K-theory of a permutative category

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## Idea

To a [[permutative category]]  $C$ is naturally associated a [[Gamma-space]], hence a [[symmetric spectrum]]. The [[generalized (Eilenberg-Steenrod) cohomology]] theory [[Brown representability theorem|represented]] by this is called the
_([[algebraic K-theory|algebraic]]) [[K-theory]]_ of (or represented by) $C$.

If the category is  even a [[bipermutative category]] then the corresponding [[K-theory of a bipermutative category]] in addition has [[E-infinity ring]] structure, hence is a [[multiplicative cohomology theory]].

## Definition
 {#Definition}

### Via topological group completion
 {#ViaTopologicalGroupCompletion}

For $\mathcal{C}$ a [[category]], write ${\vert \mathcal{C}\vert} \in$ [[classical model structure on simplicial sets|sSet]] $\simeq_{Qu}$ [[classical model structure on topological spaces|Top]] for its [[nerve]]/[[geometric realization]]. (Beware that this is often denoted $B C$, instead, but that notation clashes with that for [[delooping]], which we also need in the following.)

For $\mathcal{C}$ a [[permutative category]] its [[nerve]]/[[geometric realization]] $\vert \mathcal{C} \vert$ is naturally a [[topological monoid]] ([Quillen 70](#Quillen70) see e.g. [May 13, theorem 4.10](#May13)), hence admits a [[bar construction]]/[[classifying space]] $B {\vert \mathcal{C}\vert}$. The [[loop space]] of that

$$
  (\mathbb{K}\mathcal{C})_0
  \;\coloneqq\;
  \Omega B {\vert \mathcal{C}\vert}
  \,,
$$

being an [[∞-group]], may be regarded as the [[homotopy theory|homotopy theoretic]] [[group completion]] of the [[topological monoid]] ${\vert \mathcal{C}\vert}$.

This is the degree-0 space in the _algebraic K-theory [[spectrum]] $\mathbb{K}\mathcal{C}$ of the permutative category_  $\mathcal{C}$ (see e.g. [May 13, def 4.11](#May13)).

By ([Dwyer-Kan 80, prop. 3.7, prop. 9.2, remark 9.7](#DwyerKan80)) the operation $\Omega B (-)$ is the [[derived functor]] of [[group completion]], so that this construction ought to be a model for the [[K-theory of a symmetric monoidal (∞,1)-category]].

In particular, under [[decategorification]], its [[group]] of [[connected components]] is the actual [[Grothendieck group]] $K(-)$ of the [[isomorphism classes]] of [[objects]] in $\mathcal{C}$:

$$
  \pi_0
  \left(
    \mathbb{K} \mathcal{C}
  \right)
  \;\simeq\;
  K\left(
    \mathcal{C}/_\sim, \oplus
  \right)
$$

(recalled e.g. in [Bohmann-Osorno 14, p. 14](#BohmannOsorno14)).

In particular for $R$ a [[topological ring]] one considers $C$ a [[skeletal groupoid|skeleton]] of the [[groupoid]] of ([[finitely generated module|finitely generated]]) [[projective modules]] over $R$. Then the K-theory of $C$ is the [[algebraic K-theory]] of $R$ (e.g. [May 13, p. 25](#May13))

$$
  \mathbb{K}\left( R Mod^{fin}_{proj} \right)
  \;\simeq\;
  K R
  \,.
$$

### Via Gamma spaces

Write $FinSet^{*/}$ for the [[category]] of [[pointed objects|pointed]] [[finite sets]].

For $C$ a [[permutative category]], there is naturally a [[functor]]

$$
  \widebar {C}_{(-)}
  \;\colon\;
  FinSet^{*/} \to Cat
$$

$$
  A \mapsto \widebar C_A
$$

such that (...).

([Elmendorf-Mandell, theorem 4.2](#ElmendorfMandell06))

Accordingly, postcomposition with the [[nerve]] $N : Cat \to sSet$ produces from $C$ a [[Gamma-space]] $N \widebar C$. To this corresponds a [[spectrum]]

$$
  K^{Seg} C \coloneqq \{N \widebar C_{S_\bullet^n}\}
  \,.
$$

This is the _K-theory spectrum of $C$_.

([Elmendorf-Mandell, def. 4.3](#ElmendorfMandell06))

## Properties

### Monoidal functoriality

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

The construction of K-theory spectra of permutative categories constitutes a [[multifunctor]]

$$
  \mathbb{K}
  \;\colon\;
  PermCat
    \longrightarrow
  Spectra
$$

between [[multicategories]] of [[permutative categories]] (under [[Deligne tensor product]]) and [[spectra]] (under [[smash product of spectra]]).

Hence for a multilinear functor of permutative categories

$$
  F
    \;\colon\;
  \mathcal{A}_1 \times \cdots \times \mathcal{A}_n
  \longrightarrow
  \mathcal{B}
$$

there is a compatibly induced morphism of K-[[spectra]] out of the [[smash product of spectra|smash product]]

$$
  \mathbb{K}(\mathcal{A}_1)
    \wedge
    \cdots
    \wedge
  \mathbb{K}(\mathcal{A}_n)
  \longrightarrow
  \mathbb{K}(\mathcal{B})
  \,.
$$

This implies that the construction further extends to a [[2-functor]] from the [[2-category]] $PermCat Cat$ of [[enriched categories]] over the [[multicategory]] of [[permutative categories]] to that of [[enriched categories]] of [[spectra]]:

$$
  \mathbb{K}_\bullet
  \;\colon\;
  PermCat Cat \longrightarrow Spectra Cat
$$

which applies $\mathbb{K}$ to each [[hom-object]].

=--

([May 80, theorem 1.6, Theorem 2.1](#May80), [Elmendorf-Mandell 04, theorem 1.1](#ElmendorfMandell04), [Guillou 10, Theorem 1.1](#Guillou10)

## Examples
 {#Examples}

+-- {: .num_example #OrdinaryAlgebraicKTheoryFromPermutativeCategoryOfProjectiveModules}
###### Example
**(ordinary [[algebraic K-theory]])**

For $R$ a [[commutative ring]], let $\mathcal{C} = R Mod_{pr}$ its [[category of modules|category of]] [[finitely generated module|finitely generated]] [[projective modules]] regarded as a [[permutative category]]. Then

$$
  K (R Mod^{pr}_{fin}) \;\simeq\; K R
$$

is the classical [[algebraic K-theory]] spectrum of the ring $R$.

=--

(e.g. [Elmendorf-Mandell 04, p. 10](#ElmendorfMandell04)).

+-- {: .num_example #StableCohomotopyIsKTheoryOfFinSet}
###### Example
**([[stable cohomotopy]] is K-theory of [[FinSet]])**

Let $\mathcal{C} = $ [[FinSet]] be a [[skeleton]] of the category of [[finite sets]], regarded as a [[permutative category]]. Then

$$
  K(FinSet) \;\simeq\; \mathbb{S}
$$

is the [[sphere spectrum]], hence represents the [[cohomology theory]] called _[[stable cohomotopy]]_.

=--

(due to [Segal 74, Prop. 3.5](#Segal74), see also [Priddy 73](#Priddy73))

+-- {: .num_remark #StableCohomotopyIsAlgebraicKTheoryOverFieldWithOneElement}
###### Remark
**([[stable cohomotopy]] as [[algebraic K-theory]] over the [[field with one element]])**

Since ([[pointed sets|pointed]]) [[finite sets]] may be regarded as the modules over the "[[field with one element]]" $\mathbb{F}_1$ (see [there](field+with+one+element#Modules)),

$$
  \mathbb{F}_1 Mod
  \;=\;
  FinSet^{\ast/}
$$

one may read example \ref{StableCohomotopyIsKTheoryOfFinSet} in view of example \ref{OrdinaryAlgebraicKTheoryFromPermutativeCategoryOfProjectiveModules} as saying that [[stable cohomotopy]] is the algebraic K-theory of the [[field with one element]]:

$$
  K \mathbb{F}_1 \;=\; \mathbb{S}
  \,.
$$

This perspective is highlighted for instance in ([Deitmar 06, p. 2](#Deitmar06), [Guillot 06](#Guillot06)).

=--

## Related concepts

* [[K-theory of a bipermutative category]]

* [[algebraic K-theory of a symmetric monoidal (∞,1)-category]]

## References

* {#Quillen70} [[Daniel Quillen]], _Cohomology of groups_, Proceedings of the international congress of mathematics 1970

* {#Quillen} [[Daniel Quillen]], _On the group completion of a simplicial monoid_

* {#Priddy73} [[Stewart Priddy]], _Transfer, symmetric groups, and stable homotopy theory_, in _Higher K-Theories_, Springer, Berlin, Heidelberg, 1973. 244-255 ([pdf](https://link.springer.com/content/pdf/10.1007/BFb0067060.pdf))

* {#Segal74} [[Graeme Segal]], _Catgeories and cohomology theories_, Topology vol 13 (1974)  ([pdf](http://ncatlab.org/nlab/files/SegalCategoriesAndCohomologyTheories.pdf))

* [[Peter May]], _The spectra associated to permutative categories_, Topology 17 (1978) ([pdf](http://www.math.uchicago.edu/~may/PAPERS/23.pdf))

* {#May13} [[Peter May]], _$E_\infty$-Spaces, group completions, and permutative categories_, in [[Graeme Segal]] (ed.) _New Developments in Topology_, Cambirdge University Press 2013 ([pdf](http://www.math.uchicago.edu/~may/PAPERS/13.pdf), [doi:10.1017/CBO9780511662607.008](https://doi.org/10.1017/CBO9780511662607.008))

* {#May80} [[Peter May]], _Pairings of categories and spectra_, J. Pure Appl. Algebra, 19:299–346, 1980

* {#DwyerKan80} [[William Dwyer]], [[Daniel Kan]], _Simplicial localization of categories_, Journal of pure and applied algebra 17 (1980) 267-284

* [[Anthony Elmendorf]], [[Michael Mandell]], _Permutative categories as a model of connective stable homotopy_, in: [[Birgit Richter]] (ed.) _Structured Ring spectra_, Cambridge University Press (2004)

* {#ElmendorfMandell04} [[Anthony Elmendorf]], [[Michael Mandell]], _Rings, modules and algebras in infinite loop space theory_, Adv. in Math. 205 (2006), no. 1, 163-228 ([arXiv:math/0403403](https://arxiv.org/abs/math/0403403))

* {#ElmendorfMandell09} [[Anthony Elmendorf]], [[Michael Mandell]], _Permutative categories, multicategories, and algebraic K-theory_, Algebraic & Geometric Topology 9 (2009) 2391-2441 ([arXiv:0710.0082](http://arxiv.org/abs/0710.0082), [euclid:1513797088](https://projecteuclid.org/euclid.agt/1513797088))

* {#Guillou10} [[Bertrand Guillou]], _Strictification of categories weakly enriched in symmetric monoidal categories_, Theory Appl. Categ., 24:No. 20, 564–579, 2010.

The interpretation of [[stable cohomotopy]] as the algebraic K-theory over the [[field with one element]] is adopted in

* {#Deitmar06} [[Anton Deitmar]], _Remarks on zeta functions and K-theory over $\mathbb{F}_1$_ ([arXiv:math/0605429](https://arxiv.org/abs/math/0605429))

* {#Guillot06} [[Pierre Guillot]], _Adams operations in cohomotopy_ ([arXiv:0612327](https://arxiv.org/abs/math/0612327))

The generalization of K-theory of permutative categories to [[Mackey functors]] is discussed in

* {#BohmannOsorno14} [[Anna Marie Bohmann]], [[Angélica Osorno]], _Constructing equivariant spectra via categorical Mackey functors_, Algebraic & Geometric Topology 15.1 (2015): 537-563 ([arXiv:1405.6126](http://arxiv.org/abs/1405.6126))

Generalization to [[equivariant stable homotopy theory]] and [[G-spectra]] is discussed in

* {#GuillouMay12} [[Bert Guillou]], [[Peter May]], around section 4.4 of  _Permutative $G$-categories in equivariant infinite loop space theory_ ([arXiv:1207.3459](http://arxiv.org/abs/1207.3459))

[[!redirects K-theory of permutative categories]]

[[!redirects algebraic K-theory of a permutative category]]
[[!redirects algebraic K-theory of permutative categories]]

[[!redirects K-theory spectrum of a permutative category]]
[[!redirects K-theory spectra of permutative categories]]


K-theory of a stable (∞,1)-category

## Definition

There is a [[Waldhausen S-construction]] $w_\bullet S_\bullet C$ for [[stable (infinity,1)-categories]].
One defines the [[algebraic K-theory]] of $C$ as
  $$ K(C) = \Omega | w_\bullet S_\bullet C | $$
in the usual way.

## Properties

There is a universal characterization of the construction of the K-theory spectrum $K(A)$ of a stable $(\infty,1)$-category $A$:

there is an $(\infty,1)$-functor

$$
  U : (\infty,1)StabCat \to N
$$

to a stable $(\infty,1)$-category which is universal with the property that it respects [[filtered colimits]] and exact sequences in a suitable way. Given any stable $(\infty,1)$-category $A$, its (connective or non-connective, depending on details) algebraic K-theory spectrum is the hom-object

$$
  K(A) \simeq Hom(U(Sp), U(A))
  \,,
$$

where $Sp$ denotes the stable $(\infty,1)$-category of compact [[spectra]].

See ([Blumberg-Gepner-Tabuada 10](#BlumbergGepnerTabuada10)).

