Monster vertex operator algebra

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## Idea

The _Monster vertex algebra_ is a [[super vertex operator algebra]] whose [[automorphism group|group of]] [[automorphisms of a super VOA]] is the [[monster group]]. ([Frenkel-Lepowski-Meurman 89](#FrenkelLepowskiMeurman89), [Griess-Lam 11](#GriessLam11)).

## Related entries

* [[Moonshine]]

## References

* {#Borcherds86} [[Richard Borcherds]], _Vertex algebras, Kac-Moody algebras, and the Monster_, PNAS May 1, 1986 83 (10) 3068-3071 ([doi:10.1073/pnas.83.10.3068](https://doi.org/10.1073/pnas.83.10.3068), [pdf](https://pdfs.semanticscholar.org/a3bb/2d2772ccc0c1582f0216f9ab87c1b8a11eb6.pdf))

* {#DixonGinspargHarvey88} [[Lance Dixon]], [[Paul Ginsparg]], [[Jeffrey Harvey]], _Beauty and the Beast: Superconformal Symmetry in a  Monster Module_, Commun, Math.  Phys. 119, 221-241 (1988)  ([euclid:1104162400](https://projecteuclid.org/euclid.cmp/1104162400), [doi:10.1007/BF01217740](https://doi.org/10.1007/BF01217740),  [pdf](http://hamilton.uchicago.edu/~harvey/pdf_files/beauty&beast.pdf))

* {#FrenkelLepowskiMeurman89} [[Igor Frenkel]], [[James Lepowsky]], Arne Meurman, _Vertex operator algebras and the monster_, Pure and Applied Mathematics __134__, Academic Press, New York 1989. liv+508 pp. [MR0996026](http://www.ams.org/mathscinet-getitem?mr=996026)

* {#BorcherdsConwayQueenSloane} [[Richard Borcherds]],[[John Conway]], L. Queen, N. J. A. Sloane, _A Monster Lie Algebra?_, In: _Sphere Packings, Lattices and Groups_, Grundlehren der mathematischen Wissenschaften (A Series of Comprehensive Studies in Mathematics), vol 290. Springer 1993 ([doi:10.1007/978-1-4757-2249-9_30](https://doi.org/10.1007/978-1-4757-2249-9_30), [pdf](http://neilsloane.com/doc/Me99.pdf))

* {#GriessLam11} [[Robert Griess]] Jr., Ching Hung Lam, _A new existence proof of the Monster by VOA theory_ ([arXiv:1103.1414](https://arxiv.org/abs/1103.1414))

* Lisa Carbone, Elizabeth Jurisich, Scott H. Murray, _Constructing a Lie group analog for the Monster Lie algebra_ ([arXiv:2002.06658](https://arxiv.org/abs/2002.06658))

See also

* Wikipedia, _[Monster vertex algebra](http://en.wikipedia.org/wiki/Monster_vertex_algebra)_

* Wikipedia, _[Monster Lie algebra](https://en.wikipedia.org/wiki/Monster_Lie_algebra)_

Relation to [[quantum error correction]]:

* [[Jeffrey A. Harvey]], [[Gregory W. Moore]], *Moonshine, Superconformal Symmetry, and Quantum Error Correction*,  J. High Energ. Phys. **2020**, 146 (2020). ([arXiv:2003.13700](https://arxiv.org/abs/2003.13700))

Possible relation to [[bosonic M-theory]]:

* [[Alessio Marrani]], [[Michael Rios]], [[David Chester]], _Monstrous M-theory_ ([arXiv:2008.06742](https://arxiv.org/abs/2008.06742))

Discussion in terms of [[conformal nets]]:

* [[Sebastiano Carpi]], [[Tiziano Gaudio]], [[Robin Hillier]], *From vertex operator superalgebras to graded-local conformal nets and back* [[arXiv:2304.14263](https://arxiv.org/abs/2304.14263)]

[[!redirects monster vertex operator algebra]]
[[!redirects Monster vertex operator algebra]]

[[!redirects monster VOA]]
[[!redirects Monster VOA]]

[[!redirects Moonshine vertex operator algebra]]
[[!redirects Moonshine VOA]]

[[!redirects moonshine vertex operator algebra]]
[[!redirects moonshine VOA]]

[[!redirects Monster vertex algebra]]
[[!redirects monster vertex algebra]]

[[!redirects Griess algebra]]
[[!redirects Griess algebras]]

[[!redirects Griess vertex operator algebra]]
[[!redirects Griess vertex operator algebras]]

[[!redirects Griess super vertex operator algebra]]
[[!redirects Griess super vertex operator algebras]]

[[!redirects Monster Lie algebra]]


Montague semantics

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[[!redirects Montague grammar]]

A formal approach to [[linguistics]] and natural language [[semantics]], due to [[Richard Montague]].
In essence, it applies [[Gottlob Frege|Frege's]] principle of [[compositionality]] to natural language.

In his own words:

> There is in my opinion no important theoretical difference between natural languages and the artificial languages of logicians; indeed, I consider it possible to comprehend the syntax and semantics of both kinds of language within a single natural and mathematically precise theory. On this point I differ from a number of philosophers, but agree, I believe, with [[Noam Chomsky|Chomsky]] and his associates.

(from [Montague 70b](#Montague70b))

## Related entries

* [[dependent type theoretic methods in natural language semantics]]

## References

The original articles:

* {#Montague70a} [[Richard Montague]], _English as a formal language_, Linguaggi nella societa e nella tecnica. Edizioni di Communita. pp. 188-221 (1970) ([[MontagueEnglishFormalLanguage.pdf:file]])

* {#Montague70b} [[Richard Montague]], _Universal Grammar_, Theoria 36 (3):373--398 (1970) ([doi:10.1111/j.1755-2567.1970.tb00434.x](https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1755-2567.1970.tb00434.x))

* {#Montague73} [[Richard Montague]], _The Proper Treatment of Quantification in Ordinary English_, Approaches to Natural Language (1973) ([[MontagueQuantification.pdf:file]])

Review:

* Dowty, D., Wall, R., and Peters, S., 1981, _Introduction to Montague Semantics_ (Synthese Language Library 11), Dordrecht: Reidel.

See also:

* [SEP: Montague Semantics](http://plato.stanford.edu/entries/montague-semantics/)

* Wikipedia, _[Montague grammar](https://en.m.wikipedia.org/wiki/Montague_grammar)_

[[!redirects Montague grammar]]


Monte Carlo method

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## Idea

(...)

## References

### General

See also:

* Wikipedia, *[Monte Carlo method](https://en.wikipedia.org/wiki/Monte_Carlo_method)*

* Wikipedia, *[Monte Carlo method in statistical mechanics](https://en.wikipedia.org/wiki/Monte_Carlo_method_in_statistical_mechanics)*

In [[statistical physics]]:

* Kurt Binder, *Monte Carlo Simulations in Statistical Physics*, in: *Encyclopedia of Complexity and Systems Science*, Springer (2009) [[doi:10.1007/978-0-387-30440-3_337](https://doi.org/10.1007/978-0-387-30440-3_337)]

For [[lattice field theory|lattice]] [[QCD]]:

* Mike Peardon, *Monte Carlo Simulations of Lattice QCD*, in: *QCD and Numerical Analysis III*, Lecture Notes in Computational Science and Engineering **47** (2005) [[doi:10.1007/3-540-28504-0_4](https://doi.org/10.1007/3-540-28504-0_4)]

### In lattice field theory

The application of Monte Carlo methods in [[lattice gauge field theory]] originates with:

* {#Creutz79} [[Michael Creutz]], *Confinement and the Critical Dimensionality of Space-Time*, Phys. Rev. Lett. **43**  (1979) 553 [[doi:10.1103/PhysRevLett.43.553](https://doi.org/10.1103/PhysRevLett.43.553)]

* [[Michael Creutz]], _Monte Carlo study of quantized $SU(2)$ gauge theory_, Phys. Rev. D **21** (1980) 2308-2315 [[doi:10.1103/PhysRevD.21.2308](https://doi.org/10.1103/PhysRevD.21.2308), [pdf](http://thy.phy.bnl.gov/~creutz/mypubs/pub037.pdf)]

* [[Michael Creutz]], *Monte Carlo study of renormalization in lattice gauge theory* Phys.Rev. D **23** (1981) 1815  [[doi:10.1103/PhysRevD.23.1815](https://doi.org/10.1103/PhysRevD.23.1815), [pdf](http://thy.phy.bnl.gov/~creutz/mypubs/pub045.pdf)&rbrack

* [[Michael Creutz]], Laurence Jacobs, Claudio Rebbi, *Monte Carlo computations in lattice gauge theories*, Physics Reports **95** 4 (1983) 201-282 [doi:10.1016/0370-1573(83)90016-9, [pdf](http://thy.phy.bnl.gov/~creutz/mypubs/pub068.pdf)]

[[!redirects Monte Carlo methods]]

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[[!redirects Monte Carlo simulation]]
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Montel topological vector space

# Contents
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## Idea

A Montel space is a [[topological vector space]] such that the [[strong dual]] has a compactness property that infinite dimensional [[normed spaces]] cannot have: The weak and strong topologies coincide.

Important examples are the spaces of [[distributions]].

As a counterexample take an infinite [[Hilbert space]] and a sequence of orthonormal vectors, this will converge to zero in the [[weak topology]] but does not converge in the [[strong topology]].

## Definition

A Montel space is a [[topological vector space]] that is [[Hausdorff space|Hausdorff]], [[locally convex space|locally convex]], [[barreled topological vector space|barreled]] and where every [[closed subset|closed]] [[bounded subset|bounded]] subset is [[compact space|compact]].

## Properties

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###### Proposition
The [[strong dual]] $E^'_b$ of a Montel space is a Montel space. Furthermore, on the bounded subsets of $E^'_b$, the strong and weak topologies coincide.
=--

## Examples

A normed space is a Montel space iff it is finite dimensional, because only then the closed unit ball is compact.

For an open subset $U \subset \mathbb{R}^n$ the spaces $C^{\infty}(U)$ and $C_0^{\infty}(U)$ are Montel spaces.

[[Montel's theorem]] of classical complex analysis states that the space $\mathcal{O}(U)$ of holomorphic functions on an open set $U \subset \mathbb{C}$ is a Montel space.

## References ##

[[!redirects Montel spaces]]
[[!redirects Montel space]]
[[!redirects Montel topological vector spaces]]


monus

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## Definition

Let $\mathcal{C}$ be a [[cartesian monoidal category]] with a [[power object]] [[functor]] $\mathcal{P}:\mathcal{C}\to\mathcal{C}$, and let $M$ be a [[commutative]] [[semigroup]] object in $\mathcal{C}$. There exists a binary endorelation $R = \left( (-) \leq (-)\right)$, $R\colon \mathcal{P}(M \times M)$ such that for all $a,b\colon M$, if there exists a $c\colon M$ such that $a + c = b$, then $(a \leq b) \colon R$. If $\leq$ is a [[preorder]] object, and if for all $a,b\colon M$, there exists a unique smallest element $c$ such that $(a \leq b + c) \colon R$, then $M$ is a __commutative monoid object with monus__, and $c$ is called the __monus__ of $a$ and $b$.

## Related concepts

* [[subtraction]]

* [[difference]]

## See also

* Wikipedia, _Monus_


Moonshine

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## Idea

In [[mathematics]], *Moonshine* refers to mysterious connections between the [[Monster simple group]] and the [[modular function]] $j$, the *[[j-invariant]]*.

There were a bunch of [[conjectures]] about this connection that were proved by [[Richard Borcherds]], en passant mentioning the existence of the [[Moonshine vertex algebra]] (constructed then later in [FLM 89](#FrenkelLepowskiMeurman89)). Nowadays there is also Moonshine for other [[simple groups]], by the work of J. Duncan. Eventually there should be an entry for the general moonshine phenomenon.

The whole idea of moonshine began with [[John McKay]]'s observation that the [[Monster group]]'s first nontrivial [[irreducible representation]] has [[dimension]] 196883, and the [[j-invariant]] $j(\tau)$ has the [[Fourier series]] expansion
$$
  j(\tau) = q^{-1} + 744 + 196884q + 21493760q^{2} + \dots
$$
where $q=\exp(i2\pi\tau)$, and famously 196883+1=196884. Thompson observed in (1979) that the other coefficients are obtained from the dimensions of Monster's irreducible representations.

But the [[Monster group]] was merely _conjectured_ to exist until Griess (1982) explicitly constructed it. The construction is horribly complicated (take the sum of three irreducible representations for the [[centralizer]] of an [[involution]] of...).

[Frenkel-Lepowski-Meurman 89](#FrenkelLepowskiMeurman89) constructed an infinite-dimensional [[module]] for the [[Monster vertex algebra]]. This is by a generalized [[Kac-Moody algebra]] via [[bosonic string theory]] and the [[Goddard-Thorn theorem|Goddard-Thorn "No Ghost" theorem]]. The [[Monster group]] [[group action|acts]] naturally on this "Moonshine Module" (denoted by $V\natural$).

To cut the story short, we end up getting from the Monster group to a module it acts on which is related to "modular stuff" (namely, the modular [[j-invariant]]). The idea [[Terry Gannon]] pitches is that Moonshine is a generalization of this association, it's a sort of "mapping" from "Algebraic gadgets" to "Modular stuff".

