nilpotent completion of spectra

> under construction

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#### Stable Homotopy theory
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[[!include stable homotopy theory - contents]]
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=--

#Contents#
* table of contents
{:toc}

## Idea

Given a [[spectrum]] $X$, and a [[ring spectrum]] $E$, then the _$E$-nilpotent completion_ of $X$ at $E$ is, for any choice $X_\bullet \to X$ of $E$-[[Adams tower]], the [[homotopy limit]] $\underset{\longleftarrow}{\lim} X_\bullet$ over that tower ([Ravenel 84, def. 1.13](#Ravenel84)).

Under certain finiteness conditions (see [below](#RelationToELocalization)), but not generally, this is equivalent to the $E$-[[Bousfield localization of spectra|Bousfield localization]] $L_E X$ (which, in turn, is in special cases given by [[formal completion]], see at _[[fracture theorem]]_).

The $E$-[[Adams spectral sequence]] induced by the given [[Adams tower]] [[conditionally convergent spectral sequence|conditionally converges]] to the $E$-nilpotent completion.

## Definition

### Bousfield's definition

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

Let $(E, \mu, e)$ be a [[homotopy commutative ring spectrum]] ([def.](Introduction+to+Stable+homotopy+theory+--+1-2#HomotopyCommutativeRingSpectrum)) and $Y \in Ho(Spectra)$ any spectrum. Write $\overline{E}$ for the [[homotopy fiber]] of the unit $\mathbb{S}\overset{e}{\to} E$ as in [this def.](Adams+spectral+sequence#HomotopyFiberOfUnitOfCommutativeRingSpectrum) such that the $E$-Adams filtration of $Y$ ([def.](Adams+spectral+sequence#AdamsEAdamsSpectralSequence)) reads (according to [this lemma](Adams+spectral+sequence#Wp))

$$
  \array{
    \vdots
    \\
    \downarrow
    \\
    \overline{E}^3 \wedge Y
    \\
    \downarrow
    \\
    \overline{E}^2 \wedge Y
    \\
    \downarrow
    \\
    \overline{E} \wedge Y
    \\
    \downarrow
    \\
    Y
  }
  \,.
$$

For $n \in \mathbb{N}$, write

$$
  \overline{E}_n
   \coloneqq
  hocof( \overline{E}^{n+1} \overset{i^{n+1}}{\longrightarrow} \mathbb{S})
$$

for the homotopy cofiber. Here $\overline{E}_0 \simeq 0$.
By the [[tensor triangulated category|tensor triangulated]] structure of $Ho(Spectra)$ ([prop.](Introduction+to+Stable+homotopy+theory+--+1-2#TensorTriangulatedStructureOnStableHomotopyCategory)), this homotopy cofiber is preserved by forming [[smash product of spectra|smash product]] with $Y$, and so also

$$
  \overline{E}_n \wedge Y
   \simeq
  hocof( \overline{E}^n \wedge Y \overset{}{\longrightarrow} Y)
  \,.
$$

Now let

$$
  \overline{E}_s
    \overset{p_{s-1}}{\longrightarrow}
  \overline{E}_{s-1}
$$

be the morphism implied by the [[octahedral axiom]] of the [[triangulated category]] $Ho(Spectra)$ ([def.](Introduction+to+Stable+homotopy+theory+--+1-1#CategoryWithCofiberSequences), [prop.](Introduction+to+Stable+homotopy+theory+--+1-1#StableHomotopyCategoryIsTriangulated)):

$$
  \array{
    \overline{E}^{s+1}
      &\overset{i}{\longrightarrow}&
    \overline{E}^s
      &\longrightarrow&
    E \wedge \overline{E}^s
      &\longrightarrow&
    \Sigma \overline{E}^{s+1}
    \\
    {}^{\mathllap{=}}\downarrow
      &&
    \downarrow^{\mathrlap{i^s}}
      &&
    \downarrow^{}
      &&
    \downarrow
    \\
    \overline{E}^{s+1}
      &\longrightarrow&
    \mathbb{S}
      &\longrightarrow&
    \overline{E}_s
      &\longrightarrow&
    \Sigma \overline{E}^{s+1}
    \\
    &&
    \downarrow
      &&
    \downarrow^{\mathrlap{p_{s-1}}}
    \\
      &&
    \overline{E}_{s-1}
      &\overset{=}{\longrightarrow}&
    \overline{E}_{s-1}
    \\
    &&
    \downarrow
      &&
    \downarrow
    \\
      &&
    \Sigma \overline{E}^s
      &\longrightarrow&
     \Sigma E \wedge \overline{E}^s
  }
  \,.
$$

By the [[commuting square]] in the middle and using again the [[tensor triangulated category|tensor triangulated]] structure, this yields an inverse sequence under $Y$:

$$
  Y
    \simeq
  \mathbb{S} \wedge Y
    \longrightarrow
  \cdots
    \overset{p_3 \wedge id}{\longrightarrow}
  \overline{E}_3 \wedge Y
    \overset{p_2 \wedge id}{\longrightarrow}
  \overline{E}_2 \wedge Y
    \overset{p_1 \wedge id}{\longrightarrow}
  \overline{E}_1 \wedge Y
$$

The **[[E-nilpotent completion]]** $Y^\wedge_E$ of $Y$ is the [[homotopy limit]] over the resulting inverse sequence

$$
  Y^\wedge_E
    \coloneqq
  \mathbb{R}\underset{\longleftarrow}{\lim}_n \overline{E}_n \wedge Y
$$

or rather the canonical morphism into it

$$
  Y \longrightarrow Y^\wedge_E
  \,.
$$

Concretely, if

$$
  Y
    \simeq
  \mathbb{S} \wedge Y
    \longrightarrow
  \cdots
    \overset{p_3 \wedge id}{\longrightarrow}
  \overline{E}_3 \wedge Y
    \overset{p_2 \wedge id}{\longrightarrow}
  \overline{E}_2 \wedge Y
    \overset{p_1 \wedge id}{\longrightarrow}
  \overline{E}_1 \wedge Y
$$

is presented by a tower of fibrations between fibrant spectra in the [[model structure on topological sequential spectra]], then $Y^\wedge_E$ is represented by the ordinary [[sequential limit]] over this tower.

=--

([Bousfield 79, top, middle and bottom of page 272](#Bousfield79))

### As the totalization of the cosimplicial spectrum
 {#AsLimitOverTotTower}

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

Given a [[E-infinity ring spectrum]] $E$,  its **corresponding cosimplicial spectrum** is the augmented [[cosimplicial object|cosimplicial spectrum]]

$$
  E^\bullet
   \;\coloneqq\;
  \left(
     \mathbb{S}
       \overset{e}{\longrightarrow}
     E
       \underoverset
         {\underset{id \wedge e}{\longrightarrow}}
         {\overset{e \wedge id}{\longrightarrow}}
         {\overset{\mu}{\longleftarrow}}
     E \wedge E
        \underoverset
          {\underset{e \edge id}{\longrightarrow}}
          {\overset{id \wedge e}{\longrightarrow}}
          {
            \underoverset
              {\underset{id \wedge \mu}{\longleftarrow}}
              {\overset{\mu \wedge id}{\longleftarrow}}
              {\overset{id \wedge e \wedge id}{\longrightarrow}}
          }
      E \wedge E \wedge E
       \cdots
  \right)
  \,.
$$

(This is the [[formal dual]] of the [[Cech nerve]] of $Spec(E) \to Spec(\mathbb{S})$ in the [[opposite category]], where we write $Spec(E)$ for the object $E$ regarded in the opposite category.)

Moreover, for $X \in Ho(Spectra)$ any spectrum, then there is the corresponding augmented cosimiplicial spectrum $E^\bullet \wedge X$.

=--

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

Given an [[E-infinity ring spectrum]] $E$ and any spectrum $X$, then the $E$-nilpotent completion $X^\wedge_E$ (according to def. \ref{ENilpotentCompletion}) is equivalently the homotopy limit

$$
  \begin{aligned}
     X^\wedge_E
      &\simeq
    Tot(E^\bullet \wedge X)
      \\
     & =
      \underset{\leftarrow}{holim}_{n \in \Delta} (E^n\wedge X)
    \\
      & \simeq
     \underset{\leftarrow}{holim}_{n \in \mathbb{N}}Tot^n(E^\bullet \wedge X)
  \end{aligned}
$$

over the tower of homotopy-[[totalizations]] of the skeleta of the [[cosimplicial object|cosimplicial]] spectrum $E^\bullet \otimes Y$ (def. \ref{RingSpectrumGivesCosimplicialSpectrum}).

=--

This claim originates in ([Hopkins 99, remark 5.5 (ii)](#Hopkins99)). It is taken for granted in ([Lurie 10, lecture 8, lecture 30](#Lurie10)). The first published proof is ([Mathew-Naumann-Noel 15, prop. 2.14](#MathewNaumannNoel15)). See also ([Carlsson 07, e.g. remark 3.1](#Carlsson07)).

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

Prop. \ref{ENilpotentCompletionIsHolimOverTotTower} implies that the $E$-[[Adams spectral sequence]] may equivalently be regarded as computing [[descent]] of [[quasicoherent infinity-stacks]] in [[E-infinity geometry]] along the canonical morphisms $Spec(E)\longrightarrow $ [[Spec(S)]]. See at _[Adams spectral sequence -- As derived descent](https://ncatlab.org/nlab/show/Adams%20spectral%20sequence#DefinitionInHigherAlgebra)_.

=--

## Properties

### Relation to $E$-localization
 {#RelationToELocalization}

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

There is a canonical map

$$
  L_E X
    \overset{}{\longrightarrow}
  \underset{\leftarrow}{\lim}_n (E^{\wedge^{n+1}_S}\wedge_S X)
$$

from the $E$-[[Bousfield localization of spectra]] of $X$ into the [[totalization]].

=--

We consider now conditions for this morphism to be an [[equivalence]].

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

For $R$ a [[ring]], its _core_ $c R$ is the [[equalizer]] in

$$
  c R
    \longrightarrow
  R
    \stackrel{\longrightarrow}{\longrightarrow}
  R \otimes R
  \,.
$$

=--

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

Let $E$ be a [[connective spectrum|connective]] [[E-∞ ring]] such that the core of $\pi_0(E)$, def. \ref{CoreOfARing}, is either of

* the [[localization of a ring|localization]] of the [[integers]] at a set $J$ of [[primes]], $c \pi_0(E) \simeq \mathbb{Z}[J^{-1}]$;

* $\mathbb{Z}_n$ for $n \geq 2$.

Then the map in remark \ref{CanonicalMapFromELocalizationToTotalization} is an equivalence

$$
  L_E X \stackrel{\simeq}{\longrightarrow}
  \underset{\leftarrow}{\lim}_n (E^{\wedge^{n+1}_S}\wedge_S X)
  \,.
$$

=--

([Bousfield 79](#Bousfield79)).

## Examples

The nilpotent completion of a [[connective spectrum]] at the [[Eilenberg-MacLane spectrum]] $H \mathbb{Z}$, happens to be the spectrum itself (by a [[Postnikov tower]] argument).

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

For $X$ a [[connective spectrum]], its $H \mathbb{F}_p$-nilpotent completion is the [[formal completion]] $X^{\wedge}_p$.

The [[MU]]-nilpotent completion of any connective spectrum $X$ is $X$.

The [[BP]]-nilpotent completion at prime $p$ of any connective spectrum $X$ is $X_{(p)}$.

