}$, the [[Adams e-invariant]] with respect to [[KO]]-theory instead of [[KU]] ([Adams 66, p. 39](e-invariant#Adams66)), which, in degrees $8k + 3$, is indeed half of the e-invariant $e_{\mathbb{C}}$ for $KU$ (by [Adams 66, Prop. 7.14](e-invariant#Adams66)).

In fact, for $k = 0$ we have:

+-- {: .num_prop #AdamseRInvariantDetectsThirdStableHomotopyGroupOfSpheres}
###### Proposition
**([Adams 66, Example 7.17  and p. 46](e-invariant#Adams66))**

In degree 3, the [[KO]]-theoretic [[e-invariant]] $e_{\mathbb{R}}$ takes the value $\left[\tfrac{1}{24}\right] \in \mathbb{Q}/\mathbb{Z}$ on the [[quaternionic Hopf fibration]] $S^7 \overset{h_{\mathbb{H}}}{\longrightarrow} S^4$  and hence reflects the full [[third stable homotopy group of spheres]]:

$$
  \array{
    \pi^s_3
      &
      \underoverset{
        \simeq
      }{
        e_{\mathbb{R}}
      }{
        \;\;\longrightarrow\;\;
      }
      &
    \mathbb{Z}/24
    &
    \subset
    &
    \mathbb{Q}/\mathbb{Z}
    \\
    [h_{\mathbb{H}}]
    &&\mapsto&&
    \left[\tfrac{1}{24}\right]
  }
$$

while $e_{\mathbb{C}}$ sees only "half" of it (by [Adams 66, Prop. 7.14](e-invariant#Adams66)).

=--

## Related statements

* [[rational Todd class is Chern character of Thom class]]

## References

* {#ConnerFloyd66} [[Pierre Conner]], [[Edwin Floyd]], Theorem 16.2 in: _[[The Relation of Cobordism to K-Theories]]_, Lecture Notes in Mathematics __28__ Springer 1966 ([doi:10.1007/BFb0071091](https://link.springer.com/book/10.1007/BFb0071091), [MR216511](http://www.ams.org/mathscinet-getitem?mr=216511))

An alternative formulation via [[index theory]]:

* [[Michael Atiyah]], [[Vijay Patodi]], [[Isadore Singer]], Theorem 4.14 of: _Spectral asymmetry and Riemannian geometry. II_,  Mathematical Proceedings of the Cambridge Philosophical Society , Volume 78, Issue 3, 1975, pp. 405-432 ([doi:10.1017/S0305004100051872](https://doi.org/10.1017/S0305004100051872))

Review in:

* [[Ulrich Bunke]], [[Niko Naumann]], Section 2 of: _The f-invariant and index theory_, Manuscripta Math. 132, 365–397 (2010) ([arXiv:0808.0257](https://arxiv.org/abs/0808.0257), [https://doi.org/10.1007/s00229-010-0351-7](https://doi.org/10.1007/s00229-010-0351-7))

[[!redirects e-invariant is Todd class of cobounding (U,fr)-manifolds]]
[[!redirects e-invariant is the Todd class of cobounding (U,fr)-manifolds]]


E-k algebra > history

see [[E-k algebra]], sorry.


E-string

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

#Contents#
* table of contents
{:toc}

## Idea

In [[string theory]], specifically in [[type IIA string theory]], the [[brane intersection]] of a [[D2-brane]] with an [[O8-plane]] is called an _E-string_.

>  graphics grabbed from [KKLPV 14, p. 4](#KKLPV14)

Hence under the [[S-duality]] [[duality between type I and heterotic string theory|between type I and heterotic string theory]] the E-string is [[duality in string theory|dual]] to the [[heterotic string]].

The [[M-theory]] lift of the E-string is the [[brane intersection]] of an [[M2-brane]] with an [[M9-brane]] which [[wrapped brane|wraps]] the M-theory circle fiber, hence an $M9_I$-brane of [[Horava-Witten theory]] ([KKLPV 14, p. 4/5](#KKLPV14)), hence a [[D8-brane]] from the [[duality between type IIA string theory and M-theory|dual perspective]] of [[type I' string theory]].