## Automorphism groups of vertex operator algebras
 {#AutomorphismGroupsOfVertexOperatorAlgebras}

Realizations of [[sporadic finite simple groups]] as [[automorphism of a vertex operator algebra|automorphism groups of vertex operator algebras]] in [[heterotic string theory]] and [[type II string theory]] (mostly on [[K3-surfaces]], see [[HET - II duality]]):

* The [[Conway group]] $Co_{0}$ is the [[automorphism group|group of]] [[automorphisms of a super VOA]] of the unique chiral [[number of supersymmetries|N=1]] [[super vertex operator algebra]] of [[central charge]] $c = 12$ without fields of [[conformal weight]] $1/2$

  ([Duncan 05](#Duncan05), see also [Paquette-Persson-Volpato 17, p. 9](#PaquettePerssonVolpato17))

* similarly, there is a super VOA, the _[[Monster vertex operator algebra]]_, whose [[automorphism group|group of]] of [[automorphisms of a VOA]] is the [[monster group]]

  ([Frenkel-Lepowski-Meurman 89](#FrenkelLepowskiMeurman89), [Griess-Lam 11](#GriessLam11))

## Related concepts

* [[Monster]]

* moonshine

  * [[Mathieu moonshine]]

  * [[umbral moonshine]]

  * [[O'Nan moonshine]]

* [[automorphism of a vertex operator algebra]]

## References

### General

* [[Richard Borcherds]], _What is Moonshine?,
 Proceedings of the International Congress of Mathematicians, Vol. I (Berlin, 1998).
_Doc. Math._ 1998, Extra Vol. I, 607–615 (electronic). [MR1660657](http://www.ams.org/mathscinet-getitem?mr=1660657) [arXiv:math/9809110v1](http://arxiv.org/abs/math/9809110) [math.QA]

* John F. R. Duncan, Michael J. Griffin, Ken Ono, _Moonshine_ ([arXiv:1411.6571](http://arxiv.org/abs/1411.6571))

* {#GriessLam11} [[Robert Griess]] Jr., Ching Hung Lam, _A new existence proof of the Monster by VOA theory_ ([arXiv:1103.1414](https://arxiv.org/abs/1103.1414))

* {#FrenkelLepowskiMeurman89} [[Igor Frenkel]], [[James Lepowsky]], Arne Meurman, _Vertex operator algebras and the monster_, Pure and Applied Mathematics __134__, Academic Press, New York 1989. liv+508 pp. [MR0996026](http://www.ams.org/mathscinet-getitem?mr=996026)

* [[Terry Gannon]], *Monstrous moonshine: the first twenty-five years*, _Bull. London Math. Soc._ **38** (2006), no. 1, 1–33.  [MR2201600](http://www.ams.org/mathscinet-getitem?mr=2201600) [arXiv:math/0402345](http://arxiv.org/abs/math/0402345) [math.QA]

* [[Terry Gannon]], _Moonshine beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics_, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, Massachusetts 2006. [MR2257727](http://www.ams.org/mathscinet-getitem?mr=2257727)

* Koichiro Harada, _"Moonshine" of finite groups_.
EMS Series of Lectures in Mathematics. European Mathematical Society (EMS), Zürich, 2010. viii+76 pp. [MR2722318](http://www.ams.org/mathscinet-getitem?mr=2722318)

* Griess, Robert L., Jr.; Lam, Ching Hung *A moonshine path from E8 to the Monster*, J. Pure Appl. Algebra_ 215 (2011), no. 5, 927–948 [MR2747229](http://www.ams.org/mathscinet-getitem?mr=2747229) [arXiv:0910.2057v2](http://arxiv.org/abs/0910.2057v2) [math.GR]

* Jae-Hyun Yang "Kac-Moody algebras, the Monstrous Moonshine, Jacobi forms and infinite products." _Number theory, geometry and related topics_ (Iksan City, 1995), 13–82, Pyungsan Inst. Math. Sci., Seoul, 1996. [MR1404967](http://www.ams.org/mathscinet-getitem?mr=1404967) [arXiv:math/0612474v2](http://arxiv.org/abs/math/0612474) [math.NT]

* Vassilis Anagiannis, [[Miranda Cheng]], _TASI Lectures on Moonshine_ ([arXiv:1807.00723](https://arxiv.org/abs/1807.00723))

### Historical References

* [[John Conway]] and Simon Norton, "Monstrous moonshine." _Bull. London Math. Soc._ **11** (1979), no. 3, 308--339; [MR0554399](http://www.ams.org/mathscinet-getitem?mr=554399) (81j:20028)

* {#FrenkelLepowskiMeurman89} [[Igor Frenkel]], [[James Lepowsky]], Arne Meurman, "A natural representation of the Fischer-Griess Monster with the modular function $J$ as character." _Proc. Nat. Acad. Sci. U.S.A._ **81** (1984), no. 10, Phys. Sci., 3256--3260. [MR0747596](http://www.ams.org/mathscinet-getitem?mr=747596) (85e:20018)

* [[Robert Griess]], "The friendly giant." _Invent. Math._ **69** (1982), no. 1, 1--102. [MR671653](http://www.ams.org/mathscinet-getitem?mr=671653) (84m:20024)

* John G. Thompson, "Some numerology between the Fischer-Griess Monster and the elliptic modular function." _Bull. London Math. Soc._ **11** (1979), no. 3, 352--353. [MR0554402](http://www.ams.org/mathscinet-getitem?mr=554402) (81j:20030)

### Further developments
 {#FurtherDevelomentsReferences}

* [[Miranda Cheng]], John F. R. Duncan, Jeffrey A. Harvey, _Umbral Moonshine_ ([arXiv:1204.2779](http://arxiv.org/abs/1204.2779))

* John F. R. Duncan, Michael J. Griffin and Ken Ono, _Proof of the Umbral Moonshine Conjecture_ ([arXiv:1503.01472](http://arxiv.org/abs/1503.01472))

* [[Scott Carnahan]], _Monstrous Moonshine over Z?_ ([arXiv:1804.04161](https://arxiv.org/abs/1804.04161))

Possible relation to [[elliptic cohomology]] ([[topological modular forms]]):

* Jan Albert, Justin Kaidi, Ying-Hsuan Lin, *Topological modularity of Supermoonshine* $[$[arXiv:2210.14923](https://arxiv.org/abs/2210.14923)$]$

### Realization in superstring theory

Discussion of possible realizations in [[superstring theory]] (specifically [[heterotic string theory]] and [[type II string theory]] in [[K3-surfaces]], see [[HET - II]]) via [[automorphisms of super vertex operator algebras]]:

* S. Chaudhuri, D.A. Lowe, _Monstrous String-String Duality_, Nucl. Phys. B469 : 21-36, 1996 ([arXiv:hep-th/9512226](https://arxiv.org/abs/hep-th/9512226))

* {#Duncan05} John F. Duncan, _Super-moonshine for Conway's largest sporadic group_ ([arXiv:math/0502267](https://arxiv.org/abs/math/0502267))

* {#PaquettePerssonVolpato16} [[Natalie Paquette]], Daniel Persson, Roberto Volpato, _Monstrous BPS-Algebras and the Superstring Origin of Moonshine_ ([arXiv:1601.05412](http://arxiv.org/abs/1601.05412))

* {#KachruPaquetteVolpato16} [[Shamit Kachru]], [[Natalie Paquette]], Roberto Volpato, _3D String Theory and Umbral Moonshine_ ([arXiv:1603.07330](http://arxiv.org/abs/1603.07330))

* {#PaquettePerssonVolpato17} [[Natalie Paquette]], Daniel Persson, Roberto Volpato, _BPS Algebras, Genus Zero, and the Heterotic Monster_ ([arXiv:1701.05169](https://arxiv.org/abs/1701.05169))

* [[Shamit Kachru]], Arnav Tripathy, _The hidden symmetry of the heterotic string_ ([arXiv:1702.02572](https://arxiv.org/abs/1702.02572))

Specifically in relation to [[KK-compactifications]] of [[string theory]] on [[K3-surfaces]] ([[duality between heterotic and type II string theory]])

* {#ChengHarrisonVolpatoZimet16} [[Miranda Cheng]], Sarah M. Harrison, Roberto Volpato, Max Zimet, _K3 String Theory, Lattices and Moonshine_ ([arXiv:1612.04404](https://arxiv.org/abs/1612.04404), [doi:10.1007/s40687-018-0150-4](https://doi.org/10.1007/s40687-018-0150-4))

Possible relation to [[bosonic M-theory]]:

* [[Alessio Marrani]], [[Michael Rios]], [[David Chester]], *Monstrous M-theory*, Symmetry **15** 2 (2023) 490; [[doi:10.3390/sym15020490](https://doi.org/10.3390/sym15020490), [arXiv:2008.06742](https://arxiv.org/abs/2008.06742)]

Relation to [[Seiberg-Witten theory]]:

* Shun'ya Mizoguchi, *Seiberg-Witten Theory and Monstrous Moonshine* [[arXiv:2208.10301](https://arxiv.org/abs/2208.10301)]

* Shun'ya Mizoguchi, Takumi Oikawa, Hitomi Tashiro, Shotaro Yata, *More on Seiberg-Witten Theory and Monstrous Moonshine* [[arXiv:2211.07251](https://arxiv.org/abs/2211.07251)]

[[!redirects moonshine]]
[[!redirects monstrous moonshine]]

[[!redirects Moonshine vertex algebra]]
[[!redirects moonshine vertex algebra]]

[[!redirects Moonshine vertex algebras]]
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Moore closure

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# Moore closures
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## Idea

The concept of _Moore closure_ is a very general idea of what it can mean for a [[set]] to be _closed_ under some condition.  It includes, as special cases, the operation of [[closed subspace|closure]] in a [[topological space]], many examples of generation of structures from [[base|bases]] and even [[subbase|subbases]], and generating [[subalgebras]] from subsets of an algebra.

Secretly, it is the same thing as the collection of subsets preserved by some [[monad]] on a [[power set]] (the subset of "[[modal types]]"). In fact it is a special case of the notion of [[closure operator]] or [[modality]] in [[logic]]/[[type theory]], namely the special case where the ambient [[category]]/[[hyperdoctrine]] is the [[topos]] [[Set]].

## Definitions

We give two equivalent definitions. The first one

* [In terms of closure condition](#InTermsOfClosureCondition)

gives the explicit condition for a [[subset]] of a [[power set]] to qualify as a Moore closure, the second

* [In terms of closure operators](#InTermsOfClosureOperators)

characterizes Moore closures as the collections of [[modal types]] of suitable [[closure operators]]. More abstractly, this characterizes Moore closures

* [In terms of monads](#InTermsOfMonads)

on the [[subobject lattice]] of the given set.

### In terms of closure condition
 {#InTermsOfClosureCondition}

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

Let $X$ be a [[set]], and let $\mathcal{C} \subset \mathcal{P}X$ be a collection of [[subsets]] of $X$.  Then $\mathcal{C}$ is a __Moore collection__ if every [[intersection]] of members of $\mathcal{C}$ belongs to $\mathcal{C}$.

That is, given a family $(A_i)_i$ of sets in $X$,
$$ \forall i,\; A_i \in \mathcal{C} \;\Rightarrow\; \bigcap_i A_i \in \mathcal{C} .$$

=--

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

Given any collection $\mathcal{B}$ whatsoever of subsets of $X$, the Moore collection __generated__ by $\mathcal{B}$ is the collection of all intersections of members of $\mathcal{B}$.

=--

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

This is indeed a Moore collection, and it equals $\mathcal{B}$ if and only if $\mathcal{B}$ is a Moore collection.

=--

### In terms of closure operators
 {#InTermsOfClosureOperators}

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

Again let $X$ be a set, and now let $Cl$ be an operation on subsets of $X$.  Then $Cl$ is a __closure operation__ if $Cl$ is isotone, extensive, and idempotent.  That is,

1.  $ A \subseteq B \;\Rightarrow\; Cl(A) \subseteq Cl(B) $,
1.  $ A \subseteq Cl(A) $, and
1.  $ Cl(Cl(A)) \subseteq Cl(A) $ (the reverse inclusion follows from the previous two properties).

=--

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

If $Cl$ is a closure operation, then let $\mathcal{C}$ be the collection of sets that equal their own closures (the "[[modal types]]" or "[[local objects]]").  Then $\mathcal{C}$ is a Moore collection.

Conversely, if $\mathcal{C}$ is a Moore collection, then let $Cl(A)$ be the intersection of all closed sets that contain $A$.  Then $Cl$ is a closure operator.

Furthermore, the two maps above, from closure operators to Moore collections and vice versa, are inverses.
=--

### In terms of monads
 {#InTermsOfMonads}

Moore closures on $X$ are precisely [[monads]] on  the [[subobject lattice]] $\mathcal{P}X$.  The property (1) of a closure operator, def. \ref{ClosureOperator}, corresponds to the action of the monad on morphisms, while (2,3) are the [[unit of an adjunction|unit]] and multiplication of the monad.  (The rest of the requirements of a monad are trivial in a [[poset]], since they state the equality of various morphisms with common source and target.)

## Examples
 {#Examples}

The [[closed subsets]] in a [[topological space]] form a Moore collection; here the closure of a set $A$ is its closure in the usual sense.  In fact, a topological space can be *defined* as a set equipped with a Moore closure with either of these additional properties (which are equivalent):

*  $Cl(\empty) = \empty$ and $Cl(A \cup B) = Cl(A) \cup Cl(B)$.
*  $\empty$ is closed, and so is $A \cup B$ if $A$ and $B$ are closed.