=--

([Ravenel 84, example 1.16](#Ravenel84))

## Related concepts

* [[formal completion]]

## References

The concept originates with

* {#Bousfield79} [[Aldridge Bousfield]], _The localization of spectra with respect to homology_, Topology 18 (1979), no. 4, 257--281. ([pdf](http://www.uio.no/studier/emner/matnat/math/MAT9580/v12/undervisningsmateriale/bousfield-topology-1979.pdf))

* {#Ravenel84} [[Douglas Ravenel]], _Localization with respect to certain periodic homology theories_, American Journal of Mathematics, Vol. 106, No. 2, (Apr., 1984), pp. 351-414 ([pdf](https://www.math.rochester.edu/people/faculty/doug/mypapers/loc.pdf))

The re-interpretation in terms of totalization of the cosimplicial spectrum is briefly mentioned in

* {#Hopkins99} [[Mike Hopkins]], section 4 of _Complex oriented cohomology theories and the language of stacks_, course notes 1999 ([pdf](https://www.math.rochester.edu/people/faculty/doug/otherpapers/coctalos.pdf))

and tacitly assumed in

* {#Lurie10} [[Jacob Lurie]], _[[Chromatic Homotopy Theory]]_ (2010) lecture 30: _Localizations and the Adams-Novikov Spectral Sequence_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture30.pdf))

A proof of the equivalence of this re-interpretation appears in

* {#MathewNaumannNoel15} [[Akhil Mathew]], [[Niko Naumann]], [[Justin Noel]], _Nilpotence and descent in equivariant stable homotopy theory_ ([arXiv:1507.06869](http://arxiv.org/abs/1507.06869))

See also

* {#BakerLazarev01} [[Andrew Baker]], [[Andrey Lazarev]], _On the Adams Spectral Sequence for $R$-modules_, Algebr. Geom. Topol. 1 (2001) 173-199 ([arXiv:math/0105079](http://arxiv.org/abs/math/0105079))

* {#Carlsson07} [[Gunnar Carlsson]], _Derived completions in stable homotopy theory_ ([arXiv:0707.2585](http://arxiv.org/abs/0707.2585))

* {#Lawson09} [[Tyler Lawson]], _Completed representation ring spectra of nilpotent groups_, Algebraic & Geometric Topology 6 (2006) 253–285 ([arXiv:0902.4867](http://arxiv.org/abs/0902.4867))

[[!redirects E-nilpotent completions]]

[[!redirects nilpotent completion]]
[[!redirects nilpotent completions]]

[[!redirects E-nilpotent completion]]


nilpotent element

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#Contents#
* table of contents
{: toc}

## Idea
An element $x$ in a [[ring]] (or potentially even a [[nonassociative algebra|nonassociative]] [[rig]]) $A$ is __nilpotent__ if there exist a natural number $n$ such that $x^n = 0$.

An ring/rig/algebra is __nilpotent__ if there exists a uniform number $n$ such that any product of $n$ elements is $0$. An algebra over a [[field]] is __locally nilpotent__ if all of its finitely-generated subalgebras are nilpotent. A [[Lie algebra]] element is __[[ad-nilpotent Lie algebra|ad-nilpotent]]__ if the multiplication with any of its elements is a nilpotent linear operator. A Lie algebra $A$ is __nilpotent__ iff its [[lower central series]] $A, [A,A], [A,[A,A]], \ldots, [A,[A,[A,\ldots,[A,A]\cdots]]], \ldots$ terminates with $0$ after finitely many steps. By Engel's theorem ([English Wikipedia](http://en.wikipedia.org/wiki/Engel_theorem)) a finite-dimensional Lie algebra is nilpotent iff it is locally nilpotent. Thus sometimes locally nilpotent Lie algebras are called __Engel's Lie algebras__.

The class of locally nilpotent [[associative algebras]] is closed under extensions (defined in the category of associative algebras).  Consequently associative algebras have a largest nilpotent ideal (namely the sum of all locally nilpotent ideals), which is called __Levitskii radical__.

The structure rings of classical [[algebraic variety|algebraic varieties]] are finitely generated [[noetherian ring|noetherian]] commutative associative unital rings _without nilpotent elements_. One of the principal advantages of Grothendieck's theory of [[schemes]] is to allow for nilpotent elements in [[local rings]]. A scheme is [[reduced scheme|reduced]] if there are no nilpotent elements in stalks of the structure sheaf.

## Related pages

* [[nilpotent group]]

* [[nilpotent topological space]]

* [[nilradical]]

For [[Lie algebras]]:

* [[Jacobson-Morozov theorem]]

[[!redirects nilpotent]]
[[!redirects nilpotent element]]
[[!redirects nilpotent elements]]

[[!redirects nilpotent algebra]]
[[!redirects nilpotent algebras]]
[[!redirects locally nilpotent algebra]]
[[!redirects locally nilpotent algebras]]

[[!redirects Engel's Lie algebra]]
[[!redirects Engel's Lie algebras]]

[[!redirects Levitskii radical]]
[[!redirects Levitskii radicals]]


nilpotent group

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# Nilpotent groups
* table of contents
{: toc}

## Idea

A [[group]] is __nilpotent__ if it can be built up by [[central extensions]] from [[abelian groups]].  A __central series__ for a group is a witness to its nilpotency.

More generally, we may speak about $\mathcal{A}$-nilpotent $\pi$-groups, for any class $\mathcal{A}$ of abelian groups and any group $\pi$ acting on our groups.  A group $G$ is equivalently an [[Ab]]-nilpotent $G$-group for its [[adjoint action]].

## Definition

### Nilpotent groups and central series

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

The class of **nilpotent groups** is defined [[inductive definition|inductively]] by the following clauses:

1. The [[trivial group]] $1$ is nilpotent.

2. If $1\to G' \to G \to G''\to 1$ is a [[central extension]] (so that in particular, $G'$ is [[abelian group|abelian]]) and $G''$ is nilpotent, then $G$ is nilpotent.
=--

Phrased in this way, nilpotency is an inductive [[predicate]] on the class of groups.  If we regard the same clauses as defining an inductive [[family]] indexed over the class of groups, then we obtain the definition of a central series.

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

The set of **central series** for a group $G$ is defined [[inductive definition|inductively]] by the following clauses:

1. The trivial group $1$ has a specified central series, called the "trivial" one.

2. From any central extension $1\to G' \to G \to G''\to 1$ and any central series of $G''$, we obtain a central series of $G$, called an "extension" of the given one.
=--

Thus, central series are "witnesses" to nilpotency: a group is nilpotent if and only if it has some central series.

If we "expand out" the inductive definition of central series, and use the [[isomorphism theorems]], we see that it consists of a sequence of central extensions
$$ 1 \to G_1 \to G \to G/G_1 \to 1 $$
$$ 1 \to G_2/G_1 \to G/G_1 \to G/G_2 \to 1$$
$$ 1 \to G_3/G_2 \to G/G_2 \to G/G_3 \to 1$$
$$ \dots $$
$$ 1 \to G/G_{n-1} \to G/G_{n-1} \to 1 \to 1$$
and therefore a sequence of [[normal subgroups]]
$$ 1 = G_0 \trianglelefteq G_1 \trianglelefteq G_2 \trianglelefteq \dots \trianglelefteq G_{n-1} \trianglelefteq G_n = G $$
such that each $G_i/G_{i-1}$ is central in $G/G_{i-1}$.
This is the "usual" definition of central series.

### Nilpotency class

Every central series has a *length*, defined recursively by saying that the length of the trivial central series is $0$ and the length of an extension is one more than the length of the original.  The **nilpotency class** of a nilpotent group is the minimum length of all of its central series.

Proofs about nilpotent groups are often most naturally phrased using induction over the inductive definition of nilpotency.  However, probably due to widespread ignorance about inductive definitions, it is common to find them phrased instead using ordinary natural-number induction over the nilpotency class.

### Co-nilpotent groups and central streams

If we interpret the same defining clauses of nilpotent groups and central series [[coinductive definition|coinductively]] rather than inductively, we obtain notions that might be called *co-nilpotent groups* and *central streams* ("stream" being the standard name for the coinductive counterpart of a list).  Explicitly, a central stream is a descending countable sequence of normal subgroups, such that each successive quotient is central in the corresponding quotient of the whole group, that may or may not ever terminate with the trivial group.

In fact, *every* group admits some central stream and hence is co-nilpotent.  Two canonical central streams associated to any group are its [[lower central series]] and its [[upper central series]] (for now see [Wikipedia](http://en.wikipedia.org/wiki/Central_series#Upper_central_series)). Despite the names, these two central streams are actually central series (i.e. they terminate at the trivial group) if and only if the group is nilpotent.

### $\mathcal{A}$-nilpotent $\pi$-groups

The following generalization of nilpotent groups is sometimes useful.  Let $\mathcal{A}$ be any class of [[abelian groups]] containing $0$, and let $\pi$ be any group.  By a *$\pi$-group* we mean a group with an action of $\pi$ (through group automorphisms, of course).

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

The class of **$\mathcal{A}$-nilpotent** $\pi$-groups is defined [[inductive definition|inductively]] by the following clauses:

1. The trivial group $1$ is $\mathcal{A}$-nilpotent.

2. If $1\to G' \to G \to G''\to 1$ is a [[central extension]] of $\pi$-groups (i.e. a central extension whose maps are $\pi$-equivariant), and $G''$ is $\mathcal{A}$-nilpotent while $G'\in \mathcal{A}$ and $\pi$ acts trivially on $G'$, then $G$ is $\mathcal{A}$-nilpotent.
=--

A group $G$ is nilpotent in the original sense if and only if it is an $Ab$-nilpotent $G$-group with its [[adjoint action]].  If $\pi$ is nontrivial and/or $\mathcal{A}$ is strictly smaller than $Ab$, then the notion of $\mathcal{A}$-nilpotency can be nontrivial even for abelian groups (whereas every abelian group is obviously nilpotent in the ordinary sense).

## Properties

Every nilpotent group is an example of a [[solvable group]] (indeed, the groups in the [[lower central series]] of any group can be term-wise included into its [[derived series]]).

The [[Sylow p-subgroups]] of any nilpotent group are [[normal subgroup|normal]]. The direct product of these subgroups in such a group is its [[torsion subgroup]].

## Examples

Generally:

* Every [[abelian group]] is nilpotent.

* The [[direct product group]] of two nilpotent groups is again nilpotent.

* The [[central extension]] of any [[abelian group]] is nilpotent, a famous class of examples of such are the [[Heisenberg groups]].

Specifically:

* The multiplicative group of upper triangular unipotent $n \times n$ matrices with [[coefficients]] in any [[field]] is nilpotent.

## Related pages

* [[solvable group]]

* [[nilpotent module]]

* [[nilpotent element]]

* [[nilpotent Lie algebra]]

* [[nilpotent topological space]]

* [[localization of a space]]

## References

* [[Peter Hilton]], _Nilpotency in group theory and topology_, Publicacions de la Secció de Matemàtiques Vol. 26, No. 3 (1982), pp. 47-78 ([jstor:43741908](https://www.jstor.org/stable/43741908))

* [[Peter May]], [[Kate Ponto]], §3.1 in: _[[More concise algebraic topology]]_,   University of Chicago Press (2012) [[ISBN:9780226511795](https://press.uchicago.edu/ucp/books/book/chicago/M/bo12322308.html), [pdf](https://www.math.uchicago.edu/~may/TEAK/KateBookFinal.pdf)]

See also:

* eom [nilpotent group](http://eom.springer.de/n/n066720.htm)

* Wikipedia [nilpotent group](http://en.wikipedia.org/wiki/Nilpotent_group)

On nilpotent [[Lie groups]]:

* [[Manuel Amann]], *On quasi-isometric nilpotent Lie groups* [[arXiv:1710.04542](https://arxiv.org/abs/1710.04542)]

[[!redirects nilpotent group]]
[[!redirects nilpotent groups]]

[[!redirects central series]]


nilpotent homotopy type > history

> see at _[[nilpotent topological space]]_


nilpotent ideal

> Not to be confused with *[[nilradical]]*, the [[ideal]] of [[nilpotent elements]].

