Atiyah Lie algebroid

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##Idea

The **Atiyah Lie algebroid** associated to a $G$-[[principal bundle]] $P$ over $X$ is a [[Lie algebroid]] structure on the [[vector bundle]] $T P/ G$, the [[quotient]] of the [[tangent bundle]] of the total space $P$ by the canonical induced $G$-[[action]].

The [[Lie groupoid]] that the Atiyah Lie algebroid [[Lie integration|integrates to]] is the _[[Atiyah Lie groupoid]]_. See there for more background and discussion.

## Definition

Let $G$ be a [[Lie group]] with [[Lie algebra]] $\mathfrak{g}$ and let $P \to X$ be a $G$-[[principal bundle]]:

the **Atiyah Lie algebroid sequence** of $P$ is a sequence of [[Lie algebroid]]s

$$
  ad(P) \to at(P) \to T X
  \,,
$$

where

* $ad(P) = P \times_G \mathfrak{g}$ is the [[adjoint bundle]] of Lie algebras, associated via the [[adjoint action]] of $G$ on its Lie algebra;

* $at(P) := (T P)/G$ is the **Atiyah Lie algebroid**

* $T X$ is the [[tangent Lie algebroid]] of $X$.

The [[Lie bracket]] on the sections of $at(P)$ is that inherited from the tangent Lie algebroid of $P$.

## Relation to connections

A splitting $\nabla_{flat} : T X \to at(P)$ of the Atiyah Lie algebroid sequence in the category of [[Lie algebroid]]s is precisely a flat [[connection on a bundle|connection on]] $P$.

To get non-flat connections in the literature one often sees discussed splittings of the Atiyah Lie algebroid sequence in the category just of [[vector bundle]]s. In that case one finds the curvature of the connection precisely as the [[obstruction]] to having a splitting even in Lie algebroids.

One can describe non-flat connections without leaving the context of Lie algebroids by passing to higher Lie algebroids, namely $L_\infty$-[[L-infinity-algebroid|algebroids]], in terms of an [[horizontal categorification]] of [[nonabelian Lie algebra cohomology]]:

## Atiyah class

The $Ext^1$-cohomology class corresponding to the Atiyah exact sequence (usually in a version for vector bundles/coherent sheaves) is the __Atiyah class__.

## Related concepts

* [[Courant Lie 2-algebroid]]

## References

* [[Michael Atiyah]], _Complex analytic connections in fibre bundles_, Trans. Amer. Math. Soc. 85 (1957), 181--207, [doi](http://dx.doi.org/10.2307/1992969),[MR0086359](http://www.ams.org/mathscinet-getitem?mr=0086359)
* Pietro Tortella, _Representations of Atiyah algebroids and logarithmic connections_, [arxiv/1505.04763](http://arxiv.org/abs/1505.04763)

A discussion with an emphasis on the relation to [[connection on a 2-bundle|2-connections]] and [[Lie 2-algebras]] is on the first pages of

* {#Stevenson06} [[Danny Stevenson]], *Lie 2-algebras and the geometry of gerbes*, Unni Namboodiri Lectures 2006 [[pdf](http://math.ucr.edu/home/baez/namboodiri/stevenson_maclane.pdf), [[Stevenson-Lie2AlgebrasAndGerbes.pdf:file]]]

For Atiyah classes see:

* [[Luc Illusie]], _Complexe cotangent et déformations_ (vol. 1) IV.2.3

* [MO:atiyah-class-for-non-locally-free-sheaf](http://mathoverflow.net/questions/56405/atiyah-class-for-non-locally-free-sheaf)

* [[M. Kapranov]], _Rozansky–Witten invariants via Atiyah classes_, Compositio Math. 115 (1999), 71–113.

* [[Ugo Bruzzo]], Igor Mencattini, [[Vladimir Rubtsov]],
 _Nonabelian holomorphic Lie algebroid extensions_, Intern. J. Math. 26, No. 05, 1550040 (2015) [doi](https://doi.org/10.1142/S0129167X15500408) [arXiv:1305.2377](https://arxiv.org/abs/1305.2377)

* Zhuo Chen, Mathieu Stiénon, Ping Xu, _From Atiyah classes to homotopy Leibniz algebras_, [arXiv/1204.1075](http://arxiv.org/abs/1204.1075); _A Hopf algebra associated to a Lie pair_, [arxiv/1409.6803](http://arxiv.org/abs/1409.6803)

* R. A. Mehta, M. Stiénon, P. Xu, _The Atiyah class of a dg-vector bundle_, [arxiv/1502.03119](http://arxiv.org/abs/1502.03119)

* [[Nikita Markarian]], _The Atiyah class, Hochschild cohomology and the Riemann-Roch theorem_, J. Lond. Math. Soc. (2) 79 (2009), no. 1, 129--143 [doi](https://doi.org/10.1112/jlms/jdn064)

* F. Bottacin, _Atiyah classes for Lie algebroids_, [pdf](http://www.math.unipd.it/~bottacin/papers/liealgebroids.pdf)

* Ajay C. Ramadoss, _The big Chern classes and the Chern character_, Internat. J. Math. 19 (2008), no. 6, 699--746.

* [[Zhuo Chen]], [[Mathieu Stiénon]], [[Ping Xu]], _From Atiyah classes to homotopy Leibniz algebras_, Commun. Math. Phys. __341__ (2016) 309-349 [[arXiv:1204.1075](https://arxiv.org/abs/1204.1075), [doi:10.1007/s00220-015-2494-6](https://doi.org/10.1007/s00220-015-2494-6)]

* Stack Project 92.19 ([tag/09DF](https://stacks.math.columbia.edu/tag/09DF)) The Atiyah class of a sheaf of modules

[[!redirects Atiyah Lie algebroids]]
[[!redirects Atiyah algebroid]]
[[!redirects Atiyah algebroids]]
[[!redirects Atiyah class]]
[[!redirects Atiyah sequence]]


Atiyah Lie groupoid

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## Idea

The **Atiyah Lie groupoid** $At(P)$ of a smooth $G$-[[principal bundle]] $P \to X$ is the [[Lie groupoid]] whose [[objects]] are the [[fibers]] of the bundle, and whose [[morphisms]] are the $G$-[[equivariant maps]] between the fibers.

Schematically:

$$
  At(P)
  \,=\,
  \left\{
     P_x \stackrel{\alpha}{\to} P_y
     | x,y \in X
  \right\}
  \,.
$$

Its [[Lie algebroid]] is the [[Atiyah Lie algebroid]] $at(P)$ of $P$.

Both the Atiyah Lie groupoid and its Lie algebroid are used to characterize and are characterized by [[connection on a bundle|connections]] on $P$.

## Definition

As generally for every [[Lie algebroid]], there are different [[Lie groupoids]] [[Lie integration|integrating]] the [[Atiyah Lie algebroid]]. We describe two of them.

The Aityah Lie algebroid $at(P)$ of the [[principal bundle]] $P \to X$ comes canonically with a morphism $at(X) \to T X$ to the [[tangent Lie algebroid]].

The simplest [[Lie integration]] of the tangent Lie algebroid is the [[pair groupoid]] $X \times X$ of $X$. On the other hand, the universal integration is the [[fundamental groupoid]] $\Pi(X)$ (both coincide precisey if $X$ is a [[simply connected space]]).

Accordingly, there is a version of the Atiyah Lie groupoid over $X \times X$, and a richer version over $\Pi(X)$.