## References

* {#BlumbergGepnerTabuada10} [[Andrew Blumberg]], [[David Gepner]], [[Gonçalo Tabuada]], _A universal characterization of higher algebraic K-theory_ ([arXiv:1001.2282](http://arxiv.org/abs/1001.2282)).

[[!redirects K-theory of a stable (infinity,1)-category]]
[[!redirects K-theory of a stable infinity-category]]


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#Contents#
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## Idea

The _[[algebraic K-theory]]_ $\mathcal{K}(\mathcal{C})$ of a _[[symmetric monoidal (∞,1)-category]]_ $\mathcal{C}$ is the [[generalized (Eilenberg-Steenrod) cohomology theory]] [[Brown representability theorem|represented]] by the [[∞-group completion]] of the [[commutative ∞-monoid]] which is the [[core]] of $\mathcal{C}$.

This construction subsumes various other construction in [[algebraic K-theory]]. Specifically when restricted to 1-categories it reproduces the classical construction by ([Segal 74](#Segal74)) described at _[[K-theory of a permutative category]]_, see [below](#RelationToClassicalAlgebraicKTheory).

## Definition

Write

$$
  \mathcal{K} \;\colon\;  CMon_\infty(\infty Cat) \stackrel{CMon_\infty(core)}{\longrightarrow} CMon_\infty(\infty Gpd) \stackrel{F}{\longrightarrow} AbGrp_\infty(\infty Grpd) \hookrightarrow  Spectra
$$

for the composite of

1. the [[core]] functor from [[symmetric monoidal (∞,1)-categories]] to [[E-∞ spaces]];

1. their [[∞-group completion]] to [[abelian ∞-groups]];

1. the inclusion of "[[abelian ∞-groups]]", hence [[connective spectra]] into all [[spectra]].

This $\mathcal{K}$ is the _algebraic K-theory of symmetric monoidal (∞,1)-categories_. ([Nikolaus 13, below remark 5.3](#Nikolaus13), [Bunke-Nikolaus-Völkl 13, def.6.1](#BunkeNikolausVoelkl13))

More generally, one may start with construction with other objects that map to [[Picard ∞-groups]], such as [[(∞,1)-operads]] ([Nikolaus 13](#Nikolaus13)).
Also, instead of just group-completing one may "ring complete" to produce K-theory spectra equipped with the structure of [[E-∞ rings]] ([Bunke-Tamme 13, section 2.4](#BunkeTamme13)).

## Properties

### Relation to classical algebraic K-theory
 {#RelationToClassicalAlgebraicKTheory}

For $R$ a [[commutative ring]], and $R$[[Mod]] its [[category of modules]] ([[projective modules]]), then $\mathcal{K}(R Mod)$ is Quillen's [[algebraic K-theory]] of $R$.
More generally this reproduces the [[K-theory of a permutative category]] etc. ([Nikolaus 13, section 6](#Nikolaus13)).

In particular, applied to the [[stack]] of [[algebraic vector bundles]] this produces the [[sheaf of spectra]] of [[algebraic K-theory]] of [[schemes]] ([Bunke-Tamme 12, section 3.3](#BunkeTamme12)), see at  _[differential algebraic K-theory -- Algebraic K-theory sheaf of spectra](differential+algebraic+K-theory#AlgebraicKTheorySheafOfSpectra)_.

## Examples

### On monoidal stacks

Algebraic K-theory is traditionally applied to single [[symmetric monoidal (∞,1)-categories|symmetric monoidal]]/[[stable (∞,1)-category|stable]] [[(∞,1)-categories]], but to the extent that it is [[(∞,1)-functor|functorial]] it may just as well be applied to [[(∞,1)-sheaves]] with values in these.

Notably, applied to the monoidal [[stack]] of [[vector bundles]] (with [[connection on a bundle|connection]]) on the [[site]] of [[smooth manifolds]], the [[K-theory of a symmetric monoidal (∞,1)-category|K-theory of a monoidal category]]-functor produces a [[sheaf of spectra]] which is a form of [[differential K-theory]] and whose [[geometric realization]] is the [[topological K-theory]] spectrum. For more on this see at _[differential cohomology hexagon -- Differential K-theory](differential%20cohomology%20diagram#DifferentialKTheory)_.

## Related concepts

* [[regulator in algebraic K-theory]]

* [[K-theory of a permutative category]], [[K-theory of a bipermutative category]]

## References

### Traditional category theoretic

The functor from [[symmetric monoidal categories]] to [[connective spectra]] was originally given in

* {#Segal74} [[Graeme Segal]], _Categories and cohomology theories_, Topology 13 (1974), 293–312.

See also

* {#Thomason82} [[Robert Thomason]], _First quadrant spectral sequences in algebraic K-theory via homotopy colimits_. Comm. Algebra, 10(15):1589–1668,
1982.

The following article showed that this construction produces all [[connective spectra]], up to equivalence

* {#Thomason95} [[Robert Thomason]], Symmetric monoidal categories model all connective spectra, Theory Appl. Categ. 1 (1995), no. 5, 78–118.

and a new proof of that is in

* [[Michael Mandell]], _An Inverse K-Theory Functor_ ([arXiv:1002.3622](http://arxiv.org/abs/1002.3622))

### $\infty$-Category theoretic

The natural generalization of the construction to [[symmetric monoidal (∞,1)-categories]] appears in

* {#BunkeTamme12} [[Ulrich Bunke]], [[Georg Tamme]], section 2.1 of _Regulators and cycle maps in higher-dimensional differential algebraic K-theory_ ([arXiv:1209.6451](http://arxiv.org/abs/1209.6451))

* {#Nikolaus13} [[Thomas Nikolaus]] _Algebraic K-Theory of $\infty$-Operads_ ([arXiv:1303.2198](http://arxiv.org/abs/1303.2198))

* {#BunkeNikolausVoelkl13} [[Ulrich Bunke]], [[Thomas Nikolaus]], [[Michael Völkl]], def. 6.1 in  _Differential cohomology theories as sheaves of spectra_ ([arXiv:1311.3188](http://arxiv.org/abs/1311.3188))

* {#GepnerGrothNikolaus13} [[David Gepner]], [[Moritz Groth]], [[Thomas Nikolaus]], _Universality of multiplicative infinite loop space machines_, [arXiv:1305.4550](http://arxiv.org/abs/1305.4550).

* {#BunkeTamme13} [[Ulrich Bunke]], [[Georg Tamme]], _Multiplicative differential algebraic K-theory and applications_ ([arXiv:1311.1421](http://arxiv.org/abs/1311.1421))

[[!redirects algebraic K-theory of symmetric monoidal (∞,1)-categories]]

[[!redirects K-theory of symmetric monoidal (∞,1)-categories]]

[[!redirects algebraic K-theory of a symmetric monoidal (∞,1)-category]]
[[!redirects algebraic K-theory of a symmetric monoidal (∞,1)-categories]]

[[!redirects K-theory of a symmetric monoidal (infinity,1)-category]]
[[!redirects K-theory of a symmetric monoidal (infinity,1)-categories]]

[[!redirects algebraic K-theory of a symmetric monoidal (infinity,1)-category]]
[[!redirects algebraic K-theory of a symmetric monoidal (infinity,1)-categories]]


K-theory spectrum

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#Contents#
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## Idea

The **K-theory spectrum** $KU$ (for [[complex vector bundle|complex]] K-theory) or $KO$ (for [[real vector bundle|orthogonal]] K-theory) in the strict sense is the [[spectrum]] that represents the [[generalized (Eilenberg-Steenrod) cohomology]] theory [[topological K-theory]]. For complex [[topological K-theory]] this is periodic with period 2 (reflect Bott periodicity) of the form

$$
  \mathbb{Z} \times B U ,\; U ,\; \cdots
  \,.
$$

The [[connective cover]] is denoted in lower case: _[[ku]]_.

## Definition

As a [[symmetric spectrum]]: [Schwede 12, Example I.2.10](#Schwede12)

## Properties

### Periodicity

$KU$ is a 2-[[periodic ring spectrum]]. This is the original _[[Bott periodicity]]_.

### As a localization of an $\infty$-group $\infty$-ring

[[Snaith's theorem]] asserts that the [[K-theory spectrum]] for [[periodic complex K-theory]] is the [[∞-group ∞-ring]] of the [[circle 2-group]] localized away from the [[Bott element]] $\beta$:

$$
  KU \simeq \mathbb{S}[B U(1)][\beta^{-1}]
  \,.
$$

### Relation between $KU$, $KO$ and  $KR$-

#### $KO$ as homotopy-fixed points of $KU$

[[complex conjugation|Complex conjugation]] on [[complex vector bundles]] induces on the [[complex K-theory]] [[spectrum]] $KU$ an [[involution|involutive]] [[automorphism]]. [[KR-theory]] is the corresponding $\mathbb{Z}_2$-[[equivariant cohomology]] theory.

In particular, the [[homotopy fixed point]] of [[KU]] under this automorphism is [[KO]]

$$
  KO \simeq (KU)^{\mathbb{Z}/2}
$$

and this way where in complex K-theory one has [[KU]]-[[modules]] ([[∞-modules]]), so in KR-theory one has $KO$-modules.

#### Wood's theorem

$$
  KO \wedge \Sigma^{-2}\mathbb{CP}^2 \simeq KU
$$

a proof in terms of [[moduli stacks]] is given in [Mathew 13, section 3](#Mathew13)

### Relation to Clifford algebras

There are close relations between K-theory and [[Clifford algebras]]. One conceptual statement relating them is this:

_KO-theory is the first Weiss-derivative (in [[orthogonal calculus]]) of the K-theory of [[Clifford algebras]]._ ([[Charles Rezk]], [MO comment, Sept '13](http://mathoverflow.net/a/142091/381))

## Related concepts

* [[Pontrjagin character]]

* [[Eilenberg-MacLane spectrum]]

* [[tmf]]

* [[cohomology theory]]

* [[KO-theory]]

* [[Tate K-theory]]

* [[splitting principle]]

* [[comparison map between algebraic and topological K-theory]]

* [[KU-local stable homotopy theory]]

[[!include string theory and cohomology theory -- table]]

## References

The weak homotopy eqivalence $B U \to \Omega U$ is discussed in detail in

* {#AguilarGitlerPrieto02} Marcelo Aguilar, [[Samuel Gitler]], Carlos Prieto, appendix B.2 of _Algebraic topology from a homotopical viewpoint_, Springer (2002) ([toc pdf](http://tocs.ulb.tu-darmstadt.de/106999419.pdf))

The incarnation of $KU$ and $KO$ as a [[symmetric spectrum]] is discussed in

* {#Schwede12} [[Stefan Schwede]], Example I.2.10 in _Symmetric spectra_, 2012 ([pdf](http://www.math.uni-bonn.de/~schwede/SymSpec.pdf))

and the structure of a [[symmetric ring spectrum]] on $KO$ is discussed in

* {#Joachim01} [[Michael Joachim]], _A symmetric ring spectrum representing $KO$-theory_, Topology 40 (2001) 299-308

The structure of a homotopy [[ring spectrum]] on $KU$ is discussed in

* {#Switzer75} [[Robert Switzer]], section 13.90 of _Algebraic Topology - Homotopy and Homology_, Die  Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, Vol. 212, Springer-Verlag, New York, N. Y., 1975.

The [[E-infinity ring]] structure of $KU$ is discussed in

* [[Peter May]], section VIII §2 of _$E_\infty$-Ring spaces and $E_\infty$ ring spectra_ ([pdf](http://www.math.uchicago.edu/~may/BOOKS/e_infty.pdf))

and the underlying [[H-infinity ring spectrum]] structure in

* {#McLure86} [[James McClure]], _$H_\infty$-ring spectra via space-level homotopy theory_ ([pdf](https://web.math.rochester.edu/people/faculty/doug/otherpapers/BMMS8.pdf)), chapter VII in R. Bruner, [[Peter May]], [[James McClure]], M. Steinberger, _$H_\infty$-Ring Spectra and their Applications_, Lecture Notes in Mathematics 1176, Springer 1986 ([pdf](http://www.math.uchicago.edu/~may/BOOKS/h_infty.pdf))

The uniqueness of the [[E-infinity ring]] structure on $KU$ is due to

* [[Andrew Baker]], [[Birgit Richter]], _$\Gamma$-cohomology of rings of numerical polynomials and $E_\infty$ structures on K-theory_ ([arXiv:math/0304473](https://arxiv.org/abs/math/0304473))

Discussion of $KO$ in analogy to the construction of [[tmf]] is in

* {#Mathew13} [[Akhil Mathew]], _The homology of $tmf$_ ([arXiv:1305.6100](http://arxiv.org/abs/1305.6100))

with a summary in

* [[Akhil Mathew]], _The homotopy groups of $TMF$_ ([pdf](http://math.mit.edu/~sglasman/tmfhomotopy.pdf))

[[!redirects KU]]
[[!redirects KO]]

[[!redirects complex K-theory spectrum]]

[[!redirects KU-theory]]


K-topology

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* table of contents
{:toc}

##Idea##

The **K-topology** is a [[topological space|topology]] on the set of [[real numbers]] $\mathbb{R}$ which is [[finer topology|finer]] than the [[Euclidean space|Euclidean]] [[metric topology]], but such that the new open sets are 'piled up' on the positive side of 0 (and actually are all contained in the [[half-open interval]] $[0,1)$) and have lots of 'holes'. It is useful for constructing counterexamples in [[topology]].