---

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#Contents#
* table of contents
{:toc}

## Definition

For $n$ a positive integer and $I$ a (left) [[ideal]] of a [[ring]] $R$, let $I^n$ denote the ideal of $R$ consisting of all finite sums of $n$-tuple products $i_1\cdots i_n$ of elements in $I$.

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

A (left) [[ideal]] $I$ of a [[ring]] $R$ is __nilpotent__ if there exists a [[positive number|positive]] [[natural number]] $n$ such that $I^n$ is the zero ideal of $R$.

=--

## Related concepts

* [[infinitesimal extension]]

[[!redirects nilpotent ideals]]


nilpotent infinitesimal

## Idea

In [[infinitesimal analysis]] a _nilpotent infinitesimal_ is an [[infinitesimal quantity]] such that some power of it vanishes. This is the case in the approach of [[synthetic differential geometry]], but not for instance in what is called [[nonstandard analysis]].

[[!redirects nilpotent infinitesimals]]


nilpotent L-infinity algebra

[[!redirects nilpotent L-infinity algebras]]

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#### Higher algebra
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[[!include higher algebra - contents]]
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#### Rational homotopy theory
+--{: .hide}
[[!include differential graded objects - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

The generalization of the concept of _[[nilpotent Lie algebra]]_ from [[Lie algebras]] to [[L-infinity algebras|$L_\infty$-algebras]]..

Under the [[formal duality]] between [[L-∞ algebras|$L_\infty$-algebras]] and their [[Chevalley-Eilenberg algebra|Chevalley-Eilenberg]] [[dgc-algebras]], [[connective]] nilpotent $L_\infty$-algebras correspond [[bijection|bijectively]] to the connected [[Sullivan models]] ([Berglund 2015, Thm. 2.3](#Berglund15)).

## Properties

### Rational homotopy theory

The [[fundamental theorem of dgc-algebraic rational homotopy theory]] says that the [[homotopy theory]] of  nilpotent $L_\infty$-algebras [[of finite type]] over the [[rational numbers]] is equivalently that of [[rational topological spaces|rational]] [[nilpotent topological spaces]] of [[finite type]].

## References

* {#Getzler04} [[Ezra Getzler]], Def. 4.2 in: _Lie theory for nilpotent $L_\infty$ algebras_, Annals of Mathematics, 170 (2009), 271–301 ([math.AT/0404003](http://arxiv.org/abs/math/0404003))

* {#Berglund15} [[Alexander Berglund]], Def. 2.1 in: _Rational homotopy theory of mapping spaces via Lie theory for $L_\infty$ algebras_,  Homology, Homotopy and Applications, Volume 17 (2015) Number 2 ([arXiv:1110.6145](https://arxiv.org/abs/1110.6145), [doi:10.4310/HHA.2015.v17.n2.a16]( http://dx.doi.org/10.4310/HHA.2015.v17.n2.a16))

[[!redirects nilpotent L-∞ algebra]]
[[!redirects nilpotent L-∞ algebras]]


nilpotent Lie algebra

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#Contents#
* table of contents
{:toc}

## Idea

A [[Lie algebra]] is **nilpotent** if [[action|acting]] on any one of its elements with other elements, via the [[Lie bracket]], repeatedly eventually yields [[zero]].

## Definition

The [[lower central series]] or [[descending central series]] of a Lie algebra $\mathfrak{g}$ is a sequence of nested ideals $\mathfrak{g}^{k+1} \trianglelefteq \mathfrak{g}^{k}$ defined inductively by $\mathfrak{g}^1 \coloneqq \mathfrak{g}$,  $\mathfrak{g}^{k+1} \coloneqq [\mathfrak{g}, \mathfrak{g}^k]$. The Lie algebra is said to be **nilpotent** if $\mathfrak{g}^{k} = 0$ for some $k \in \mathbb{N}$.

In other words, a Lie algebra $\mathfrak{g}$ is nilpotent if and only the improper ideal $\mathfrak{g}$ is a [[nilpotent element]] in the ideal lattice with respect to the ideal product $[-,-]$.

## Properties

* Every abelian Lie algebra is nilpotent, and every nilpotent Lie algebra is [[solvable Lie algebra|solvable]].

### Relation to Sullivan models

A finite-dimensional Lie algebra $\mathfrak{g}$ is nilpotent precisely if its [[Chevalley-Eilenberg algebra]] $CE(\mathfrak{g})$ is a [[Sullivan algebra]] (necessarily minimal). See also at _[[rational homotopy theory]]_ for more on this.

## Related concepts

* [[nilpotent element]]

* [[nilpotent group]]

* [[nilpotent topological space]]

* [[nilpotent L-infinity algebra]]

## References

* [[Victor Kac]] (notes by Marc Doss), *Nilpotent and Solvable Lie algebras* (2010) [[pdf](https://math.mit.edu/classes/18.745/Notes/Lecture_4_Notes.pdf), [[KacDoss-NilpotentLieAlebras.pdf:file]]]

* _Nilpotent Lie algebras_ ([pdf](http://relaunch.hcm.uni-bonn.de/fileadmin/perrin/chap5.pdf))

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

On the [[Sullivan models]] which are [[Chevalley-Eilenberg algebra]] of nilpotent Lie algebras:

* [[Manuel Amann]], pp. 4 of: *On quasi-isometric nilpotent Lie groups* [[arXiv:1710.04542](https://arxiv.org/abs/1710.04542)]

[[!redirects ad-nilpotent Lie algebra]]
[[!redirects ad-nilpotent Lie algebras]]
[[!redirects nilpotent Lie algebra]]
[[!redirects nilpotent Lie algebras]]

[[!redirects ad-nilpotent Lie algebra]]


nilpotent module

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

#Contents#
* table of contents
{:toc}

## Idea

Let $G$ be a [[group]], $N$ an [[abelian group]] and $(-)\cdot (-) \colon G \times N \to N$ an [[action]] of $G$ on $N$ by [[linear maps]], thus making $N$ a [[module]] over $G$.

Then this is called a _nilpotent module_ if the [[sequence]] of abelian subgroups

$$
  \Gamma_0 N
    \supset
  \Gamma_1 N
    \supset
  \Gamma_2 N
    \supset
  \cdots
  \,,
$$

given [[recursion|recursively]] by

$$
  \begin{aligned}
  \Gamma_0 N & \coloneqq\; N
  \\
  \Gamma_{k+1} N
    & \coloneqq\;
  \big\{
     g \cdot n - n
     \;\vert\;
     g \in G,\;
     n \in \Gamma_k N
  \big\}
  \,,
  \end{aligned}
$$

terminates, in that there is $k_{max} \in \mathbb{N}$ with
$\Gamma_{k_{max}}N = 0$.

## Related concepts

* [[nilpotent group]]

* [[nilpotent topological space]]

* [[nilpotent Lie algebra]]

* [[nilpotent element]]

\linebreak

* [[rational homotopy theory]]

  * [[Sullivan model]]

  * [[rational fiber lemma]]

## References

* [[Peter Hilton]], _Nilpotency in group theory and topology_, Publicacions de la Secció de Matemàtiques Vol. 26, No. 3 (1982), pp. 47-78 ([jstor:43741908](https://www.jstor.org/stable/43741908))

[[!redirects nilpotent modules]]


nilpotent topological space

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#### Homotopy theory
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[[!include homotopy - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A [[connected topological space|connected]] [[topological space]] $X$ (or rather its [[homotopy type]]) is called _nilpotent_ if

1. its [[fundamental group]] $\pi_1(X)$ is a [[nilpotent group]];

1. the [[action]] of $\pi_1(X)$ on the higher [[homotopy groups]] is a [[nilpotent module]] in that the [[sequence]] $N_{0,n} \coloneqq \pi_n(X)$, $N_{k+1,n} \coloneqq \{g n - n | n \in N_{k,n}, g \in \pi_1(X)\}$ terminates.

## Examples

Directly from the definition we have that:

* every [[simply connected topological space]] is nilpotent;

and more generally

* every [[simple space]] is nilpotent.

As a special case of this

* every [[connected topological space|connected]] [[H-space]] is nilpotent

and thus

* every [[loop space]] is nilpotent

  (since all its connected components are homotopy equivalent to the unit component, which is a connected H-space).

(See [May-Ponto 12, p. 49 (77 of 542)](#MayPonto12))

* For $G$ a [[nilpotent group|nilpotent]] [[Lie group]], the [[classifying space]] $B G$ is a nilpotent space.

(See [Hilton 82, Section 3](#Hilton82)).

## Properties
 {#Properties}

Nilpotency is involved in sufficient conditions for many important constructions in ([[stable homotopy theory|stable]]) [[homotopy theory]], see for instance at

* [[localization of a space]];

* [[fracture theorem]].

* [[fundamental theorem of dg-algebraic rational homotopy theory]]

## Related concepts

* [[nilpotent element]]

* [[nilpotent Lie algebra]], [[nilpotent L-infinity algebra]]

* [[nilpotent group]], [[nilpotent module]]

\linebreak

* [[connected topological space]]

* [[simply-connected topological space]]

* [[simple topological space]]

\linebreak

* [[rational homotopy theory]]

  * [[Sullivan model]]

  * [[rational fiber lemma]]

\linebreak

* [[finite homotopy type]]

* [[p-local homotopy type]]

* [[p-complete homotopy type]]

## References

* {#Hilton82} [[Peter Hilton]], _Nilpotency in group theory and topology_, Publicacions de la Secció de Matemàtiques Vol. 26, No. 3 (1982), pp. 47-78 ([jstor:43741908](https://www.jstor.org/stable/43741908))

* {#MayPonto12} [[Peter May]], [[Kate Ponto]], _More Concise Algebraic Topology_, University of Chicago Press (2012) ([pdf](https://www.math.uchicago.edu/~may/TEAK/KateBookFinal.pdf))

* {#Riehl14} [[Emily Riehl]], def. 14.4.9 in: _[[Categorical Homotopy Theory]]_, New Mathematical Monographs 24, Cambridge University Press 2014 ([pdf](http://www.math.jhu.edu/~eriehl/cathtpy.pdf), [doi:10.1017/CBO9781107261457](https://doi.org/10.1017/CBO9781107261457))

See also

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

See also

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

The [[rational homotopy theory]] of nilpotent topological spaces is discussed in

* {#BousfieldGugenheim76} [[Aldridge Bousfield]], V. K. A. M. Gugenheim, section 9.1 of _On PL deRham theory and rational homotopy type_ , Memoirs of the AMS, vol. 179 (1976)

* {#Neisendorfer78} [[Joseph Neisendorfer]], _Lie algebras, coalgebras and rational homotopy theory for nilpotent spaces_, Pacific J. Math. Volume 74, Number 2 (1978), 429-460. ([euclid](http://projecteuclid.org/euclid.pjm/1102810284))

Discussion in [[homotopy type theory]]:

* {#Scoccola19} Luis Scoccola, _Nilpotent Types and Fracture Squares in Homotopy Type Theory_ ([arXiv:1903.03245](https://arxiv.org/abs/1903.03245))

[[!redirects nilpotent topological spaces]]

[[!redirects nilpotent homotopy type]]
[[!redirects nilpotent homotopy types]]

[[!redirects nilpotent space]]
[[!redirects nilpotent spaces]]

[[!redirects nilpotent spectrum]]
[[!redirects nilpotent spectra]]


nilradical

> Not to be confused with *[[nilpotent ideal]]*, which is a [[nilpotent element]] in the [[lattice]] of [[ideals]] of a [[ring]].