### Over the pair groupoid

For $G$ a Lie group and $p \colon P \to X$ a $G$-principal bundle, the **Atiyah groupoid** $At(P)$ -- also called the **gauge groupoid** or **transport groupoid** -- of $P$ is the [[Lie groupoid]] with

* the [[smooth manifold]] of [[objects]] is $Obj(At(P)) \coloneqq X$;
* the [[smooth manifold]] of [[morphisms]] $Mor(At(P)) = (P \times P)/G$, where the [[quotient object|quotient]] is taken with respect to the [[diagonal action]] of $G$ on $P \times P$;

* the [[source]]/[[target]] maps are those induced by the bundle projection $p$;

  notice that a point $f \colon * \to (P \times P)/G$ over $(x_1 = s(f), x_2 = t(f))$, being an [[equivalence class]] of a pair $(s_1, s_2) \in P \times P$ is canonically identified with the unique $G$-equivariant [[function]] $f \colon P_{x_1} \to P_{x_2}$ which sends $s_1$ to $s_2$;

* [[composition]] is the given by ordinary composition of these functions.

### The integrated Atiyah sequence

The Atiyah groupoid sits in a sequence of groupoids

$$
  Ad(P) \to At(P) \to Pair(X)
$$

where

* $Ad(P) = P \times_G G$ is the **adjoint bundle** of groups associated via the adjoint action of $G$ on itself; regarded as a smooth union $\coprod_{x \in X} \mathbf{B} P_x \times_G G$ of one-object groupoids coming from [[group]]s;

* $Pair(X) = (X \times X \rightrightarrows X)$ is the [[pair groupoid]] of $X$

* the [[functor]] $Ad(P) \to At(P)$ is the identity on objects and on morphisms given by the canonical identification $P_x \times_G G \stackrel{\simeq}{\to} (P_x \times P_x)/G$, where again we use the diagonal action of $G$ on $P_x \times P_x$.

* the functor $At(P) \to Pair(X)$ is the unique one that is the identity on objects.

Notice that a splitting (a [[section]])

$$
  Pair(X) \to At(P)
$$

of the Atiyah groupoid is a trivialization of $P$. On the other hand, locally on [[contractible space|contractible]] $U \subset X$ we have $Pair(U) \simeq \Pi_1(U)$ with $U$ the [[fundamental groupoid]] of $U$, and a splitting $Pair(U) \simeq \Pi_1(U) \to At(P)|_U$ is still a trivialization over $U$ but indicates now that one may want to interpret it as giving rise to a flat [[connection]].

### Over a path groupoid

We have the sequence of  surjective and [[full functor]]s of
[[path category|path categories]]

$$
  P_1(X) \to \Pi_1(X) \to Pair(X)
$$

with $\Pi_1(X)$ the [[fundamental groupoid]] and $P_1(X)$ the smooth [[path groupoid]] and may refine the Atiyah groupoid by pulling back along these.

Write therefore $At'(P) := At(P) \times_{Pair(X)} \Pi_1(X)$ for the [[pullback]]

$$
  \array{
     At'(P) &\to& \Pi_1(X)
     \\
     \downarrow && \downarrow
     \\
     At(P) &\to& Pair(X)
  }
  \,.
$$

A splitting $\Pi_1(X) \to At'(P)$ of the top row is now precisely a flat [[connection]] on $P$.

If we pull back further to $A''$

$$
  \array{
     At''(P) &\to& P_1(X)
     \\
     \downarrow && \downarrow
     \\
     At'(P) &\to& \Pi_1(X)
     \\
     \downarrow && \downarrow
     \\
     At(P) &\to& Pair(X)
  }
  \,.
$$

then splittings of $P_1(X) \to At''(X)$ are precisely (not necessarily flat) connections on $P$.

All this is more well known in terms of the [[Lie algebroid]] underlying the Atiyah Lie groupoid, i.e. the **Atiyah Lie algebroid** sequence

$$
  ad(P) \to at(P) \to T X
  \,,
$$

where

* $ad(P) = P \times_g Lie(G)$ is the adjoint bundle of Lie algebras, associated via the adjoint action of $G$ on its Lie algebra;

* $at(P) = (T P)/G$ is the **Atiyah Lie algebroid**

* $T X$ is the [[tangent Lie algebroid]].

Indeed, a splitting $\nabla_{flat} : T X \to at(P)$ of this sequence in the category of [[Lie algebroid]]s is precisely again a flat connection on $P$ and integrates under [[Lie integration]] to the splitting of $At'(P)$ discussed above.

To get non-flat connections in the literature one often sees discussed splittings of the Atiyah Lie algebroid sequence in the category just of vector bundles. In that case one finds the curvature of the connection precisely as the obstruction to having a splitting even in Lie algebroids.

One can describe non-flat connections without leaving the context of Lie algebroids by passing to higher Lie algebroids, namely $L_\infty$-[[L-infinity-algebroid|algebroids]].

## Related concepts

* [[higher Atiyah groupoid]]

  * [[Atiyah Lie groupoid]], [[Atiyah Lie algebroid]]

  * [[Courant Lie 2-groupoid]], [[Courant Lie 2-algebroid]]

  * [[quantomorphism group]], [[quantomorphism n-group]]

* [[Schauenburg bialgebroid]] (analogue for affine noncommutative principal bundles)

[[!include higher Atiyah groupoid - table]]

[[!redirects Atiyah Lie groupoids]]

[[!redirects Atiyah Lie-groupoid]]

[[!redirects Atiyah groupoid]]
[[!redirects Atiyah groupoids]]


Atiyah Lie-groupoid > history

< [[Atiyah Lie-groupoid]]
[[!redirects Atiyah Lie-groupoid -- history]]


Atiyah-Bott fixed point formula

> under construction

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## Related entries

* [[fixed point]]

* [[Lefschetz fixed point theorem]]

* [[Lawvere's fixed point theorem]]

* [[Kleene's fixed point theorem]]

* [[Brouwer's fixed point theorem]]

## References

* [Getzler:bott.pdf](http://www.math.northwestern.edu/~getzler/Papers/bott.pdf)

* [[Loring Tu]], bio web on R. Bott, node18:[Atiyah-Bott fixed point theorem](http://www.math.harvard.edu/history/bott/bottbio/node18.html)

[[!redirects Atiyah-Bott fixed point theorem]]


Atiyah-Bott-Shapiro isomorphism

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## Idea

The _Atiyah-Bott-Shapiro isomorphism_ (due to [Atiyah-Bott-Shapiro 63](#AtiyahBottShapiro)) is an [[isomorphism]] between the real/complex [[topological K-theory]] ring of the point in degree $q$ and the [[quotient]] of [[Clifford modules]] of rank $q$ by those that have an extension to Clifford modules of rank $q+1$.

The generalization of this to [[Clifford module bundles]] is the content of [[Karoubi K-theory]].

## Related concepts

* [[Karoubi K-theory]]

* [[K-orientation]]

## References

The orginal reference is

* {#AtiyahBottShapiro} [[Michael Atiyah]], [[Raoul Bott]], [[Arnold Shapiro]], _Clifford modules_, Topology 3(Suppl 1):3–38 (1963) ([pdf](http://dell5.ma.utexas.edu/users/dafr/Index/ABS.pdf))

The result is reviewed as theorem I 9.27 in

* [[H. Blaine Lawson]], [[Marie-Louise Michelsohn]], chapter I, section 9 of _[[Spin geometry]]_, Princeton University Press (1989)


Atiyah-Segal completion theorem

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## Idea

By the general discussion at _[[equivariant K-theory]]_, given a suitable [[topological group]] $G$ with an [[action]] on a [[topological space]] $X$ there is a canonical map

$$
  K_G(X) \to K_G^{Bor}(X) \simeq K_G(X \times E G) \simeq K(X \!\sslash\! G)
$$

from the [[equivariant K-theory]] of $X$ to the ordinary [[topological K-theory]] of the [[homotopy quotient]] ([[Borel construction]]).