For instance the K-topology is an example of a [[topological space]] that is [[Hausdorff topological space|Hausdorff]] $(T_2)$ but not [[normal Hausdorff space|normal]] ($T_4$).

##Definition ##

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

Write

$$
  K \coloneqq \{1/n \,\vert\, n \in \mathbb{N}_{\geq 1}\} \subset \mathbb{R}
$$

for the [[subset]] of [[natural number|natural]] [[fractions]] inside the [[real numbers]].

Define a [[topological basis]] $\beta \subset P(\mathbb{R})$ on $\mathbb{R}$ consisting of all the [[open intervals]] as well as the [[complements]] of $K$ inside them:

$$
  \beta
    \;\coloneqq\;
  \left\{
    (a,b),
    \,\vert\,
    a\lt b \in \mathbb{R}
  \right\}
   \,\cup\,
  \left\{
    (a,b) \backslash K,
    \,\vert\,
    a\lt b \in \mathbb{R}
  \right\}
  \,.
$$

The [[topological space|topology]] $\tau_{\beta} \subset P(\mathbb{R})$ which is generated from this [[topological basis]] is called the _K-topology_.

We may denote the resulting [[topological space]] by

$$
  \mathbb{R}_K
   \;\coloneqq\;
  ( \mathbb{R}, \tau_{\beta})
  \,.
$$

=--

## Properties

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

The real numbers with their K-topology (def. \ref{KTopology}) $\mathbb{R}_K$ are a [[Hausdorff topological space]] which is not a [[regular Hausdorff space]] (hence in particular not a [[normal Hausdorff space]]).

=--

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

By construction the K-topology is [[finer topology|finer]] than the usual [[Euclidean space|euclidean]] [[metric topology]]. Since the latter is Hausdorff, so is $\mathbb{R}_K$. It remains to see that $\mathbb{R}_K$ contains a point and a [[disjoint subset|disjoint]] closed subset such that do not have [[disjoint subset|disjoint]] [[open neighbourhoods]].

But this is the case essentially by construction: Observe that

$$
  \mathbb{R} \backslash K
  \;=\;
  (-\infty,-1/2)
    \cup
  \left(
    (-1,1) \backslash K
  \right)
    \cup
  (1/2, \infty)
$$

is an open subset in $\mathbb{R}_K$, whence

$$
  K = \mathbb{R} \backslash ( \mathbb{R} \backslash K  )
$$

is a [[closed subset]] of $\mathbb{R}_K$.

But every [[open neighbourhood]] of $\{0\}$ contains at least $(-\epsilon, \epsilon) \backslash K$ for some positive real number $\epsilon$. There exists then $n \in \mathbb{N}_{\geq 0}$ with $1/n \lt \epsilon$ and $1/n \in K$. An open neighbourhood of $K$ needs to contain an open interval around $1/n$, and hence will have non-trivial intersection with $(-\epsilon, \epsilon)$. Therefore $\{0\}$ and $K$ may not be separated by disjoint open neighbourhoods, and so $\mathbb{R}_K$ is not normal.

=--

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

The set of [[rational numbers]], with the [[subspace topology]] inherited from the inclusion $\mathbb{Q} \hookrightarrow \mathbb{R}_K$ into the real numbers with their K-topology, is an example of a non-[[regular space|regular]], [[path-connected space|totally path-disconnected]] [[Hausdorff space|Hausdorff]] space. This space can be used to construct a [[quasitopological groupoid]] which isn't a [[internal groupoid|topological groupoid]].

=--

## Related concepts

* [[Dowker space]]

[[!redirects K-topologies]]


k-tuply associative n-groupoid

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###Context###
#### Algebra
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[[!include higher algebra - contents]]
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#### Type theory
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[[!include type theory - contents]]
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#### Categorification
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[[!include categorification - contents]]
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#### Category theory
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[[!include monoid theory - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

A version of the [[k-tuply monoidal n-groupoid]] but for [[An-spaces|$A_n$ spaces]].

## Definition

### Homotopy-theoretic

Invoking the homotopy hypothesis, we define a __$k$-tuply associative $n$-groupoid__ to be an _$A_k$-$n$-truncated type_: a [[topological space]] which is a [[homotopy n-type]] with a multiplication that is associative up to higher homotopies involving up to nn variables.

## Periodic table
 {#PeriodicTable}

There is a [[periodic table]] of $k$-tuply monoidal $n$-groupoids:

| $k \,\backslash\, n$ | $-1$ | $0$ | $1$ | $2$ | $\cdots$ | $\infty$ |
|--|--|--|--|--|--|--|
| $0$ | trivial | [[pointed set]] | [[pointed object|pointed]] [[groupoid]] | [[pointed object|pointed]] [[2-groupoid]] | $\cdots$ | [[pointed infinity-groupoid|pointed $\infty$-groupoid]] |
| $1$ | trivial | [[pointed set]] | [[pointed object|pointed]] [[groupoid]] | [[pointed object|pointed]] [[2-groupoid]] | $\cdots$ | [[pointed infinity-groupoid|pointed $\infty$-groupoid]] <br/> [[A1-space|$A_1$-space]] |
| $2$ | trivial | [[unital magma]] | [[H-groupoid|magmoidal groupoid with unit]] | [[magmoidal 2-groupoid with unit]] | $\cdots$ | [[H-space]]/[[A2-space]] |
| $3$ | " | [[monoid]] | [[H-monoidal groupoid]] | [[H-monoidal 2-groupoid]] | $\cdots$ | [[H-monoid]]/[[A3-space]] |
| $4$ | " | " | [[monoidal groupoid]] | [[A4 2-groupoid]] | $\cdots$ | [[A4-space]] |
| $5$ | " | " | " | [[monoidal 2-groupoid]] | $\cdots$ | [[A5-space]] |
| $\vdots$ | " | " | " | " | $\ddots$ | $\vdots$ |
| $\infty$ | trivial | [[monoid]] | [[monoidal groupoid]] | [[monoidal 2-groupoid]] | $\cdots$ | [[A-infinity space|$A_\infty$-space]] |

## See also

* [[k-tuply monoidal n-groupoid]]


k-tuply groupal n-groupoid

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

# Contents
* table of contents
{: toc}

## Idea

A $k$-tuply groupal $n$-groupoid is an $n$-[[n-groupoid|groupoid]] in which objects can be multiplied in $k$ invertible ways, all of which interchange with each other up to isomorphism.  By the [[Eckmann-Hilton argument]], this implies that these $k$ ways all end up being equivalent, but that the single resulting operation is more and more commutative as $k$ increases.  The [[stabilization hypothesis]] states that by the time we reach $k = n + 2$, the multiplication has become "maximally commutative."

There is as yet no general category-theoretic definition of $k$-tuply groupal $n$-groupoid, but the [[delooping hypothesis]] says that a $k$-tuply groupal $n$-groupoid can be interpreted as a special kind of $(n+k)$-groupoid, and if we wish we can take this hypothesis as a definition.  A $k$-tuply groupal $n$-groupoid can also be thought of as a $k$-tuply monoidal $(n,-1)$-[[k-tuply monoidal (n,r)-category|category]].  An alternate approach is to invoke instead the [[homotopy hypothesis]] to identify $n$-groupoids with [[homotopy n-type|homotopy n-types]], in which case we can apply classical homotopy theory.

## Definitions

### Category-theoretic

Invoking the delooping hypothesis, we define a __$k$-tuply groupal $n$-groupoid__ to be a [[pointed object|pointed]] $(n+k)$-groupoid such that any two parallel $j$-morphisms are [[equivalence|equivalent]], for $j \lt k$.  One usually relabels the $j$-morphisms as $(j-k)$-morphisms.  You may interpret this definition as weakly or strictly as you like, by starting with weak or strict notions of $(n+k)$-groupoid.

The given point serves as an equivalence between $(-1)$-morphisms (for now, see $(n,r)$-[[(n,r)-category|category]] for these), so there is nothing to say if $k \leq 0$ except that the category is pointed.  Thus we may as well assume that $k \geq 0$.  Also, according to the [[stabilization hypothesis|stabilisation hypothesis]], every $k$-tuply groupal $n$-groupoid for $k \gt n + 2$ may be reinterpreted as an $(n+2)$-tuply groupal $n$-groupoid.  Unlike the restriction $k \geq 0$, this one is not trivial.

### Homotopy-theoretic

Invoking the homotopy hypothesis, we define a __[[k-tuply monoidal n-groupoid]]__ to be an _$E_k$-$n$-type_: a [[topological space]] which is a [[homotopy n-type]] and which is equipped with an action by the [[little n-cubes operad|little k-cubes]] [[operad]] (or some operad equivalent to it).  The $k$-tuply groupal $n$-groupoids can then be identified with the [[grouplike topological monoid|grouplike]] $E_k$-[[E-k-space|spaces]].

It is a classical theorem of homotopy theory that grouplike $E_k$-spaces are the same as $k$-fold [[loop space]]s (see J.P. May, _The Geometry of Iterated Loop Spaces_).  This is the topological version of the delooping hypothesis (from which, of course, it takes its name).

### Special cases

A $0$-tuply groupal $n$-groupoid is simply a pointed $n$-groupoid, that is an $n$-groupoid equipped with a chosen object (or a space equipped with a chosen basepoint).  A $1$-tuply groupal $n$-groupoid may be called simply a __groupal $n$-groupoid__, or an __$(n+1)$-[[n-group|group]]__; topologically these can be identified with grouplike monoids that are $n$-types.

A __stably groupal $n$-groupoid__, or __symmetric groupal $n$-groupoid__, is an $(n+2)$-tuply groupal $n$-groupoid.  This is also called a __symmetric $(n+1)$-group__, with the numbering off as before.  Although the general definition above won\'t give it, there is a notion of stably groupal $\infty$-groupoid, basically an $\infty$-groupoid that can be made $k$-tuply groupal for any value of $k$ in a consistent way.  Topologically, these are called _grouplike $E_\infty$-[[E-infinity-space|spaces]]_ and can be identified with infinite [[loop space]]s.

## The periodic table
 {#PeriodicTable}

There is a [[periodic table]] of $k$-tuply groupal $n$-groupoids:

| $k \,\backslash\, n$ | $-1$ | $0$ | $1$ | $2$ | $\cdots$ | $\infty$ |
|--|--|--|--|--|--|--|
| $0$ | trivial | [[pointed set]] | [[pointed object|pointed]] [[groupoid]] | [[pointed object|pointed]] [[2-groupoid]] | $\cdots$ | [[pointed infinity-groupoid|pointed $\infty$-groupoid]] |
| $1$ | trivial | [[group]] | [[2-group]] | [[3-group]] | $\cdots$ | [[infinity-group|$\infty$-group]] |
| $2$ | " | [[abelian group]] | [[braided 2-group]] | [[braided 3-group]] | $\cdots$ | [[braided infinity-group|braided $\infty$-group]] |
| $3$ | " | " | [[symmetric 2-group]] | [[sylleptic 3-group]] | $\cdots$ | groupal [[E3-algebra|$E_3$-space]] |
| $4$ | " | " | " | [[symmetric 3-group]] | $\cdots$ | groupal  [[E4-algebra|$E_4$-space]] |
| $\vdots$ | " | " | " | " | $\ddots$ | $\vdots$ |
| $\infty$ | " | " | " | " | $\cdots$ | groupal [[E-infinity space|$E_\infty$-space]]
 / [[abelian infinity-group|abelian $\infty$-group]]  |

## See also

* [[k-tuply monoidal n-groupoid]]

## References

The [[homotopy theory]] of $k$-tuply groupal $n$-groupoids is discussed in

* [[A.R. Garzón]], J.G. Miranda, _Serre homotopy theory in subcategories of simplicial groups_, Journal of Pure and Applied Algebra Volume 147, Issue 2, 24 March 2000, Pages 107-123 (doi:10.1016/S0022-4049(98)00143-1)

[[!redirects k-tuply groupal n-groupoid]]
[[!redirects k-tuply groupal n-groupoids]]


k-tuply monoidal (n,r)-category

+-- {: .rightHandSide}
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###Context###
#### Higher category theory
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[[!include higher category theory - contents]]
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#### Monoidal categories
+--{: .hide}
[[!include monoidal categories - contents]]
=--
=--
=--

# $k$-tuply monoidal $(n,r)$-categories
* table of contents
{: toc}

## Idea

Two important [[periodic table]]s are the table of $k$-[[k-tuply monoidal n-category|tuply monoidal]] $n$-categories and the table of $(n,r)$-[[(n,r)-category|categories]]. These can actually be combined into a single 3D table, which surprisingly also includes $k$-[[k-tuply groupal n-groupoid|tuply groupal]] $n$-groupoids.

## Definition

A __$k$-tuply monoidal $(n,r)$-category__ is a [[pointed object|pointed]] $\infty$-[[infinity-category|category]] (which you may interpret as weakly or strictly as you like) such that:

* any two parallel $j$-morphisms are equivalent, for $j \lt k$;
* any $j$-morphism is an equivalence, for $j \gt r + k$;
* any two parallel $j$-morphisms are equivalent, for $j \gt n + k$.

Keep in mind that one usually relabels the $j$-morphisms as $(j-k)$-morphisms, which explains the usage of $r + k$ and $n + k$ instead of $r$ and $n$. As explained below, we may assume that $n \geq -1$, $-1 \leq r \leq n + 1$, $0 \leq k \leq n + 2$, and (if convenient) $r + k \geq 0$.