---

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

#Contents#
* table of contents
{:toc}

## Definition

For $R$ a [[commutative ring]], its **nilradical** $I \subset R$ is the [[ideal]] of [[nilpotent elements]]: the collection of those elements $a \in R$ such that there is $n \in \mathbb{N}$ with $a^n = 0$.

The [[quotient]] $R/I$ is also called the _[[reduced object|reduced]]_ part of $R$.

(If $R$ is not [[commutative ring|commutative]] there are different generalization of the notion of nilradical. See Wikipedia, for the moment.)

## Examples

Every [[local Artinian ring]] is a commutative ring whose set of non-invertible elements is a [[nilradical]].

With rings regarded as [[Isbell duality|formal dual]]s of [[affine scheme]]s, the canonical inclusion

$$
  Spec R/I \to Spec R
$$

is to be thought of as exhibiting the inclusion of $Spec R/I$ into an [[infinitesimal object|infinitesimal thickening]] of itself.

For $X : CRing \to Set$ a [[presheaf]] on the [[category]] of [[commutative rings]], the presheaf

$$
  X_{dR} : Spec R \mapsto X(Spec R/I)
$$

is called the [[de Rham space]] of $X$.

## Related concepts

* [[infinitesimal extension]]

* [[reduced ring]]

* [[Artinian local ring]]

## References

See also:

* Wikipedia, *[Nilradical](https://en.m.wikipedia.org/wiki/Nilradical_of_a_ring)*

[[!redirects nilradical]]
[[!redirects nilradicals]]


Nils A. Danielsson

* [personal page](https://www.cse.chalmers.se/~nad/)

* [Institute page](https://www.gu.se/en/about/find-staff/nilsdanielsson)

## Selected writings

On the [[structure identity principle]]:

* [[Thierry Coquand]], [[Nils Anders Danielsson]], *Isomorphism is equality*, Indagationes Mathematicae **24** 4 (2013) 1105-1120 [[doi:10.1016/j.indag.2013.09.002](https://doi.org/10.1016/j.indag.2013.09.002)]

On [[quotient inductive types]]:

 * {#ADK2017} [[Thorsten Altenkirch]], [[Nils A. Danielsson]], [[Nicolai Kraus]], _Partiality, Revisited: The Partiality Monad as a Quotient Inductive-Inductive Type_, FoSSaCS 2017, Lecture Notes in Computer Science **10203** Springer (2017) [[arXiv:1610.09254](https://arxiv.org/abs/1610.09254)]

category: people

[[!redirects Nils Danielsson]]
[[!redirects Nils Anders Danielsson]]


Nils Baas

Nils Baas is a Norwegian mathematician working in Trondheim.

* [Norwegian wikipedia entry](http://no.wikipedia.org/wiki/Nils_Baas)

## Selected writings

Interpretation of [[algebraic K-theory]] of [[topological K-theory]] ([[iterated algebraic K-theory]]) as the [[K-theory]] of [[BDR 2-vector bundles]]:

* [[Nils Baas]], [[Bjørn Ian Dundas]], [[John Rognes]], _Two-vector bundles and forms of elliptic cohomology_, London Math. Soc. Lecture Note Ser., 308, Cambridge Univ. Press, Cambridge, 2004 ([arXiv:0909.1742](https://arxiv.org/abs/0909.1742), [doi:10.1017/CBO9780511526398.005](https://doi.org/10.1017/CBO9780511526398.005))

* [[Nils Baas]], [[Bjørn Ian Dundas]], [[Birgit Richter]], [[John Rognes]], _Stable bundles over rig categories_, Journal of Topology, Volume 4, Issue 3, September 2011, Pages 623–640 ([doi:10.1112/jtopol/jtr016](https://doi.org/10.1112/jtopol/jtr016))

## Related $n$Lab entries

* [[hyperstructure]]

* [[2-vector space]]

* [[BDR 2-vector bundle]]

category: people

[[!redirects Nils A. Baas]]


Nils Carqueville

* [webpage](http://nils.carqueville.net/)

## Selected writings

On a [[bicategory]] of [[Landau-Ginzburg models]] with [[matrix factorizations]] as [[1-morphisms]]:

* [[Nils Carqueville]], [[Daniel Murfet]], *Adjunctions and defects in Landau-Ginzburg models*, Advances in Mathematics **289** (2016) 480-566 [[doi:10.1016/j.aim.2015.03.033](https://doi.org/10.1016/j.aim.2015.03.033), [arXiv:1208.1481](https://arxiv.org/abs/1208.1481)]

Discussion of [[Rozansky-Witten theory|Rozansky-Witten]] [[defect QFT]] as an [[extended TQFT]] via the [[cobordism hypothesis]]:

* [[Ilka Brunner]], [[Nils Carqueville]], [[Daniel Roggenkamp]], *Truncated affine Rozansky--Witten models as extended TQFTs*,  Commun. Math. Phys. (2023) [[arXiv:2201.03284](https://arxiv.org/abs/2201.03284), [doi:10.1007/s00220-022-04614-4](https://doi.org/10.1007/s00220-022-04614-4)]

reviewed in:

* [[Nils Carqueville]], *Extended defect TQFTs*, talk at *[QFT and Cobordism](https://nyuad.nyu.edu/en/events/2023/march/quantum-field-theories-and-cobordisms.html)*, [[CQTS]] (Mar 2023) [[web](Center+for+Quantum+and+Topological+Systems#CarquevilleMar2023), [[Carqueville-RW.pdf:file]], video:[YT](https://www.youtube.com/watch?v=bwDJSitenVE)]

A discussion of the [[swampland cobordism conjecture]]:

* David Andriot, [[Nils Carqueville]], Niccolò Cribiori, _Looking for structure in the cobordism conjecture_ ([arXiv:2204.00021](https://arxiv.org/abs/2204.00021))

On [[QFT with defects]]:

* [[Nils Carqueville]], [[Michele Del Zotto]], [[Ingo Runkel]], *Topological defects*, in: *[[Encyclopedia of Mathematical Physics 2nd ed]]* [[arXiv:2311.02449](https://arxiv.org/abs/2311.02449)]

## Related entries

* [[QFT with defects]]

category: people


Nima Arkani-Hamed

* [webpage](http://www.sns.ias.edu/~arkani/)

## Selected writings

On [[configuration spaces of points]] in general poisition inside [[complex projective spaces]]:

* [[Nima Arkani-Hamed]], [[Thomas Lam]], [[Marcus Spradlin]], *Positive configuration space*, Commun. Math. Phys. **384** (2021) 909–954  [[arXiv:2003.03904](https://arxiv.org/abs/2003.03904), [doi:10.1007/s00220-021-04041-x](https://doi.org/10.1007/s00220-021-04041-x)]

On [[celestial amplitudes]]:

* [[Nima Arkani-Hamed]], Monica Pate, Ana-Maria Raclariu, [[Andrew Strominger]], *Celestial Amplitudes from UV to IR*, JHEP 08 (2021) 062 [[arXiv:2012.04208](https://arxiv.org/abs/2012.04208)]

## Related entries

* [[split supersymmetry]]

* [[Randall-Sundrum model]]

category: people


Nima Rasekh

* [webpage](http://guests.mpim-bonn.mpg.de/rasekh/)

## Selected writings

On [[elementary (infinity,1)-toposes]]:

* [[Nima Rasekh]], _A Theory of Elementary Higher Toposes_, ([arXiv:1805.03805](https://arxiv.org/abs/1805.03805))

* [[Nima Rasekh]], _Yoneda Lemma for Elementary Higher Toposes_, ([arXiv:1809.01736](https://arxiv.org/abs/1809.01736))

Specifically on [[natural number objects]]:

* [[Nima Rasekh]], _Every Elementary Higher Topos has a Natural Number Object_, Theory and Applications of Categories **37** 13 (2021) 337-377.  ([arXiv:1809.01734](https://arxiv.org/abs/1809.01734), [tac:37-13](http://www.tac.mta.ca/tac/volumes/37/13/37-13abs.html))

Specifically on [[n-truncated objects in an (infinity,1)-category|n-truncation]]:

* [[Nima Rasekh]], _An Elementary Approach to Truncations_ ([arXiv:1812.10527](https://arxiv.org/abs/1812.10527))

Discussion of straightening and unstraightening for [[(∞,n)-categories]]:

* [[Lyne Moser]], [[Nima Rasekh]], [[Martina Rovelli]], _An (∞,n)-categorical straightening-unstraightening construction_ ([arXiv:2307.07259](https://arxiv.org/abs/2307.07259)).

On [[fractured (infinity,1)-toposes|fractured $\infty$-toposes]] in [[condensed mathematics]] (cf. [[condensed local contractibility]]):

* [[Nima Rasekh]], *What is a topological structure?*, talk notes (April 2023) [[pdf](https://guests.mpim-bonn.mpg.de/rasekh/Condensed.pdf), [[Rasekh-TopologicalStructure.pdf:file]]]

## Related entries

* [[elementary (infinity,1)-topos]]

category: people


ninebrane 10-group

#Contents#
* table of contents
{:toc}

## Idea

The [[n-group|10-group]] which is the [[looping]] of the [[homotopy fiber]] of (some fractional multiple of) the third [[Pontryagin class]] on the [[delooping]] of the [[fivebrane 6-group]].

## Related concepts

* [[M9-brane]]

[[!include image of J -- table]]
[[!include higher spin structure - table]]

## References

* {#Sati14} [[Hisham Sati]], _Ninebrane structures_ ([arXiv:1405.7686](http://arxiv.org/abs/1405.7686))

[[!redirects ninebrane 10-groups]]

[[!redirects Ninebrane 10-group]]


ninebrane group

#Contents#
* table of contents
{:toc}

## Idea

The low degrees  in the [[Whitehead tower]] of the [[orthogonal group]] are closely linked to the [[physics]] of [[branes]] appearing in [[string theory]]/[[M-theory]]. Accordingly the low stages are called

 $\cdots \to$ [[fivebrane group]] $\to$ [[string group]] $\to$ [[spin group]] $\to$ [[special orthogonal group]] $\to$ [[orthogonal group]]

To which extent and in which sense this matching with [[brane]] [[physics]] continues is not entirely clear. [[Hisham Sati]] gives arguments that co-killing the next [[integer]] [[homotopy group]] $\pi_{11}(O)$ corresponds to the [[M9-brane]]. Therefore one might call the corresponding stage in the [[Whitehead tower]] the _ninebrane group_. This suggestion appeared in ([Sati-Schreiber-Stasheff 08](#SatiSchreiberStasheff08))

## Related concepts

[[!include image of J -- table]]

## References

* [[Hisham Sati]], [[Urs Schreiber]], [[Jim Stasheff]], _[[schreiber:L-∞ algebra connections]]_, in -Quantum Field Theory_, Birkhäuser (2009), 303-424, DOI: 10.1007/978-3-7643-8736-5_17 ([arXiv:0801.3480](http://arxiv.org/abs/0801.3480))
 {#SatiSchreiberStasheff08}

[[!redirects ninebrane groups]]

[[!redirects Ninebrane group]]
[[!redirects Ninebrane groups]]


ninebrane structure

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

> $\cdots \to$ **ninebrane 10-group** $\to$ [[Fivebrane group]] $\to$ [[string group]] $\to$ [[spin group]] $\to$ [[special orthogonal group]] $\to$ [[orthogonal group]].

#Contents#
* table of contents
{:toc}

## Idea

On a space with [[Fivebrane structure]] a _ninebrane structure_ is a trivialization of (some fractional multiple of) the 3rd [[Pontryagin class]]. This is supposed to be related to parts of the [[quantum anomaly]] cancellation of the [[M9-brane]].