While this map is never an [[isomorphism]] unless $G$ is the [[trivial group]], the _Atiyah-Segal completion theorem_ says that this map exhibits $K(X//G)$ as the [[formal completion]] of the ring $K_G(X)$ at the [[augmentation ideal]] of the [[representation ring]] of $G$ (hence, regarded as a [[ring of functions]], the restriction to an [[infinitesimal neighbourhood]] of the base point).

See also at _[formal completion -- Examples -- Atiyah-Segal theorem](completion+of+a+ring#AtiyahSegalTheorem)_.

The analog stable for [[stable cohomotopy]] is the _[[Segal-Carlsson completion theorem]]_:

[[!include Segal completion -- table]]

## Consequences

In the case where $X=*$ (i.e. a point), we have that $K_G(*) \simeq R(G)$ and $K(*//G)=KU^0(B G)$, thus we conclude that

$$
  KU^0(BG) \simeq R(G)\hat{_I}
$$

where $I$ is the [[augmentation ideal]] of the [[representation ring]] of $G$.

## Examples and applications
 {#ExamplesAndApplications}

\begin{example}\label{ComplexTopologicalKTheoryOfBS1}
**(complex topological K-theory of $B S^1$)** \linebreak

The [[complex topological K-theory]] of the [[classifying space]]
$B S^1$ of the [[circle group]] is the [[power series ring]]:
$$
  KU^0
  \big(
    B S^1
  \big)
  \;\simeq\;
  \mathbb{Z}
  [ [ c_1^{KU} ] ]
  \,,
$$
where $c_1^{K U}$ is any [[complex oriented cohomology theory|complex orientation]] of [[KU]] (see [this Prop](complex+oriented+cohomology+theory#CohomologyRingOfBU1)).

On the other hand, the [[representation ring]] of the [[circle group]] is (see [this Example](representation+ring#RepresentationRingOfUnitaryGroups))

$$
  R_{\mathbb{C}}(S^1) \;\simeq\; \mathbb{Z}[x, x^{-1}]
  \,.
$$

Here $x$ is the class of the 1-dimensional [[irrep]] $U(1) \hookrightarrow \mathbb{C}$, the [[augmentation ideal]] is clearly generated by $c \coloneqq x - 1$. The corresponding completion is again

$$
  \begin{aligned}
    \big(
      R_{\mathbb{C}}(S^1)
    \big)\hat{_{(c)}}
    &
    \;\simeq\;
    \mathbb{Z}[x, x^{-1}][ [ c ] ] / (c - x + 1)
    \\
    &
    \;\simeq\;
    \mathbb{Z}[ [ c ] ][ (1 + c)^{-1} ]
    \\
    &
    \;\simeq\;
    \mathbb{Z}[ [ c ] ]
    \,,
  \end{aligned}
$$

where the first step is [this example](completion+of+a+ring#CompletionOfNoetherianRingByPowerSeries) and the third step observes that $1 + c$ is invertible in the $c$-power series ring (by [this Prop.](power+series#InvertibleElements)).

(e.g. [Buchholtz 08, Sec. 8.2](#Buchholtz08), also [Math.SE:a/3282578](https://math.stackexchange.com/a/3282578/58526))

\end{example}

compare also [Greenlees 1994, p. 74](https://www.sciencedirect.com/science/article/pii/002240499490006X)

## Related theorems

* [[Baum-Connes conjecture]], [[Green-Julg theorem]]

* As explained in _[[Equivariant stable homotopy theory]]_, there is a related characterization of the [[K-homology]] of $B G$ as a [[localization]].

## References

The original articles:

* [[Michael Atiyah]], _Characters and cohomology of finite groups_, Publications Mathématiques de l'IHÉS, Volume 9 (1961) , p. 23-64 ([numdam:PMIHES_1961__9__23_0]( http://www.numdam.org/item?id=PMIHES_1961__9__23_0))

* [[Michael Atiyah]], [[Friedrich Hirzebruch]], _Vector bundle and homogeneous spaces_, Proc. Sympos. Pure Math., Vol. III, American Mathematical Society, Providence, R.I., 1961, 3, 7–38 ([[AtiyahHirzebruch61.pdf:file]])

* [[Michael Atiyah]], [[Graeme Segal]], _Equivariant $K$-theory and completion_,  J. Differential Geom. Volume 3, Number 1-2 (1969), 1-18. ([euclid:jdg/1214428815](http://projecteuclid.org/euclid.jdg/1214428815))

Review and survey:

* {#Buchholtz08} [[Ulrik Buchholtz]], _The Atiyah-Segal completion theorem_, Master Thesis 2008 ([pdf](http://www.math.ku.dk/~jg/students/buchholtz.ms.2008.pdf), [[Buchholtz_AtiyahSegalCompletion.pdf:file]])

* Wikipedia, _[Atiyah-Segal completion theorem](http://en.wikipedia.org/wiki/Atiyah%E2%80%93Segal_completion_theorem)_

[[!redirects Atiyah-Segal completion]]
[[!redirects Atiyah-Segal completions]]


Atiyah-Singer index theorem

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## Statement

[[Michael Atiyah]] and [[Isadore Singer]] associated two differently defined numbers to an [[elliptic operator]] on a manifold: the **[[topological index]]**  and the **[[analytical index]]**. The index theorem asserts that the two are equal.

The index theorem generalizes earlier results such as the [[Riemann-Roch theorem]].

## References

### General

The original articles:

* [[Michael Atiyah]], [[Isadore Singer]], _The index of elliptic operators I_, Ann. of Math. (2) __87__ (1968)  pp. 484–530; _III_,  Ann. of Math. (2) __87__  (1968)  pp. 546–604; _IV_,  Ann. of Math. (2) __93__  (1971)  pp. 119–138; _V_,  Ann. of Math. (2) __93__  (1971)  pp. 139–149

* [[Michael Atiyah]], [[Graeme Segal]], _The index of elliptic operators II_,  Ann. of Math. (2) __87__ (1968)  pp. 531–545

Review:

* {#HirzebruchGergerJung92} [[Friedrich Hirzebruch]], Thomas Berger, Rainer Jung, chapter 6 of: *Manifolds and Modular Forms*, Aspects of Mathematics **20**, Viehweg (1992), Springer (1994) [[doi:10.1007/978-3-663-10726-2](https://doi.org/10.1007/978-3-663-10726-2), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/hirzjung.pdf)]

* {#Gilkey95} [[Peter Gilkey]], Section 5.2 of: _Invariance Theory: The Heat Equation and the Atiyah-Singer Index Theorem_, 1995 ([pdf](https://pages.uoregon.edu/gilkey/dirPDF/InvarianceTheory1Ed.pdf))

* Rafe Mazzeo, _The Atiyah-Singer Index theorem: what it is and why you should care_, ([slides](https://docplayer.net/52573131-The-atiyah-singer-index-theorem-what-it-is-and-why-you-should-care-rafe-mazzeo-october-10-2002.html))

* [[Daniel Freed]], _The Atiyah-Singer index theorem_, ([slides](https://web.ma.utexas.edu/users/dafr/cmsa.pdf))

See also

* Wikipedia, _[Atiyah-Singer index theorem](http://en.wikipedia.org/wiki/Atiyah%E2%80%93Singer_index_theorem)_

A proof of the Atiyah-Singer index theorem in terms of  [[KK-theory]]/[[E-theory]] has been given by [[Nigel Higson]], an account is in

* {#HigsonRoe} [[Nigel Higson]], [[John Roe]], _Lectures on operator K-theory and the Atiyah-Singer Index Theorem_ (2004) ([pdf](http://folk.uio.no/rognes/higson/Book.pdf))

A lightning review of the proof is on the last pages of

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

### From the point of view of physics
 {#ReferencesPhysics}

The index theorem has an interpretation in terms of the [[quantum field theory]] of the [[superparticle]] on the given space. See also at [supersymmetric quantum mechanics](https://ncatlab.org/nlab/show/supersymmetric+quantum+mechanics#RelationToIndexTheory).