To interpret this correctly for low values of $j$, assume that all [[object]]s ($0$-morphisms) in a given $\infty$-category are parallel, which leads one to speak of the two $(-1)$-morphisms that serve as their common source and target and to accept any object as an equivalence between these. In particular, any $j$-morphism is an equivalence for $j \lt 1$, so if $r + k = 0$, then the condition is satisfied for any smaller value of $r + k$. Thus, we may assume that $r + k \geq 0$. Similarly, since there is a chosen object (the basepoint), any parallel $j$-morphisms are equivalent for $j \lt 1$,

The conditions that $j \lt k$ and that $j \gt n + k$ will overlap if $n \lt - 1$, so we don\'t use such values of $n$. In other words, any $k$-tuply monoidal $(-1,r)$-category is also a $k$-tuply monoidal $(n,r)$-category for any $n \lt - 1$.

If any two parallel $j$-morphisms are equivalent, then any $j$-morphism between equivalent $(j-1)$-morphisms is an equivalence (being parallel to an equivalence for $j \gt 0$ and automatically for $j \lt 1$). Accordingly, any $k$-tuply monoidal $(n,0)$-category is automatically also a $k$-tuply monoidal $(n,r)$-category for any $r \lt 0$, and any $k$-tuply monoidal $(n,r)$-category for $r \gt n + 1$ is also a $k$-tuply monoidal $(n,n+1)$-category. Thus, we don\'t need $r \lt -1$ or $r \gt n + 1$.

According to the [[stabilization hypothesis|stabilisation hypothesis]], every $k$-tuply monoidal $(n,r)$-category for $k \gt n + 2$ may be reinterpreted as an $(n+2)$-tuply monoidal $(n,r)$-category. Unlike the other restrictions on values of $n, r, k$, this one is not trivial.

## Special cases

A $0$-tuply monoidal $(n,r)$-category is simply a pointed $(n,r)$-category. The restriction that $r + k \geq 0$ becomes that $r \geq 0$. This is why $(n,r)$-categories use $0 \leq r \leq n + 1$ rather than the restriction on $r$ given before.

A $k$-tuply monoidal $(n,0)$-category is a $k$-tuply monoidal $n$-[[n-groupoid|groupoid]]. A $k$-tuply monoidal $(n,-1)$-category is a $k$-[[k-tuply groupal n-groupoid|tuply groupal]] $n$-groupoid. This is why [[groupal category|groupal categories]] don\'t come up much; the progression from [[monoidal category|monoidal categories]] to [[monoidal groupoid|monoidal groupoids]] to [[groupal groupoid|groupal groupoids]] is a straight line up one column of the periodic table of [[monoidal (n,r)-category|monoidal]] $(n,r)$-categories. (But if we moved to a 4D table that required all $j$-morphisms to be equivalences for sufficiently low values of $j$, then groupal categories would appear there.)

A $k$-tuply monoidal $(n,n)$-category is simply a $k$-tuply monoidal $n$-category. A $k$-tuply monoidal $(n,n+1)$-category is a $k$-tuply monoidal $(n+1)$-[[n-poset|poset]]. Note that a $k$-tuply monoidal $\infty$-category and a $k$-tuply monoidal $\infty$-poset are the same thing.

A __stably monoidal $(n,r)$-category__, or __symmetric monoidal $(n,r)$-category__, is an $(n+2)$-tuply monoidal $(n,r)$-category. Although the general definition above won\'t give it, there is a notion of stably monoidal $(\infty,r)$-category, basically an $(\infty,r)$-category that can be made $k$-tuply monoidal for any value of $k$ in a consistent way.

## Related concepts

* [[k-tuply monoidal n-groupoid]]

* [[k-tuply groupal n-groupoid]]

* [[stabilization hypothesis]]

[[!redirects k-tuply monoidal (n,r)-category]]
[[!redirects k-tuply monoidal (n,r)-categories]]


k-tuply monoidal n-category

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

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

## Idea

A $k$-tuply monoidal $n$-category is an $n$-[[n-category|category]] in which [[objects]] can be multiplied in $k$ different ways, all of which [[interchange law|interchange]] with each other up to equivalence.  By the [[Eckmann-Hilton argument]], this implies that these $k$ ways all end up being equivalent, but that the single resulting operation is more and more commutative as $k$ increases.  The [[stabilization hypothesis]] states that by the time we reach $k = n + 2$, the multiplication has become "maximally commutative."

While there is maybe no generally accepted definition of $k$-tuply monoidal $n$-category yet, it seems that defining it to be an $n$-category with an action of the [[little k-cubes operad]] makes good sense, as discussed further below. With this definition the stabilization hypothesis is a theorem at least for $(n,1)$-[[(n,1)-category|categories]] (see below).

By the [[delooping hypothesis]] a $k$-tuply monoidal $n$-category can be interpreted as a special kind of $(n+k)$-category. One may take this hypothesis as a definition, but it has been verified in many low-dimensional cases; see below.

## Definition

For purposes of this page, a __$k$-tuply monoidal $n$-category__ is a [[pointed object|pointed]] $(n+k)$-[[n-category|category]] such that any two parallel $j$-[[k-morphism|morphisms]] are [[equivalence|equivalent]] for $j \lt k$.  One usually relabels the $j$-morphisms as $(j-k)$-morphisms.  You may interpret this definition as weakly or strictly as you like, by starting with weak or strict notions of $(n+k)$-category.

The given point serves as an equivalence between $(-1)$-[[(-1)-morphism|morphisms]] (for now, see $(n,r)$-[[(n,r)-category|category]] for these), so there is nothing to say if $k \leq 0$ except that the category is pointed.  Thus we may as well assume that $k \geq 0$.  Also, according to the [[stabilization hypothesis|stabilisation hypothesis]], every $k$-tuply monoidal $n$-category for $k \gt n + 2$ may be reinterpreted as an $(n+2)$-tuply monoidal $n$-category.  Unlike the restriction $k\ge 0$,  this one is not trivial.

### Special cases

* A $0$-tuply monoidal $n$-category is simply a [[pointed object|pointed]] $n$-category, that is an $n$-category equipped with a chosen object.

* A $1$-tuply monoidal $n$-category may be called simply a __[[monoidal n-category]]__. See

  * [[monoidal category]]

* A **doubly monoidal** $n$-category is a __[[braided monoidal n-category]]. See

  * [[braided monoidal category]],
  * [[braided monoidal 2-category]].

* A __symmetric monoidal $n$-category__ is an $k$-tuply monoidal $n$-category for $k \geq n+2$. (This also called a _stably monoidal $n$-category_, but never a stable monoidal $n$-category.)

  That this is indeed independent of $k$, as soon as $k$ is large enough, is the statement of the [[stabilization hypothesis]], more on which below.

  See

  * [[symmetric monoidal category]],
  * [[symmetric monoidal (∞,1)-category]].

## The periodic table

There is a [[periodic table]] of $k$-tuply monoidal $n$-categories:

$k$↓\$n$→	$-1$	$0$	$1$	$2$	...

$0$	trivial	[[pointed set]]	[[pointed object|pointed]] [[category]]	[[pointed object|pointed]] [[2-category]]	...

$1$	trivial	[[monoid]]	[[monoidal category]]	[[monoidal 2-category]]	...

$2$	"	[[abelian monoid]]	[[braided monoidal category]]	[[braided monoidal 2-category]]	...

$3$	"	"	[[symmetric monoidal category]]	[[sylleptic monoidal 2-category]]	...

$4$	"	"	"	[[symmetric monoidal 2-category]]	...

⋮	"	"	"	"	⋱

### Historical notes

Originally the importance of pointedness was not fully appreciated, so any $n$-category was accepted as $0$-tuply monoidal, and $k$-tuply monoidal $n$-categories were identified simply with $(k-1)$-[[k-tuply connected n-category|simply connected]] $(n+k)$-categories (those in which any two parallel $j$-morphisms are equivalent for $j \lt k$).  See [[periodic table]] for this original.

## Low dimensions

### $k=0$

As remarked above, a $0$-tuply monoidal $n$-category is just a pointed one, and functors and transformations between such are required to preserve the chosen object, at least up to specified coherent isomorphism.  (In other words, the $(n+1)$-category of $0$-tuply monoidal $n$-categories is the co-slice $(n+1)$-category $1/n Cat$, where the slicing happens in a suitably weak $(n+1)$-sense.

### $k=1$, $n=0$

A 1-tuply monoidal 0-category is a pointed 0-connected 1-category, or a 1-category with a chosen object in which all objects are isomorphic.  Thus we might as well as assume there is exactly one object, in which case we just have a [[monoid]].  A functor between one-object categories, which preserves the basepoint automatically, is exactly a monoid [[homomorphism]].

More interestingly, a natural transformation between functors $f,g:X\to Y$ between one-object categories is just an object $y\in Y$ (its component at the single object) such that $f(x) y = y g(x)$ for all $x\in X$.  So the 2-category of 0-connected 1-categories is not equivalent to the 1-category of monoids.  However, a _pointed_ natural transformation must have its component at the basepoint being the identity; thus $y=1$ and so the only such natural transformations are identities $f=g$.  Therefore, the 2-category of pointed 0-connected 1-categories (that is, 1-tuply monoidal 0-categories) is equivalent to the 1-category of monoids.

### $k=2$, $n=0$

A 2-tuply monoidal 0-category is a pointed 1-connected 2-category.  Interpreting things as weakly as possible, we are talking about a [[bicategory]] $B$ with one object $*$ and one 1-cell (its identity).  By the usual [[Eckmann-Hilton argument]], the set $B(1_*,1_*)$ is a commutative monoid, but there is also additional structure: the associatior and unitors of the bicategory.  The pentagon identity implies that the associator is the identity, and the unitor axioms imply that the two unitors are the same, but they are not necessarily the identity.  Therefore, a 1-connected 2-category (if by 2-category we mean bicategory) is a commutative monoid $X$ equipped with a chosen invertible element $d_X$.  This was apparently first observed by [[Tom Leinster]].

In similar vein, one can work out (see Cheng--Gurski):

* a (weak) functor between 1-connected bicategories is a monoid homomorphism $F:X\to Y$ equipped with a distinguished invertible element $m_F\in Y$.
* a (weak) natural transformation between two monoid homomorphisms is just the assertion that they are equal (and thus, every such transformation is invertible).
* a modification between two such transformations is a distinguished not-necessarily-invertible element $\Gamma\in Y$.

Note that the invertible elements $d_X$ and $m_F$ play no role in the definition of the higher morphisms, so they might as well not be there, up to equivalence.  However, the nonidentity modifications do screw things up, so the tricategory of 1-connected bicategories is not equivalent to the 1-category of commutative monoids.  But if we add in the basepoints, then we get:

* a pointed 1-connected bicategory is one equipped with a functor from $1$, the terminal bicategory.  This is just a monoid homomorphism $1\to X$, which of course is unique, together with a distinguished invertible element in $X$ which we can ignore.  Thus every 1-connected bicategory can be pointed in an essentially unique way.

* a pointed functor between two pointed 1-connected bicategories is a functor $F:X\to Y$ together with a weak natural equivalence, say $t_F$, between $1\to X\to Y$ and $1\to Y$.  Since these are always equal as monoid homomorphisms, there is always a unique (invertible) transformation connecting them, so such a pointed functor is just a monoid homomorphism $F:X\to Y$.

* a pointed transformation from $F$ to $G$ is a transformation $a$ from $F:X\to Y$ to $G:X\to Y$ together with an invertible modification, say $c_a$, relating $a t_F$ to $t_G$.  In other words, it is an assertion that $F=G$ together with a distinguished invertible element $\Gamma_a$ of $Y$.

* a pointed modification from $a$ to $b$ is a modification $m$ such that $m c_a = c_b$.  In other words, it is a distinguished element $\Lambda\in Y$ such that $\Lambda \Gamma_a = \Gamma_b$, or $\Lambda = \Gamma_a^{-1} \Gamma_b$ since $\Gamma_a$ is invertible.  Thus, any two pointed transformations $F\to G$ are related by a unique invertible modification.

We conclude that the tricategory of pointed 1-connected bicategories is equivalent to the category of commutative monoids and monoid homomorphisms, so again the delooping hypothesis is verified.

One might complain that in addition of the single weak natural equivalence $t_F$, $F$ ought also to be equipped with an inverse _adjoint_ equivalence for it.  The modifications involved in this would introduce two distinguished invertible elements in $Y$, which (by the triangle identities) would have to be each other's inverses.  But these elements would again play no role in the higher morphisms, so they might as well be identities.

### $k=1$, $n=1$

A 1-tuply monoidal 1-category is a pointed 0-connected 2-category, which we can identify with a bicategory with one object.  It is well-known that this is precisely the data of a monoidal category. Likewise, (weak) functors between such bicategories correspond precisely (strong) monoidal functors.  However, again the transformations and modifications screw things up in the merely connected case, but by using pointed objects instead we can remedy the situation.

### $k=1$, $n=(\infty,0)$

If we identify $\infty$-[[infinity-groupoid|groupoids]] with spaces, then a 1-tuply monoidal $(\infty,0)$-category, or a monoidal $\infty$-groupoid, can be identified intuitively with an $A_\infty$-[[A-infinity-space|space]].  This is a space equipped with a multiplication which is associative and unital up to all higher homotopies; see [[operad]] for one way to encode these data.