## Related concepts

[[!include higher spin structure - table]]

## References

* {#Sati14} [[Hisham Sati]], _Ninebrane structures_ ([arXiv:1405.7686](http://arxiv.org/abs/1405.7686))

[[!redirects ninebrane structures]]

[[!redirects Ninebrane structure]]
[[!redirects Ninebrane structures]]


Ning Bao

* [personal page](https://sites.google.com/view/ning-bao/home)

## Selected writings

On [[holographic tensor networks]] and [[quantum error correcting codes]] in full [[AdS-CFT]]:

* [[Ning Bao]], [[Geoffrey Penington]], [[Jonathan Sorce]], [[Aron C. Wall]], _Beyond Toy Models: Distilling Tensor Networks in Full AdS/CFT_,  JHEP 2019:69 ([arXiv:1812.01171](https://arxiv.org/abs/1812.01171))

* [[Ning Bao]], [[Geoffrey Penington]], [[Jonathan Sorce]], [[Aron C. Wall]], _Holographic Tensor Networks in Full AdS/CFT_ ([arXiv:1902.10157](https://arxiv.org/abs/1902.10157))

On [[holographic Renyi entropy]]:

* [[Ning Bao]], Mudassir Moosa, Ibrahim Shehzad, *The holographic dual of Rényi relative entropy*,  J. High Energ. Phys. 2019, 99 (2019) ([arXiv:1904.08433](https://arxiv.org/abs/1904.08433))

On understanding [[quantum error correcting codes]] in terms of the [[eigenstate thermalization hypothesis]]:

* [[Ning Bao]], Newton Cheng, *Eigenstate Thermalization Hypothesis and Approximate Quantum Error Correction*, JHEP 08 (2019) 152 ([arXiv:1906.03669](https://arxiv.org/abs/1906.03669))

category: people


nInsights

This page aims to gather priceless snippets of mathematical wisdom about specific subjects, garnered from around the n-categorical blogosphere!

[Weighted colimits](http://golem.ph.utexas.edu/category/2007/02/day_on_rcfts.html#c007688), explained nicely by John Baez as the _categorification of weighted sums_.

[Bernoulli numbers](http://golem.ph.utexas.edu/category/2008/02/metric_spaces.html#c014936), link to a problem sheet by John Baez with solution by Jeff Morton on everything you need to know.

[Framed cobordisms and TQFT's](http://golem.ph.utexas.edu/category/2008/02/new_hire_at_ucr.html#c016709), link to a classic comment by John Baez explaining how stable homotopy theory and extended TQFT become one and the same thing in the limit $n \rightarrow \infty$.

[The link between duals for objects and adjoint functors](http://golem.ph.utexas.edu/category/2008/02/logicians_needed_now.html#c018557). Here Bruce described what extra condition you need, besides $V \otimes -$ being left adjoint to $V^* \otimes -$, to make sure that $V^*$ is a right dual of $V$ in the normal sense, and gave an example showing why this condition was necessary.

+-- {: .query}

[[Urs Schreiber|Urs]]: To my mind, the purpose of this page here should be to remind us to eventually create separate $n$Lab entries that actually describe these "$n$Insights", if only by collecting the material that is currently just being linked to.

Everybody with a minute of free time should feel encouraged to create an entry for one of the above links and start putting in material there, so that we can begin hyperreferencing it, etc. and it becomes part of the big web.

For instance the material on weighted colimits should go to [[weighted colimit]] parallel to [[weighted limit]]. Notice that some of the aspects that John mentions behind the above link do appear currently at [[Day convolution]].

Similarly the stuff about framed cobordisms should go to either the existing [[generalized tangle hypothesis]], or, better yet, should motivate us to finally create [[framed bordism]] and [[cobordism hypothesis]].

The stuff about duals and adjoints might get its own page or should be added to [[adjoint functor]] or else we possibly want a separete page [[dual object]] anyway, where this should go.

=--


Nir Gadish

* [personal page](http://www-personal.umich.edu/~gadish/)

* [institute page](https://lsa.umich.edu/math/people/postdoc-faculty/gadish.html)

## Selected writings

On [[FI-representations]] and [[representation stability]]:

* [[Nir Gadish]], *Categories of FI type: a unified approach to generalizing representation stability and character polynomials*, Journal of Algebra **480** (2017) 450-486 [[arXiv:1608.02664](https://arxiv.org/abs/1608.02664), [doi:10.1016/j.jalgebra.2017.03.010](https://doi.org/10.1016/j.jalgebra.2017.03.010)]

* [[Christin Bibby]], [[Nir Gadish]], *A generating function approach to new representation stability phenomena in orbit configuration spaces*, Trans. Amer. Math. Soc. Ser. B **10** (2023) 241-287 [[ams:S2330-0000-2023-00130-3](https://www.ams.org/journals/btran/2023-10-09/S2330-0000-2023-00130-3), [arXiv:1911.02125](https://arxiv.org/abs/1911.02125)]

category: people


Nirmalya Kajuri

* [personal page](https://sites.google.com/view/nirmalya-kajuri/?pli=1)

## Selected writings

On the "entanglement island"-proposal for resolving the [[black hole information paradox]]:

* [[Nirmalya Kajuri]], *[Of Islands, Holograms and Saving Quantum Physics From a Black Hole Paradox](https://science.thewire.in/the-sciences/unitarity-entanglement-islands-holography-information/)*, Science -- The Wire (Nov 2022)

  > [...] a few sceptics have argued that while the calculations are correct, they don't help resolve the black hole information-loss paradox. The troubles stem from the reservoir attached to the anti-de Sitter universe. The physicists who authored the island papers assumed that gravity stopped at the boundary of the anti-de Sitter space and didn’t enter the reservoir. This is not an innocuous assumption. [...] The key takeaway is that the island way to recover information and save unitarity works perfectly well – if you slightly modify Einstein’s theory of gravity. These criticisms have been around for some two years now, and physicists are yet to resolve them in print. [...] physicists continue to publish papers by the hundreds about the entanglement islands but few attempt to answer whether the islands are compatible with the Einsteinian gravity of our universe. [...]

category: people


Nisnevich site

#Contents#
* table of contents
{:toc}

## Idea

The *Nisnevich topology* is a certain [[Grothendieck topology]] on the [[category]] of [[scheme]]s which is finer than the [[Zariski topology]] but coarser than the [[étale topology]]. It retains many desirable properties from both topologies:

* The Nisnevich [[cohomological dimension]] (and even the [[homotopy dimension]]) of a scheme is bounded by its [[Krull dimension]] (like Zariski)
* [[field|Fields]] have [[shape of an (∞,1)-topos|trivial shape]] for the Nisnevich topology (like Zariski)
* [[algebraic K-theory|Algebraic K-theory]] satisfies [[descent]] over the Nisnevich site -- as is true for the [[Zariski site]] but not in full generality for the [[etale site]], see at _[algebraic K-theory -- Descent](algebraic%20K-theory#Descent)_ for more;

* For $Z\subset X$ a [[closed subscheme|closed immersion]] between affine schemes that are smooth over a base $S$, the Nisnevich sheaf $X/(X-Z)$ is isomorphic to $N_{X,Z}/(N_{X,Z}-Z)$, where $N_{X,Z}$ is the normal bundle of $Z$ in $X$ (like étale)
* Pushforward along a [[finite morphism]] is an [[exact functor]] on Nisnevich sheaves of abelian groups (like étale)
* Nisnevich cohomology can be computed using [[?ech cohomology]] (like étale)

The Nisnevich topology plays a central rôle in [[motivic homotopy theory]].

## Definition

An family of morphisms of [[Noetherian scheme]]s $\{p_i:V_i\to U\}$ is a **Nisnevich cover** if each $p_i$ is an [[étale map]] and if every field-valued point $Spec k\to U$ lifts to one of the $V_i$. This is a [[pretopology]] on the category of Noetherian schemes, and the associated topology is the **Nisnevich topology**.

The **Nisnevich site** over a [[Noetherian scheme]] $S$ usually refers to the [[site]] given by the [[category]] of [[smooth scheme]]s of finite type over $S$ equipped with the Nisnevich topology. The **small Nisnevich site** of $S$ is the subsite consisting of [[étale map|étale]] $S$-schemes.

### For non-Noetherian schemes

For a general [[affine scheme]] $X$, one defines a [[sieve]] $S$ on $X$ to be a covering sieve for the Nisnevich topology if there exist a Noetherian affine scheme $Y$, a morphism $f: X\to Y$, and a Nisnevich covering sieve $T$ on $Y$ such that $f^\ast(T)\subset S$. On an arbitrary [[scheme]] $X$, a sieve $S$ is a Nisnevich covering sieve if there exists an open cover $\{U_i\to X\}$ by affine schemes such that $S_{/U_i}$ is a Nisnevich covering sieve on $U_i$ for all $i$.

### As an excision property

Let $Et/S$ be the category of [[étale scheme]]s of [[morphism of finite presentation|finite presentation]] over a [[quasi-compact morphism|quasi-compact]] [[quasi-separated morphism|quasi-separated]] scheme $S$. An [[(∞,1)-presheaf]] $F$ on $Et/S$ is said to satisfy _Nisnevich excision_ if the following conditions hold:

* $F(\emptyset)$ is [[contractible]].
* If $Z$ is a closed subscheme of $X\in Et/S$ and if $X'\to X$ is a morphism in $Et/S$ which is an isomorphism over $Z$, then the square

$$
  \array{
     F(X) &\to& F(X-Z)
     \\
     \downarrow && \downarrow
     \\
     F(X') &\to& F(X'-Z)
  }
$$

is an [[(∞,1)-pullback]] square. Intuitively, this says that the space of sections of $F$ over $X$ with support in $Z$ (i.e., the [[homotopy fiber]] of $F(X) \to F(X-Z)$) does not depend on $X$. This is Definition 2.5 in [DAG XI](#DAGXI).

+-- {: .num_prop}
###### Proposition
An [[(∞,1)-presheaf]] on $Et/S$ is an [[(∞,1)-sheaf]] for the Nisnevich topology if and only if it satisifes Nisnevich excision.

=--

This is [Morel-Voevosky, Prop. 1.16](#MV99) or [DAG XI, Thm. 2.9](#DAGXI).

## Properties

### Homotopy dimension

+-- {: .num_prop}
###### Proposition
If $S$ is a [[Noetherian scheme]] of finite [[Krull dimension]], then the [[(∞,1)-topos]] of [[(∞,1)-sheaves]] on the small Nisnevich site of $S$ has [[homotopy dimension]] $\leq\dim(S)$.

=--

This is [DAG XI, Theorem 2.24](#DAGXI). As a consequence, [[Postnikov tower]]s are convergent in the [[(∞,1)-topos]] of [[(∞,1)-sheaves]] on the Nisnevich site over $S$, and in particular that (∞,1)-topos is [[hypercomplete]].

+-- {: .num_prop}
###### Proposition
More generally, if $S$ is a pro-algebraic space limit of a cofiltered diagram of qcqs algebraic spaces of Krull dimension $\leq d$, then the [[(∞,1)-topos]] of [[(∞,1)-sheaves]] on the small Nisnevich site of $S$ has [[homotopy dimension]] $\leq d$.

=--

This is Clausen and A. Mathew, Hyperdescent and etale K-theory, 2019, arXiv:1905.06611, Cor.3.11, Thm.3.12, and Thm.3.17.

## Related concepts

* [[fpqc-site]] $\to$ [[fppf-site]] $\to$ [[syntomic site]] $\to$ [[étale site]] $\to$ **Nisnevich site** $\to$ [[Zariski site]]

* [[cd-structure]]

## References

A quick overview is at the beginning of the talk slides

* Jardine, _Motivic spaces and the motivic stable category_ ([pdf](http://www.aimath.org/WWN/motivesdessins/motivic.pdf)) .