The other end of the [[M2-brane]]-lift of the E-string is [[M2-M5 brane bound state|on]] an [[M5-brane]]:

> graphics grabbed from [GHKKLV 15](#GHKKLV15)

> graphics grabbed from [HLV 14](#HLV14)

In contrast, if both ends of the [[M2-brane]] are on an [[M5-brane]], some authors speak of "[[M-string|M-strings]]":

> graphics grabbed from [HLV 14](#HLV14)

Finally, an [[M2-brane]] stretching between two [[MO9-planes]] gives the [[heterotic string]] at its boundary:

> from [Kashima 00](Horava-Witten+theory#Kashima00)

## Properties

### E-string elliptic genus

See at _[[E-string elliptic genus]]_.

## Related concepts

* [[D=6 N=(1,0) SCFT]]

[[!include KK-compactifications of M-theory -- table]]

\linebreak

[[!include brane bound states -- table]]

## References

### General

E-strings as [[M5-branes]] [[wrapped brane|wrapped]] on a [[half K3]]:

* J. A. Minahan, D. Nemeschansky, [[Nicholas Warner]], _Partition Functions for BPS States of the Non-Critical $E_8$ String_, Adv. Theor. Math. Phys.1:167-183, 1998 ([arXiv:hep-th/9707149](https://arxiv.org/abs/hep-th/9707149))

* J. A. Minahan, D. Nemeschansky, [[Cumrun Vafa]], [[Nicholas Warner]], _E-Strings and $N=4$ Topological Yang-Mills Theories_, Nucl. Phys. B527 (1998) 581-623 ([arXiv:hep-th/9802168](https://arxiv.org/abs/hep-th/9802168))

* [[Amer Iqbal]], _A note on E-strings_, Adv. Theor. Math. Phys. 7 (2003) 1-23 ([arXiv:hep-th/0206064](https://arxiv.org/abs/hep-th/0206064))

E-strings as M2-branes stretched between M5 and MO9:

* {#HLV14} [[Babak Haghighat]], [[Guglielmo Lockhart]], [[Cumrun Vafa]], _Fusing E-string to heterotic string: $E + E \to H$_, Phys. Rev. D **90** 126012 (2014) [[arXiv:1406.0850](https://arxiv.org/abs/1406.0850), [doi:10.1103/PhysRevD.90.126012](https://doi.org/10.1103/PhysRevD.90.126012)]

* {#GHKKLV15} [[Abhijit Gadde]], [[Babak Haghighat]], [[Joonho Kim]], [[Seok Kim]], [[Guglielmo Lockhart]], [[Cumrun Vafa]], _6d String Chains_, JHEP 1802 (2018) 143 ([arXiv:1504.04614](https://arxiv.org/abs/1504.04614))

On their [[Seiberg-Witten curve]]:

* [[Tohru Eguchi]], Kazuhiro Sakai, _Seiberg-Witten Curve for the E-String Theory_, JHEP 0205:058, 2002 ([arXiv:hep-th/0203025](https://arxiv.org/abs/hep-th/0203025))

and on their [[quantum Seiberg-Witten curve]] in relation to [[D6-D8-brane bound states]]:

* [[Jin Chen]], [[Babak Haghighat]], [[Hee-Cheol Kim]], [[Marcus Sperling]], [[Xin Wang]], *E-string Quantum Curve*, Nuclear Physics B **973** (2021) 115602 [[arXiv:2103.16996](https://arxiv.org/abs/2103.16996), [doi:10.1016/j.nuclphysb.2021.115602](https://doi.org/10.1016/j.nuclphysb.2021.115602)]

On the [[E-string elliptic genus]]:

* [[Kenji Mohri]], _Exceptional String: Instanton Expansions and Seiberg-Witten Curve_, Rev. Math. Phys. 14 (2002) 913-975 ([arXiv:hep-th/0110121](https://arxiv.org/abs/hep-th/0110121))