It is a well-known fact in homotopy theory that the homotopy theories (that is, $(\infty,1)$-[[(infinity,1)-category|categories]]) of based connected spaces and of grouplike $A_\infty$-spaces are equivalent, via the [[loop space]] and [[classifying space]] constructions.  This can be regarded as another version of the delooping hypothesis.  The "grouplike" restriction (meaning that $\pi_0$ is a group, or that the multiplication has inverses up to homotopy) is because we consider only based connected $(\infty,0)$-categories, whereas we would need based connected $(\infty,1)$-categories to recover all $A_\infty$-spaces.  It would be interesting to verify the hypothesis in this case using one of the known models for $(\infty,1)$-categories.

### $n = (\infty,0)$ and $\infty$-stacks {#InfStackCase}

A $k$-tuply monoidal [[∞-groupoid]] may be identified, under $k$-fold [[delooping]], with a $(k-1)$-[[connected]] $\infty$-groupoid. This statement holds in fact even for _parameterized $\infty$-groupoids_, i.e. for [[∞-stack]]s.

+-- {: .un_theorem}
###### Theorem (k-tuply monoidal $\infty$-stacks)

Let $k \gt 0$, let $\mathcal{X}$ be an [[(∞,1)-category of (∞,1)-sheaves]] and let $\mathcal{X}_*^{\geq k}$ denote the full [[sub-(∞,1)-category]] of the category $\mathcal{X}_{*}$ of pointed objects, spanned by those pointed objects thar are $k-1$-[[connected]] (i.e. their first $k$ [[homotopy groups in an (∞,1)-topos|∞-stack homotopy groups]]) vanish. Then there is a canonical equivalence of [[(∞,1)-category|(∞,1)-categories]]

$$
  \mathcal{X}_*^{\geq k} \simeq Mon^{gp}_{\mathbb{E}[k]}(\mathcal{X}) \,.
$$

=--

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

This is [[Ek-Algebras|EKAlg, theorem 1.3.6.]].

=--

Specifically for $\mathcal{X} = Top$, this refines to the classical theorem by [[Peter May]]

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

Let $Y$ be a [[topological space]] equipped with an action of the [[little cubes operad]] $\mathcal{C}_k$ and suppose that $X$ is grouplike. Then $Y$ is homotopy equivalent to a $k$-fold loop space $\Omega^k X$ for some pointed topological space $X$.

=--

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

This is [[Ek-Algebras|EkAlg, theorem 1.3.16.]]

=--

### $n = (n,1)$ and the stabilization hypothesis {#Inf1Case}

An [[(n,1)-category]] is a $n$-[[truncated]] [[(∞,1)-category]]

In [John Francis' PhD thesis](http://dspace.mit.edu/handle/1721.1/43792) (reference _EnAction_ below ) the theory of [[(∞,1)-categories]] equipped with an action of the [[little k-cubes operad]] is established, so that

* $(\infty,1)$-categories with an $E_1$-action are precisely [[monoidal (∞,1)-categories]]  -- 1-fold monoidal $(\infty,1)$-categories;

* $(\infty,1)$-categories with an $E_\infty$-action are precisely [[symmetric monoidal (∞,1)-categories]] -- $\infty$-tuply monoidal $(\infty,1)$-categories;

* $(\infty,1)$-categories with an $E_n$-action for $1 \lt n \lt \infty$ are the corresponding $k$-tuply monoidal $(\infty,1)$-categories in between.

+-- {: .un_prop}
###### Proposition
**(stabilization for $(n,1)$-categories)

Equipping an [[(n,1)-category]] of $k$ compatible monoidal structures for $k \geq n + 2$ (which is the same as equipping it with a [[little k-cubes operad]] action) is the same as equipping it with the structure of a [[symmetric monoidal (∞,1)-category|symmetric monoidal (n,1)-category]].

=--

An aspect of this was demonstrated in terms of [[Tamsamani n-categories]] in

* [[Carlos Simpson]], _On the Breen-Baez-Dolan stabilization hypothesis for Tamsamani's weak $n$-categories_ ([arXiv:math/9810058](http://arxiv.org/abs/math/9810058))

A proof of the full statement in terms of [[quasi-categories]] is sketched in
[section 43.5](http://www.math.uchicago.edu/~may/IMA/Joyal.pdf#page=187) of

* [[André Joyal]], _Notes on quasi-categories_ ([pdf](http://www.math.uchicago.edu/~may/IMA/Joyal.pdf)).

Probably the first full proof in print is given in

* [[Jacob Lurie]], _[[Ek-Algebras]]_ ,

where it appears in example 1.2.3 as a direct consequence of a more general statement, corollary 1.1.10.

### Other low-dimensional cases

One expects that

* pointed 2-connected tricategories can be identified with commutative monoids (again),
* pointed 1-connected tricategories can be identified with braided monoidal categories,
* pointed 0-connected tricategories can be identified with monoidal bicategories,
* and so on.

## References

Some discussion of the peridodic table is in

* [[Eugenia Cheng]], [[Nick Gurski]],  The periodic table of n-categories for low dimensions I: degenerate categories and degenerate bicategories.  [arXiv:0708.1178](http://arxiv.org/abs/0708.1178).

* [[John Baez]] and [[Mike Shulman]],  Lectures on $n$-categories and cohomology.  [arXiv:math.CT/0608420](http://arxiv.org/abs/math.CT/0608420).

The theory of $k$-tuply monoidal [[(∞,1)-categories]] was maybe first studied in

* **EnAction** [[John Francis]] PhD thesis ([web](http://dspace.mit.edu/handle/1721.1/43792))

and later further developed in

* [[Jacob Lurie]], _[[Ek-Algebras]]_

where also the proof of the [[stabilization hypothesis]] in this context is noticed.

Related discussion can be found in the theory of [[iterated monoidal category|iterated monoidal categories]]. See for example

* C. Balteanu, Z. Fiedorowicz, R. Schwänzl, [[Rainer Vogt]], _Iterated monoidal categories_ ([pdf](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.45.290&rep=rep1&type=pdf))

* [[Clemens Berger]], _Double loop spaces, braided monoidal categories and algebraic 3-type of space_ ([pdf](http://math.unice.fr/~cberger/bmonoid.pdf))

## Discussion

A previous version of this entry led to the following discussion

+--{: .query}

[[Mike Shulman]]: I would like to suggest that we switch to using _symmetric monoidal_ rather than _stably monoidal_, and especially avoid calling these just _stable_.  One advantage of "symmetric monoidal" is that it has a well-established meaning in low-dimensions; if I say "symmetric monoidal $n$-category" then people who are familiar with symmetric monoidal 1-categories are more likely to have an intuitive understanding of what I mean than if I say "stably monoidal $n$-category."

Use of the word "stable" here also creates confusion with its other meanings (see [here](http://golem.ph.utexas.edu/category/2009/01/benzvi_on_geometric_function_t.html#c021217) and [here](http://golem.ph.utexas.edu/category/2009/01/categories_logic_and_physics_i_2.html#c021205)).  Algebraic topologists often use "stable" to mean "related to [[spectrum|spectra]]," and spectra are related to, but distinct from, symmetric monoidal $\infty$-groupoids.  (Connective spectra can be identified with symmetric _groupal_ $\infty$-groupoids.)  Lurie is also using "stable $(\infty,1)$-category" to mean "an $(\infty,1)$-category which behaves like the $(\infty,1)$-category of spectra."  One might not like this, but it is not original with him; several other algebraic topologists use "stable model category" in the same sense.  And since we have the perfectly good alternative term "symmetric monoidal" to use here, which has other things to recommend it as well, why create needless confusion?

_Toby_:  Hopefully John will admit that saying 'stable' instead of 'stably monoidal' (or 'stably groupal') was a slip of the tongue ... pen ... fingers.  I\'m used to 'stably monoidal', and I don\'t think that it should cause confusion ---if used in full.  Also, I think there\'s some historical confusion about 'symmetric monoidal $2$-category' or maybe 'symmetric monoidal $3$-category' that 'stably monoidal' isn\'t subject to, although that\'s a bit parochial.

=--

[[!redirects k-tuply monoidal n-category]]
[[!redirects k-tuply monoidal n-categories]]


k-tuply monoidal n-groupoid

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### Monoid theory
+-- {: .hide}
[[!include monoid theory - contents]]
=--
#### Categorification
+-- {: .hide}
[[!include categorification - contents]]
=--
#### Category theory
+-- {: .hide}
[[!include monoid theory - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Definition

### Homotopy-theoretic

Invoking the [[homotopy hypothesis]], we define a __$k$-tuply monoidal $n$-groupoid__ to be an _$E_k$-$n$-truncated type_: a [[topological space]] which is a [[homotopy n-type]] and which is equipped with an [[algebra over an operad|action]] by the [[little n-cubes operad|little k-cubes]] [[operad]] (or some operad equivalent to it).

## The periodic table
 {#PeriodicTable}

There is a [[periodic table]] of $k$-tuply monoidal $n$-groupoids:

$k$↓\$n$→	$-1$	$0$	$1$	$2$	...	$\infty$

$0$	trivial	[[pointed set]]	[[pointed object|pointed]] [[groupoid]]	[[pointed object|pointed]] [[2-groupoid]] 	...	[[pointed object|pointed]] [[∞-groupoid]]

$1$	trivial	[[monoid]]	[[monoidal groupoid]]	[[monoidal 2-groupoid]]	...	[[A-∞-space]]/[[E1-algebra]]

$2$	\"	[[commutative monoid]]	[[braided monoidal groupoid]]	[[braided monoidal 2-groupoid]]	...	[[E2-algebra]]

$3$	\"	\"	[[symmetric monoidal groupoid]]	[[sylleptic monoidal 2-groupoid]]	...	[[E3-algebra]]

$4$	\"	\"	\"	[[symmetric monoidal 2-groupoid]]	...	[[E4-algebra]]

⋮	\"	\"	\"	\"	⋱	⋮

$\infty$	trivial	[[commutative monoid]]	[[symmetric monoidal groupoid]]	[[symmetric monoidal 2-groupoid]] 	...	[[E-∞-algebra]]

## See also

* [[k-tuply associative n-groupoid]]
* [[k-tuply groupal n-groupoid]]
* [[k-tuply monoidal (n,r)-category]]

[[!redirects k-tuply monoidal n-groupoids]]


K. Jaakko J. Hintikka

* [Wikipedia entry](https://en.wikipedia.org/wiki/Jaakko_Hintikka)

## Selected writings

Early discussion of [[epistemic modal logic]] and introducing the (terminology of) [[K modal logic]] for its formalization:

* [[K. Jaakko J. Hintikka]], *Knowledge and belief: An introduction to the logic of the two notions*, Cornell University Press (1962) [[ark:/13960/t9k437s65](https://archive.org/details/knowledgebeliefi00hint_0)]

category: people

[[!redirects Jaakko Hintikka]]


K. P. Yogendran

* [institute page](https://web.iisermohali.ac.in/dept/physics/K_P_Yogendran.html)

## Selected writings

On the [[AdS-QCD correspondence]] (embedding the "hard-wall model" into [[type IIB string theory]]):

* [[Akash Singh]], [[K. P. Yogendran]], *Phases of a 10-D Holographic hard wall model* [[arXiv:2208.09387](https://arxiv.org/abs/2208.09387)]

category: people


K3 > history

see [[K3 surface]]


K3 surface

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

#Contents#
* table of contents
{:toc}

## Definition

A **K3 surface** is a [[Calabi-Yau variety]] of [[dimension]] $2$ (a Calabi-Yau [[algebraic surface]]/[[complex surface]]). This means that the [[canonical bundle]] $\omega_X=\wedge^2\Omega_X\simeq \mathcal{O}_X$ is trivial and $H^1(X, \mathcal{O}_X)=0$.

The term "K3" is

> in honor of Kummer, Kähler, Kodaira, and the beautiful K2 mountain in Kashmir

([Weil 79, p. 546](#Weil79))

## Examples

* A cyclic cover of $\mathbb{P}^2$ branched over a curve of degree $6$.

* A nonsingular degree $4$ hypersurface in $\mathbb{P}^3$, such as the [[Fermat hypersurface|Fermat quartic]] $\{[w,x,y,z] \in \mathbb{P}^3 | w^4 + x^4 + y^4 + z^4 = 0\}$ (in fact every K3 surface over $\mathbb{C}$ is [[diffeomorphism|diffeomorphic]] to this example).

* The [[flat orbifold]] quotient of the [[4-torus]] by the sign [[involution]] on all four canonical [[coordinates]] is the flat compact 4-dimensional orbifold known as the _Kummer surface_ $T^4 \sslash \mathbb{Z}_2$, a singular [[K3-surface]] (e.g. [Bettiol-Derdzinski-Piccione 18, 5.5](Riemannian+orbifold#BettiolDerdzinskiPiccione18))

## Properties

### Compact hyperkähler structure

Over the [[complex numbers]] [[K3 surfaces]] are all [[Kähler manifold|Kähler]], and even [[hyperkähler manifold|hyperkähler]].

The only known examples of [[compact hyperkähler manifolds]] are [[Hilbert schemes of points]] $X^{[n+1]}$ (for $n \in \mathbb{N}$) for $X$ either

1. a [[K3-surface]]

1. a [[4-torus]] (in which case the [[compact hyperkähler manifolds]] is really the [[fiber]] of $(\mathbb{T}^4)^{[n]} \to \mathbb{T}^4$)

([Beauville 83](compact+hyperkähler+manifold#Beauville83))
and two exceptional examples ([O'Grady 99](compact+hyperkähler+manifold#OGrady99), [O'Grady 03](compact+hyperkähler+manifold#OGrady03)
), see [Sawon 04, Sec. 5.3](compact+hyperkähler+manifold#Sawon04).