(However, these properties may fail in [[constructive mathematics]]; in fact, a topology cannot be constructively recovered from its closure operation.)

The first pair of properties is equivalent to the following weaker ones given that $A \subseteq Cl(A)$:

* $Cl(\empty) \subseteq \empty$ and $Cl(A \cup B) \subseteq Cl(A) \cup Cl(B)$.

The closure operator of a topological space with underlying set $X$, and thus in classical mathematics, a topology on a set $X$, can thus be described as an [[monoidal monad | oplax monoidal monad]] on the [[join semilattice]] $\mathcal{P}X$ considered as a cocartesian monoidal category.

Notice that a preclosure in a [[pretopological space]], is not a closure in the above sense, even though some authors do call this 'closure'.

Here are some algebraic examples:

*  The [[subgroup]]s of a [[group]] $G$ form a Moore collection; the closure of a subset $B$ of $G$ is the subgroup generated by $B$.

*  The subrings of a [[ring]] $R$ form a Moore collection; the closure of a subset $B$ of $R$ is the subring generated by $B$.

*  The subspaces of a [[vector space]] $V$ form a Moore collection; the closure of a subset $B$ of $V$ is the subspace spanned by $B$.

*  OK, you get the idea.  This applies to any [[algebraic theory]]. For a [[Lawvere theory|finitary algebraic theory]], the lattice of closed elements is an [[algebraic lattice]].

But also:

*  The [[normal subgroup]]s of $G$ form a Moore collection; the closure of $B$ is the normal subgroup generated by $B$.

*  The [[ideal]]s of a ring $R$ form a Moore collection; the closure of $B$ is the ideal generated by $B$.

*  The (topologically) closed subspaces of a [[Hilbert space]] $H$ form a Moore collection; the closure of $B$ is the closed subspace generated by $B$.

*  And many further examples.

Here are some examples on [[power set]]s:

*  The topologies on $X$ form a Moore collection on $\mathcal{P}X$; the closure of a subset $\mathcal{B}$ of $\mathcal{P}X$ is the topology generated by $\mathcal{B}$ as a [[subbase]].

*  The [[filter]]s on $X$ form a Moore collection on $\mathcal{P}X$; the closure of $\mathcal{B}$ is the filter generated by $\mathcal{B}$ as a [[subbase]].  (The *proper* filters on $X$ do *not* form a Moore collection; not every $\mathcal{B}$ generates a proper filter.)

*  The $\sigma$-[[sigma-algebra|algebras]] on $X$ form a Moore collection on $\mathcal{P}X$; the closure of $\mathcal{B}$ is the $\sigma$-algebra generated by $\mathcal{B}$.  (This is the 'abstract nonsense' way to generate a $\sigma$-algebra; else you have to do transfinite induction on countable [[ordinal number|ordinals]].)

*  And so on.

Topping off these, the Moore collections on $X$ form a Moore collection on $\mathcal{P}X$; the closure of $\mathcal{B}$ is the Moore collection generated by $\mathcal{B}$ as described in the definitions.

See also at _[[matroid]]_.

## Generalisations

The definition of Moore collection really makes sense in any [[inflattice]]; even better, the definition of closure operator makes sense in any [[partial order|poset]].  This context is the generic meaning of [[closure operator]]; here are some examples:

*  Instead of $\mathcal{P}X$, work in the [[opposite category|opposite poset]] $\mathcal{P}^{op}X$.  Then the open sets in a [[topological space]] $X$ form a Moore collection whose closure operator is the usual interior operation.  Now we can define a topological space as a set equipped with a Moore closure operator on $\mathcal{P}^{op}X$ that preserves [[join]]s (which here are [[intersection]]s); this definition is even valid constructively.

*  Let $f \dashv g$ be a [[Galois connection]] between [[partial order|posets]] $A$ and $B$.  Call an element of $A$ _normal_ if $g(f(a)) \leq a$ (the reverse is always true).  Then $g \circ f$ is a closure operator.  This generalises the case of the normal subgroups of $G$ when $G$ is the [[Galois group]] of an extension of [[field]]s.

Since Galois connections are simply [[adjunction]]s between posets, the concept of Moore closure cries out for [[categorification]].  And in fact, the answer is well known in category theory: it is a [[monad]].

## Related conceots

* [[matroid]]

* [[modal operator]], [[modality]]

* [[modal logic]], [[modal type theory]], [[modal homotopy type theory]]

## References

*  Erich Schechter, Section 4.1-4.12 in: *[[Handbook of Analysis and its Foundations]]*

See also:

* Wikipedia, _[Closure operator](http://en.wikipedia.org/wiki/Closure_operator)_

[[!redirects Moore closure]]
[[!redirects Moore closures]]
[[!redirects Moore clsoure]]
[[!redirects Moore collection]]
[[!redirects Moore collections]]


Moore complex

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Homological algebra
+--{: .hide}
[[!include homological algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _Moore complex_ of a [[simplicial group]] -- also known in its normalized version as the **complex of normalized chains** -- is a [[chain complex]] whose [[differential]] is built from the face maps of the simplicial group.

The operation of forming the Moore complex of chains of a [[simplicial group]] is one part of the [[Dold-Kan correspondence]] that relates [[simplicial object|simplicial]] (abelian) [[groups]] with [[chain complexes]].

Recall that a [[simplicial group]] $G$, being in particular a [[Kan complex]], may be thought of, in the sense of the [[homotopy hypothesis]], as a combinatorial space equipped with a group structure. The Moore complex of $G$ is a [[chain complex]]

* whose $n$-cells are the "$n$-disks with basepoint on their boundary" in this space, with the basepoint sitting on the identity element of the space;

* the boundary map on which acts literally like a boundary map should: it sends an $n$-disk to its boundary, read as an $(n-1)$-disk whose entire boundary is concentrated at the identity point.

This is entirely analogous to how a [[crossed complex]] is obtained from a [[strict ∞-groupoid]]. In fact it is a special case of that, as discussed at [[Dold-Kan correspondence]] in the section on the nonabelian version.

## Definition

### For general simplicial groups
 {#ForGeneralSimplicialGroups}

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

Given a [[simplicial group]] $G$, its _[[normalized chain complex]]_ or _Moore complex_ is the $\mathbb{N}$-graded [[chain complex]] $((N G)_\bullet,\partial )$ of (possibly [[nonabelian group|nonabelian]]) groups which

* is in degree $n$ the joint [[kernel]]

  $$
    (N G)_n=\bigcap_{i=1}^{n}ker\,d_i^n
  $$

  of all face maps except the 0-face;

* with differential given by the remaining 0-face map

  $$
   \partial_n \coloneqq d_0^n|_{(N G)_n} : (N G)_n \rightarrow (N G)_{n-1}
   \,.
  $$

=--

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

In def. \ref{NormalizedChainComplexOnGeneralGroup} one may equivalently take the  joint [[kernel]] of all but the $n$-face map and take that remaining face map,  $d_n^n$, to be the differential.

=--

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

We may think of the elements of the complex $N G$, def. \ref{NormalizedChainComplexOnGeneralGroup}, in degree $k$
as being $k$-dimensional [[disks]] in $G$ all of whose [[boundary]] is captured by a single face:

* an element $g \in N G_1$ in degree 1 is a 1-disk

  $$
    1 \stackrel{g}{\to} \partial g
    \,,
  $$

* an element $h \in N G_2$ is a 2-disk

  $$
    \array{
       && 1
       \\
       & {}^1\nearrow &\Downarrow^h& \searrow^{\partial h}
       \\
       1
       &&\stackrel{1}{\to}&&
       1
    }
    \,,
  $$

* a degree 2 element in the kernel of the boundary map is such a 2-disk that is closed to a 2-[[sphere]]

  $$
    \array{
       && 1
       \\
       & {}^1\nearrow &\Downarrow^h& \searrow^{\partial h = 1}
       \\
       1
       &&\stackrel{1}{\to}&&
       1
    }
    \,,
  $$

etc.

=--

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

For every [[simplicial group]] $G$ the normalized chain complex $(N G)_\bullet$ in def. \ref{NormalizedChainComplexOnGeneralGroup} is a [[normal complex of groups]],

=--

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

This means that is easy to take the homology of the complex, even though the groups involved may be non-abelian.

=--

### For simplicial abelian groups
 {#ForSimplicialAbelianGroups}

Let now $A$ be a [[simplicial abelian group]]. Then its [[normalized chain complex]] $(N A)_\bullet \in Ch_\bullet^+$ of def. \ref{NormalizedChainComplexOnGeneralGroup} is an ordinary [[connective chain complex|connective]] [[chain complex]] in the [[abelian category]] [[Ab]].

In this abelian cases are two other chain complexes naturally associated with $A$:

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

For $A$ a [[simplicial abelian group]] its **alternating face map complex** $(C A)_\bullet$ of $A$ is the [[chain complex]] which

* in degree $n$ is given by the group $A_n$ itself

  $$
    (C A)_n \coloneqq A_n
  $$

* with [[differential]] given by the alternating sum of face maps (using the abelian group structure on $A$)

  \[
    \label{DifferentialOnChains}
    \partial_n \coloneqq \sum_{i = 0}^n (-1)^i d_i  \;\colon\; (C A)_n \to (C A)_{n-1}
    \,.
  \]

  (see lemma \ref{AlternatingSumOfFacesInNilpotent}).

=--

+-- {: .num_lemma #AlternatingSumOfFacesInNilpotent}
###### Lemma

The differential in def. \ref{AlternatingFaceMapComplex} is well-defined in that it indeed squares to 0.

=--

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

Using the [[simplicial identities|simplicial identity]] $d_i \circ d_j = d_{j-1} \circ d_i$ for $i \lt j$ one finds:

$$
  \begin{aligned}
    \partial_n \partial_{n+1}
    & =
    \sum_{i, j}
      (-1)^{i+j}
      d_i \circ d_{j}
    \\
    &=
    \sum_{i  \geq j} (-1)^{i+j} d_i \circ d_j
    +
    \sum_{i \lt j} (-1)^{i+j}
      d_i \circ d_j
    \\
    &=
    \sum_{i  \geq j} (-1)^{i+j} d_i \circ d_j
    +
    \sum_{i \lt j} (-1)^{i+j}
      d_{j-1} \circ d_i
    \\
    &=
    \sum_{i  \geq j} (-1)^{i+j} d_i \circ d_j
    -
    \sum_{i \leq k} (-1)^{i+k}
      d_{k} \circ d_i
    \\
    &=
    0
  \end{aligned}
  \,.
$$

=--

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

Given a [[simplicial group]] $A$ (or in fact any [[simplicial set]]), then an element $a \in A_{n+1}$ is called _degenerate_  if it is in the [[image]] of one of the simplicial degeneracy maps $s_i \colon A_n \to A_{n+1}$. All elements of $A_0$ are regarded a non-degenerate.
Write

$$
  D (A_{n+1}) \coloneqq \langle \cup_i s_i(A_{n}) \rangle \hookrightarrow A_{n+1}
$$

for the [[subgroup]] of $A_{n+1}$ which is generated by the degenerate elements (i.e. the smallest subgroup containing all the degenerate elements). Elements of $D(A)_{n}$ are often called _[[thin element|thin]]_ $n$-simplices.

=--

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

For $A$ a [[simplicial abelian group]] its **alternating face maps chain complex modulo degeneracies**, $(C A)/(D A)$ is the [[chain complex]]

* which in degree 0 equals is just $((C A)/D(A))_0 \coloneqq A_0$;

* which in degree $n+1$ is the [[quotient]] group obtained by dividing out the group
  the degenerate elements, def. \ref{DegenerateElement}:

  $$
    ((C A)/D(A))_{n+1} \coloneqq A_{n+1} / D(A_{n+1})
  $$

* whose [[differential]] is the induced action of the alternating sum of faces on the quotient (which is well-defined by lemma \ref{LeftCosetsDisjoint}).

=--

+-- {: .num_lemma #LeftCosetsDisjoint}
###### Lemma

Def. \ref{ComplexModuloDegeneracies} is indeed well defined
in that the  alternating face map differential
respects the degenerate subcomplex.

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

Using the mixed [[simplicial identities]] we find that for $s_j(a) \in A_n$ a degenerate element, its boundary is

$$
  \begin{aligned}
     \sum_i (-1)^i d_i s_j(a)
     &=
     \sum_{i \lt j} (-1)^i  s_{j-1} d_i(a)
     +
     \sum_{i = j, j+1} (-1)^i  a
     +
     \sum_{i \gt j+1} (-1)^i s_j d_{i-1}(a)
     \\
     &=
     \sum_{i \lt j} (-1)^i  s_{j-1} d_i(a)
     +
     \sum_{i \gt j+1} (-1)^i s_j d_{i-1}(a)
  \end{aligned}
$$

which is again a combination of elements in the image of the degeneracy maps.

=--

## Properties

### Normalization

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

Given a [[simplicial abelian group]] $A$, the evident composite of natural morphisms

$$
  N A \stackrel{i}{\hookrightarrow} CA
  \stackrel{p}{\to}
  (C A)/(D A)
$$

from the normalized chain complex, def. \ref{NormalizedChainComplexOnGeneralGroup}, into the alternating face map complex modulo degeneracies, def. \ref{ComplexModuloDegeneracies}, (inclusion followed by projection to the quotient) is a [[natural isomorphism]] of chain complexes.