Traditional physics arguments along these lines include for instance

* P. Windey,  _Supersymmetric quantum mechanics and the Atiyah–Singer index theorem_, Acta Physica Polonica, __B15__ (1984). ([PDF](http://th-www.if.uj.edu.pl/~acta/vol15/pdf/v15p0435.pdf))

* [[Luis Alvarez-Gaumé]], _Supersymmetry and the Atiyah-Singer index theorem_, Comm. Math. Phys. Volume 90, Number 2 (1983), 161-173. ([EUCLID](http://projecteuclid.org/euclid.cmp/1103940278))

* Florian Hanisch, Matthias Ludewig, _A Rigorous Construction of the Supersymmetric Path Integral Associated to a Compact Spin Manifold_, ([arXiv:1709.10027](https://arxiv.org/abs/1709.10027))

[[!redirects Atiyah-Singer index theorems]]


Atiyah-Sutcliffe conjecture

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## Idea
 {#Idea}

In the course of providing a [[geometry|geometric]] [[proof]] of the [[spin-statistics theorem]], [Berry & Robbins 1997](#BerryRobbins97) asked, at each [[natural number]] $n \in \mathbb{N}$, for a [[continuous function|continuous]] and [[symmetric group|$Sym(n)$]]-[[equivariant function]]

\[
  \label{TheBerryRobbinsMap}
  \underset{ {}^{\{1,\cdots, n\}}  }{Conf}\big(\mathbb{R}^3\big)
  \xrightarrow{\phantom{AAA}}
  \mathrm{U}(n)/\big(\mathrm{U}(1)\big)^n
\]

* from the [[configuration space of points|configuration space of $n$ points]] (ordered and unlabeled) in [[Euclidean space]] $\mathbb{R}^3$

* to the [[coset space]] of the [[unitary group]] $\mathrm{U}(n)$ by its [[maximal torus]], hence the complete [[flag manifold]] of flags in $\mathbb{C}^n$,

both equipped with the evident [[group action]] by the [[symmetric group]] $Sym(n)$.

For the first non-empty case $n = 2$ this readily reduces to asking for a [[continuous map]] of the form $\mathbb{R}^3 \setminus \{0\} \xrightarrow{\;\;} \mathbb{C}P^1 \simeq S^2$ which is [[equivariant function|equivariant]] with respect to passage to antipodal points. This is immediately seen to be given by the radial projection.
But this special case turns out not to be representative of the general case, as this simple construction idea does not generalize to $n \gt 2$.

That a continuous and $Sym(n)$-equivariant Berry-Robbins map (eq:TheBerryRobbinsMap) indeed exists for all $n$ was proven in [Atiyah 2000](#Atiyah00).

In this article, [[Michael Atiyah|Atiyah]] turned attention to the stronger question asking for a function (eq:TheBerryRobbinsMap) which is [[smooth function|smooth]] and $Sym(n) \times $[[SO(3)|$SO(3)$]]-[[equivariant]] and provided an elegant proof strategy for this stronger statement, which however hinges on some [[conjecture|conjectural]] positivity properties of a certain [[determinant]] (discussed in more detail and with first numerical evidence in [Atiyah 2001](#Atiyah01)), interpreted as the [[electromagnetism|electrostatic]] [[energy]] of $n$-[[particles]] in $\mathbb{R}^3$.

Extensive numerical checks of this stronger but conjectural construction was recorded, up to $n \lt 30$ , in [Atiyah & Sutcliffe 2002](#AtiyahSutcliffe02), together with a refined formulation of the [[conjecture]], whence it came to be known as the *Atiyah-Sutcliffe conjecture*.

The Atiyah-Sutcliffe conjecture has been [[proof|proven]] for $n = 3$ in [Atiyah 2000](#Atiyah00)/[01](#Atiyah01) and for $n = 4$ by [Eastwood & Norbury 01](#EastwoodNorbury01).

## References

The origin of the question in investigation of the [[spin-statistics theorem]]:

* {#BerryRobbins97} [[Michael V. Berry]], [[Jonathan M. Robbins]], *Indistinguishability for quantum particles: spin, statistics and the geometric phase*, Proceedings of the Royal Society A **453** 1963 (1997) 1771-1790 ([doi:10.1098/rspa.1997.0096](https://doi.org/10.1098/rspa.1997.0096))

First form and first checks of the conjecture:

* {#Atiyah00} [[Michael F. Atiyah]], _The geometry of classical particles_, in Surveys in Differential Geometry, Surv. Differ. Geom. __7__, Int. Press 2000, 1–15 ([doi:10.4310/SDG.2002.v7.n1.a1](https://doi.org/10.4310/SDG.2002.v7.n1.a1))

* {#Atiyah01} [[Michael F. Atiyah]], _Configurations of points_, R. Soc. Lond. Philos. Trans. Ser. A, Math. Phys. Eng. Sci. 359 (2001) 1375–1387 ([doi:10.1098/rsta.2001.0840](https://doi.org/10.1098/rsta.2001.0840))

Generalization of the [[codomain]] to [[flag manifolds]] of other [[compact Lie groups]]:

* [[Michael F. Atiyah]], [[Roger Bielawski]], _Nahm’s equations, configuration spaces and flag manifolds_, Bull. Braz. Math. Soc. (N.S.) 33 (2002), 157–176 ([math.RT/0110112](https://arxiv.org/abs/math.RT/0110112), [doi:10.1007/s005740200007](https://doi.org/10.1007/s005740200007))

  > (using [[Nahm's equation]])

Full formulation of the Atiyah-Sutcliffe conjecture:

* {#AtiyahSutcliffe02} [[Michael F. Atiyah]], [[Paul M. Sutcliffe]], _The geometry of point particles_, Proc. Roy. Soc. London Ser. A 458 (2002), 1089–1115 ([hep-th/0105179](https://arxiv.org/abs/hep-th/0105179), [doi:10.1098/rspa.2001.0913](https://doi.org/10.1098/rspa.2001.0913))

Proof for $n = 4$:

* {#EastwoodNorbury01} [[Michael Eastwood]], [[Paul Norbury]], _A proof of Atiyah’s conjecture of four points in Euclidean three-space_, Geom. Topol. 5 (2001) 885–893 ([arXiv:math/0109161](https://arxiv.org/abs/math/0109161), [doi:10.2140/gt.2001.5.88510.2140/gt.2001.5.885](https://msp.org/gt/2001/5-2/p12.xhtml))

Further discussion:

* Dragutin Svrtan, Igor Urbiha, _Atiyah-Sutcliffe conjectures for almost collinear configurations and some new conjectures for symmetric functions_, ([math.AG/0406386](https://arxiv.org/abs/math/0406386))