* {#KKLPV14} Joonho Kim, Seok Kim, Kimyeong Lee, [[Jaemo Park]], [[Cumrun Vafa]], _Elliptic Genus of E-strings_, JHEP 1709 (2017) 098 ([arXiv:1411.2324](https://arxiv.org/abs/1411.2324))

* Wenhe Cai, Min-xin Huang, Kaiwen Sun, _On the Elliptic Genus of Three E-strings and Heterotic Strings_, J. High Energ. Phys. 2015, 79 (2015).
([arXiv:1411.2801](https://arxiv.org/abs/1411.2801), doi:10.1007/JHEP01(2015)079)

On the [[Green-Schwarz anomaly cancellation]] in [[D=6 N=(1,0) SCFT]]:

* [[Kantaro Ohmori]], [[Hiroyuki Shimizu]], [[Yuji Tachikawa]], _Anomaly polynomial of E-string theories_, J. High Energ. Phys. 2014, 2 (2014) ([arXiv:1404.3887](https://arxiv.org/abs/1404.3887))

See also:

* Sung-Soo Kim, Yuji Sugimoto, Futoshi Yagi, _Surface defects on E-string from 5-brane webs_, JHEP 12(2020) 183 ([arXiv:2008.06428](https://arxiv.org/abs/2008.06428))

### In F-theory

Discussion in [[F-theory]]:

* Jie Gu, Babak Haghighat, [[Albrecht Klemm]], Kaiwen Sun, Xin Wang, _Elliptic Blowup Equations for 6d SCFTs. III: E--strings, M--strings and Chains_ ([arXiv:1911.11724](https://arxiv.org/abs/1911.11724))

### In heterotic M-theory on ADE-orbifolds

Discussion in [[heterotic M-theory on ADE-orbifolds]], where the [[M5-branes]] parallel to an [[MO9]] sit on an [[MK6]]/[[D6-brane]] on an [[ADE-singularity]]:

* {#GKSTY02} E. Gorbatov, [[Vadim Kaplunovsky]], [[Jacob Sonnenschein]], [[Stefan Theisen]], [[Shimon Yankielowicz]], around (6.13) in _On Heterotic Orbifolds, M Theory and Type I' Brane Engineering_, JHEP 0205:015, 2002 ([arXiv:hep-th/0108135](https://arxiv.org/abs/hep-th/0108135))

### Relation to D$p$-D$(p+2)$-brane bound states

The lift of [[Dp-D(p+2)-brane bound states]] in [[string theory]] to [[M2-M5-brane bound states]]/[[E-strings]] in  [[M-theory]], under [[duality between M-theory and type IIA string theory]]+[[T-duality]], via generalization of [[Nahm's equation]] (eventually motivating the [[BLG model]]/[[ABJM model]]):

* {#BasuHarvey05} [[Anirban Basu]], [[Jeffrey Harvey]], _The M2-M5 Brane System and a Generalized Nahm's Equation_, Nucl.Phys. B713 (2005) 136-150 ([arXiv:hep-th/0412310](https://arxiv.org/abs/hep-th/0412310))

* {#BaggerLambertMukhiPapageorgakis13} [[Jonathan Bagger]], [[Neil Lambert]], [[Sunil Mukhi]], [[Constantinos Papageorgakis]], Section 2.2.1 of _Multiple Membranes in M-theory_, Physics Reports, Volume 527, Issue 1, 1 June 2013, Pages 1-100 ([arXiv:1203.3546](http://arxiv.org/abs/1203.3546), [doi:10.1016/j.physrep.2013.01.006](https://doi.org/10.1016/j.physrep.2013.01.006))

[[!redirects E-strings]]


E-theory

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### Operator algebra
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
#### Functional analysis
+--{: .hide}
[[!include functional analysis - contents]]
=--
#### Noncommutative geometry
+--{: .hide}
[[!include noncommutative geometry - contents]]
=--
=--
=--

# $E$-theory
* table of contents
{: toc}

## Idea

_$E$-Theory_ is the name of a [[category]] whose [[objects]] are [[C*-algebras]] and whose [[hom-sets]] are homotopy classes of slightly generalized $C*$-homomorphisms, called _[[asymptotic C*-homomorphisms]]_. These hom-sets have the structure of an [[abelian group]] and are also called the _E-groups_ of their arguments. Since under [[Gelfand duality]] [[C*-algebras]] may be thought of as exhibiting [[noncommutative topology]], one also speaks of _[[noncommutative stable homotopy theory]]_.