A detailed discussion is in [section 3.1.1](http://www.math.uiuc.edu/K-theory/0305/nowmovo.pdf#page=61) of

* [[Fabien Morel]], [[Vladimir Voevodsky]], _$\mathbb{A}^1$-homotopy theory of schemes_ ,  K-theory, 0305 ([web](http://www.math.uiuc.edu/K-theory/0305/) [pdf](http://www.math.uiuc.edu/K-theory/0305/nowmovo.pdf))
{#MV99}

or in the lecture notes

* [[Eric Friedlander]], _[Algebraic Cycles and algebraic K-theory, II](http://www.math.northwestern.edu/~eric/lectures/)_ ([lecture 6 (pdf)](http://www.math.northwestern.edu/~eric/lectures/ihp/ihplec6.pdf))

A self-contained account of the Nisnevich $(\infty,1)$-topos including the non-Noetherian case is in

* [[Jacob Lurie]], _Derived Algebraic Geometry XI: Descent Theorems_ ([pdf](http://www.math.harvard.edu/~lurie/papers/DAG-XI.pdf))
{#DAGXI}

[[!redirects Nisnevich topology]]
[[!redirects Nisnevich topos]]
[[!redirects Nisnevich (∞,1)-topos]]

[[!redirects Nisnevich sheaf]]
[[!redirects Nisnevich sheaves]]

[[!redirects Nisnevich descent]]


Nissan Itzhaki

* [webpage](https://english.m.tau.ac.il/profile/nitzhaki)

## Selected writings

On [[supergravity]], [[D-branes]] and the [[large N limit]] of [[super Yang-Mills theories]]:

* [[Nissan Itzhaki]], [[Juan Maldacena]], [[Jacob Sonnenschein]], [[Shimon Yankielowicz]], _Supergravity and The Large $N$ Limit of Theories With Sixteen Supercharges_, Phys. Rev. D 58, 046004 (1998) ([arXiv:hep-th/9802042](https://arxiv.org/abs/hep-th/9802042))

category: people


Nitin Nitsure

* [MathematicsGenealogy page](https://www.mathgenealogy.org/id.php?id=35427)

* [institute page](https://www.icts.res.in/people/nitin-nitsure)

## Selected writings

On basics of [[algebraic geometry]] following [[Grothendieck]]'s [[FGA]]:

* [[Barbara Fantechi]], [[Lothar Göttsche]], [[Luc Illusie]], [[Steven L. Kleiman]], [[Nitin Nitsure]], [[Angelo Vistoli]]:

  **[[FGA explained]]** (ICTP, Trieste 2003--2005)

  Mathematical Surveys and Monographs __123__

  Amer. Math. Soc. 2005. x+339 pp.

  * [ISBN:978-0-8218-4245-4](https://bookstore.ams.org/surv-123-s)

    [MR2007f:14001](http://www.ams.org/mathscinet-getitem?mr=2007f:14001)

    [lecture notes](http://indico.ictp.it/event/a0255/other-view?view=ictptimetable)

category: people


nitrogen

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### Context
#### Fields and quanta
+--{: .hide}
[[!include fields and quanta - table]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The [[chemical element]] with [[atomic number]] 7.

## Related entries

* [[nitrogen-vacancy center in diamond]]

## References

See also:

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


nitrogen-vacancy center in diamond

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### Context
#### Solid state physics
+-- {: .hide}
[[!include solid state physics -- contents]]
=--
#### Quantum systems
+--{: .hide}
[[!include quantum systems -- contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[solid state physics]], an impure form of [[diamond]] may be obtained where, in the center of the [[Bravais lattice]], [[carbon]] [[nuclei]] are replaced by a [[nitrogen]] [[nucleus]] and a vacancy, referred to as the *nitrogen-vacancy center in diamond* (often short: *NV center*).

In such a material, the [[spin]] of one of the [[electron]] orbitals bound to the [[nitrogen]] nucleus has a long [[quantum decoherence]] time and lends itself to manipulation by [[spin resonance]].

<img src="/nlab/files/NVCenterSchematics.jpg" width="600">

> from [AHWZ18, Fig 1](#AHWZ18)

Therefore NV center spins serve as a platform for [[spin resonance qbits]], used in [[quantum sensing]], [[quantum information theory|quantum information processing]] and simple [[quantum computing]].

## Related concepts

* [[nuclear magnetic resonance]]

* [[quantum sensing]]

## References

Original articles:

* J. M. Taylor et al., *High-sensitivity diamond magnetometer with nanoscale resolution*, Nature Phys **4** (2008) 810–816 [[doi:10.1038/nphys1075](https://doi.org/10.1038/nphys1075)]

Review and further discussion

* {#AHWZ18} [[David D. Awschalom]], Ronald Hanson, Jörg Wrachtrup & Brian B. Zhou, *Quantum technologies with optically interfaced solid-state spins*, Nature Photonics **12** (2018) 516–527 [[doi:10.1038/s41566-018-0232-2](https://doi.org/10.1038/s41566-018-0232-2)]

* [Awschalom group](https://pme.uchicago.edu/group/awschalom-group), *The Nitrogen-Vacancy Center in Diamond* [[web](https://pme.uchicago.edu/awschalom-group/nitrogen-vacancy-center-diamond)]

Application to [[quantum sensing]]:

* Jixing Zhang, Lixia Xu, Guodong Bian, Pengcheng Fan, Mingxin Li, Wuming Liu, Heng Yuan, *Diamond Nitrogen-Vacancy Center Magnetometry: Advances and Challenges* [[arXiv:2010.10231](https://arxiv.org/abs/2010.10231)]

* [[Mouzhe Xie]] et al., *Biocompatible surface functionalization architecture for a diamond quantum sensor*, PNAS **119** 8 (2022) e2114186119 [[arXiv:2108.04843](https://arxiv.org/abs/2108.04843), [doi:10.1073/pnas.2114186119](https://doi.org/10.1073/pnas.2114186119)]

* Nick R. Neuling, Robin D. Allert, [[Dominik B. Bucher]], *Prospects of Single-Cell NMR Spectroscopy with Quantum Sensors*, Current Opinion in Biotechnology **83** (2023) 102975 [[arXiv:2306.07593](https://arxiv.org/abs/2306.07593), [doi:10.1016/j.copbio.2023.102975](https://doi.org/10.1016/j.copbio.2023.102975)]

Survey:

* [[Mouzhe Xie]], *Diamond-based quantum sensor for molecular analytics*, [talk at](Center+for+Quantum+and+Topological+Systems#XieApr2023) [[CQTS]] (Apr 2023) [slides: [[Xie-CQTS-Apr2023.pdf:file]], video: [YT](https://www.youtube.com/watch?v=f4qv-H5vkR0)]

See also:

* Wikipedia, *[Nitrogen-vacancy center](https://en.wikipedia.org/wiki/Nitrogen-vacancy_center)*

* {#BSERTRH23} Santiago Bussandri, Daphna Shimon, [[Asif Equbal]], Yuhang Ren, Susumu Takahashi, [[Chandrasekhar Ramanathan]], Songi Han,  *P1 center electron spin clusters are prevalent in type Ib diamond* [[arXiv:2311.05396](https://arxiv.org/abs/2311.05396)]

[[!redirects nitrogen-vacancy center]]

[[!redirects NV center]]
[[!redirects NV centers]]

[[!redirects NV-center]]
[[!redirects NV-centers]]


Nitu Kitchloo

* [website](http://www.math.jhu.edu/~nitu/)

## Selected writings

* [[Nitu Kitchloo]], _The Stable Symplectic Category and Geometric Quantization_ ([arXiv:1204.5720](http://arxiv.org/abs/1204.5720v1))

* [[Nitu Kitchloo]], [[Jack Morava]], _The Grothendieck--Teichmüller group and the stable symplectic category_, 2012.  ([arxiv](http://arxiv.org/abs/1212.6905))

On [[link homology]] via [[topological quantum field theory]]:

* [[Nitu Kitchloo]], *Symmetry Breaking and Link Homologies I* [[arXiv:1910.07443](https://arxiv.org/abs/1910.07443)]

* [[Nitu Kitchloo]], *Symmetry Breaking and Link Homologies II* [[arXiv:1910.07444](https://arxiv.org/abs/1910.07444)]

* [[Nitu Kitchloo]], *Symmetry Breaking and Link Homologies III* [[arXiv:1910.07516](https://arxiv.org/abs/1910.07516)]

Relation to [[gauge theory]]:

* [[Nitu Kitchloo]], *Symmetry breaking and homotopy types for link homologies*, talk at *[QFT and Cobordism](https://nyuad.nyu.edu/en/events/2023/march/quantum-field-theories-and-cobordisms.html)*, [[CQTS]] (Mar 2023) [[web](Center+for+Quantum+and+Topological+Systems#KitchlooMar2023), video:[YT](https://www.youtube.com/watch?v=StRyIjpER9I)]

## Related entries

* [[Weinstein symplectic category]]

category: people


nitwit > history

< [[nitwit]]


nitwits > history

< [[nitwits]]


no boundary proposal

## References

* S.W.Hawking, section 3 of _The Nature of Space and Time_ ([arXiv:hep-th/9409195](http://arxiv.org/abs/hep-th/9409195))


no-cloning theorem

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### Context
#### Quantum systems
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[[!include quantum systems -- contents]]
=--
#### Monoidal categories
+--{: .hide}
[[!include monoidal categories - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[quantum physics]] the _no-cloning theorem_ is the statement that there cannot be physical processes which produce "copies" of [[quantum states]] in the way known from [[classical physics]].

Quantum theoretically, this effect is ultimately due to the non-[[cartesian monoidal category|cartesian]] nature of the [[tensor product]] [[tensor product of vector spaces|of vector]]/[[tensor product of Hilbert spaces|Hilbert spaces]] [[space of quantum states|of quantum states]] ([[category theory|category theoretically]] it refers to this tensor product lacking [[natural transformation|natural]] [[diagonal morphisms]]) and as such is a direct cousin of the fundamental phenomenon of [[quantum entanglement]].

In other words when [[quantum physics]] is axiomatized by [[quantum logic]] in the guise of [[linear logic]]/[[linear type theory]], the content of "no-cloning" and "no-deleting" is (see [there](linear+logic#AsAQuantumLogic)) the very "linearity" of this logic, the absence of a [[diagonal]] map and of a [[projection]] map for the non-[[cartesian monoidal category|cartesian]] [[tensor product]] in the [[categorical semantics]] given by non-[[cartesian monoidal category|cartesian]] [[symmetric monoidal categories]] such as that of Hilbert spaces. See also at _[[quantum information theory via dagger-compact categories]]_.

## Details

#### Elementary explanation
 {#ElementaryExplanation}

In its original formulation, the statement of the "no-cloning theorem" is that given a quantum system with [[Hilbert space]] $H$ and with a chosen initial pure [[quantum state]] $e \in H$, then there is no [[unitary operator]] on the [[tensor product]] $H \otimes H$ which would take states of the form $(\psi, e)$ to $(\psi,\psi)$.

Of course this cannot exist, because such a map would not even be [[linear map|linear]].

Often the statement is relaxed to observing that if there is a unitary that takes $(\psi,e)$ to $(\psi,\psi)$ and $(\phi,e)$ to $(\phi,\phi)$ for any two states $\phi,\psi \in H$, then either $\phi = \psi$ or $\phi \perp \psi$ (since it follows from unitarity that in this case $\langle\phi | \psi \rangle  = \langle\phi | \psi \rangle^2 $).