### Homotopy

* All K3 surfaces are [[simply connected]].

### Cohomology

+-- {: .num_prop #IntegralCohomology}
###### Proposition
**([[integral cohomology]] of [[K3-surface]])**

The [[integral cohomology]] of a K3-surface $X$ is

$$
  H^n(X,\mathbb{Z})
  \;\simeq\;
  \left\{
  \array{
    \mathbb{Z} &\vert& n = 0
    \\
    0 &\vert& n = 1
    \\
    \mathbb{Z}^{22} &\vert& n = 2
    \\
    0 &\vert& n = 3
    \\
    \mathbb{Z} &\vert& n = 4
  }
  \right.
$$

=--

(e.g. [Barth-Peters-Van den Ven 84, VIII Prop. 3.2](#BarthPetersVandenVen84))

+-- {: .num_prop #BettiNumbers}
###### Proposition
**([[Betti numbers]] of a [[K3-surface]])**

The [[Hodge diamond]] is completely determined (even in positive [[characteristic]]) and hence the  [[Hodge-de Rham spectral sequence]] degenerates at $E_1$. This also implies that the [[Betti numbers]] are completely determined as $1, 0, 22, 0, 1$:

$$
  \array{
    && h^{0,0}
    \\
    & h^{1,0} && h^{0,1}
    \\
    h^{2,0} & & h^{1,1} & & h^{0,2}
    \\
    & h^{2,1} & & h^{1,2}
    \\
    && h^{2,2}
  }
  \;\;\;=\;\;\;
  \array{
    && 1
    \\
    & 0 && 0
    \\
    1 & & 20 & & 1
    \\
    & 0 & & 0
    \\
    &&
    1
  }
$$

=--

(e.g. [Barth-Peters-Van den Ven 84, VIII Prop. 3.3](#BarthPetersVandenVen84))

\linebreak

### SU-Bordism
 {#SUBordism}

+-- {: .num_prop #K3SurfaceSpansSUBordismRingInDegree4}
###### Proposition
**([[K3-surface]] spans [[SU-bordism ring]] in degree 4)**

The canonical degree-4 generator $y_4 \in \Omega^{SU}_4$ in the [[SU-bordism ring]] ([this Prop.](MSU#SUBordismRingAwayFromTwo)) is represented by minus the class of any (non-[[torus]]) [[K3-surface]]:

$$
  \Omega^{SU}_4
    \;\simeq\;
  \mathbb{Z}\big[ \tfrac{1}{2}\big]\big\langle -[K3] \big\rangle
  \,.
$$

=--

([LLP 17, Lemma 1.5, Example 3.1](MSU#LLP17), [CLP 19, Theorem 13.5a](MSU#CLP19), see at _[[Calabi-Yau manifolds in SU-bordism theory]]_)

\linebreak

### Characteristic classes
  {#CharacteristicClasses}

We discuss some [[characteristic classes]] of (the [[tangent bundle]] of) [[K3]], and their evaluation on (the [[fundamental class]] of) $K3$ (i.e. their [[integration]] over $K3$).

#### Of $K3$

+-- {: .num_prop #EulerCharacteristicOfK3}
###### Proposition
**([[Euler characteristic]] of [[K3]])**

The [[Euler characteristic]] of [[K3]] is 24:

$$
  \chi_4[K3] \;=\; 24
  \,.
$$

=--

+-- {: .num_prop #FirstChernClassOfK3}
###### Proposition
**([[first Chern class]] of [[K3]])**

The [[first Chern class]] of [[K3]] vanishes:

$$
  c_1(K3) \;=\; 0
  \,.
$$

=--

+-- {: .num_prop #SecondChernClassOfK3}
###### Proposition
**([[second Chern class]] of [[K3]])**

The [[second Chern class]] of [[K3]] evaluates to 24:

$$
  c_2[K3] \;=\; 24
  \,.
$$

=--

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

For every [[closed manifold|closed]] [[complex manifold]] the evaluation of the top degree [[Chern class]] equals the [[Euler characteristic]]. Hence the statement follows from Prop. \ref{EulerCharacteristicOfK3}.

=--

+-- {: .num_prop #FirstPontryaginClassOfK3}
###### Proposition
**([[first Chern class]] of [[K3]])**

The [[first Pontryagin class]] of [[K3]] evaluates to 48:

$$
  p_1[K3] \;=\; -48
  \,.
$$

=--

(see also e.g. [Duff-Liu-Minasian 95 (5.10)](I8#DuffLiuMinasian95))

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

For a [[complex manifold]] the [[first Pontryagin class]] is the following [[polynomial]] in the [[first Chern class]] and the [[second Chern class]].
By Prop. \ref{FirstChernClassOfK3} the first Chern class vanishes,
and by Prop. \ref{SecondChernClassOfK3} the second Chern class evaluates to 24:

$$
  \begin{aligned}
    p_1[K3]
      & =\;
    \underset{= 0}{\underbrace{c_1 \cup c_1}}[K3]
    - 2
    \underset{24}{\underbrace{c_2[K3]}}
    \\
    & = - 48
  \end{aligned}
  \,.
$$

=--

#### Of $K3 \times K3$

Now consider the [[Cartesian product|Cartesian]] [[product space]] $K3 \times K3$.

See also at _[[C-field tadpole cancellation]]_ the section _[Integrality on $K3 \times K3$](C-field+tadpole+cancellation#IntegralityOnK3TimesK3)_.

+-- {: .num_prop #EulerCharacteristicOfK3TimesK3}
###### Proposition
**([[Euler characteristic]] of [[K3]]$\times$[[K3]])**

The [[Euler characteristic]] of $K3 \times K3$ is $24^2$:

$$
  \chi_8[K3\times K3] \;=\; 24^2
  \,.
$$

=--

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

By the [Whitney sum formula for the Euler class](Euler%20class#EulerClassOfWhitneySumIsCupProductOfEulerClasses) we have $\chi_8[K3 \times K3] = (\chi_4[K3])^2$. Hence the statement follows by Prop. \ref{EulerCharacteristicOfK3}.

=--

+-- {: .num_prop #FirstPontryaginClassOfK3TimesK3}
###### Proposition
**([[first Pontryagin class]] of [[K3]]$\times$[[K3]])**

The [[first Pontryagin class]] evaluated on $K3 \times K3$ is:

$$
  p_1[K3 \times K3]
  \;=\;
  - 2 \times 48
$$

=--

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

By the general formula for [[Pontryagin classes]] of [[product spaces]] we have

$$
  \begin{aligned}
    p_1(K3 \times K3)
    & =\;
    p_0(K3) \smile p_1(K3)
    +
    p_1(K3) \smile p_0(K3)
    \\
    & =
    2 p_1(K3)
  \end{aligned}
$$

With this, the statement follows by Prop. \ref{FirstPontryaginClassOfK3}.

=--

#### Of $K3 \times X^4$

Now consider the [[Cartesian product|Cartesian]] [[product space]] $K3 \times X^4$ of [[K3]] with some [[4-manifold]].

+-- {: .num_prop #EulerCharacteristicOfK3TimesK3}
###### Proposition
**([[Euler characteristic]] of [[K3]]$\times X^4$)**

The [[Euler characteristic]] of $K3 \times K3$ is $24$ times the Euler characteristic of $X^4$:

$$
  \chi_8[K3\times X^4] \;=\; 24 \cdot \chi_4[X^4]
  \,.
$$

=--

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

By the [Whitney sum formula for the Euler class](Euler%20class#EulerClassOfWhitneySumIsCupProductOfEulerClasses) we have $\chi_8[K3 \times X^4] = \chi_4[K3] \cdot \chi_4[X^4]$. Hence the statement follows by Prop. \ref{EulerCharacteristicOfK3}.

=--

+-- {: .num_prop #FirstPontryaginClassOfK3TimesK3}
###### Proposition
**([[first Pontryagin class]] of [[K3]]$\times X^4$)**

The [[first Pontryagin class]] evaluated on $K3 \times X^4$ is:

$$
  p_1[K3 \times X^4]
  \;=\;
  p_1[X^4] - 48
$$

=--

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

By the general formula for [[Pontryagin classes]] of [[product spaces]] we have

$$
  \begin{aligned}
    p_1[K3 \times X^4]
    & =\;
    p_0[K3] \smile p_1[X^4]
    +
    p_1[K3] \smile p_0[X^4]
    \\
    & =
    p_1[X^4] + p_1[K3]
  \end{aligned}
$$

With this, the statement follows by Prop. \ref{FirstPontryaginClassOfK3}.

=--

### Moduli of higher line bundles and deformation theory
 {#ModuliOfHigherLineBundles}

In [[positive number|positive]] [[characteristic]] $p$:

The [[Néron-Severi group]] of a K3 is a [[free abelian group]]

The [[formal Brauer group]] is

* either the formal [[additive group]], in which case it has [[height of a formal group|height]] $h = \infty$, by definition;

* or its [[height of a formal group|height]] is $1 \leq h \leq 10$, and every value may occur

([Artin 74](#Artin74)), see also ([Artin-Mazur 77, p. 5 (of 46)](#ArtinMazur77))

[[!include moduli of higher lines -- table]]

### Relation to third stably framed bordism group

The [[third stable homotopy group of spheres]] (the third [[stable stem]]) is the [[cyclic group]] of [[order of a group|order]] 24:

$$
  \array{
    \pi_3^s &\simeq& \mathbb{Z}/24
    \\
    [h_{\mathbb{H}}] &\leftrightarrow& [1]
  }
$$

where the generator $[1] \in \mathbb{Z}/24$ is represented by the [[quaternionic Hopf fibration]] $S^7 \overset{h_{\mathbb{H}}}{\longrightarrow} S^4$.

Under the [[Pontrjagin-Thom isomorphism]], identifying the [[stable homotopy groups of spheres]] with the [[bordism ring]] $\Omega^{fr}_\bullet$ of [[stable framing|stably framed]] manifolds (see at _[[MFr]]_), this generator is represented by the [[3-sphere]] (with its left-invariant framing induced from the identification with the [[Lie group]] [[SU(2)]] $\simeq$ [[Sp(1)]] )

$$
  \array{
    \pi_3^s & \simeq & \Omega_3^{fr}
    \\
    [h_{\mathbb{H}}] & \leftrightarrow & [S^3]
    \,.
  }
$$

Moreover, the relation $2 4 [S^3] \,\simeq\, 0$ is represented by the [[bordism]] which is the [[complement]] of 24 [[open balls]] inside [[generalized the|the]] [[K3]]-manifold ([Wang-Xu 10, Sec. 2.6](third+stable+homotopy+group+of+spheres#WangXu10)).

### Elliptic fibration

See at _[[elliptically fibered K3-surface]]_.

### As a fiber space in string compactifications

See

* [[duality between heterotic and type II string theory]]

* [[F-theory on K3]]

* [[M-theory on Sp(1)-manifolds]]

## Related concepts

* [[formal Brauer group]]

* [[elliptic fibration]], [[elliptically fibered K3-surface]]

* [[K3-cohomology]]

* [[F-theory]], [[F-theory on K3]]

## References

### General

Original sources:

* {#Artin74} [[Michael Artin]], _Supersingular K3 Surfaces_, Annal. Sc. d, l'Éc Norm. Sup.  4e  séries, T. 7, fasc.  4, 1974, pp. 543-568

* {#Weil79} [[Andre Weil]], Final report on contract AF 18 (603)-57. In Scientific works. Collected papers. Vol. II (1951-1964). 1979.