This construction may be understood as the universal improvement of [[KK-theory]] under [[excision]] ([Higson 90](#Higson90)). Accordingly, the $E$-groups behave like groups of a [[K-theory]]-like [[generalized cohomology theory]].

In terms of [[noncommutative topology]] (regarding, in view of [[Gelfand duality]], noncommutative [[C*-algebras]] as [[algebras of functions]] on "noncommutative topological spaces") one may understand this as dealing with "locally badly behaved space" such as certain [[quotients]] of [[foliations]] ([Connes-Higson 90](ConnesHigson90))
in a way that resembles a noncommutative version of [[shape theory]] ([Dādārlat 94](#Dadarlat94)).

## Definition

First some notation and terminology.

For $A \in $ [[C*Alg]], we write

$$
  \Sigma A \coloneqq C_0((0,1),A)
$$

for the $C^\ast$-algebra of continuous $A$-valued functions on the open inverval [[vanishing at infinity]]. This is also called the _suspension_ of $A$.

For $A,B \in $ [[C*Alg]], write $[A,B]$ for the set of [[homotopy]]-[[equivalence classes]] of [[asymptotic C*-homomorphisms]]  $A \to B$. As discussed there

1. there is a natural [[composition operation]] $[A,B] \times [B,C] \to [A,C]$;

1. $[A,\Sigma B]$ is naturally an [[abelian group]].

Finally, write $\mathcal{K} \in $ [[C*Alg]] for the $C^\ast$-algebra of [[compact operators]] on an infinite-dimensional [[separable Hilbert space]]. For $A \in C^\ast Alg$ the [[tensor product of C*-algebras]] $A \otimes \mathcal{K}$ is also called the _stabilization_ of $A$.

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

For $A,B \in $ [[C*Alg]], the **E-group** of $A$ with [[coefficients]] in $B$ is

$$
  E(A,B) \coloneqq [(\Sigma A )\otimes \mathcal{K}, (\Sigma B) \otimes \mathcal{K}]
  \in Ab
  \,.
$$

Under the induced [[composition]] operation this yields an [[additive category]] $E$ whose [[objects]] are [[C*-algebras]], and whose [[hom-objects]] are $E(-,-)$.

=--

## Properties

### Universal characterization

E-theory is the [[universal construction|universal]]
localization [[C*Alg]] $\to E$ which is homotopy-invariant, stable and preserves [[exact sequences]] in the middle.

(...)

### Relation to KK-theory
 {#RelationToKKTheory}

Because KK-theory is the universal _split exact_ (stable and homotopy-invariant) localization of [[C*Alg]], and E-theory the universal half-exact localization, and since every [[split exact sequence]] is in particular exact, there is a universal [[functor]]

$$
  KK \to E
$$

from the [[KK-theory]] [[homotopy category]] to that of $E$-theory.

Restricted to [[nuclear C*-algebras]] this is a [[full and faithful functor]]. ([Higson 90](#Higson90)) (...)

If in the definition of E-theory by [[asymptotic C*-homomorphisms]] one restricts to those which take values in contractive [[completely positive maps]], then the results is isomorphic to KK-theory again. (K. Thomsen, [Introduction, p. 34](#Introduction)). The above universal functor $KK \to E$ is then just the corresponding [[forgetful functor]].

It follows that the Kasparov product in [[KK-theory]] is equivalently given by the composition of the corresponding completely positive [[asymptotic C*-homomorphisms]].