=--

([Goerss-Jardine, theorem III 2.1](#GoerssJardine), see also [Schwede-Shipley 03, Section 2.1](#SchwedeShipley03)).

+-- {: .num_cor #SplittingOffDegenerateCells}
###### Corollary

For $A$ a [[simplicial abelian group]], there is a [[split exact sequence|splitting]]

$$
  C_\bullet(A) \simeq N_\bullet(A) \oplus D_\bullet(A)
$$

of the alternating face map complex, def. \ref{AlternatingFaceMapComplex} as a [[direct sum]],
where the first direct summand is [[natural isomorphism|naturally isomorphic]] to the [[normalized chain complex]] of def. \ref{NormalizedChainComplexOnGeneralGroup} and the second is the degenerate cells from def. \ref{ComplexModuloDegeneracies}.

=--

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

By prop. \ref{NormalizedIntoModuloDegeneraciesIsIsomorpism} there is an [[inverse]] to the diagonal composite in

$$
  \array{
    C A &\stackrel{p}{\longrightarrow}& (C A)/(D A)
    \\
    {}^{\mathllap{i}}\uparrow & \nearrow
    \\
    N A
  }
  \,.
$$

This hence exhibits a [[split exact sequence|splitting]] of the [[short exact sequence]] given by the quotient by $D A$.

$$
  \array{
    0 &\to& D A &\hookrightarrow & C A &\stackrel{p}{\longrightarrow}& (C A)/(D A) &\to & 0
    \\
    && && {}^{\mathllap{i}}\uparrow & \swarrow_{\mathrlap{\simeq}_{iso}}
    \\
    && && N A
  }
  \,.
$$

=--

\begin{theorem}
\label{EMTheorem}
**(Eilenberg-MacLane)**
\linebreak
Given a [[simplicial abelian group]] $A$, then the inclusion

$$
  i \colon N A \hookrightarrow C A
$$

of the normalized chain complex, def. \ref{NormalizedChainComplexOnGeneralGroup} into the full alternating face map complex, def. \ref{AlternatingFaceMapComplex},
is a [[natural transformation|natural]] [[quasi-isomorphism]] and in fact a natural chain [[homotopy equivalence]], i.e. the complex $D_\bullet(X)$ is [[null homotopy|null-homotopic]] (a [[contractible chain complex]]).
\end{theorem}

([Goerss-Jardine, theorem III 2.4](#GoerssJardine))

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

Following the proof of ([Goerss-Jardine, theorem III 2.1](#GoerssJardine)) we look for each $n \in \mathbb{N}$ and each $j \lt n$ at the  groups

$$
  N_n(A)_j \coloneqq  \cap_{i=0}^j ker (d_i) \subset A_n
$$

and similarly at

$$
  D_n(A)_j = \{s_{i}\}_{i \leq j}(A_{n-1}) \subset A_n
  \,,
$$

the subgroup generated by the first $j$ degeneracies.

For $j= n-1$ these coincide with $N_n(A)$ and with $D_n(A)$, respectively.  We show by induction on $j$ that the composite

$$
  N_n(A)_j \hookrightarrow A_n \stackrel{}{\to} A_n/D_n(A)_j
$$

is an isomorphism of all $j \lt n$. For $j = n-1$ this is then the desired result.

(...)

=--

+-- {: .num_cor}
###### Corollary

Given a [[simplicial abelian group]] $A$, then the projection [[chain map]]

$$
  (C A) \longrightarrow (C A)/(D A)
$$

from its alternating face maps complex, def. \ref{AlternatingFaceMapComplex}, to the alternating face map complex modulo degeneracies, def. \ref{ComplexModuloDegeneracies}, is a [[quasi-isomorphism]].

=--

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

Consider the pre-composition of the map with the inclusion of the
normalized chain complex, def. \ref{NormalizedChainComplexOnGeneralGroup}.

$$
  \array{
    C A &\stackrel{p}{\longrightarrow}& (C A)/(D A)
    \\
    {}^{\mathllap{i}}\uparrow & \nearrow
    \\
    N A
  }
$$

By theorem \ref{EMTheorem} the vertical map is a [[quasi-isomorphism]] and by prop. \ref{NormalizedIntoModuloDegeneraciesIsIsomorpism} the composite diagonal map is an [[isomorphism]], hence in particular also a [[quasi-isomorphism]]. Since quasi-isomorphisms satisfy the [[two-out-of-three]] property, it follows that also the map in question is a quasi-isomorphism.

=--

### Equivalence of categories

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

The [[normalized chain complex]] [[functor]] of def. \ref{NormalizedChainComplexOnGeneralGroup} restricts on [[simplicial abelian groups]] to an [[equivalence of categories]]

$$
  N \colon
  sAb \stackrel{\simeq}{\longrightarrow} Ch_\bullet^+(A)
$$

between [[sAb]] and the [[category of chain complexes]] in non-negative degree.

=--

This is the statement of the _[[Dold-Kan correspondence]]_. See there for details.

### Homology and homotopy groups

Notice that the [[simplicial set]] underlying any [[simplicial group]] $G$ (as described there) is a [[Kan complex]]. Write

$$
  \pi_n(G) \;\;\; n \in \mathbb{N}
$$

for the $n$-th [[simplicial homotopy group]] of $G$. Notice that due to the group structure of $G$ in this case also $\pi_0(G)$ is indeed canonically a group, not just a set.

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

For $A$ a simplicial abelian group there are [[natural isomorphism]]s

$$
  \pi_n(A,0) \simeq H_n(N A) \simeq H_n(A)
$$

between the [[simplicial homotopy group]]s and the [[chain homology]] groups of the unnormalized and of the normalized chain complexes.

=--

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

The first isomorphism follows with the [[Eckmann-Hilton argument]]. The second directly from the [Eilenberg-MacLane theorem](#EMTheorem) above.

=--

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

Both $sAb$ as well as $Ch_\bullet^+$ are naturally [[categories with weak equivalences]] given by those morphisms that induce [[isomorphism]]s on all [[simplicial homotopy group]] and on all [[chain homology]] groups, respectively. So the above statement says that the Moore complex functor $N$ respects these weak equivalences.

In fact, it induces an equivalence of categories also on the corresponding [[homotopy categories]]. And even better, it induces a [[Quillen equivalence]] with respect to the standard [[model category]] structures that refine the structures of categories of weak equivalences. All this is discussed at [[Dold-Kan correspondence]].

=--

### Hypercrossed complex structure {#HypercrossedComplex}

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

The Moore complex of a [[simplicial group]] is naturally a [[hypercrossed complex]].

=--

This has been established in ([Carrasco-Cegarra](#Carrasco-Cegarra)). In fact, the analysis of the Moore complex and what is necessary to rebuild the simplicial group from its Moore complex is the origin of the abstract motion of hypercrossed complex, so our stated proposition is almost a tautology!

Typically one has pairings  $N G_p \times N G_q \to N G_{p+q}$. These use the [[Conduché decomposition theorem]], see the discussion at [[hypercrossed complex]].

These Moore complexes are easily understood in low dimensions:

* Suppose that $G$ is a simplicial group with Moore complex $N G$, which satisfies $N G_k = 1$ for $k\gt 1$, then $(G_1,G_0,d_1,d_0)$ has the structure of a [[2-group]]. The [[interchange law]] is satisfied since the corresponding equation in $G_1$ is always the image of an element in $N G_2$, and here that must be trivial. If one thinks of the 2-group as being specified by a [[crossed module]] $(C,P,\delta, a)$, then in terms of the original simplicial group, $G$,  $N G_0 = G_0 = P$, $N G_1 \cong C$, $ \partial = \delta$ and the action of $P$ on $C$ translates to an action of $N G_0$ on $N G_1$ using conjugation by $s_0(p)$, i.e., for $p\in G_0$ and $c\in N G_1$,
$$a(p)(c) = s_0(p)c s_0(p)^{-1}.$$

* Suppose next that $N G_k = 1$ for $k \gt 2$, then the Moore complex is a [[2-crossed module]].

## Examples
 {#Examples}

\begin{example}\label{ChainsOnThe1Simplex}
**(chain on the 1-simplex)**
\linebreak
Consider the 1-[[simplex]] $\Delta[1]$ regarded as a [[simplicial set]], and write $\mathbb{Z}[\Delta[1]]$ for the [[simplicial abelian group]] which in each degree is the [[free abelian group]] on the simplices in $\Delta[1]$.

This simplicial abelian group starts out as

$$
  \mathbb{Z}[\Delta[1]]
  =
  \left(
     \cdots
     \stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}}}
     \mathbb{Z}^4 \stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}}
     \mathbb{Z}^3 \stackrel{\overset{\partial_0}{\longrightarrow}}{\underset{\partial_1}{\longrightarrow}} \mathbb{Z}^2
  \right)
$$

(where we are indicating only the face maps for notational simplicity).

Here the first $\mathbb{Z}^2 = \mathbb{Z}\oplus \mathbb{Z}$, the [[direct sum]] of two copies of the [[integers]], is the group of 0-chains generated from the two endpoints $(0)$ and $(1)$ of $\Delta[1]$, i.e. the abelian group of formal linear combinations of the form

$$
  \mathbb{Z}^2 \simeq \left\{ a \cdot (0) + b \cdot (1) | a,b \in \mathbb{Z}\right\}
  \,.
$$

The second $\mathbb{Z}^3 \simeq \mathbb{Z}\oplus \mathbb{Z}\oplus \mathbb{Z}$ is the abelian group generated from the three (!) 1-simplicies in $\Delta[1]$, namely the non-degenerate edge $(0\to 1)$ and the two degenerate cells $(0 \to 0)$ and $(1 \to 1)$, hence the abelian group of formal linear combinations of the form

$$
  \mathbb{Z}^3 \simeq \left\{ a \cdot (0\to 0) + b \cdot (0 \to 1) + c \cdot (1 \to 1) | a,b,c \in \mathbb{Z}\right\}
  \,.
$$

The two face maps act on the basis 1-cells as

$$
  \partial_1 \colon (i \to j) \mapsto (i)
$$

$$
  \partial_0 \colon (i \to j) \mapsto (j)
  \,.
$$

Now of course most of the (infinitely!) many simplices inside $\Delta[1]$ are degenerate. In fact the only non-degenerate simplices are the two 0-cells $(0)$ and $(1)$ and the 1-cell $(0 \to 1)$. Hence the alternating face maps complex modulo degeneracies, def. \ref{ComplexModuloDegeneracies}, of $\mathbb{Z}[\Delta[1]]$ is simply this:

$$
  (C (\mathbb{Z}[\Delta[1]])) / D (\mathbb{Z}[\Delta[1]]))
  =
  \left(
    \cdots
    \to
    0
    \to
    0
    \to
    \mathbb{Z}
    \stackrel{\left(1 \atop -1\right)}{\longrightarrow}
    \mathbb{Z}^2
  \right)
  \,.
$$

Notice that alternatively we could consider the topological 1-simplex $\Delta^1 = [0,1]$ and its [[singular simplicial complex]] $Sing(\Delta^1)$ in place of the smaller $\Delta[1]$, then the free simplicial abelian group $\mathbb{Z}(Sing(\Delta^1))$ of that. The corresponding alternating face map chain complex $C(\mathbb{Z}(Sing(\Delta^1)))$ is "huge" in that in each positive degree it has a free abelian group on uncountably many generators. Quotienting out the degenerate cells still leaves uncountably many generators in each positive degree (while every singular $n$-simplex in $[0,1]$ is "thin", only those whose parameterization is as induced by a degeneracy map are actually regarded as degenerate cells here). Hence even after normalization the singular simplicial chain complex is "huge". Nevertheless it is quasi-isomorphic to the tiny chain complex found above.
\end{example}

\begin{example}\label{NormalizedChainsOnEZTwo}
**(normalized chain complex of $E\mathbb{Z}_2$)**
\linebreak
We write $\mathbb{Z}_2 \,\equiv\, \mathbb{Z}/2\mathbb{Z}$ for the [[cyclic group of order 2]], whose [[underlying]] [[set]] we denote by $\{0, 1\}$.