* Dragutin Svrtan, Igor Urbiha, _Verification and strengthening of the Atiyah--Sutcliffe conjectures for several types of configurations_ ([math.MG/0609174](https://arxiv.org/abs/math/0609174))

* Marcin Mazur, Bogdan V. Petrenko, _On the conjectures of Atiyah and Sutcliffe_, Geom Dedicata **158**  (2012) 329–342
([doi:10.1007/s10711-011-9636-6](https://doi.org/10.1007/s10711-011-9636-6), [arxiv:1102.4662](https://arxiv.org/abs/1102.4662))

* [[Joseph Malkoun]], *Root Systems and the Atiyah-Sutcliffe Problem*, Journal of Mathematical Physics 60, 101702 (2019) ([arXiv:1903.00325](https://arxiv.org/abs/1903.00325))

* [[Joseph Malkoun]], _The Atiyah-Sutcliffe determinant_ ([arXiv:1903.05957](https://arxiv.org/abs/1903.05957))


Atiyah–Hirzebruch spectral sequence

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

#Contents#
* table of contents
{:toc}

## Idea

The _Atiyah-Hirzebruch spectral sequence_ (AHSS) is a type of [[spectral sequence]] that generalizes the [[Serre spectral sequence]] from [[ordinary cohomology]] $H^\bullet$ to any [[generalized (Eilenberg-Steenrod) cohomology]] theory $E^\bullet$.

For any ([[finite homotopy type|finite]]) [[homotopy fiber sequence]]

$$
  \array{
    F &\longrightarrow& P
    \\
    && \downarrow
    \\
    && X
  }
$$

then the corresponding $E$-Atiyah-Hirzebruch spectral sequence has on its second page the [[ordinary cohomology]] of $X$ with [[coefficients]] in the $E$-[[cohomology groups]] of the [[fiber]] and converges to the proper $E$-cohomology of the total space:

$$
  E_2^{p,q} =
  H^p(X,E^q(F))
  \Rightarrow
  E^{\bullet}(P)
  \,.
$$

This is of interest already for $F \simeq \ast$, as then it expresses generalized cohomology in terms of ordinary cohomology with coefficients in the base cohomology ring.

+-- {: .num_remark #NoteOnTerminology}
###### Remark
**(note on terminology)**

Often the terminology "Atiyah-Hirzebruch spectral sequence" is taken to refer to only this case with $F = \ast$, while the general case is then referred to as "Serre spectral sequence for generalized cohomology" or similar. In ([Atiyah-Hirzebruch 61,p. 17](#AtiyahHirzebruch61)) the case $F = \ast$ is labeled "Theorem", while the general case, stated right after the theorem, is labeled "2.2 Remark".

The proof of the theorem that is given is very short, it just says that since topological K-theory satisfies the [exactness axiom](generalized+%28Eilenberg-Steenrod%29+cohomology#ExactnessUnreduced)  of a generalized cohomology theory, it is immediate that the conditions for a spectral sequence stated as Axioms (SP.1)-(SP.5) in ([Cartan-Eilenberg 56, section XV.7](#CartanEilenberg56)) are met. Indeed Example 2 in ([Cartan-Eilenberg 56, section XV.7](#CartanEilenberg56)) observes that the spectral sequence in question exists for $E$  "some fixed [[cohomology theory]]"  because "Axioms (SP.1)-(SP.4) are consequences of usual properties of cohomology groups".

In view of this, the contribution of ([Atiyah-Hirzebruch 61](#AtiyahHirzebruch61)) would not be so much the observation of what is now called the AHSS, rather than the proof that K-theory satisfies the axioms of a generalized cohomology theory. Indeed, according to ([Adams 74, p. 127-128, 215](#Adams74)), the AHSS was earlier observed by [[George Whitehead]] and "then became a folk-theorem" which was "eventually published by Atiyah and Hirzebruch".

Maybe it should be called the "Cartan-Eilenberg-Whitehead spectral sequence".

=--

There is a generalization to [[equivariant cohomology theory]] ([Davis-Lueck 98, theorem 4.7](#DavisLueck98)).

For [[genuine G-spectrum|genuine G-equivariance]], with [[RO(G)-grading]] for [[representation spheres]] $S^V$, then for $P \to X$ an $F$-fibration of [[topological G-spaces]] and for $A$ any $G$-[[Mackey functor]], the equivariant Serre spectral sequence looks like ([Kronholm 10, theorem 3.1](#Kronholm10)):

$$
  E_2^{p,q} = H^p(X, H^{V+q}(F,A)) \,\Rightarrow\, H^{V+p+q}(P,A)
  \,,
$$

where on the left in the $E_2$-page we have [[ordinary cohomology]] with [[coefficients]] in the genuine equivariant cohomology groups of the fiber.

## Construction
 {#Construction}

### Statement
 {#Statement}

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

Let $A^\bullet$ be a an [additive](#UnreducedAdditivity)  unreduced  [[generalized (Eilenberg-Steenrod) cohomology|generalized cohomology functor]]  ([def.](Introduction+to+Stable+homotopy+theory+--+S#ReducedGeneralizedCohomologyHomotopyHomotopicalFunctor)). Let $B$ be a [[CW-complex]] and let $X \stackrel{\pi}{\to} B$ be a [[Serre fibration]], such that all its [[fibers]] are [[weakly contractible topological space|weakly contractible]] or such that $B$ is [[simply connected topological space|simply connected]]. In either case all [[fibers]] are identified with a typical fiber $F$ up to [[weak homotopy equivalence]] by connectedness ([this example](Introduction+to+Stable+homotopy+theory+--+P#FibersOfSerreFibrations)), and well-defined up to unique iso in the homotopy category by simply-connectedness:

$$
  \array{
    F &\longrightarrow& X
    \\
    &&
    \Big\downarrow\mathrlap{{}^{\in Fib_{cl}}}
    \\
    && B
  }
  \,.
$$

If at least one of the following two conditions is met

* $B$ is [[finite number|finite]]-dimensional as a [[CW-complex]];

* $A^\bullet(F)$ is bounded below in degree and the sequences $\cdots \to A^p(X_{n+1}) \to A^p(X_n) \to \cdots$ satisfy the [[Mittag-Leffler condition]] ([def.](Introduction+to+Stable+homotopy+theory+--+S#MittagLefflerCondition)) for all $p$;

then there is a cohomology [[spectral sequence]], ([def.](Introduction+to+Stable+homotopy+theory+--+S#CohomologySpectralSequence)), whose $E_2$-page is the [[ordinary cohomology]] $H^\bullet(B,A^\bullet(F))$ of $B$ with [[coefficients]] in the $A$-[[cohomology groups]] $A^\bullet(F)$ of the fiber, and which converges to the $A$-cohomology groups of the total space

$$
  E_2^{p,q}
    =
  H^p(B, A^q(F))
   \;
    \Rightarrow
   \;
  A^\bullet(X)
$$

with respect to the filtering given by

$$
  F^p A^\bullet(X)
   \coloneqq
  ker\left(
    A^\bullet(X)
      \to
    A^\bullet(X_{p-1})
  \right)
  \,,
$$

where $X_{p} \coloneqq \pi^{-1}(B_{p})$ is the fiber over the $p$th stage of the [[CW-complex]] $B = \underset{\longleftarrow}{\lim}_n B_n$.