## Related concepts

* [[KK-bootstrap category]]

[[!include noncommutative motives - table]]

## References

### General

The idea of E-theory was introduced in

* [[Alain Connes]], [[Nigel Higson]], _Déformations, morphismes asymptotiques et $K$-théorie bivariante_, C. R. Acad. Sci. Paris Sér. I Math. __311__ (1990), no. 2, 101–106, [MR91m:46114](http://www.ams.org/mathscinet-getitem?mr=1065438), [pdf](ftp://ftp.bnf.fr/578/N5781521_PDF_107_112DM.pdf)
 {#ConnesHigson90}

* [[Nigel Higson]], _Categories of fractions and excision in KK-theory_ J. Pure Appl. Algebra, 65(2):119–138, (1990) ([pdf](http://www.math.psu.edu/higson/math/Papers_files/Higson%20-%201990%20-%20Categories%20of%20fractions%20and%20excision%20in%20KK-theory.pdf))
 {#Higson90}

Reviews and surveys include

* _Introduction to KK-theory and E-theory_, Lecture  notes (Lisbon 2009) ([pdf slides](http://oaa.ist.utl.pt/files/cursos/courseD_Lecture4_KK_and_E1.pdf))
 {#Introduction}

A standard textbook account is in section 25 of

* [[Bruce Blackadar]], _[[K-Theory for Operator Algebras]]_

The [[stable homotopy theory|stable]] [[homotopy theory]] aspects are further discussed in

* Martin Grensing, _Noncommutative stable homotopy theory_ ([arXiv:1302.4751](http://arxiv.org/abs/1302.4751))

* Rasmus Bentmann, _Homotopy-theoretic E-theory and n-order_ ([arXiv:1302.6924](http://arxiv.org/abs/1302.6924))

See also

* web page of a project [Noncommutative topology - homotopy functors and E-theory ](http://www.math.ku.dk/~jg/papers/etheory.html)

* [[Snigdhayan Mahanta]], _Higher nonunital Quillen $K'$-theory, KK-dualities and applications to topological $\mathbb{T}$-duality_  [pdf](http://wwwmath.uni-muenster.de/u/snigdhayan.mahanta/papers/KQ.pdf)
 {#Mahanta}

* [pdf](http://faculty.tcu.edu/epark/papers/ETheory_JFA.pdf)

### Relation to shape theory

Relation to [[shape theory]] is discussed in

* [[Marius Dādārlat]], _Shape theory and asymptotic morphisms for $C^\ast$-algebras_, Duke Math. J. __73__ (3):687-711, 1994, [MR95c:46117](http://www.ams.org/mathscinet-getitem?mr=1262931), [pdf](http://www.math.purdue.edu/~mdd/Publications/shape.pdf)
 {#Dadarlat94}

* Vladimir Manuilov, Klaus Thomsen, _Shape theory and extensions of $C^\ast$-algebras_, ([arxiv/1007.1663](http://arxiv.org/abs/1007.1663))

[[!redirects E-theory]]

[[!redirects asymptotic morphism]]
[[!redirects asymptotic morphisms]]


E-∞ geometry

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

#Contents#
* table of contents
{:toc}

## Idea

What may be called **$E_\infty$-geometry**, or **[[spectral algebraic geometry]]** is the "full" version of [[derived algebraic geometry]] where the spaces are locally modelled on [[E-infinity rings]] (as opposed to [[simplicial commutative rings]] or [[dg-algebras]]).

(In view of existing terms "[[arithmetic geometry]]" and "[[higher geometry]]" one might also tend call this **higher  [[arithmetic geometry]]**, but notice that the term _[[higher arithmetic geometry]]_ already has a different traditional usage. Maybe **$E_\infty$-arithmetic geometry** works well.)

That is to say, it is the [[higher geometry]] in the [[(∞,1)-topos]] over the [[(∞,1)-site]] of [[Isbell duality|formal duals]] of [[E-∞ rings]], equipped with the [[etale topology]].

An [[E-∞ scheme]] ( _spectral scheme_ ) in this sense is a [[structured (∞,1)-topos]] in $E_\infty$-geometry.

## Terminology

The term _spectral algebraic geometry_ is used in the literature.  On the [[nLab]] we use the term _$E_\infty$-geometry_ due to a possible terminology clash discussed at [[spectral geometry]].