Consider the [[simplicial group]] (see [here](simplicial+classifying+space#eq:SimplicialClassifyingSpaceAsNerve))
$$
  W \mathbb{Z}_2
  \;=\;
  N\big(
    \mathbf{E} \mathbb{Z}_2
  \big)
  \;\equiv\;
  N\big(
    \mathbb{Z}_2 \times \mathbb{Z}_2
    \underoverset
      {pr_2}
      {(\text{-})\cdot(-)}
      {\rightrightarrows}
    \mathbb{Z}_2
  \big)
  \,.
$$
Forming degree-wise [[linear spans]] gives a [[simplicial abelian group]]
$$
  \mathbb{Z}\big[
    W \mathbb{Z}_2
  \big]
  \;\;
  \in
  \;\;
  Ab^{\Delta^{op}}
  \,.
$$
with [[normalized chain complex]] denoted
\[
  \label{NormalizedChainsOnWZTwo}
  N_\bullet
  \big(
    \mathbb{Z}
    [
      W \mathbb{Z}_2
    ]
  \big)
  \;\;
  \in
  \;\;
  Ch_{\geq 0}
  \,.
\]

Now, since the only non-[[identity morphism|identity]] morphisms in $\mathbf{E}\mathbb{Z}_2$ are the two morphsism $0 \to 1$ and $1 \to 0,$ the non-degenerate [[n-simplex|$n$-simplices]] of (eq:NormalizedChainsOnWZTwo) are alternating sequences on $n+1$-elements in $\{0,1\}$. By the alternating property, these are fully determined by their first element (in particular), whence there are exactly two non-degenerate $n$-simplices for all $n \in \mathbb{N}$. On these, the differential is given as follows, using  (eq:DifferentialOnChains) and Prop. \ref{NormalizedIntoModuloDegeneraciesIsIsomorpism}, according to which only the 0th and $n$th [[face maps]] contribute in degree $n$, the latter with sign $(-1)^n$:
\[
  \label{NormalizedChainsOnWZTwo}
  N_\bullet
  \mathbb{Z}
  \big[
    W \mathbb{Z}_2
  \big]
  \;\simeq\;
  \Bigg[
    \cdots
    \to
    \xrightarrow{
      \left[
      \array{
        +1  & +1
        \\
        +1  & +1
      }
      \right]
    }
    \mathbb{Z}_{0101} \oplus \mathbb{Z}_{1010}
    \xrightarrow{
      \left[
      \array{
        -1  & +1
        \\
        +1  & -1
      }
      \right]
    }
    \mathbb{Z}_{010} \oplus \mathbb{Z}_{101}
    \xrightarrow{
      \left[
      \array{
        +1  & +1
        \\
        +1  & +1
      }
      \right]
    }
    \mathbb{Z}_{01} \oplus \mathbb{Z}_{10}
    \xrightarrow{
      \left[
      \array{
        -1  &  +1
        \\
        +1  &  -1
      }
      \right]
    }
    \mathbb{Z}_0 \oplus \mathbb{Z}_1
  \Bigg]
  \,.
\]
One may think of the two generators in degree $n+1$ as corresponding to two $n+1$-dimensional [[hemispheres]] with common [[boundary]] an [[equator]] similarly formed by two $n$-dimensional [[hemispheres]], and so on. Thereby this chain complex is seen to be  isomorphically that for the [[cellular homology]] of the standard [[CW-complex]]-[[structure]] on the [infinite-dimensional sphere](sphere#InfiniteDimensionalSphere) $S^\infty$ (cf. also at *[[real projective space]]* the section *[Relation to $\mathbb{Z}/2$-classifying space](real+projective+space#RelationToClassifyingSpace)*).

Indeed, one sees immediately that (eq:NormalizedChainsOnWZTwo) has vanishing [[chain homology]] in all [[positive number|positive]] degrees and homology $\simeq \mathbb{Z}$ in degree 0, whence the canonical map to the normalized chain complex on the point is a [[quasi-isomorphism]]:

$$
  N_\bullet\,\mathbb{Z}[W \mathbb{Z}_2]
  \xrightarrow[\;qiso\;]{}
  N_\bullet\,\mathbb{Z}[\ast]
  \,.
$$

This reflects the [[contractible homotopy type]] of both $E \mathbb{Z}_2$ (the total space of the [universal principal $\mathbb{Z}_2$-bundle](simplicial+principal+bundle#UniversalSimplicialBundle)) and of $S^\infty$.
\end{example}

\begin{example}\label{NormalizedChainDGAlgebraOnEZTwo}
**(normalized chain dg-algebra of $E\mathbb{Z}_2$)**

On the normalized chain complex of Ex. \ref{NormalizedChainsOnEZTwo}, the
[[group]] [[structure]] on $W \mathbb{Z}_2$ induces a [[simplicial ring]]-[[structure]], the "simplicial [[group ring]]")
$$
  \mathbb{Z}\big[
    W \mathbb{Z}_2
  \big]
  \;\;
  \in
  \;\;
  Ring^{\Delta^{op}}
  \,.
$$

Since the [[normalized chain complex]]-functor $N_\bullet \,\colon\, Ab^{\Delta^{op}} \to Ch_{\geq 0}$ is (see [here](monoidal+Dold-Kan+correspondence#ChainsIsLaxMonoidal)) a [[lax monoidal functor]] via the [[Eilenberg-Zilber map]], this induces on $N_\bullet\big( \mathbb{Z}[W \mathbb{Z}_2] \big)$ the [[structure]] of a [[dg-algebra]].

To write this out, denote the two generators in each degree by
$$
  (0), (1)
  \;\in\;
  \mathbb{Z}_{0\cdots}
  \oplus
  \mathbb{Z}_{1\cdots}
  \;=\;
  N_n\big( \mathbb{Z}[W \mathbb{Z}_2] \big)
  \,.
$$
The non-degenerate cells of the tensor product simplicial group are similarly labeled (via [this Prop.](product+of+simplices#NonDegenerateSimplicesInProductOfSimplices)) by

1. a pair of first elements of $n+1$-sequences in $\{0,1\}$

1. a $(p, q)$-[[shuffle]],

hence:
$$
  \big(
    (g, g'), sh
  \big)
  \;\in\;
  \mathbb{Z}
  \big[
    \{0,1\}^2
      \times
    \sqcup_{p+q = n}
    Sh(p,q)
  \big]
  \;\simeq\;
  N_{n}
  \Big(
    \mathbb{Z}\big[
      W \mathbb{Z}_2
      \times
      W \mathbb{Z}_2
    \big]
  \Big)
  \;\simeq\;
  N_{n}
  \Big(
    \mathbb{Z}\big[
      W \mathbb{Z}_2
    \big]
    \otimes
    \mathbb{Z}\big[
      W \mathbb{Z}_2
    \big]
  \Big)
  \,.
$$
One immediately finds that on these generators the induced product map is just the group operation, independent of the shuffle:
$$
  \array{
    N_\bullet
    \big(
      \mathbb{Z}[W \mathbb{Z}_2]
      \otimes
      \mathbb{Z}[W \mathbb{Z}_2]
    \big)
    &
    \xrightarrow{
      N_\bullet
      \big(
        \mathbb{Z}[(\text{-})\cdot(\text{-})]
      \big)
    }
    &
    N_\bullet
    \big(
      \mathbb{Z}[W \mathbb{Z}_2]
    \big)
    \\
    \big(
      g, g', sh
    \big)
    &\mapsto&
    (g \cdot g')
    \mathrlap{\,.}
  }
$$
Composed with the [[Eilenberg-Zilber map]] this gives
$$
  \array{
    \big(
      N_\bullet
      \,
      \mathbb{Z}[W \mathbb{Z}_2]
    \big)
    \otimes
    \big(
      N_\bullet
      \,
      \mathbb{Z}[W \mathbb{Z}_2]
    \big)
    &\xrightarrow{\phantom{----}}&
    N_\bullet
    \big(
      \mathbb{Z}[W \mathbb{Z}_2]
      \otimes
      \mathbb{Z}[W \mathbb{Z}_2]
    \big)
    &
    \xrightarrow{
      N_\bullet
      \big(
        \mathbb{Z}[(\text{-})\cdot(\text{-})]
      \big)
    }
    &
    N_\bullet
    \big(
      \mathbb{Z}[W \mathbb{Z}_2]
    \big)
    \\
    (g)_p
      \otimes
    (g')_q
    &\mapsto&
    \displaystyle{
      \underset{
        \mathclap{
          sh \in Sh(p,q)
        }
      }{\sum}
    }
    \;
    sgn(sh)
    \,
    \big(
      g,g', sh
    \big)
    &\mapsto&
    \Big(
      \displaystyle{
        \underset{
          \mathclap{
            sh \in Sh(p,q)
           }
        }{\sum}
      }
      \;
     sgn(sh)
   \Big)
   \,
   (g \cdot g')_{p + q}
   \mathrlap{\,.}
  }
$$

\end{example}

## References

Original sources:

* [[John Moore]], _Homotopie des complexes monoïdaux, I._  Séminaire Henri Cartan **7** 2 (1954-1955), Exposé **18**  ([numdam:SHC_1954-1955__7_2_A8_0](http://www.numdam.org/item?id=SHC_1954-1955__7_2_A8_0))

 * [[John Moore]], _Semi-simplicial complexes and Postnikov systems_, Symposium international de topologia algebraica, Mexico (1958) p. 243

* [[John Moore]], *Semi-simplicial Complexes, seminar notes*, Princeton University (1956)

There is also a never published

* [[John C. Moore]], _Algebraic homotopy theory_.  Princeton 1956. Mimeographed notes [[pdf](https://dmitripavlov.org/scans/moore-algebraic-homotopy-theory.pdf)]

A proof by Cartan is in

* Cartan, _Quelques questions de topologies_ seminar, 1956-57

A standard textbook reference for the abelian version is

* {#GoerssJardine} [[Paul Goerss]], [[Rick Jardine]], chapter III.2 of _[[Simplicial homotopy theory]]_

Notice that these authors write "normalized chain complex" for the complex that elsewhere in the literature would be called just "Moore complex", whereas what Goerss--Jardine call "Moore complex" is sometime maybe just called "alternating sum complex".

See also

* {#SchwedeShipley03} [[Stefan Schwede]], [[Brooke Shipley]], Section 2.1 of: _Equivalences of monoidal model categories_ , Algebr. Geom. Topol. 3 (2003), 287--334 ([arXiv:math.AT/0209342](http://arxiv.org/abs/math.AT/0209342), [euclid:euclid.agt/1513882376](https://projecteuclid.org/euclid.agt/1513882376))

A discussion with an emphasis of the generalization to non-abelian simplicial groups is in section 1.3.3 of

* [[Tim Porter]], _[[Crossed Menagerie]]_

The discusson of the [[hypercrossed complex]] structure on the Moore complex of a general simplicial group is in

* {#Carrasco-Cegarra} [[Pilar Carrasco|P. Carrasco]], [[Antonio Cegarra|A. M. Cegarra]], _Group-theoretic Algebraic Models for Homotopy Types_ , J. Pure Appl. Alg., 75, (1991), 195--235 <https://doi.org/10.1016/0022-4049(91)90133-M>

[[!redirects Moore complexes]]
[[!redirects Moore-complex]]
[[!redirects alternating face map complex]]

[[!redirects normalized chain complex]]
[[!redirects normalized chain complexes]]
[[!redirects normalized chains complex]]
[[!redirects normalized chains complexes]]
[[!redirects normalized chains functor]]
[[!redirects normalized chains functors]]

[[!redirects normalized cochain complex]]
[[!redirects normalized cochain complexes]]
[[!redirects normalized cochains complex]]
[[!redirects normalized cochains complexes]]
[[!redirects normalized cochains functor]]
[[!redirects normalized cochains functors]]


Moore decomposition

the [[Moore space]]-analog of a [[Postnikov tower]]


Moore machine

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

#Contents#
* table of contents
{:toc}

## Idea

_Moore machines_ are a special type of [[finite state automata]] and in particular a special case of *[[Mealy machines]]* (see there for more).

## Definition

A finite _Moore machine_, $\mathbf{A}$, consists of

* $Q$, a finite set of _states_;

* $A$, an _input alphabet_;

* $B$, an _output alphabet_;

* $q_0$, an _initial state_;

* $\delta: Q\times A\to Q$, a _transition function_;

and

* $\lambda:Q\to B$, the _state output function_.

These determine an _output response function_, $\omega_\mathbf{A}:A^*\to B^*$ and a _characteristic function_ $\chi_\mathbf{A}:A^*\to B$ that picks out the language recognised by the machine.

Here we have, as usual, denoted by $A^*$ the [[free monoid]] (of all strings of symbols) over the 'alphabet' $A$, etc.

##Related entries

* [[Mealy machine]]

* [[automaton]]

##References

* [[Mark V. Lawson]], _[Finite automata](http://www.ma.hw.ac.uk/~markl/books.html)_, CRC Press, see also [here](http://www.ma.hw.ac.uk/~markl/teaching/AUTOMATA/kleene.pdf) for a shorter version in the form of Course Notes.

* Guido Boccali, Bojana Femić, Andrea Laretto, Fosco Loregian, Stefano Luneia, _The semibicategory of Moore automata_ ([arXiv:2305.00272](https://arxiv.org/abs/2305.00272))

[[!redirects Moore machines]]


Moore path category

# Moore path categories
* table of contents
{: toc}

## Idea

The classical  _Moore path category_ of a [[topological space]] $X$ is a variant on the usual space of [[paths]] $I \to X$, but one which yields a [[strict category]] (even an [[internal category|internal]] [[dagger category|$\dagger$-category]] in [[Top]]).

## Definition

\begin{definition}
Let $X$ be a [[topological space]]. Its **Moore path category** $M X$ has

* as space of objects $M_0 X \coloneqq (M X)_0 \coloneqq X$;

* as space of morphisms $M_1 X \coloneqq (M X)_1 \subseteq X^{[0,\infty]} \times [0,\infty)$ the subspace of pairs $(\gamma, len(\gamma))$ where $\gamma \colon [0, \infty) \to X$ is continuous and constant on $[len(\gamma), \infty]$;
 the source $s(\gamma)$ of $\gamma$ is $\gamma(0)$ and the target $t(\gamma)$ of $\gamma$ is $\gamma(len(\gamma))=\gamma(\infty)$;

* as composition ${\circ} \colon M_1 X \times_{M_0 X} M_1 X \to M_1 X$, given by $len(\gamma' \circ \gamma) = len (\gamma) + len (\gamma')$, $(\gamma' \circ \gamma)(t)=\gamma(t)$ for $t\in[0, len (\gamma)]$ and $(\gamma' \circ \gamma)(t)=\gamma'(t - len (\gamma))$ otherwise;

* as identity-assigning map $X \to M_1 X$ the map sending $x\in X$ to the pair $(t \mapsto x, 0)$;

* as [[dagger-category|path-reversal]] map $(\gamma, len (\gamma))^\dagger \coloneqq (t \mapsto \gamma(len (\gamma) - t), len (\gamma))$.