Textbook accounts include

* {#BarthPetersVandenVen84} W. Barth, C. Peters, A. Van den Ven, chapter VII of _Compact complex surfaces_, Springer 1984

Lecture notes:

* {#Huybrechts16} [[Daniel Huybrechts]], _Lectures on K3-surfaces_,  Cambridge University Press  2016 ([pdf](http://www.math.uni-bonn.de/people/huybrech/K3Global.pdf), [[HuybrechtsLecturesOnK3.pdf:file]], [doi:10.1017/CBO9781316594193]( https://doi.org/10.1017/CBO9781316594193))

* [[Claire Voisin]], _Geometry of the Moduli Space of K3 surfaces_

* [[David Morrison]], _[The geometry of K3 surfaces](http://www.cgtp.duke.edu/ITP99/morrison/cortona.pdf)_ Lecture notes (1988)

* Viacheslav Nikulin, _Elliptic fibrations on K3 surfaces_ ([arXiv:1010.3904](http://arxiv.org/abs/1010.3904))

Discussion of the [[deformation theory]] of K3-surfaces (of their [[Picard schemes]]) is (see also at _[[Artin-Mazur formal group]]_) in

* {#ArtinMazur77} [[Michael Artin]], [[Barry Mazur]], _Formal Groups Arising from Algebraic Varieties_, Annales scientifiques de l'École Normale Supérieure, Sér. 4, 10 no. 1 (1977), p. 87-131  [numdam](http://www.numdam.org/item?id=ASENS_1977_4_10_1_87_0), [MR56:15663](http://www.ams.org/mathscinet-getitem?mr=56:15663)

Systematic construction of [[Ricci tensor|Ricci flat]] [[Riemannian metrics]] on [[K3]] [[orbifolds]]:

* [[Shamit Kachru]], [[Arnav Tripathy]], [[Max Zimet]], _K3 metrics_ ([arXiv:2006.02435](https://arxiv.org/abs/2006.02435))

* [[Arnav Tripathy]], [[Max Zimet]], _A plethora of K3 metrics_ ([arXiv:2010.12581](https://arxiv.org/abs/2010.12581))

### Elliptic fibrations

* {#Lecacheux19} O. Lecacheux, _Weierstrass Equations for the Elliptic Fibrations of a K3 Surface_ In: Balakrishnan J., Folsom A., Lalín M., Manes M. (eds.) _Research Directions in Number Theory Association for Women in Mathematics Series, vol 19. Springer (2019) ([doi:10.1007/978-3-030-19478-9_4](https://doi.org/10.1007/978-3-030-19478-9_4))

* Marie Bertin, _Elliptic Fibrations on K3 surfaces_, 2013 ([pdf](https://webusers.imj-prg.fr/~marie-jose.bertin/WINEurope13.pdf))

### In string theory
  {#InStringTheory}

In [[string theory]], the [[KK-compactification]] of [[type IIA string theory]]/[[M-theory]]/[[F-theory]] on K3-[[fibers]] is supposed to exhibit te [[duality between M/F-theory and heterotic string theory]], originally due to

* [[Chris Hull]], [[Paul Townsend]], section 6 of _Unity of Superstring Dualities_, Nucl.Phys.B438:109-137,1995 ([arXiv:hep-th/9410167](http://arxiv.org/abs/hep-th/9410167))

* {#Witten95} [[Edward Witten]], section 4 of _[[String Theory Dynamics In Various Dimensions]]_, Nucl.Phys.B443:85-126,1995 ([arXiv:hep-th/9503124](http://arxiv.org/abs/hep-th/9503124))

Review includes

* {#Aspinwall96} [[Paul Aspinwall]], _K3 Surfaces and String Duality_, in  [[Shing-Tung Yau]] (ed.): _Differential geometry inspired by string theory_ 1-95 ([arXiv:9611137](https://arxiv.org/abs/hep-th/9611137), [spire:426102](http://inspirehep.net/record/426102))

Further discussion includes

* [[Paul Aspinwall]], [[David Morrison]], _String Theory on K3 Surfaces_, in [[Brian Greene]], [[Shing-Tung Yau]] (eds.), _Mirror Symmetry II_, International Press, Cambridge, 1997, pp. 703-716 ([arXiv:hep-th/9404151](https://arxiv.org/abs/hep-th/9404151))

* [[Paul Aspinwall]], _Enhanced Gauge Symmetries and K3 Surfaces_, Phys.Lett. B357 (1995) 329-334 ([arXiv:hep-th/9507012](http://arxiv.org/abs/hep-th/9507012))

Specifically in relation to [[orbifold]] [[string theory]]:

* {#Wendland01} [[Katrin Wendland]], _Orbifold Constructions of K3: A Link between Conformal Field Theory and Geometry_, in _[[Orbifolds in Mathematics and Physics]]_ ([arXiv:hep-th/0112006](https://arxiv.org/abs/hep-th/0112006))

Specifically in relation to the putative [[K-theory]]-classification of [[D-brane charge]]:

* {#GarciaUranga05} Inaki Garcia-Etxebarria, [[Angel Uranga]], _From F/M-theory to K-theory and back_, JHEP 0602:008,2006 ([arXiv:hep-th/0510073](https://arxiv.org/abs/hep-th/0510073))

Specifically in [[M-theory on G2-manifolds]]:

* {#AtiyahWitten01} [[Michael Atiyah]], [[Edward Witten]] section 6.4 of _$M$-Theory dynamics on a manifold of $G_2$-holonomy_, Adv. Theor. Math. Phys. 6 (2001) ([arXiv:hep-th/0107177](http://arxiv.org/abs/hep-th/0107177))

Specifically in relation to [[Moonshine]]:

* [[Miranda Cheng]], Sarah M. Harrison, Roberto Volpato, Max Zimet, _K3 String Theory, Lattices and Moonshine_ ([arXiv:1612.04404](https://arxiv.org/abs/1612.04404))

Specifically in relation to [[little string theory]]:

* [[Shamit Kachru]], Arnav Tripathy, Max Zimet, _K3 metrics from little string theory_ ([arXiv:1810.10540](https://arxiv.org/abs/1810.10540))

[[!redirects K3 surfaces]]

[[!redirects K3]]

[[!redirects K3-surface]]
[[!redirects K3-surfaces]]

[[!redirects K3-manifold]]
[[!redirects K3-manifolds]]

[[!redirects Kummer surface]]
[[!redirects Kummer surfaces]]

[[!redirects generalized Kummer surface]]
[[!redirects generalized Kummer surfaces]]


K3-spectrum

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## Idea

In higher dimensional analogy of how the [[formal Picard group]] of an [[elliptic curve]] gives the [[formal group]] of an [[elliptic spectrum]] representing an [[elliptic cohomology theory]], so the [[formal Brauer group]] of a [[K3 surface]] gives the formal group of an [[complex oriented cohomology theory]] given by a [[spectrum]] hence called a _K3-spectrum_ [[Brown representability theorem|representing]] _K3-cohomology_ ([Szymik 10, section 4.2](#Szymik10)).

## Properties

### Existence

The [[formal Brauer groups]] $\Phi^2_{X}$ of [[K3 surfaces]] $X$ have [[height of a formal group|height]] in $\{1,2,3,4,5,6,7,8,9,10,\infty\}$, and all values appear. ([Artin 74](#Artin74), [Artin-Mazur 77](#ArtinMazur77)).

By the [[Landweber exact functor theorem]] there is a K3-spectrum associated with $\Phi^2_X$ if it is Landweber exact.

([Szymik 10, theorem 1](#Szymik10)) gives sufficient conditions for this to be the case and ([Szymik 10, prop. 7, prop. 8](#Szymik10)) say that these condition are satisfied for enough K3 surfaces to realize all formal Brauer groups (...add details...).

## Related concepts

[[!include moduli of higher lines -- table]]

[[!include chromatic tower examples - table]]

## References

Discussion of the [[formal groups]] given as the [[formal Brauer groups]] of [[K3-surfaces]] originates in

* {#ArtinMazur77} [[Michael Artin]], [[Barry Mazur]], _Formal Groups Arising from Algebraic Varieties_, Annales scientifiques de l'École Normale Supérieure, Sér. 4, 10 no. 1 (1977), p. 87-131  [numdam](http://www.numdam.org/item?id=ASENS_1977_4_10_1_87_0), [MR56:15663](http://www.ams.org/mathscinet-getitem?mr=56:15663)

based on

* {#Artin74} [[Michael Artin]], _Supersingular K3 Surfaces_, Annal. Sc. d, l'Éc Norm. Sup.  4e  séries, T. 7, fasc.  4, 1974, pp. 543-568

The general idea of [[Calabi-Yau cohomology]] apparently appears in

* [[Michael Hopkins]], lectures at Midwest Topology Seminar, 1992

Textbook account:

* {#Thomas02} [[Charles Thomas]], Chapter 10 of: _Elliptic cohomology_, Kluwer Academic, 2002 ([doi:10.1007/b115001](https://link.springer.com/book/10.1007/b115001), [pdf](https://link.springer.com/content/pdf/10.1007%2Fb115001.pdf))

Lecture notes:

* [[Paul VanKoughnett]], _K3 cohomology theories_, 2014 ([pdf](http://math.northwestern.edu/~dwilson/k3notes/Lecture1-K3Spectra.pdf))

See also:

* {#Nogami10} Jumpei Nogami, _On derived Calabi-Yau varieties_, University of Illinois at Chicago 2010 ([proquest](https://search.proquest.com/docview/814798658))

The suggestion that from the point of view of [[string theory]]/[[F-theory]] K3-cohomology, and more generally Calabi-Yau cohomology, is the required generalization of [[elliptic cohomology]] appears in

* {#Sati05} [[Hisham Sati]], p. 18 of: _The Elliptic curves in gauge theory, string theory, and cohomology_, JHEP 0603 (2006) 096 ([arXiv:hep-th/0511087](http://arxiv.org/abs/hep-th/0511087))

A discussion of some kind of K3-cohomology in terms of [[differential geometry]] appears in

* Jorge Devoto, _Quaternionic elliptic objects and K3-cohomology_ London Mathematical Society Lecture Note Series (No. 342) 05/2007 ([doi:10.1017/CBO9780511721489.004](https://doi.org/10.1017/CBO9780511721489.004))

The concepts of K3-spectrum as such as considered in

* {#Szymik10} [[Markus Szymik]], _K3 spectra_, Bull. Lond. Math. Soc. 42 (2010) 137-148 ([pdf](http://www.math.ku.dk/~xvd217/szymik.K3spectra.pdf), [publisher](http://blms.oxfordjournals.org/content/early/2010/01/12/blms.bdp106.abstract))

* [[Markus Szymik]], _Crystals and derived local moduli for ordinary K3 surfaces_, Adv. Math. 228 (2011) 1-21

See also:

* {#Propp18} Oron Y. Propp, _Constructing explicit K3-spectra_ ([arXiv:1810.08953](https://arxiv.org/abs/1810.08953))

[[!redirects K3-spectra]]

[[!redirects K3 spectrum]]
[[!redirects K3 spectra]]

[[!redirects K3-cohomology]]
[[!redirects K3 cohomology]]


Kac algebra

## Idea

Kac algebras are one of the standard setups for [[Tannaka duality]] involving functional analysis. In a spirit of generalizing [[Pontrjagin duality]] to nonabelian locally compact groups they generalize locally compact groups and their Tannaka duals in a framework of [[von Neumann algebra]]s with additional structure where a [[multiplicative unitary]] plays a central role. They are named after [[George Kac]].

## Literature

The standard monograph on [[Kac algebra]]s is

* M. Enock, J. M. Schwartz, _Kac algebras and duality of locally compact groups_,  Springer-Verlag, 1992, , x+257 pp. [gBooks](http://books.google.com/books/about/Kac_algebras_and_duality_of_locally_comp.html?id=U6e6aD1gj3oC), [MR94e:46001](https://mathscinet.ams.org/mathscinet-getitem?mr=1215933)

See also

* Takehiko Yamanouchi, _An example of Kac algebra actions on Von Neumann algebra_, Proc. Amer. Math. Soc.. __119__, No. 2 (1993) 503-511
[jstor](https://www.jstor.org/stable/2159935)
* M. Enock, L. Vaînerman, _Deformation of a Kac algebra by an abelian subgroup_, Commun.Math. Phys. __178__ (1996) 571--595 [doi](https://doi.org/10.1007/BF02108816)
* Leonid Vaĭnerman, _2-cocycles and twisting of Kac algebras_, Commun. Math. Phys. __191__ (1998) 697-721 [doi](https://doi.org/10.1007/s002200050284)
* Teodor Banica, _Compact Kac algebras and commuting squares_, J. Funct. Anal. __176__ (2000) 80-99 arXiv:[math.QA/9909156](https://arxiv.org/abs/math/9909156)

A related, but different setup of Hopf-von Neumann algebras is in

* L. I. Vaĭnerman, [[G. I. Kac]], _Nonunimodular ring groups and Hopf-von Neumann algebras_,[doi](https://doi.org/10.1070/SM1974v023n02ABEH002176)
Math. USSR Sbornik __23__:2 (1974) 185
* [[Shahn  Majid]], _Hopf-von Neumann algebra bicrossproducts, Kac algebra bicrossproducts and
classical Yang-Baxter equations_, J. Funct. Anal. 95 (1991) 291-319

category: algebra, functional analysis

[[!redirects Kac algebras]]


Kac character formula

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## Idea

A formula for the [[Kac-Weyl character]] of [[loop group]] [[representations]].

([Pressley-Segal 88 (14.3.4)](#PressleySegal88), [Ando 00, 11.4](#Ando2000), [Ganter 12, corollary 8.6](#Ganter12))

## Related concepts

* [[Weyl character formula]]

## References

Originally due to [[Victor Kac]].

* Andrew Pressley, [[Graeme Segal]], _Loop groups_ Oxford University Press (1988)
  {#PressleySegal88}

* {#Ando2000} [[Matthew Ando]], _Power operations in elliptic cohomology and representations of loop groups_, Trans. Amer. Math. Soc., 352(12):5619-5666, 2000 ([pdf](http://www.math.uiuc.edu/~mando/papers/POECLG/poeclg.pdf))

Derivation from [[fiber integration in generalized cohomology|fiber integration]] in [[equivariant elliptic cohomology]] is in

* [[Nora Ganter]], _The elliptic Weyl character formula_ ([arXiv:1206.0528](http://arxiv.org/abs/1206.0528))
 {#Ganter12}

[[!redirects Kac character formulas]]

[[!redirects Kac-Weyl character formula]]
[[!redirects Weyl-Kac character formula]]


Kac-Moody algebra

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## Idea

The notion of _Kac-Moody Lie algebra_ is a generalization of that of _[[semisimple Lie algebra]]_ to infinite [[dimension]] of the underlying [[vector space]].

## Definition

(...)

## Examples

The sequence of [[exceptional Lie group|exceptional semisimple Lie algebras]] [[E7]], [[E7]], [[E8]] may be continued to the Kac-Moody algebras:

* affine: [[E9]]

* hyperbolic: [[E10]],

* Lorentzian: [[E11]], ...