## Properties

### Higher analogs of Fermat derivations and Frobenius maps

The refinement to $E_\infty$-arithmetic geometry of the power map $(-)^p$ which gives the [[Fermat quotient]] [[p-derivation]] and the [[Frobenius homomorphism]] in plain [[arithmetic geometry]] is the [[power operations]] on [[multiplicative cohomology theory]] ([Lurie "Adic", remark 2.2.9](#LurieAdic)).

### Cohesion

see at _[[differential cohesion and idelic structure]]_

[[!include arithmetic cohesion -- table]]

## Examples

* [[Spec(S)]]

## References

The basic definitions are in

* {#Lurie} [[Jacob Lurie]], _[[Spectral Schemes]]_

Fundamental properties of $E_\infty$-geometry are discussed in

* [[Jacob Lurie]], _[[Quasi-Coherent Sheaves and Tannaka Duality Theorems]]_

* [[Jacob Lurie]], _[[Closed Immersions]]_

* [[Jacob Lurie]], _[[Descent Theorems]]_

* [[Jacob Lurie]], _[[Proper Morphisms, Completions,
and the Grothendieck Existence Theorem]]_

* [[Jacob Lurie]], _[[Representability Theorems]]_

* {#LurieAdic} [[Jacob Lurie]], _[[Rational and p-adic Homotopy Theory]]_

* [[Jacob Lurie]], _[[Spectral Algebraic Geometry]]_ ([pdf]( http://math.harvard.edu/~lurie/papers/SAG-rootfile.pdf))

See also the overview

* [[Clark Barwick]], _Applications of derived algebraic geometry to homotopy theory_, lecture notes, mini-course in Salamanca, 2009, [pdf](https://dl.dropboxusercontent.com/u/1741495/papers/salamanca.pdf).

[[!redirects E-infinity geometry]]
[[!redirects spectral algebraic geometry]]

[[!redirects E-∞ arithmetic geometry]]
[[!redirects E-infinity arithmetic geometry]]

[[!redirects E? geometry]]


E-∞ scheme

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#Contents#
* table of contents
{:toc}

## Idea

A [[locally representable structured (∞,1)-topos]] in [[E-∞ geometry]].

The refinement of the notion of [[scheme]] from [[algebraic geometry]] to [[E-∞ geometry]].

## Examples

* [[spectral affine line]]

## Related concepts

* [[spectral super-scheme]]

## References

* [[Jacob Lurie]], _[[Spectral Schemes]]_.

[[!redirects E-∞ schemes]]

[[!redirects E-infinity scheme]]
[[!redirects E-infinity schemes]]

[[!redirects spectral scheme]]
[[!redirects spectral schemes]]


E. B. Curtis > history

see _[[Edward Curtis]]_


E. C. George Sudarshan

[[!redirects E. C. G. Sudarshan]]

Ennackal Chandy George Sudarshan

* [Wikipedia entry](https://en.wikipedia.org/wiki/E._C._George_Sudarshan)

## Selected writings

On [[tachyons]]:

* Olexa‐Myron Bilaniuk, V. K. Deshpande, [[E. C. G. Sudarshan]], _"Meta" Relativity_, American Journal of Physics 30, 718 (1962) ([doi:10.1119/1.1941773](https://doi.org/10.1119/1.1941773))

* Olexa‐Myron Bilaniuk, [[E. C. George Sudarshan]], _Particles beyond the light barrier_, Physics Today 22, 5, 43 (1969) ([doi:10.1063/1.3035574](https://doi.org/10.1063/1.3035574))

## Selected publications

* [[E. C. G. Sudarshan]] (ed.), _A Gift of Prophecy -- Essays in Celebration of the Life of [[Robert Eugene Marshak]]_, World Scientific  (1995)  ([doi:10.1142/2580](https://doi.org/10.1142/2580))

## Related $n$Lab entries

* [[tachyon]]

[[!redirects Ennackal Chandy George Sudarshan]]
[[!redirects George Sudarshan]]

category: people