\end{definition}

The reference below defines $M_*(X)$ as a strict [[cubical omega-category|cubical]] $\omega$-category. It also has connections, which satisfy all the laws except cancellation of $\Gamma^-_i$ and $\Gamma^+_i$ under composition. This structure seems a sensible home for $n$-paths in $X$ for all $n \geq 0$, and has the advantage  over simplicial or globular versions of "$\infty$-groupoids" of easily encompassing multiple compositions.

\begin{definition}
Suppose $f \colon X \to Y$ is a continuous map. The **Moore mapping path space** $\Gamma f$ is the [[pullback]]

\begin{tikzcd}
  \Gamma f \arrow[r] \arrow[d] \arrow[dr, phantom, "\lrcorner" , very near start]
& M_1 Y \arrow[d, "s"] \\
  X \arrow[r, "f"]
& Y\text.
\end{tikzcd}

\end{definition}
([Barthel–Riehl 2013, definition 3.2](#BarthelRiehl2013))

## Properties

### Functorial factorization

\begin{proposition}
The Moore mapping path space construction $\Gamma\colon Top^\to \to Top$ yields a [[functorial factorization]] of maps $f\colon X\to Y$ as a composition

$$
  X \stackrel{L f}{\to} \Gamma f \stackrel{R f}{\to} Y
$$

of a [[Strøm model structure|(closed) trivial Hurewicz cofibration]] $L f$ and a [[Hurewicz fibration]] $R f$.
\end{proposition}
([Barthel–Riehl 2013, section 3.1 and corollary 3.12](#BarthelRiehl2013))

\begin{proposition}
A morphism of $Top$ is

* a Hurewicz fibration iff it admits an [[algebra for an endofunctor|algebra structure]] for the [[pointed endofunctor]] $R$;

* a [[Strøm model structure|(closed) trivial Hurewicz cofibration]] iff it admits a [[coalgebra for an endofunctor|coalgebra structure]] for the [[copointed endofunctor]] $L$ (a _Moore strong deformation retraction_).

\end{proposition}
([Barthel–Riehl 2013, section 3.3](#BarthelRiehl2013))

## Related concepts

* [[schedule]]

## Reference

* {#BarthelRiehl13} [[Tobias Barthel]], [[Emily Riehl]], _On the construction of functorial factorizations for model categories_, Algebr. Geom. Topol. 13 (2013) 1089-1124 ([arXiv:1204.5427](http://arxiv.org/abs/1204.5427), [doi:10.2140/agt.2013.13.1089](https://doi.org/10.2140/agt.2013.13.1089), [euclid:agt/1513715550](https://projecteuclid.org/euclid.agt/1513715550))

* [[Ronnie Brown|R. Brown]], _Moore hyperrectangles on a space form a strict cubical omega-category_, [arXiv 0909.2212v2](http://arxiv.org/abs/0909.2212)

[[!redirects Moore path space]]
[[!redirects Moore path spaces]]


Moore space

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

#Contents#
* table of contents
{:toc}

## Idea

The **Moore space** $M(G, n)$, where $G$ is an [[abelian group]] and $n \geq 1$, is a [[topological space]] which has non-trivial (reduced) [[homology group]]  $G$ precisely only in dimension $n$ and is simply connected if $n \geq 2$.

(This is somewhat dual to the notion of [[Eilenberg-MacLane space]], which instead has nontrivial [[homotopy group]] in one single dimension.)

## Definition

(the following is based on [Hatcher](#Hatcher))

Let $G$ be an abelian group. Take a [[presentation]] of $G$ i.e. a [[short exact sequence]]
$$
  0\rightarrow H \rightarrow F\rightarrow G\rightarrow 0
$$

where $F$ a [[free abelian group]]. Since $H=ker(F\rightarrow G)$, it is a free abelian group as well and we choose [[bases]] $\{f_\alpha\}_\alpha$ for $F$ and $\{h_\beta\}_\beta$ for $H$. We then construct a [[CW complex]]

$$
  X=M(G,n)
$$

by taking the $n$-th [[skeleton]] to be $X^n:=\vee_\alpha S^n_\alpha$ and for each $\beta$ we attach an $n+1$-cell as follows:

Write $h_\beta=\Sigma_\alpha d_{\alpha\beta}f_\alpha$ and let $\delta_{d_{\alpha\beta}}$ be $0$ if $d_{\alpha\beta}= 0$ and be $1$ otherwise. Define an attaching map $S^n_\beta\rightarrow X^n$ by contracting $\ell_\beta:=(\Sigma_\alpha \delta_{d_{\alpha\beta}})-1$ $(n-1)-$spheres in $S^n$ thus defining a map $S^n_\beta\rightarrow \vee_{\ell_\beta} S^n_{\alpha\beta}$ and then map each $S^n_{\alpha\beta}$ to $S^n_{\alpha}$ by a degree $d_{\alpha\beta}$.

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

The ([[homotopy type]] of) the [[topological space]] $M(G,n)$ constructed this way we call the _Moore space_ of $G$ in degree $n$.

=--

The resulting CW-complex can be seen to have the desired properties via [[cellular homology]].

## Properties

### Homotopy type

The homotopy type of $M(G,n)$ is determined by specifying $G$ and $n$.

### (Non-)Functoriality of the construction

The construction above is not functorial in $G$ because of the choice of bases (see more below). However, it does give a [[functor]] to the [[homotopy category]] $M(-,n):Ab\rightarrow Ho(Top)$.

The functoriality problem of the construction above cannot be corrected. That is, there is no functor $Ab\rightarrow Top$ that lifts $M(-,n)$.
This can be seen as a corollary of a counterexample of Carlsson which gives a negative answer to a conjecture of Steenrod:

+-- {: .num_remark}
###### Conjecture
**(Steenrod)**

Given a group $G$ a $G$-module $M$ and a natural number $n$, there is a $G$-space $X$ which has only one non-zero reduced homology G-module in dimension $n$ that satisfy $\tilde{H}_n(X;\mathbb{Z}) \cong M$ as $G$-modules.

=--

Carlsson provides counter examples for such "equivariant Moore spaces" for all non-cyclic groups.

+-- {: .num_cor}
###### Corollary

There is thus no [[functor]] [[Ab]]$\rightarrow$ [[Top]] that lifts $M(-,n)\colon Ab\rightarrow Ho(Top)$ since if there was such, it would induce, for any group $G$ a functor $Ab^G\rightarrow Top^G$ and in particular a positive answer to the Steenrod conjecture.

Moreover, there can also not be an [[(∞,1)-functor]] $Ab\rightarrow L_{whe} Top$ that lifts $M(-,n)$ since this will similarly yield an $\infty$-functor $Ab^G\rightarrow Top^{hG}$ where $Top^{hG}$ is the [[(∞,1)-category]] of [[∞-actions]] of $G$ on spaces. Since there is a "rigidification" functor $Top^{hG}\rightarrow Top^G$ this would yield an (ordinary) functor $Ab^G\rightarrow Top^G$ which does not exist by our previous observation.

=--

## Related concepts

### Co-Moore spaces

There is also a cohomology analogue known as a **co-Moore space** or a **Peterson space**, but this is not defined for all abelian $G$. [[sphere|Spheres]] are both Moore and co-Moore spaces for $G = \mathbb{Z}$.

Co-Moore spaces are the [[Eckmann-Hilton duality|Eckmann–Hilton duals]] of [[Eilenberg–Mac Lane spaces]].

According to Baues, Moore spaces are $H \pi$-duals to Eilenberg--Mac Lane spaces. This leads to an extensive duality for connected [[CW complexes]].

### Moore decomposition

Just as there is a [[Postnikov decomposition]] of a space as a tower of fibrations, so there is a [[Moore decomposition]] as a tower of cofibrations.

## Related concepts

* [[Peterson space]]

* [[Eilenberg–MacLane space]]

##References##

* Example 2.40 of Hatcher [Algebraic Topology](http://www.math.cornell.edu/~hatcher/AT/ATch2.pdf).
 {#Hatcher}

* [[Marek Golasinski]] and  [[Daciberg Lima Gonçalves]], [On Co-Moore Spaces](https://www.mscand.dk/article/view/13841/11841)
* [[Hans J. Baues]], Homotopy types, in Handbook of Algebraic Topology, (edited by I.M. James), North Holland, 1995.

* Gunnar Carlsson "A counterexample to a conjecture of Steenrod"
Invent. Math. 64 (1981), no. 1, 171–174.

[[!redirects Moore spaces]]


Moore spectrum

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

#Contents#
* table of contents
{:toc}

## Definition

For $G$ an [[abelian group]], then the _Moore spectrum_ $S G$ (often $M G$) of $G$ is the [[spectrum]] characterized by having the following [[homotopy groups]]:

1. $\pi_{\lt 0} S G = 0$;

1. $\pi_0(S G) = G$;

1. $H_{\gt 0}(S G,\mathbb{Z}) = 0$.

Here $H \mathbb{Z}$ is the [[Eilenberg-MacLane spectrum]] of the [[integers]] and $H_\ast(S G,\mathbb{Z})= \pi_{\ast}(S G \wedge H \mathbb{Z})$.

## Properties

### Bousfield localization at Moore spectra

A basic special case of $E$-[[Bousfield localization of spectra]] is given by $E = S A$ the Moore spectrum of an [[abelian group]] $A$. For $A = \mathbb{Z}_{(p)}$ this is [[p-localization]] and for $A = \mathbb{F}_p$ this is [[p-completion]].

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

For $A_1$ and $A_2$ two [[abelian groups]] then the following are equivalent

1. the [[Bousfield localization of spectra|Bousfield localizations]] at their Moore spectra are equivalent

   $$
     L_{S A_1} \simeq L_{S A_2}
     \,;
   $$

1. $A_1$ and $A_2$ have the same _type of acyclicity_, meaning that

   1. every [[prime number]] $p$ is invertible in $A_1$ precisely if it is in $A_2$;

   1. $A_1$ is a [[torsion group]] precisely if $A_2$ is.

=--

([Bousfield 79, prop. 2.3](#Bousfield79)) recalled e.g. in ([VanKoughnett 13, prop. 4.2](#VanKoughnett13)).

This means that given an abelian group $A$ then

* either $A$ is not torsion, then

  $$
    L_{S A} \simeq L_{S \mathbb{Z}[I^{-1}]}
    \,,
  $$

  where $I$ is the set of primes invertible in $A$ and $\mathbb{Z}[I^{-1}] \hookrightarrow \mathbb{Q}$ is the [[localization of a ring|localization]] at these primes of the [[integers]];

* or $A$ is torsion, then

  $$
   L_{S A }\simeq L_{S(\oplus_q \mathbb{F}_q )  }
   \,,
  $$

  where the [[direct sum]] is over all [[cyclic groups]] of order $q$ for $q$ a prime not invertible in $A$.

### Serre's theorem
  {#SerreTheorem}

* For $\mathbb{Q}$ the [[rational numbers]] there is an equivalence $S \mathbb{Q} \stackrel{\simeq}{\longrightarrow} H \mathbb{Q}$ between the Moore spectrum and the [[Eilenberg-MacLane spectrum]]. (e.g. [Banagl 10, p. 6](#Banagl10))

## Related concepts

* [[Eilenberg-MacLane spectrum]]

* [[Bousfield localization of spectra]]

## References

* [[Frank Adams]], Part III.6 of _[[Stable homotopy and generalised homology]]_, 1974

* {#Schwede12} [[Stefan Schwede]], section II.6.3 of _[[Symmetric spectra]]_, 2012 ([pdf](http://www.math.uni-bonn.de/people/schwede/SymSpec-v3.pdf))

Lecture notes include

* {#Strickland} [[Neil Strickland]], p. 9 of _An introduction to the category of spectra_ ([pdf](https://neil-strickland.staff.shef.ac.uk/research/stableintro.pdf))

* {#VanKoughnett13} [[Paul VanKoughnett]], _Spectra and localization_, 2013 ([[VanKoughnettLocalization.pdf:file]])

* {#Rognes12} [[John Rognes]], section 4.6 of _The Adams spectral sequence_, 2012 ([pdf](https://www.uio.no/studier/emner/matnat/math/MAT9580/v12/undervisningsmateriale/notes.050612.pdf))

Discussion in the context of [[Bousfield localization of spectra]] is in

* {#Bousfield79} [[Aldridge Bousfield]], section 2 of _The localization of spectra with respect to homology_ , Topology vol 18 (1979) ([pdf](http://www.uio.no/studier/emner/matnat/math/MAT9580/v12/undervisningsmateriale/bousfield-topology-1979.pdf))

See also

* {#Banagl10} [[Markus Banagl]], *Rational generalized intersection homology theories*, Homology, Homotopy and Applications, **12** 1 (2010) 157-185  [[pdf](http://www.mathi.uni-heidelberg.de/~banagl/pdfdocs/spectrumqih.pdf), [euclid](https://projecteuclid.org/journals/homology-homotopy-and-applications/volume-12/issue-1/Rational-generalized-intersection-homology-theories/hha/1296223826.full)]

[[!redirects Moore spectra]]


Moore-Seiberg data

Moore--Seiberg data are structure constants for a [[modular tensor category]], seen as a [[Frobenius algebra]] in the 2-category of $Vect_k$-enriched abelian categories. More explicitely, Moore--Seiberg data for the modular tensor category $\mathcal{C}$ are the collections of $k$-vector spaces
$$
\langle X_1,\dots,X_n\rangle=Hom_\mathcal{C}(\mathbf{1},X_1\otimes\cdots\otimes X_n),
$$
where $\mathbf{1}$ is the unit object of $\mathcal{C}$.