## Related concepts

* [[Kac-Moody group]]

* [[higher Kac-Moody algebra]]

* [[Borcherds algebra]]

## References

### General

On the [[Wess-Zumino-Witten model]] [[2d CFT]] via [[Kac-Moody algebra]] and [[Virasoro algebra]]:

* [[Peter Goddard]], [[David Olive]], *Kac-Moody and Virasoro algebras in relation to quantum physics*,  International Journal of Modern Physics A **01** 02 (1986) 303-414 [[doi:10.1142/S0217751X86000149](https://doi.org/10.1142/S0217751X86000149), [spire:18583](https://inspirehep.net/literature/18583)]

See also:

* wikipedia, _[Kac-Moody algebra](http://en.wikipedia.org/wiki/Kac%E2%80%93Moody_algebra)_

Lecture notes include

* {#Wassermann11} [[Antony Wassermann]], _Kac-Moody and Virasoro algebras_, course notes (2011) ([pdf](https://www.dpmms.cam.ac.uk/~ajw/course11.pdf))

* [[Hermann Nicolai]], _Infinite dimensional symmetries_ (2009) ([pdf](http://www.itp.uni-hannover.de/~lechtenf/Events/Lectures/nicolai.pdf))

The standard textbook is

* [[Victor G. Kac]], *Infinite Dimensional Lie Algebras*,  Progress in Mathematics **44** Springer 1983 [[doi:10.1007/978-1-4757-1382-4](https://link.springer.com/book/10.1007/978-1-4757-1382-4)], Cambridge University Press (1990) [[doi:10.1017/CBO9780511626234](https://doi.org/10.1017/CBO9780511626234)]

Collections of articles include

* N. Sthanumoorty, K. Misra (eds.), _Kac-Moody Lie algebras and related topics_, Contemporary Mathematics 343 AMS (2002)

### The $E$-series

* [[E7]], [[E8]], [[E9]], [[E10]], [[E11]], ...

Surveys include

* wikipedia, _[En](http://en.wikipedia.org/wiki/En_%28Lie_algebra%29)_

The fact that every simply laced hyperbolic Kac-Moody algebra appears as a subalgebra of [[E10]] is in

* Sankaran Viswanath, _Embeddings of hyperbolic Kac-Moody algebras into $E_{10}$_,
Letters in Mathematical Physics **83** (2008) Issue 2, pp 139-148, doi:[10.1007/s11005-007-0214-7](http://dx.doi.org/10.1007/s11005-007-0214-7), [arXiv:0801.2586](http://arxiv.org/abs/0801.2586).

### Affine Lie algebras

As far as applications this is the most important class.
See $n$Lab entry [[affine Lie algebra]] and

* [[David Hernandez]], _An introduction to affine Kac-Moody algebras_ ([pdf](http://www.ctqm.au.dk/events/2006/October/Week42/Masterclassnotes.pdf))

### In supergravity

The following references discuss aspects of the Kac-Moody [[exceptional geometry]] of [[supergravity]] theories.

* [[Hermann Nicolai]], _Infinite dimensional symmetries_ (2009) ([pdf](http://www.itp.uni-hannover.de/~lechtenf/Events/Lectures/nicolai.pdf))

* [[Paul Cook]], _Connections between Kac-Moody algebras and M-theory_ PhD thesis ([arXiv:0711.3498](http://arxiv.org/abs/0711.3498))

* Daniel Persson, Nassiba Tabti, _Lectures on Kac-Moody algebras with applications in (Super-)Gravity_ ([pdf](http://www.ulb.ac.be/sciences/ptm/pmif/Rencontres/KMModaveLectures2007.pdf))

[[!redirects Kac-Moody algebras]]

[[!redirects Kac-Moody Lie algebra]]
[[!redirects Kac-Moody Lie algebras]]


Kac-Moody group

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## Idea

A [[group]] corresponding to a [[Kac-Moody algebra]] is a Kac-Moody group.

This subsumes the case of [[affine Lie algebras]] in which case the corresponding Kac-Moody group is a centrally extended [[loop group]] (see there for more) and often the words "Kac-Moody group" are used synonomously with "centrally extended loop group".

## Properties

### Classifying spaces

For $G$ a [[topological group]], write $B G$ for its [[delooping]], the [[classifying space]] for topological $G$-[[principal bundles]].

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

The classifying space $B G$ of every Kac-Moody group is a [[homotopy colimit]] over classifying spaces $B G_i$, $i \in I$ of [[compact topological space|compact]] [[connected topological space|connected]] [[Lie groups]] $G_i$:

$$
  B G \simeq hocolim_i B G_i
$$

in [[Top]] $\simeq$ [[∞Grpd]].

Moreover, the [[diagram]] $I$ may be taken to be a [[sieve]] in the [[poset of subobjects]] of the $n$-element set, for some $n \in \mathbb{B}$.

=--

This is due to [[Nitu Kitchloo]], 1998, see for instance [Kitchloo's survey, p. 9](#KitchlooSurvey).

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

This means that the classifying space of every Kac-Moody group has a smooth refinement to a [[smooth infinity-groupoid|smooth]] [[moduli stack]] given by forming

$$
  \mathbf{B}G \coloneqq hocolim_i \mathbf{B}G_i
$$

in [[Smooth∞Grpd]].

=--

## Related entries

* [[Kac-Moody algebra]], [[loop group]]

* [[G2]], [[F4]],

  [[E6]], [[E7]],  [[E8]], [[E9]], [[E10]], [[E11]], $\cdots$

Kay moody groups appear as [[U-duality]] groups in [[11-dimensional supergravity]] [[Kaluza-Klein mechanism|compactified]] to low dimensions.

## References

A standard textbook is

* Shrawan Kumar, _Kac-Moody groups, their flag varieties and representation theory_, Progress in Mathematics, 204. Birkhauser Boston, Inc., Boston, MA, (2002)

A survey is

* Nitu Kitchloo, _Kac-Moody groups over the last decade_ ([pdf](http://www.math.ucsd.edu/~nkitchlo/papers/MPI%20Bonn.pdf))
 {#KitchlooSurvey}

Original articles include

* Nitu Kitchloo, _On the Topology of Kac-Moody groups_ ([arXiv](http://arxiv.org/abs/0810.0851)), ([Phd thesis](http://www.math.jhu.edu/~nitu/papers/Thesis.pdf))

* Carles Broto, Nitu Kitchloo, _Classifying spaces of Kac-Moody groups_, Math. Z. 240,621–649 (2002) ([pdf](http://www.math.jhu.edu/~nitu/papers/fulltext-2.pdf))

* Udo Baumgartner, Jacqueline Ramagge, Bertrand Remy, _Contraction groups in complete Kac-Moody groups_ (2008) ([pdf](http://ro.uow.edu.au/cgi/viewcontent.cgi?article=2260&context=infopapers))

* Andreas Mars, _On the topology and geometry of Kac-Moody groups_, PhD thesis (2011) ([web](http://tuprints.ulb.tu-darmstadt.de/2496/))

* Christof Geiss, Bernard Leclerc, Jan Schröer, _Kac-Moody groups and cluster algebras_ ([arXiv:1001.3545](http://arxiv.org/abs/1001.3545))

[[!redirects Kac-Moody groups]]


Kac-Weyl character

#Contents#
* table of contents
{:toc}

## Idea

A kind of [[character]] for [[loop group representations]] (e.g. [Gordon 08, def 5.11](#Gordon08)).

The _[[Weyl-Kac character formula]]_ expresses it for integrable highest weigth representations in terms of the weights (e.g. [Gordon 08, theorem 18.16](#Gordon08)).

## Properties

### Jacobi form and relation to Witten genus

On an integral highest weight representation the KW character is a [[Jacobi form]], closely related to the [[Witten genus]] ([KL 95](#KL95)). See also at _[[equivariant elliptic cohomology]]_.

## References

* {#Gordon08} [[Iain Gordon]], _Infinite-dimensional Lie algebras_ (2008/9) ([pdf](http://www.maths.ed.ac.uk/~igordon/LA1.pdf)

* {#KL95} [[Kefeng Liu]], section 2.2 of _On modular invariance and rigidity theorems_, J. Differential Geom. Volume 41, Number 2 (1995), 247-514 ([EUCLID](http://projecteuclid.org/euclid.jdg/1214456221), [pdf](http://www.math.ucla.edu/~liu/Research/loja.pdf))

* {#Wassermann11} [[Antony Wassermann]], _Kac-Moody and Virasoro algebras_, course notes (2011) ([pdf](https://www.dpmms.cam.ac.uk/~ajw/course11.pdf))

[[!redirects Kac-Weyl characters]]


Kadeishvili's theorem

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## Idea

Kadeishvili's theorem says that every [[A-∞-algebra]] (in [[chain complexes]]) is equivalent, as an $A_\infty$-algebra, to its [[cohomology]] ring.

## Statement

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

Let $A$ be an [[A-∞-algebra]] in [[chain complexes]] and let $H^\bullet(A)$ be the [[cohomology]] ring of $A$. There is an $A_\infty$-algebra structure on $H^\bullet(A)$ with $m_1 = 0$ and $m_2$ induced by the multiplication on $A$, constructed from the $A_\infty$-structure of $A$, such that there
is a [[quasi-isomorphism]] of $A_\infty$-algebras $H^\bullet(A) \to A$ lifting the identity of $H^\bullet(A)$. This $A_\infty$-algebra structure on $H^\bullet(A)$ is unique up to quasi-isomorphism.

=--

This is due to ([Kadeishvili](#Kadeishvili)). A clear English exposition with applications to Kähler manifolds is in ([Merkulov](#Merkulov)).

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

The $A_\infty$-brackets on $H^\bullet(A)$ by theorem \ref{KadeishviliTheorem} are up to a sign equal to the [[Massey products]] on cohomology, whenever the latter are defined.

=--

References for this statement are listed at _[Massey product -- Relation to A-infinity algebra](Massey+product#ReferencesRelationToAInfinity)_.

## References

* [[Tornike Kadeishvili|T. V. Kadeishvili]], _On the theory of homology of fiber spaces_, (Russian) International Topology Conference (Moscow State Univ., Moscow, 1979). Uspekhi Mat. Nauk 35 (1980), no. 3 (213), 183--188. Translated in Russ. Math. Surv. 35 (1980), no. 3 (238) 231--211
 {#Kadeishvili}

* [[S. A. Merkulov]], _Strong homotopy algebras of a Kähler manifold_, Internat. Math. Res. Notices 1999, no. 3, 153–164.
 {#Merkulov}


Kadison-Singer problem

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## Idea

Given a [[separable Hilbert space]] $\mathcal{H}$, such as the [[sequence space]] $\mathcal{l}_2$, write $B(\mathcal{H})$ for its [[C*-algebra]] of [[bounded linear operators]] and $D(\mathcal{H})$ for the subalgebra of diagonal operators.

The _Kadison-Singer problem_ is the question:

_Does every [[pure state]] $\psi$ on $D(\mathcal{H})$ [[extension|extend]] to a [[state]] $\rho$ on $B(\mathcal{H})$?_

$$
  \array{
     D(\mathcal{H}) &\overset{\psi}{\longrightarrow}& \mathbb{C}
     \\
     \downarrow  & \nearrow_{\mathrlap{\exists \rho}}
     \\
     B(\mathcal{H})
  }
$$

This was proven to be the case in [Marcus-Spielman-Srivastava 13](#MarcusSpielmanSrivastava13).

## Related entries

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

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

  * [[Kochen-Specker theorem]]

  * [[Alfsen-Shultz theorem]]

  * [[Harding-Döring-Hamhalter theorem]]

* [[Fell's theorem]]

* [[Gleason's theorem]]

* [[Wigner theorem]]

* [[Bell's theorem]]

* [[Bub-Clifton theorem]]

## References

The problem was stated in

* [[Richard V. Kadison]], [[Isadore Singer]], _Extensions of pure states_, American Journal of Mathematics, 81(2):383–400, 1959 ([jstor:2372748](https://www.jstor.org/stable/2372748))

and its solution was proven in

* {#MarcusSpielmanSrivastava13} Adam Marcus, Daniel A. Spielman, and Nikhil Srivastava, _Interlacing Families II: Mixed Characteristic Polynomials and The Kadison-Singer Problem, 2013. ([arXiv:1306.3969](https://arxiv.org/abs/1306.3969))

Review includes

* Nicholas Harvey, _An introduction to the Kadison-Singer conjecture and the paving conjecture_ ([pdf](http://www.cs.ubc.ca/~nickhar/Publications/KS/KS.pdf))


Kae Nemoto

* [institute page](https://qis1.ex.nii.ac.jp/quantumCenter/p_nemoto.html)

* [Wikipedia entry](https://en.wikipedia.org/wiki/Kae_Nemoto)

## Selected writings

On "continuous variable" or *analog* [[quantum computation]]:

* [[Vivien M. Kendon]], [[Kae Nemoto]], [[William J. Munro]], *Quantum Analogue Computing*, Phil. Trans. R. Soc. A **368** 1924 (2010) 3621-3632 [[arXiv:1001.2215](https://arxiv.org/abs/1001.2215), [doi:10.1098/rsta.2010.0017](https://doi.org/10.1098/rsta.2010.0017)]

On [[quantum error correction]]:

* {#DevittNemotoMunro09} [[Simon J. Devitt]], [[Kae Nemoto]], [[William J. Munro]], *Quantum Error Correction for Beginners*, Rep. Prog. Phys. **76** (2013) 076001 [[arXiv:0905.2794](https://arxiv.org/abs/0905.2794), [doi:10.1088/0034-4885/76/7/076001](https://iopscience.iop.org/article/10.1088/0034-4885/76/7/076001)]

category: people