Generally, without assumptions on the connectivity of $B$, there is a spectral sequence of this form with ordinary cohomology with coefficients in $A^\bullet(F)$ replaced by ordinary cohomology with [[local coefficients]] $(b \mapsto A^\bullet(F_b))$.

=--

### Construction by filtering the base space
 {#ConstructionByFilteringTheBaseSpace}

The [following proof](#ProofOfAHSS) is the standard and original argument due to ([Atiyah-Hirzebruch 61, p. 17](#AtiyahHirzebruch61)).

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

The [exactness axiom](#ExactnessUnreduced) for $A$ gives an [[exact couple]], ([def.](Introduction+to+Stable+homotopy+theory+--+S#ExactCoupleAndDerivedExactCouple)), of the form

$$
  \array{
    \underset{s,t}{\prod} A^{s+t}(X_{s})
    &&
      \stackrel{}{\longrightarrow}
    &&
    \underset{s,t}{\prod} A^{s+t}(X_{s})
    \\
    & \nwarrow && \swarrow
    \\
    && \underset{s,t}{\prod} A^{s+t}(X_{s}, X_{s-1})
  }
  \;\;\;\;\;\;\;
  \left(
      \array{
        A^{s+t}(X_s) & \longrightarrow & A^{s+t}(X_{s-1})
        \\
        \uparrow && \downarrow_{\mathrlap{\delta}}
        \\
        A^{s+t}(X_s, X_{s-1}) && A^{s+t+1}(X_{s}, X_{s-1})
      }
  \right)
  \,,
$$

where we take $X_{\gg 1} = X$ and $X_{\lt 0} = \emptyset$.

In order to determine the $E_2$-page, we analyze the $E_1$-page: By definition

$$
  E_1^{s,t}
   =
  A^{s+t}(X_s, X_{s-1})
$$

Let $C(s)$ be the set of $s$-dimensional cells of $B$, and notice that for $\sigma \in C(s)$ then

$$
  (\pi^{-1}(\sigma), \pi^{-1}(\partial \sigma)) \simeq (D^n, S^{n-1}) \times F_\sigma
  \,,
$$

where $F_\sigma$ is [[weak homotopy equivalence|weakly homotopy equivalent]] to $F$ ([exmpl.](Introduction+to+Stable+homotopy+theory+--+P#FibersOfSerreFibrations)).

This implies that

$$
  \begin{aligned}
    E_1^{s,t}
    & \coloneqq
    A^{s+t}(X_s, X_{s-1})
    \\
    & \simeq \tilde A^{s+t}(X_s/X_{s-1})
    \\
    & \simeq \tilde A^{s+t}(\underset{\sigma \in C(n)}{\vee} S^s \wedge F_+)
    \\
    & \simeq \underset{\sigma \in C(s)}{\textstyle{\prod}} \tilde A^{s+t}(S^s \wedge F_+)
    \\
    & \simeq \underset{\sigma \in C(s)}{\textstyle{\prod}} \tilde A^t(F_+)
    \\
    & \simeq \underset{\sigma \in C(s)}{\textstyle{\prod}} A^t(F)
    \\
    & \simeq C^s_{cell}(B,A^t(F))
  \end{aligned}
  \,,
$$

where we used the relation to [[reduced cohomology]] $\tilde A$, ([prop.](Introduction+to+Stable+homotopy+theory+--+S#ReducedToUnreducedGeneralizedCohomology)) together with ([lemma](Introduction+to+Stable+homotopy+theory+--+S#EvaluationOfCohomologyTheoryOnGoodPairIsEvaluationOnQuotient)),
then the [wedge axiom](Introduction+to+Stable+homotopy+theory+--+S#WedgeAxiom) and the [suspension isomorphism](Introduction+to+Stable+homotopy+theory+--+S#SuspensionIsomorphismForReducedGeneralizedCohomology) of the latter.

The last group $C^s_{cell}(B,A^t(F))$ appearing in this sequence of isomorphisms is that of [[cellular cohomology|cellular cochains]] ([def.](Introduction+to+Stable+homotopy+theory+--+I#CellularChainComplex)) of degree $s$ on $B$ with [[coefficients]] in the group $A^t(F)$.

Since [[cellular cohomology]] of a [[CW-complex]] agrees with its [[singular cohomology]] ([thm.](Introduction+to+Stable+homotopy+theory+--+I#CelluarEquivalentToSingularFromSpectralSequence)), hence with its [[ordinary cohomology]], to conclude that the $E_2$-page is as claimed, it is now sufficient to show that the differential $d_1$ coincides with the differential in the [[cellular cochain complex]] ([def.](Introduction+to+Stable+homotopy+theory+--+I#CellularChainComplex)).

We discuss this now for $\pi = id$, hence $X = B$ and $F = \ast$. (The general case works the same, just with various factors of $F$ replacing the point.)

Consider the following diagram, which [[commuting diagram|commutes]] due to the [[natural transformation|naturality]] of the [connecting homomorphism](#ConnectingHomomorphismOfUnreducedCohomology) $\delta$ of $A^\bullet$:

$$
  \array{
    \partial^\ast
     \colon
    & C^{s-1}_{cell}(X,A^t(\ast))
      & =&  \underset{i \in I_{s-1}}{\prod} A^t(\ast)
      && \longrightarrow &&
    \underset{i \in I_s}{\prod} A^t(\ast)
    & = &
    C_{cell}^{s}(X,A^t(\ast))
    \\
    && & {}^{\mathllap{\simeq}}\downarrow && && \downarrow^{\mathrlap{\simeq}}
    \\
    && & \underset{i \in I_{s-1}}{\prod} \tilde A^{s+t-1}(S^{s-1})
      && &&
    \underset{i \in I_s}{\prod} \tilde A^{s+t}(S^{s})
    \\
    && & {}^{\mathllap{\simeq}}\downarrow && && \downarrow^{\mathrlap{\simeq}}
    \\
    && d_1 \colon &
    A^{s+t-1}(X_{s-1}, X_{s-2})
      &\overset{}{\longrightarrow}&
    A^{s+t-1}(X_{s-1})
      &\overset{\delta}{\longrightarrow}&
    A^{s+t}(X_s, X_{s-1})
    \\
    && &
    \downarrow && \downarrow && \downarrow
    \\
    && &
    A^{s+t-1}(S^{s-1}, \emptyset)
      &\overset{}{\longrightarrow}&
    A^{s+t-1}(S^{s-1})
      &\overset{\delta}{\longrightarrow}&
    A^{s+t}(D^s , S^{s-1})
  }
  \,.
$$

Here the bottom vertical morphisms are those induced from any chosen cell inclusion $(D^s , S^{s-1}) \hookrightarrow (X_s, X_{s-1})$.

The differential $d_1$ in the spectral sequence is the middle horizontal composite. From this the vertical isomorphisms give the top horizontal map. But the bottom horizontal map identifies this top horizontal morphism componentwise with the restriction to the boundary of cells. Hence the top horizontal morphism is indeed the coboundary operator $\partial^\ast$ for the [[cellular cohomology]] of $X$ with coefficients in $A^\bullet(\ast)$ ([def.](Introduction+to+Stable+homotopy+theory+--+I#CellularChainComplex)). This cellular cohomology coincides with [[singular cohomology]] of the [[CW-complex]] $X$ ([thm.](https://ncatlab.org/nlab/show/Introduction+to+Stable+homotopy+theory+--+I#CelluarEquivalentToSingularFromSpectralSequence)), hence computes the [[ordinary cohomology]] of $X$.