###References###
Bojko Bakalov and Alexander Kirillov, Lectures on Tensor Categories and Modular Functors, University Lecture Series 21, AMS.

Gregory Moore and Nathan Seiberg, Classical and quantum conformal field theory, Comm. Math. Phys. 123 (1989), 177--254.

[[!redirects Moore-Seiberg data]]
[[!redirects Moore–Seiberg data]]
[[!redirects Moore--Seiberg data]]
[[!redirects Moore-Seiberg datum]]
[[!redirects Moore–Seiberg datum]]
[[!redirects Moore--Seiberg datum]]


Morava E-theoretic Chern character

[[!redirects E-theoretic Chern character]]

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### Differential cohomology
+--{: .hide}
[[!include differential cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A [[higher chromatic Chern character]] for [[Morava E-theory]].

The construction in ([Torii 03](#Torii03)) produces a map from [[Morava E-theory]] at [[chromatic level]] $(n+1)$ to [[Morava K-theory]] of [[chromatic level]] $n$, tensored over something.  This is induced by
degeneration of the corresponding [[formal group laws]].

## Related concepts

* [[transchromatic character]]

## References

* [[Matthew Ando]], [[Jack Morava]], Hal Sadofsky, _Completions of Z/(p)-Tate cohomology of periodic spectra_, Geom. Topol. 2 (1998), 145-174 ([arXiv:math/9808141](http://arxiv.org/abs/math/9808141))

* [[Takeshi Torii]], _On degeneration of one-dimensional formal group laws and applications to stable homotopy theory_, Amer. J. Math. 125 (2003) 1037–1077 ([web](http://muse.jhu.edu/journals/ajm/summary/v125/125.5torii.html))
 {#Torii03}

* [[Takeshi Torii]], _Milnor operations and the generalized Chern character_, Geom. Topol. Monogr. 10 (2007) 383-421 ([arXiv:0903.4708](http://arxiv.org/abs/0903.4708))
 {#Torii09}

* [[Takeshi Torii]], _Equivariance of generalized Chern characters_, ([arXiv:0904.1647](http://arxiv.org/abs/0904.1647))

[[!redirects E-theoretic Chern characters]]

[[!redirects Morave E-theoretic Chern character]]
[[!redirects Morave E-theoretic Chern characters]]


Morava E-theory

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

#Contents#
* table of contents
{:toc}

## Idea

There are several [[cohomology theories]] that are being called _Morava E-theory_ at times:

* $B P\langle n\rangle$, the truncated [[Brown-Peterson spectrum]];

* $E(n)$, the [[Johnson-Wilson spectrum]], a localization of $B P \langle n\rangle$ at $v_n$;

* $\widehat{E(n)}$ the complete Johnson-Wilson spectrum

* $E(k,\Gamma)$ the [[Lubin-Tate spectrum]] associated to the universal deformation of a [[formal group law]] $\Gamma$ over $k$.

## Definition

Choose

* $k$ be a [[perfect field]] of [[characteristic]] $p$;

* $f$ be a [[formal group]] of [[height of a formal group|height]] $n$ over $k$.

Write

$$
  R \coloneqq W(k)[ [ v_1, \cdots, v_{n-1} ] ]
$$

for the [[Lubin-Tate ring]] of $f$, classifying its universal deformation.

By the discussion there, this is [[Landweber exactness|Landweber exact]], hence defines a [[cohomology theory]]. Therefore by the [[Landweber exact functor theorem]] there is an [[even periodic cohomology theory]] $E(n)^\bullet$ [[Brown representability theorem|represented by]] a [[spectrum]] $E(n)$ with the property that its [[homotopy groups]] are

$$
  \pi_\bullet(E(n))
  \simeq
  W(k)[ [v_1, \cdots, v_{n-1} ] ] [ \beta^{\pm 1} ]
$$

for $\beta$ of degree 2. This is called alternatively $n$th _Morava E-theory_, or _[[Lubin-Tate theory]]_ or _[[Johnson-Wilson theory]]_.

(e.g. [Lurie, lect 22](#LurieLect22))

## Properties

### As a localization of the $\infty$-group $\infty$-ring on $B^{n+1}\mathbb{Z}_p$

There is a [[Snaith theorem]] for the [[homotopy fixed points]] of
the Morava E-theory spectrum $E_n$ for the canonical [[infinity-action|action]] of a certain group, which identifies these with a localization of the [[∞-group ∞-ring]] on the [[n-group|(n+1)-group]] $B^{n+1} \mathbb{Z}_p$. ([Westerland 12, theorem 1.2](#Westerland12))

See at _[Snaith-like theorem for Morava E-theory](Snaith+theorem#ForMoravaETheory)_ for more.

### Bousfield localization and chromatic filtration
 {#BousfieldLocalizationAndChromaticFiltration}

The [[Bousfield localization of spectra]] $L_{E(n)}$ at $n$th Morava E-theory is called _[[chromatic localization]]. It_ behaves on [[complex oriented cohomology theories]] like the restriction to the closed substack

$$
  \mathcal{M}_{FG}^{\leq n+1}
  \hookrightarrow
  \mathcal{M}_{FG} \times Spec \mathbb{Z}_{(p)}
$$

of the [[moduli stack of formal groups]] on those of [[height of a formal group|height]] $\geq n+1$.

(e.g. [Lurie, lect 22, above theorem 1](#LurieLect22))

In this way the localization tower at the Morava E-theories
exhibits the [[chromatic filtration]] in [[chromatic homotopy theory]].

[[!include chromatic tower examples - table]]

### Smash product theorem

A version of the [[smash product theorem]]

For $X$ a [[homotopy type]]/[[spectrum]] and for all $n$, there is a [[homotopy pullback]]

$$
  \array{
    L_{E(n)}X &\longrightarrow& L_{K(n)}X
    \\
    \downarrow && \downarrow
    \\
    L_{E(n-1)}X &\longrightarrow& L_{E(n-1)}L_{K(n)}X
  }
  \,,
$$

where $L_{K(n)}$ denotes the [[Bousfield localization of spectra]] at $n$th [[Morava K-theory]] and similarly $L_{E(n)}$ denotes localization at [[Morava E-theory]].

 ([Lurie 10, lect 23, theorem 4](#LurieLect23))

### Bousfield equivalence class
 {#BousfieldEquivalenceClass}

For all $n$, $E(n)$ is [[Bousfield equivalence|Bousfield equivalent]] to $E(n-1) \times K(n)$, where the last factor is $n$th [[Morava K-theory]].

([Lurie 10, lect. 23, prop. 1](#LurieLect23))

## Related concepts

* [[Morava K-theory]]

* orthogonal form: [[EO(n)]]

* [[real-oriented cohomology theory|real]] form [[ER-theory]]

* [[Morava E-theoretic Chern character]]

* [[chromatic homotopy theory]]

Not to be confused with [[C*-algebra]]-[[E-theory]].

## References

Named after [[Jack Morava]] (see at _[[Morava K-theory]]_).

Relevant background lecture notes include

* [[Mike Hopkins]], _[[Complex oriented cohomology theories and the language of stacks]]_, course notes ([pdf](https://people.math.rochester.edu/faculty/doug/otherpapers/coctalos.pdf))

and more specifically see the lectures

* [[Jacob Lurie]], _[[Chromatic Homotopy Theory]]_ Lecture notes, ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture22.pdf)), Lecture 22 _Morava E-theory and Morava K-theory_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture22.pdf))
 {#LurieLect22}

* [[Jacob Lurie]], _[[Chromatic Homotopy Theory]]_ Lecture notes, ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture22.pdf))  Lecture 23 _The Bousfield Classes of $E(n)$ and $K(n)$_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture23.pdf))
 {#LurieLect23}

* [[Jacob Lurie]], _[[Chromatic Homotopy Theory]]_ Lecture notes, ([pdf]  Lecture 24 _Uniqueness of Morava K-theory_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture24.pdf))

also

* _Report of $E$-theory conjectures seminar_ (2013) ([pdf](https://people.math.rochester.edu/faculty/doug/otherpapers/peterson-mit-etheory.pdf))

Discussion of the $E_\infty$-algebra structure over $B P$ is in

* [[Andrew Baker]], _Brave new Hopf algebroids_ ([pdf](http://www.maths.gla.ac.uk/~ajb/dvi-ps/brave-ha.pdf))
 {#Baker}

based on

* [[Neil Strickland]], _Products on $MU$-modules_, Trans. Amer. Math. Soc. 351 (1999), 2569-2606.
 {#Strickland99}

Discussion of [[twisted cohomology|twists]] of Morava E-theory is in

* [[Hisham Sati]], [[Craig Westerland]], _Twisted Morava K-theory and E-theory_ ([arXiv:1109.3867](http://arxiv.org/abs/1109.3867))
 {#SatiWesterland11}

A [[Snaith theorem]]-like characterization of Morava E-theory is given in

* {#Westerland12} [[Craig Westerland]], _A higher chromatic analogue of the image of J_ ([arXiv:1210.2472](http://arxiv.org/abs/1210.2472))

Morava E-theory of [[configuration spaces of points]]:

* [[Lukas Brantner]], [[Jeremy Hahn]], [[Ben Knudsen]], *The Lubin-Tate Theory of Configuration Spaces: I* ([arXiv:1908.11321](https://arxiv.org/abs/1908.11321))

[[!redirects Morava E-theories]]


Morava K-theory

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

#Morava K-theory
* table of contents
{:toc}

## Idea

For each [[prime]] $p$, the Morava K-theories are a tower $\{K(n)\}_{n \in \mathbb{N}}$ of [[complex oriented cohomology theories]] whose [[coefficient]] [[ring]] is

$$
  \pi_\bullet\left(K\left(n\right)\right)
  \simeq
  \mathbb{F}_p [v_n, v_n^{-1}]
$$

where $v_n$ is in degree $2(p^n-1)$.

Hence with $p = 2$ for $n = 1$ $v_1$ is a [[Bott element]] of degree 2 and $K(1)$ is closely related to [[complex K-theory]], while for $n= 2$ $v_2$ is then a [[Bott element]] of degree 6 and $K(2)$ is closely related to [[elliptic cohomology]].

There is also [[integral Morava K-theory]] which instead has coefficient ring

$$
  \pi_\bullet\left(K\left(n\right)\right)
  \simeq
  \mathbb{Z}_{(p)} [v_n, v_n^{-1}]
  \,,
$$

where $\mathbb{Z}_{(p)}$ is the [[p-adic integers]].

Integral Morava K-theory can be obtained as a localization of a quotient $MU/I$ of [[complex cobordism cohomology theory]] $MU$ ([Buhné 11](#Buhne11)).

## Definition

We need the following standard notation throughout this entry.

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

For $p \in \mathbb{N}$ a [[prime number]], we write

* $\mathbb{F}_p = \mathbb{Z}/(p)$ for the [[field]] with $p$ elements;

* $\mathbb{Z}_{(p)}$ for the [[localization of a ring|localization ring]] of the [[integers]] at $p$;

* $\mathbb{Z}_p$ for the [[p-adic integers]].

=--

### Construction from complex cobordism

(e.g. [Lurie 10, lecture 22, def. 5](#LurieLecture))

### Axiomatic characterization

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

For each [[prime number|prime integer]] $p$ there exists a sequence of [[multiplicative cohomology theory|multiplicative]] [[generalized cohomology theories|generalized cohomology]]/[[generalized homology theories|homology theories]]

$$
  \{K(n)\}_{n \in \mathbb{N}}
$$

with the following properties:

1. $K(0)_\ast(X)=H_\ast(X;\mathbb{Q})$ and $\overline{K(0)}_\ast(X)=0$ when $\overline{H}_\ast(X)$ is all [[torsion]].

1. $K(1)_\ast(X)$ is one of $p-1$ isomorphic summands of mod-$p$ complex [[topological K-theory]].

1. $K(0)_\ast(pt.)=\mathbb{Q}$ and for $n\neq 0$, $K(n)_\ast(pt.)=\mathbb{F}_p[v_n,v_n^{-1}]$ where $\vert v_n\vert=2p^n-2$.

   (This [[ring]] is a graded [[field]] in the sense that every graded [[module]] over it is [[free module|free]].  $K(n)_\ast(X)$ is a module over $K(n)_\ast(pt.)$, see [below](#AsInfinityFields))

1. There is a [[Künneth isomorphism]]: $K(n)_\ast(X\times Y)\cong K(n)_\ast(X)\otimes_{K(n)_\ast(pt.)}K(n)_\ast(Y).$

1. Let $X$ be a p-local finite [[CW-complex]]. If $\overline{K(n)}_\ast(X)$ vanishes then so does $\overline{K(n-1)}_\ast(X)$.

1. If $X$ as above is not contractible then $\overline{K(n)}_\ast(X)=K(n)_\ast(pt.)\otimes \overline{H}_\ast(X;\mathbb{Z}/(p))$.

These are called the _Morava K-theories_.

=--

Due to the third point one may regard $K(n)$ as a [[∞-field]] among the [[A-infinity rings]]. See [below](#AsInfinityFields).