#### Category theoretic explanation

More [[category theory|category theoretically]], the no-cloning theorem comes down to the statement that the [[tensor product of vector spaces]] (and similarly its refinement  [[tensor product of Hilbert spaces|to Hilbert spaces]]) which makes [[VectorSpaces]] a [[symmetric monoidal category|symmetric]] [[monoidal category]]

$$
  (-) \otimes (-)
  \;\colon\;
  Vect \times Vect
  \longrightarrow
  Vect
$$

does not admit a *[[natural transformation|natural]]* transformation of the form

\[
  \label{WouldBeDiagonal}
  A \xrightarrow{\;\;} A \otimes A
\]

satisfying basic properties expected of the [[diagonal morphisms]] in a [[cartesian monoidal category]].

Without even requiring [[projection]] maps and hence "deleting" operations (which are part of a [[cartesian product]]) two such properties that the would-be diagonal (eq:WouldBeDiagonal) would have to satisfy are [[coassociativity]] and [[cocommutativity]] -- and one finds that such cannot hold in [[VectorSpaces]] [[Abramsky (2009), Thm. 11](#Abramsky09)].

\begin{remark}\label{RelationToQuantumMeasurement}
**(relation to classical outcomes of [[quantum measurement]])**
\linebreak
Notice that the condition that (eq:WouldBeDiagonal) be a *[[natural transformation]]* is crucial for the above conclusion of "no cloning":

Namely, for a *fixed* [[vector space]] $\mathscr{H}$, any choice of [[linear basis]] $\big\{ \vert b \rangle \big\}_{b \colon B}$ does induce a single [[linear map]] of the required form, given by duplicating (hence "cloning") the *fixed basis* elements:

\[
  \label{CoalgebraStructureInducedByLinearBasis}
  \array{
    \mathscr{H}
      &\longrightarrow&
    \mathscr{H}
    \otimes
    \mathscr{H}
    \\
    \vert b \rangle
      &\mapsto&
    \vert b \rangle
      \otimes
    \vert b \rangle
  }
\]

On the other hand, beware that, by the laws of [[linear maps]], this means that *no other* elements except the basis elements are exactly cloned even by this operation; for example the [[superposition]] of two distinct basis elements goes to:

\[
  \label{ExampleOfNonCloningOfCoalgebraMap}
  \begin{array}{lcl}
  \vert b \rangle + \vert b' \rangle
  &\mapsto&
  \vert b \rangle \otimes \vert b \rangle
  +
  \vert b' \rangle \otimes \vert b' \rangle
  \\
  &=&
  \big(
    \vert b \rangle + \vert b' \rangle
  \big)
  \otimes
  \big(
    \vert b \rangle + \vert b' \rangle
  \big)
  \,-\,
  \underset{correction}{
    \underbrace{
      \vert b \rangle \otimes \vert b' \rangle
      \,+\,
      \vert b' \rangle \otimes \vert b \rangle
    }
  }
  \end{array}
\]

Nonetheless, the linear map (eq:CoalgebraStructureInducedByLinearBasis) does satisfy [[coassociativity]] and [[cocommutativity]] in the evident sense; in fact it also satisfies [[counitality]] with respect to the [[unit]] element $1 \,\coloneqq\, \underset{b \colon B}{\sum} \vert b \rangle \;\in\; \mathscr{H}$. In summary this means that (eq:CoalgebraStructureInducedByLinearBasis) defines a [[cocommutative coalgebra]]-[[structure]] on $\mathscr{H}$, which hence certainly exists on *any* ([[finite-dimensional vector space|finite-dimensional]]) [[space of quantum states]].

While such structures exist, they do not represent a *general* process of "cloning" of quantum states: Due to effects as in (eq:ExampleOfNonCloningOfCoalgebraMap), these structures are not [[natural transformations]] and hence they "clone" not arbitrary quantum states, but just the chosen [[linear basis]]-states that define them.

Conversely, such [[coalgebra]]-[[structure]] (at least when regarded as the special symmetric [[Frobenius algebra]]-[[structure]] to which they always extend) in fact *define* a choice of [[linear basis]] of a quantum space of states. Since such a choice may encode the set of possible classical state outcomes under a [[quantum measurement]], these "restricted cloning"-operations are also known as "[[classical structures]]" (at least in the literature on [[quantum information theory in terms of dagger-compact categories]]).

In summary: While a *natural* cloning operation on quantum states does not exist, *enforcing* one on a select [[linear basis]] of quantum states is tantamount to "turning these into classical states" (in a sense that can be made more precise, see discussion at *[[quantum reader monad]]* and *[[quantum circuits via dependent linear types]]*).

\end{remark}

\begin{remark}
**(Converse statement)**
\linebreak
One may turn the issue around asks which conditions are imposed on a [[symmetric monoidal category]] if its [[tensor product]] is assumed to admit a [[diagonal map]] (for "cloning") or [[projection]] maps (for "deleting") or both. These turn out to be very strong conditions, asserting that the category must be "essentially classical":

For instance in order for a [[diagonal]] to exist for the tensor product of a [[compact closed category]] implies that every [[endomorphism]] on every [[object]] in the category is a multiple of the [[identity]] [[Abramsky (2009), Thm. 11](#Abramsky09) in generalization of "[Joyal's lemma](dualizing+object+in+a+closed+category#InACartesianClosedCategory)"]. This is clearly false in any category in which one could find interesting [[quantum mechanics]], certainly in that of (finite dimensional) [[Hilbert spaces]].
\end{remark}

### No broadcasting

There is a  slightly more substantial generalization of the no-cloning theorem to [[mixed quantum states]], then called the _no-broadcasting theorem_. Dually, there is also a _no-deleting theorem_.

## Related concepts

* [[deferred measurement principle]]

* [[quantum superposition]]

* [[entanglement]]

* [[Bell's inequalities]]

* [[Gleason's theorem]]

* [[interpretation of quantum mechanics]]

* [[quantum cryptography]]

## References

### No-cloning

The original articles on the "no-cloning theorem":

* [[Dennis Dieks]], _Communication in EPR devices_, Physics Letters A 92, 271-272 (1982)

* [[William Wooters]], [[Wojciech Zurek]], *A single quantum cannot be cloned*, Nature **299** (1982) 802-803 [[doi:10.1038/299802a0](https://doi.org/10.1038/299802a0)]

See also:

* Wikipedia, _[No-cloning theorem](http://en.wikipedia.org/wiki/No-cloning_theorem)_

Discussion from the point of view of [[monoidal category]]-theory ([[quantum information theory via dagger-compact categories]]):

* {#Abramsky09} [[Samson Abramsky]], *No-Cloning in categorical quantum mechanics*, in: I. Mackie, S. Gay (eds.), *Semantic Techniques for Quantum Computation*, Cambridge University Press (2009) 1-28 [[arXiv:0910.2401](http://arxiv.org/abs/0910.2401), [doi:10.1017/CBO9781139193313.002](https://doi.org/10.1017/CBO9781139193313.002)]

The original suggestion to use the no-cloning theorem for [[quantum cryptography]]:

* [[Stephen Wiesner]], *Conjugate Coding* circulated in the 1960s, finally published in ACM SIGACT News **15** 1 (1083) 78–88 ([original pdf](http://users.cms.caltech.edu/~vidick/teaching/120_qcrypto/wiesner.pdf), [doi:10.1145/1008908.1008920](https://doi.org/10.1145/1008908.1008920))

### No-deleting

The original statement of the "no-deleting theorem":

* [[Arun Kumar Pati]], [[Samuel L. Braunstein]], *Impossibility of deleting an unknown quantum state*, Nature **404** (2000) 164-165 [[doi:10.1038/404130b0](https://doi.org/10.1038/404130b0), [arXiv:quant-ph/9911090](https://arxiv.org/abs/quant-ph/9911090)]

See also:

* Wikipedia, *[No-deleting theorem](https://en.wikipedia.org/wiki/No-deleting_theorem)*

Discussion via [[quantum information theory via dagger-compact categories]]:

* [Abramsky 2009, Section 5](*Abramsky09)

[[!redirects non-broadcasting theorem]]
[[!redirects no-deleting theorem]]


no-go theorem

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### Context
#### Physics
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[[!include physicscontents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A no-go theorem is generally understood to be any theorem, usually in theoretical physics, that shows that an idea is not possible even though it may appear as if it should be. Many no-go theorems include hidden, implicit assumptions.

Examples of no-go theorems in [[theoretical physics]] include

in foundations of [[quantum physics]]:

* [[Bell's theorem]],

* [[Kochen-Specker theorem]]

* [[Gleason's theorem]].

in [[quantum field theory]]:

* [[Coleman-Mandula theorem]]

in [[supergravity]]:

* [no de Sitter spacetime (?)](de+Sitter+spacetime#ReferencesInStringTheory)

* [no ekpyrotic cosmology (?)](ekpyrotic+cosmology#ReferencesEmbeddingIntoStringTheory)

[[!redirects no-go theorems]]


no-hair theorem

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###Context###
#### Gravity
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[[!include gravity contents]]
=--
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In the context of [[gravity]] ([[general relativity]] and its variants, such as [[supergravity]]) what are called _no-hair theorems_ are statements saying that or to which extent [[black hole]] [[spacetimes]] and more generally [[black brane]] spacetimes are characterized by their [[mass]], [[angular momentum]] and [[charge]], hence that they look much like macroscopic [[fundamental particles]] or fundamental [[branes]] ("[[black branes]]").

## Examples

### In 4d Einstein-Maxwell theory

The archetypical example is 4-dimensional [[Einstein-Maxwell theory]], where the no-hair theorem asserts that all [[black hole]] solutions are entirely characterized by their [[mass]], [[angular momentum]] and [[electric charge]].

[[!include charged and rotating black holes -- table]]

### In 4d Einstein-Yang-Mills theory

Generalizing to 4-dimensional [[Einstein-Yang-Mills theory]] with [[nonabelian group|nonabelian]] [[gauge group]], the situation becomes much more subtle. (...)

### In higher supergravity

...

## References

* {#MisnerThorneWheeler73} [[Charles Misner]], [[Kip Thorne]], [[John Archibald Wheeler]], pp. 875–876 of _Gravitation_, San Francisco: W. H. Freeman. (1973)

* Pawel O. Mazur, _Black Uniqueness Theorems_, Proceedings of the 11th International Conference on General Relativity and Gravitation, ed. M. A. H. MacCallum, Cambridge University Press, Cambridge 1987, pp. 130-157 ([arXiv:hep-th/0101012](https://arxiv.org/abs/hep-th/0101012))

* J. Lopes Costa, _On black hole uniqueness theorems_, 2010 ([pdf](http://homepage.univie.ac.at/piotr.chrusciel/papers/TeseFinal.pdf))

* [[Stefan Hollands]], Akihiro Ishibashi, _Black hole uniqueness theorems in higher dimensional spacetimes_, Classical and Quantum Gravity, Volume 29, Number 16 ([arXiv:1206.1164](https://arxiv.org/abs/1206.1164))

See also

* Wikipedia, _[No-hair theorem](https://en.wikipedia.org/wiki/No-hair_theorem)_


Noah Snyder

[[!redirects NoahSnyder]]

* [website](http://math.berkeley.edu/~nsnyder/)

## Selected writings

On [[4d TQFTs]] from [[fully dualizable object|fully dualizable]] [[braided monoidal category|braided]] [[tensor categories]], via the [[cobordism hypothesis]]:

* [[Adrien Brochier]], [[David Jordan]], [[Pavel Safronov]], [[Noah Snyder]], _Invertible braided tensor categories_ ([arXiv:2003.13812](https://arxiv.org/abs/2003.13812))

category: people


Noam Chomsky

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#### Linguistics
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=--
=--
=--

# Contents
* table of contents
{:toc}

**Noam Chomsky** is an American [[linguistics|linguist]], [[philosophy|philosopher]], cognitive scientist, historian, political critic and activist.