Now to see the convergence. If $B$ is finite dimensional then the convergence condition as stated in [this prop.](Introduction+to+Stable+homotopy+theory+--+S#CohomologicalSpectralSequenceOfAnExactCouple) is met.  Alternatively, if $A^\bullet(F)$ is bounded below in degree, then by the above analysis the $E_1$-page has a horizontal line below which it vanishes. Accordingly the same is then true for all higher pages, by each of them being the cohomology of the previous page. Since the differentials go right and down, eventually they pass beneath this vanishing line and become 0. This is again the condition needed in the proof of [this prop.](Introduction+to+Stable+homotopy+theory+--+S#CohomologicalSpectralSequenceOfAnExactCouple) to obtain convergence.

By that proposition the convergence is to the [[inverse limit]]

$$
  \underset{\longleftarrow}{\lim}
  \left(
    \cdots \stackrel{}{\to} A^\bullet(X_{s+1}) \longrightarrow A^\bullet(X_{s})
    \to \cdots
  \right)
  \,.
$$

If $X$ is finite dimensional or more generally if the sequences that this limit is over satisfy the [[Mittag-Leffler condition]] ([def.](Introduction+to+Stable+homotopy+theory+--+S#MittagLefflerCondition)), then this limit is $A^\bullet(X)$, by [this prop.](Introduction+to+Stable+homotopy+theory+--+S#Lim1VanihesUnderMittagLeffler).

=--

### Construction by filtering the coefficient spectrum
 {#ConstructionByFilteringTheCoefficientSpectrum}

One also gets the Atiyah-Hirzebruch spectral sequence, up to isomorphism, by instead using the filtering given by the [[Postnikov tower]] of an [[Omega-spectrum]] [[Brown representability theorem|representing]] the given generalized cohomology theory.

This is due to ([Maunder 63, theorem 3.3](#Maunder63))

This alternative construction should be the one that is discussed in ([Shulman 13](#Shulman13)) from the perspective of [[homotopy type theory]].

[[!include Lurie spectral sequences -- table]]

## Properties

### Multiplicative structure

for $E$ a [[ring spectrum]], then the AHSS is multiplicative...

([MO discussion](http://mathoverflow.net/q/225579/381))

### Kronecker pairing

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

For $E$ a [[ring spectrum]] and $X$ a [[CW complex]] of finite [[dimension]], then the [[Kronecker pairing]] $\langle -,-\rangle \colon E^\bullet(X)\otimes E_\bullet(X)\to \pi_\bullet(E)$ passes to a page-wise pairing of the corresponding Atiyah-Hirzebruch spectral sequences for $E$-cohomology/homology

$$
  \langle-,-\rangle_r
  \;\colon\;
  \mathcal{E}_r^{n,-s}
  \otimes
  \mathcal{E}^r_{n,t}
  \longrightarrow
  \pi_{s+t}(E)
$$

such that

1. on the $\mathcal{E}_2$-page this restricts to the Kronecker pairing for [[ordinary cohomology]]/[[ordinary homology]] with [[coefficients]] in $\pi_\bullet(E)$;

1. the [[differentials]] act as [[derivations]]

   $$
     \langle d_r(-),-\rangle = \langle -, d^r(-)\rangle
     \,,
   $$

1. The pairing on the $\mathcal{E}_\infty$-page is compatible with the Kronecker pairing.

=--

([Kochman 96, prop. 4.2.10](#Kochman96))

## Examples and Applications

### To complex oriented cohomology theory

...([Adams 74](#Adams74))....

### For topological K-theory
 {#ForKTheory}

For $E = $ [[KU]], hence for [[topological K-theory]], the differential $d_3$ of the Atiyah-Hirzebruch spectral sequence with $F = \ast$ is given by a [[Steenrod square]] operation

$$
  d_3 = Sq^3
$$

([Atiyah-Hirzebruch 61](#AtiyahHirzebruch61)).

For [[twisted K-theory]] this picks up in addition the [[cup product]] with the 3-class $H$ of the twist:

$$
  d_3 = Sq^3 + H \cup(-)
$$

([Rosenberg 82](#Rosenberg82), [Atiyah-Segal 05 (4.1)](#AtiyahSegal05)). The higher differentials $d_5$ and $d_7$ here are given by higher [[Massey products]] with the twisting class ([Atiyah-Segal 05 sections 5-7](#AtiyahSegal05)).

### D-brane charges in string theory
 {#ApplicationDBraneChargesInStringTheory}

In [[string theory]] [[D-brane charges]] are classes in $E = KU$-cohomology, i.e. in [[K-theory]]. The second page of of the corresponding Atiyah-Hirzebruch spectral sequence (see [above](#ForKTheory)) for $F = \ast$ hence expresses ordinary cohomology in all even or all odd degrees, and being in the kernel of all the differentials is hence the constraint on such ordinary cohomology data to lift to genuine K-theory classes, hence to genuine D-brane charges. In this way the Atiyah-Hirzebruch spectral sequences is used in ([Maldacena-Moore-Seiberg 01](#MaldacenaMooreSeiberg01), [Evslin-Sati 06](#EvslinSati06))

## References

### General

The statement of the existence of the spectral sequence first appears in print (with $E = $ [[topological K-theory]]) in

* {#AtiyahHirzebruch61} [[M. F. Atiyah]], [[F. Hirzebruch]], _Vector bundles and homogeneous spaces_, 1961, Proc. Sympos. Pure Math., Vol. III pp. 7–38 American Mathematical Society, Providence, R.I. ([web](http://hirzebruch.mpim-bonn.mpg.de/87/), doi:10.1142/9789814401319_0008, [MR 0139181](http://www.ams.org/mathscinet-getitem?mr=0139181))

but the proof given consists essentially in pointing to section XV.7 ("a more general setting in which the theory of spectral sequences may be developed") of

* {#CartanEilenberg56} [[Henri Cartan]], [[Samuel Eilenberg]], _Homological algebra_, Princeton Univ. Press (1956)

This is by filtering over the stages of the base space CW-complex. That one gets an isomorphic spectral sequence by instead filtering over the [[Postnikov tower]] of any [[Omega-spectrum]] [[Brown representability theorem|representing]] the given [[generalized (Eilenberg-Steenrod) cohomology|generalized cohomology]] theory is due to

* {#Maunder63} [[C. R. F. Maunder]], *The spectral sequence of an extraordinary cohomology theory*, Mathematical Proceedings of the Cambridge Philosophical Society **59** 3 (1963) 567- 574 $[$[doi:10.1017/S0305004100037245](https://doi.org/10.1017/S0305004100037245)$]$

Early lecture notes:

* {#Adams74} [[Frank Adams]], part III section 7 of _[[Stable homotopy and generalised homology]]_, 1974

where the idea is attributed to [[George Whitehead]]:

> the Atiyah-Hirzebruch spectral sequence, which was really invented by G. W. Whitehead but not published by him ([Adams 74, p. 127-128](#Adams74))

> These spectral sequences were probably first invented by G. W. Whitehead, but he got them just after he wrote the paper $[$Whitehead 56 'Homotopy groups of joins and unions'$]$ in which they ought to have appeared. They then became a folk-theorem and were eventually published by Atiyah and Hirzebruch ([Adams 74, p. 215](#Adams74))

A more detailed account of the proof is in

* {#Kochman96} [[Stanley Kochman]], theorem 4.2.5 of _[[Bordism, Stable Homotopy and Adams Spectral Sequences]]_, AMS 1996

Further discussion of the case of [[twisted K-theory]]:

* {#Rosenberg82} [[Jonathan Rosenberg]], _Homological Invariants of Extensions of $C^\ast$-algebras_, Proc. Symp. Pure Math 38 (1982) 35.