## Properties

### Universal characterization

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

For each [[prime number]] $p$ and each $n \in \mathbb{N}$, the Morava K-theory $K(n)$ is, up to [[equivalence in an (∞,1)-category|equivalence]], the unique [[spectrum]] underlying an [[H-space|homotopy associative]] spectrum which is

1. [[complex oriented cohomology theory|complex oriented]];

1. whose [[formal group]] has [[height of a formal group|height]] exactly $n$;

1. whose [[homotopy groups]] are $\pi_\bullet \simeq \mathbb{F}_p[v_n^\pm]$. (with $v_n$ defined as at _[[height of a formal group|height]]_).

=--

For instance ([Lurie, lecture 24, prop. 11](#LurieLecture)).

### Ring structure

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

$K(n)$ admits the structure of an [[A-∞ algebra]], in fact of an $MU_{(p)}$-[[A-∞ algebra]].

=--

Due to Robinson (and [[Andrew Baker]] at $p = 2$). (See e.g. [Lurie 10, lecture 22, lemma 2](#LurieLecture))

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

With the exception of the extreme case of $n=0$, the fields $K(n)$ do not admit [[E-∞-ring]] multiplicative structures. However, when $p\neq 2$, the multiplication is homotopy commutative. For $p = 2$ it is _not_ even homotopy commutative. Nevertheless, for many spaces $X$, the $K(n)$-[[generalized (Eilenberg-Steenrod) cohomology|generalized cohomology]] at the prime $2$ of $X$ forms a commutative ring.

=--

(e.g. [Lurie 10, lecture 22, warning 6](#LurieLecture))

### As $A_\infty$-fields
 {#AsInfinityFields}

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

If $E$ is an [[∞-field]] then $E \otimes K(n) \neq 0$ and $E$ admits the structure of a $K(n)$-[[module spectrum|module]].

=--

This appears for instance as ([Lurie, lecture 24, prop. 9, remark 13](#LurieLecture))

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

This means that the Morava $A_\infty$-rings $K(n)$ are essentially the only [[∞-fields]] in the [[stable homotopy category]].

=--

See ([Lurie, lecture 24, remark 13](#LurieLecture))

### As the primes in the $\infty$-category of spectra

The Morava K-theories label the [[prime spectrum of a symmetric monoidal stable (∞,1)-category]] of the [[(∞,1)-category of spectra]] for [[p-localization|p-local]] and [[finite spectra]] . This is the content of the _[[thick subcategory theorem]]_.

### Relation to chromatic homotopy theory

The layers in the [[chromatic homotopy theory|chromatic tower]] capture periodic phenomena in [[stable homotopy theory]], corresponding to the Morava K-theory $E_\infty$-fields.

Specifically the [[Bousfield localization of spectra]] $L_{K(n)}$ acts on [[complex oriented cohomology theories]] like completion along the locally closed substack

$$
  \mathcal{M}^n_{FG}
  \hookrightarrow
  \mathcal{M}_{FG}
$$

of the [[moduli stack of formal groups]] at those of [[height of a formal group|height]] $n$.

([Lurie 10, lecture 29](#LurieLecture))

[[!include chromatic tower examples - table]]

### Relation to Bousfield lattice

It is known that in the [[Bousfield lattice]] of the [[stable homotopy category]], the Bousfield classes of the Morava K-theories are minimal. It is conjectured by [[Mark Hovey]] and John Palmieri that the [[Boolean algebra]] contained in the Bousfield lattice is atomic and generated by the Morava K-theories and the spectra $A(n)$ which measure the failure of the [[telescope conjecture]].

### Orientation
 {#Orientation}

The [[orientation in generalized cohomology|orientation]] of [[integral Morava K-theory]] is discussed in ([Sati-Kriz 04](#SatiKriz04), [Buhné 11](#Buhne11)). It is essentially given by the vanishing of the seventh [[integral Stiefel-Whitney class]] $W_7$.

Notice that this is in higher analogy to how [[orientation in generalized cohomology|orientation]] in [[complex K-theory]] is given by the vanishing third [[integral Stiefel-Whitney class]] $W_3$ ([[spin^c-structure]]).

### $\infty$-Group rings and twists

Write $gl_1(K(n))$ for the [[∞-group of units]] of the (a) Morava K-theory spectrum

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

For $p = 2$ and all $n \in \mathbb{N}$, there is an [[equivalence in an (infinity,1)-category|equivalence]]

$$
  Maps(B^{n+1}U(1), B gl_1(K(n)))
  \simeq
  \mathbb{Z}_2
$$

between

* the [[mapping space]] from the [[classifying space]] for [[circle n-bundle|circle (n+1)-bundles]] to the [[delooping]] of the [[∞-group of units]] of $K(n)$

and

* the [[p-adic integers|2-adic integers]].

=--

([Sati-Westerland 11, theorem 1](#SatiWesterland11))

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

By the discussion at [[(∞,1)-vector bundle]] this means that for each such map there is a type of [[twisted cohomology|twist]] of Morava K-theory (at $p = 2$).

=--

## Related concepts

* [[equivariant Morava K-theory]]

* [[2-periodic Morava K-theory]]

* [[K-theory]], [[topological K-theory]], [[algebraic K-theory]]

* [[chromatic homotopy theory]]

* [[K(n)-local stable homotopy theory]]

* [[integral Morava K-theory]]

* [[Morava E-theory]]

  * [[Lubin-Tate theory]]

## References

Morava K-theory originates in unpublished preprints by [[Jack Morava]] in the early 1970s.

A first published account appears in (see at _[[Johnson-Wilson spectrum]]_):

* [[David Copeland Johnson]], [[W. Stephen Wilson]], _BP operations and Morava's extraordinary K-theories_, Math. Z. 144 (1): 55−75,  (1975) ([pdf](http://www.math.jhu.edu/~wsw/papers2/math/08-BP-Kn-J2W-1975.pdf))

see also

* [[Doug Ravenel]], _Nilpotence and Periodicity in Stable Homotopy Theory_, Annals of Mathematics Studies **128**, Princeton University Press (1992) [[ISBN:9780691025728](https://press.princeton.edu/books/paperback/9780691025728/nilpotence-and-periodicity-in-stable-homotopy-theory-am-128-volume), [pdf](https://people.math.rochester.edu/faculty/doug/mybooks/nilpb.pdf), [webpage](https://people.math.rochester.edu/faculty/doug/nilp.html)]

Textbook account:

* {#Rudyak98} [[Yuli Rudyak]], Section IX.7 in: _On Thom Spectra, Orientability, and Cobordism_, Springer 1998 ([doi:10.1007/978-3-540-77751-9](https://doi.org/10.1007/978-3-540-77751-9))

A discussion with an eye towards [[category theory|category theoretic]] general abstract properties of localized stable homotopy theory is in

* [[Mark Hovey]], [[Neil Strickland]], _Morava K-theories and localization_, American Mathematical Soc., Jan 1, 1999 ([pdf](http://www.math.rochester.edu/people/faculty/doug/otherpapers/kn.pdf))

A survey of the theory is in

* Urs Würgler, _Morava K-theories: a survey_, Algebraic topology Poznan 1989, Lecture Notes in Math. 1474, Berlin: Springer, pp. 111 138 (1991) ([doi:10.1007/BFb0084741](https://doi.org/10.1007/BFb0084741))

In

* [[Jacob Lurie]], _[[Chromatic Homotopy Theory]]_ Lecture notes, ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture22.pdf)){#LurieLecture}

  Lecture 22 _Morava E-theory and Morava K-theory_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture22.pdf))

  Lecture 23 _The Bousfield Classes of $E(n)$ and $K(n)$_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture23.pdf))

  Lecture 24 _Uniqueness of Morava K-theory_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture24.pdf))

the explicit definition via [[formal group laws]] is in lecture 22 and the abstract characterization in lecture 24.

The $E_\infty$-algebra structure over $\widehat{E(n)}$ is comment on in

* {#Baker} [[Andrew Baker]], _Brave new Hopf algebroids_ ([pdf](http://www.maths.gla.ac.uk/~ajb/dvi-ps/brave-ha.pdf))

based on

* {#Strickland99} [[Neil Strickland]], _Products on $MU$-modules_, Trans. Amer. Math. Soc. 351 (1999), 2569-2606.

Discussion in relation to the [[Arnold conjecture]] in [[symplectic topology]]:

* [[Mohammed Abouzaid]], [[Andrew J. Blumberg]], _Arnold Conjecture and Morava K-theory_, ([arXiv:2103.01507](https://arxiv.org/abs/2103.01507))

On the Morava K-theory of [[iterated loop spaces]] of [[n-spheres]]:

* [[Douglas Ravenel]], *What we still don't know about loop spaces of spheres*, in: [[Mark Mahowald]], [[Stewart Priddy]] (eds.), *Homotopy Theory via Algebraic Geometry and Group Representations*, Contemporary Mathematics **220**, AMS 1998  ([pdf](https://people.math.rochester.edu/faculty/doug/mypapers/loop.pdf), [[Ravenel_LoopSpacesOfSpheres.pdf:file]], [doi:10.1090/conm/220](http://dx.doi.org/10.1090/conm/220))

The [[orientation in generalized cohomology|orientation]] of [[integral Morava K-theory]] is discussed in

* {#SatiKriz04} [[Igor Kriz]], [[Hisham Sati]], _M-theory, type IIA superstrings, and elliptic cohomology_, Adv. Theor.
Math. Phys. 8 (2004), no. 2, 345–394 ([arXiv:hep-th/0404013](http://arxiv.org/abs/hep-th/0404013))

* {#Buhne11} [[Lukas Buhné]], _[[Properties of Integral Morava K-Theory and the Asserted Application to the Diaconescu-Moore-Witten Anomaly]]_, Diploma thesis Hamburg (2011)

Some [[twisted cohomology|twists]] of Morava K-theory/maps into its [[∞-group of units]] as well as the [[Atiyah-Hirzebruch spectral sequence]] for Morava $K$ and Morava $E$ are discussed in

* {#SatiWesterland11} [[Hisham Sati]], [[Craig Westerland]], _Twisted Morava K-theory and E-theory_ ([arXiv:1109.3867](http://arxiv.org/abs/1109.3867))

For a review in the context of [[M-theory]] see

* [[Hisham Sati]], _[[Geometric and topological structures related to M-branes]]_ (2010)

[[!redirects Morava K-theories]]


Morava stabilizer group

#Contents#
* table of contents
{:toc}

## Idea

The [[moduli stack of formal groups]] $\mathcal{M}_{FG}$ (1-dimensional commutative formal groups) admits a natural [[stratification]] whose open [[strata]] are labeled by a [[natural number]] called the _[[height of a formal group|height of formal groups]]_.

For $p$ a [[prime number]], write $\overline{\mathbb{F}}_{\mathrm{p}}$ for the [[algebraic closure]] of the [[prime field]] $\mathbb{F}_p$.

The [[stratum]] $\mathcal{M}_{FG}^n$ can be identified with the [[homotopy quotient]] $Spec (\overline{\mathbb{F}}_{\mathrm{p}})// \mathbb{G}$, where the [[group]] $\mathbb{G}$ is the [[automorphism group]]  over $\mathbb{F}_p$ of the unique formal group law $f$ of [[height of a formal group|height]] $n$,

$$
  \mathbb{G} \coloneqq Aut(\overline{\mathbb{F}}_{\mathrm{p}}, f)
$$

This is called the _Morava stabilizer group_. Essentially its [[group algebra]] ([[Hopf algebra]]) is called the _Morava stabilizer algebra_.

This is discussed around ([Lurie 10, lect. 19, prop. 1](#LurieLect19)), see also the beginning of [Lurie 10, lect 21](#LurieLect21), and in ([Ravenel, chapt. 6](#RavenelCh6)).

The [[deformation theory]] around these [[strata]] is [[Lubin-Tate theory]].

## References

* {#LurieLect19} [[Jacob Lurie]], _Morava Stabilizer Groups_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture19.pdf)), lecture 19 in _[[Chromatic Homotopy Theory]]_, Lecture series 2010 ([web](http://www.math.harvard.edu/~lurie/252x.html))

* {#LurieLect21} [[Jacob Lurie]], _Lubin-Tate Theory _ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture21.pdf)), lecture 21 in  _[[Chromatic Homotopy Theory]]_, Lecture series 2010 ([web](http://www.math.harvard.edu/~lurie/252x.html))

* {#RavenelCh6} [[Doug Ravenel]], _[[Complex cobordism and stable homotopy groups of spheres]]_, Chapter 6 _Morava stabilizer algebra_ ([pdf](http://www.math.rochester.edu/people/faculty/doug/mybooks/ravenel6.pdf))

* [[Niko Naumann]], _Torsors under smooth group-schemes and Morava stabilizer groups_ ([web](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.131.5497))

[[!redirects Morava stabilizer groups]]

[[!redirects Morava stabilizer algebra]]
[[!redirects Morava stabilizer algebras]]


Mordehai Milgrom

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

* [Institute webpage](http://www.weizmann.ac.il/particle/milgrom/)

## Selected writings

* [[Mordehai Milgrom]], *Broader view of bimetric MOND*, Physical Review D, Volume 106, Issue 8. Published 10 October 2022. ([doi:10.1103/PhysRevD.106.084010](https://doi.org/10.1103/PhysRevD.106.084010), [arXiv:2208.10882](https://arxiv.org/abs/2208.10882))

## related $n$Lab entries

* [[MOND]]

category: people