He is often regarded as the greatest linguist of the 20th century and has indisputably been the most influential one.

His seminal early work on [[language|formal language]] theory and computational [[complexity theory]] is by now an established part of the foundations of [[computer science]].

* [personal page](https://chomsky.info/)

* [institute page](https://linguistics.arizona.edu/user/noam-chomsky)

For what it's worth, there is of course also a:

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

## Notable Ideas

* [[universal grammar]]

* [[context-free grammar]]

* [[Chomsky hierarchy]]

## References

Introduction of the hierarchy of formal languages via automata models of grammar:

* Noam Chomsky, _Three models for the description of language_, IRE Transactions on Information Theory (1956). ([draft](https://www.its.caltech.edu/~matilde/Chomsky3Models.pdf))

THE classic text of modern (generative) linguistics introducing [[tree|tree representations]]:

* Noam Chomsky, _Syntactic structures_, Mouton The Hague (1957).

Introduction of features into syntax and first general constraints on grammar transformations:

* Noam Chomsky, _Aspects of the Theory of Syntax_, MIT Press (1965).

The founding text of modern generative [[phonology]] carrying a heavy debt to structuralist linguistics:

* N. Chomsky, M. Halle, _The Sound Pattern of English_ , MIT Press 1966.

Taking issue with the generative semantics "heresy" as well as laying down the foundations of contemporary phrase structure theory:

* Noam Chomsky, _Studies on Semantics in Generative Grammar_ , Mouton The Hague 1972.

Containing the early "debates" with [[Montague semantics]]:

* Noam Chomsky, _Essays on Form and Interpretation_ , North-Holland Amsterdam 1977.

The mature and genuine framework of generative linguistics as it evolved from 25 years of research:

* Noam Chomsky, _Lectures on Government and Binding_ , Mouton de Gruyter Berlin 1993[1981].

Many well established concepts are tossed overboard when optimality is adopted as central design feature of natural language in

* Noam Chomsky, _The Minimalist Program_ , MIT Press 1995.

The preliminarily last word is here

* Noam Chomsky, _The UCLA Lectures_ , ms. October 2020.  ([lingbuzz](https://ling.auf.net/lingbuzz/005485))

## Selected publications

[[Hopf algebras]] in generative linguistics:

* [[Matilde Marcolli]], [[Noam Chomsky]], Robert Berwick, *Mathematical Structure of Syntactic Merge* ([arXiv:2305.18278](https://arxiv.org/abs/2305.18278))

* [[Matilde Marcolli]], Robert Berwick, [[Noam Chomsky]], *Old and New Minimalism: a Hopf algebra comparison* ([arXiv:2306.10270](https://arxiv.org/abs/2306.10270))

category:people

[[!redirects Noam Chomsky]]
[[!redirects Noam Chomksy]]


Noam Nisan

* [personal page](https://www.cs.huji.ac.il/~noam/)

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

## Selected writings

On [[quantum circuits]] with [[mixed quantum states]]/[[density matrices]]:

* [[Dorit Aharonov]], [[Alexei Kitaev]], [[Noam Nisan]], *Quantum Circuits with Mixed States*, *Proceedings of the Thirtieth Annual ACM Symposium on Theory of Computation (STOC)* (1998) 20-30 [[arXiv:quant-ph/9806029](https://arxiv.org/abs/quant-ph/9806029), [doi:10.1145/276698.276708](https://doi.org/10.1145/276698.276708)]

category: people


Noam Zeilberger

[website](http://noamz.org/)

category: people


Noboyuki Ishibashi

* [webpage](https://trios.tsukuba.ac.jp/en/researcher/0000000575)

## Selected writings

Introducing the [[IKKT matrix model]]:

* {#IKKT96} [[Noboyuki Ishibashi]], [[Hikaru Kawai]], Y. Kitazawa, A. Tsuchiya, _A Large-$N$ Reduced Model as Superstring_, Nucl. Phys. B498 (1997) 467-491 ([arXiv:hep-th/9612115](http://arxiv.org/abs/hep-th/9612115), doi:10.1016/S0550-3213(97)00290-3)

and relating it to [[Yang-Mills theory]]:

* H. Aoki, [[Noboyuki Ishibashi]], S. Iso, [[Hikaru Kawai]], Y. Kitazawa, T. Tada, _Noncommutative Yang-Mills in IIB Matrix Model_, Nucl.Phys. B565 (2000) 176-192 ([arXiv:hep-th/9908141](http://arxiv.org/abs/hep-th/9908141))

Relation of the $SL(2,\mathbb{R})$-[[gauged WZW-model]] and [[analytically continued Wess-Zumino-Witten theory]] to [[Liouville theory]]:

* [[Noboyuki Ishibashi]], _Extra Observables in Gauged WZW Models_, Nucl.Phys. B379 (1992) 199-219 ([arXiv:hep-th/9110071](http://arxiv.org/abs/hep-th/9110071))

category: people


Nobuo Yoneda

Nobuo Yoneda (kanji: 米田, 信夫, katakana: ヨネダ, ノブオ)
was a Japanese mathematician.

He got his PhD degree in 1961 from the University of Tokyo,
supervised by Shokichi Iyanaga.

* [Wikipedia entry](http://en.wikipedia.org/wiki/Nobuo_Yoneda)

* Yoshiki Kinoshita, *Nobuo Yoneda (1930-1996)*

  &

  [[Saunders MacLane]], *The Yoneda Lemma*,

  Math. Japonica, **47**, No. 1 (1998) 155-156

  ([pdf](https://dmitripavlov.org/scans/yoneda.pdf) [[MathJaponica_YonedaObituary.pdf:file]], also: [posting to catlist in 1996](http://www.mta.ca/~cat-dist/catlist/1999/yoneda))

<img src="http://ncatlab.org/nlab/files/YonedaObituary.jpg">

## Selected writings
 {#SelectedWritings}

* [[Nobuo Yoneda]], *On ext and exact sequences* (PhD thesis), Journal of the Faculty of Science, Section I. **8** University of Tokyo (1960) 507–576 [[pdf](http://alpha.math.uga.edu/~lorenz/YonedaExtExactSequences.pdf), [CiNii:naid/500000325773](https://ci.nii.ac.jp/naid/500000325773)]

## Related entries

* [[Yoneda lemma]]

* [[end]]

category: people

[[!redirects Yoneda]]


Nobuyuki Oda

## Selected writings

On the [[Toda bracket]] understood via [[homotopy coherent diagram|homotopy-coherent]] [[pasting diagrams]] in a [[pointed (infinity,1)-category|pointed]] [[homotopy 2-category]]:

* [[Keith Hardie]], [[Howard Marcum]], [[Nobuyuki Oda]], _Bracket operations in the homotopy theory of a 2-category_, Rend. Ist. Mat. Univ. Trieste 33, 19–70 (2001) ([rendiconti:33/02](https://rendiconti.dmi.units.it/volumi/33/02.pdf))

* [[Howard Marcum]], [[Nobuyuki Oda]], _Long Box Bracket Operations in Homotopy Theory_, Appl Categor Struct 19, 137–173 ([doi:10.1007/s10485-009-9186-3](https://doi.org/10.1007/s10485-009-9186-3))

category: people


nodal curve

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#Contents#
* table of contents
{:toc}

## Idea

A _nodal_  [[singular point of an algebraic variety|singularity]] of an [[algebraic curve]] is one of the form parameterized by the [[equation]] $x y = 0$. A _nodal curve_ is a curve with a nodal singularity.

(e.g. [Hain 08, p. 45](#Hain08))

## The nodal cubic

The nodal [[cubic curve]] (over some base) is (see at _[elliptic curve -- Nodal curves and cuspidal curves](elliptic+curve#EllipticCurvesNodalCurvesCuspidalCurves)_ for notation and background) the solution to the [[Weierstrass equation]] for which the [[discriminant]] vanishes, but the modular invariant $c_4$ does not.

This is equivalently the limit in which the [[j-invariant]] $j = \frac{c_4^3}{\Delta}$ goes to $\infty$.

## Properties of the nodal cubic

### Compactified moduli stack of elliptic curves and the Tate curve

The nodal [[cubic curve]] is not an [[elliptic curve]], as it is singular, but adding it to the [[moduli stack of elliptic curves]] $\mathcal{M}_{ell}$ produces the [[Deligne-Mumford compactification|compactification]] $\mathcal{M}_{\overline{ell}}$ which is often relevant.

The [[formal neighbourhood]] of the nodal curve in $\mathcal{M}_{\overline{ell}}$ is the [[Tate curve]].

### Over the complex numbers
 {#OverTheComplexNumbers}

Over the [[complex numbers]], the nodal cubic $E_0$ is the [[Riemann sphere]]/complex [[projective space]] $\mathbb{P}^1$ with the pole points 0 and $\infty$ identified (hence is a "[[complex manifold|complex]] [[torus]] with one cycle shrunk away"). Precisely: there is a [[holomorphic function]]

$$
  \mathcal{P}^1 \to \mathcal{P}^2
$$

which is onto $E_0 \subset \mathcal{P}_2$, sends the unit of the [[multiplicative group]] $1 \in \mathbb{C}^\times \hookrightarrow \mathbb{P}^1$ to the unit of $E_0$, maps $0,\infty \in \mathbb{P}^1$ both to the nodal singular double point of $E_0$ and is [[injective]] away from these points (e.g. [Hain 08, exercise 47, p. 45](#Hain08))

$$
  \array{
     \mathbb{C}^\times &\hookrightarrow& E_0
     \\
     \downarrow^\mathrlap{=} && \downarrow
     \\
     \mathbb{P}^1-\{0,\infty\} &\longrightarrow& \mathbb{P}^2
  }
  \,.
$$

<img src="http://ncatlab.org/nlab/files/ADE2Cycle.jpeg" width="220" alt="ADE 2Cycle" />

### Formal group and height

The [[formal group]] associated with a nodal cubic curve is of [[height of a formal group|height]] 1.  Indeed, passing to the point of the nodal curve in $\mathcal{M}_{\overline{ell}}$ connects [[elliptic cohomology]] (of [[chromatic level]] 2) to [[topological K-theory]] (of chromatic level 1). For more on this see at _[[moduli stack of tori]]_ and at _[tmf -- Properties -- Maps to K-theory and to Tate K-theory](tmf#MapToTateKTheory)_.

### Relation to gauge enhancement in F-theory

In [[F-theory]] the nodal singularity locus of the given [[elliptic fibration]] is interpreted as the locus of [[D7-branes]], see at _[F-brane scan](F-theory#Fbranescan)_.

## Related concepts

* [[cusp]]

* [[augmented Teichmüller space]]

* in [[F-theory]] the points where the fibers of the [[elliptic fibration]] degenerate to the nodal curve are where the [[D7-branes]] are located

## References

Discussion over the [[complex numbers]] is in

* {#Hain08} [[Richard Hain]], section 5.2 of _Lectures on Moduli Spaces of Elliptic Curves_ ([arXiv:0812.1803](http://arxiv.org/abs/0812.1803))

[[!redirects nodal curves]]

[[!redirects nodal cubic curve]]
[[!redirects nodal cubic curves]]

[[!redirects nodal cubic]]
[[!redirects nodal cubics]]

[[!redirects nodal singularity]]
[[!redirects nodal singularities]]


Noether identity

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> for the moment see at _[[geometry of physics -- A first idea of quantum field theory]]_ [this prop.](geometry+of+physics+--+A+first+idea+of+quantum+field+theory#NoetherIdentities)

[[!redirects Noether identities]]

[[!redirects Noether operator]]
[[!redirects Noether operators]]