* {#AtiyahSegal05} [[Michael Atiyah]], [[Graeme Segal]], _Twisted K-theory and cohomology_, Nankai Tracts Math. 11, World Sci. Publ., Hackensack, NJ, pp. 5–43,  ([arXiv:math/0510674](http://arxiv.org/abs/math/0510674))

* [[Dale Husemöller]], [[Michael Joachim]], [[Branislav Jurčo]], [[Martin  Schottenloher]], Section 21 of: _[[Basic Bundle Theory and K-Cohomology Invariants]]_, Springer Lecture Notes in Physics __726__, 2008, ([pdf](http://www.mathematik.uni-muenchen.de/~schotten/Texte/978-3-540-74955-4_Book_LNP726.pdf), [doi:10.1007/978-3-540-74956-1](https://link.springer.com/book/10.1007/978-3-540-74956-1))

An analogous spectral sequence for higher [[twisted de Rham cohomology]] is discussed in [Li, Liu & Wang 2009](twisted+de+Rham+cohomology#LiLiuWang09).

A discussion of the $E$-AHSS as the spectral sequence of a tower induced by forming [[mapping spectra]] $[X,-]$ into the [[Postnikov tower]] is due to

* {#Shulman13} [[Mike Shulman]], _[[homotopytypetheory:spectral sequences]]_ (2013)

### Equivariant version
 {#EquivariantVersion}

Discussion of the AHSS in [[Bredon cohomology|Bredon]] [[equivariant stable homotopy theory]]/[[equivariant cohomology]] includes

* {#DavisLueck98} [[James Davis]], [[Wolfgang Lueck]], _Spaces over a category and assembly maps in isomorphism conjectures in K- and L-theory_, K-Theory 15: 201–252, 1998 ([pdf](http://www.indiana.edu/~jfdavis/papers/assembly.pdf), [K-Theory archive](http://www.math.uiuc.edu/K-theory/0131/))

Discussion in genuine equivariant cohomology, i.e. including [[RO(G)-grading]], is in

* {#Kronholm10} [[William Kronholm]], _The $RO(G)$-graded Serre spectral sequence_, Homology Homotopy Appl. Volume 12, Number 1 (2010), 75-92. ([pdf](http://www.swarthmore.edu/NatSci/wkronho1/serre.pdf), [Euclid](https://projecteuclid.org/euclid.hha/1296223823))

### Examples

Application for the case of [[K-theory]] to [[D-brane charges]] in [[string theory]] is discussed in

* {#MaldacenaMooreSeiberg01} [[Juan Maldacena]], [[Gregory Moore]], [[Nathan Seiberg]], _D-Brane Instantons and K-Theory Charges_, JHEP 0111:062,2001 ([arXiv:hep-th/0108100](http://arxiv.org/abs/hep-th/0108100))

* {#EvslinSati06} [[Jarah Evslin]], [[Hisham Sati]], _Can D-Branes Wrap Nonrepresentable Cycles?_, JHEP0610:050,2006 ([arXiv:hep-th/0607045](http://arxiv.org/abs/hep-th/0607045))

and detailed review of this is in

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

Discussion for the $\mathbb{Z}/2$-equivariant [[KR cohomology theory]] (relevant for D-branes in [[orientifolds]]) includes

* [[Daniel Dugger]], _An Atiyah-Hirzebruch spectral sequence for KR-theory_ ([arXiv:math/0304099](http://arxiv.org/abs/math/0304099)

Discussion for the case of [[Morava K-theory]] and [[Morava E-theory]] with comments on application to charges of [[M-branes]] is in

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

Application to [[motivic homotopy theory|motivic]] [[cobordism cohomology theory]] is discussed in

* Nobuaki Yagita, _Applications of the Atiyah-Hirzebruch spectral sequences for motivic cobordisms_ ([pdf](http://www.math.uiuc.edu/K-theory/0627/ah.pdf))

Application to the [[K-theory classification of topological phases of matter]]:

* [[Ken Shiozaki]], [[Masatoshi Sato]], [[Kiyonori Gomi]], *Atiyah-Hirzebruch Spectral Sequence in Band Topology: General Formalism and Topological Invariants for 230 Space Groups*, Phys. Rev. B **106** (2022) 165103 [[arXiv:1802.06694](https://arxiv.org/abs/1802.06694), [doi:10.1103/PhysRevB.106.165103](https://doi.org/10.1103/PhysRevB.106.165103)]

[[!redirects Atiyah-Hirzebruch spectral sequence]]
[[!redirects Atiyah-Hirzebruch spectral sequences]]

[[!redirects Atiyah-Hirzeburch spectral sequence]]


atlas

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###Context###
#### Higher geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
#### Manifolds and cobordisms
+--{: .hide}
[[!include manifolds and cobordisms - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[general topology|basic topology]] and [[differential geometry]], by an _atlas_ of/for a [[topological manifold|topological]]-, [[differentiable manifold|differentiable]]- or [[smooth manifold]] $X$ one means a collection of [[coordinate charts]] $U_i \subset X$ which form an [[open cover]] of $X$.

If one considers here the [[disjoint union]] $\mathcal{U} \coloneqq \underset{i}{\sqcup} U_i$ of all the [[coordinate charts]], then the separate chart embeddings $U_i \subset X$ give rise to a single [[map]] ([[continuous function|continuous]]/[[differentiable function]])

$$
  \mathcal{U} \longrightarrow X
$$

and now the condition for an atlas is that this is a [[surjective function|surjective]] [[étale map]]/[[local diffeomorphism]].

If, next, one regards this morphism, under the [[Yoneda embedding]], inside the [[topos]] of [[formal smooth sets]], then these conditions on an atlas say that this morphism is

1. an [[effective epimorphism]];

1. a [[formally étale morphism]].

In this abstract form the concept of an atlas generalizes to any [[cohesion|cohesive]] [[higher geometry]] ([KS 17, Def. 3.3](#KhavkineSchreiber17), [Wellen 18, Def 4.13](#Wellen18), [Sati & Schreiber 2020, p. 27](#SatiSchreiber20)).

Next, for a [[geometric stack]] $\mathcal{X}$, an atlas is a [[smooth manifold]] $\mathcal{U}$ (for [[differentiable stacks]]) or [[scheme]] $\mathcal{U}$ (for [[algebraic stacks]]) or similar, equipped with a morphism

$$
  \mathcal{U} \longrightarrow \mathcal{X}
$$

that is an [[effective epimorphism]] and [[formally étale morphism]]
in the corresponding [[(infinity,1)-topos|higher topos]] (for instance in that of [[formal smooth infinity-groupoids]]).

Here the terminology has a bifurcation:

1. In the general context of [[geometric stacks]] one typically drops the second condition and calls any [[effective epimorphism]] from a [[smooth manifold]] or [[scheme]] to a [[differentiable stack]] or [[algebraic stack]], respectively, an _atlas_ (e.g. [Leman 10, 4.4](#Leman10)).

1. If in addition the condition is imposed that such an effective epimorphism exists which is also [[formally étale morphism|formally étale]], then the [[geometric stack]] is called an _[[orbifold]]_ or _[[Deligne-Mumford stack]]_ (often with various further conditions imposed).

From here, the terminology generalizes to [[infinity-stacks|$\infty$-stacks]] in general [[(infinity,1)-toposes|$\infty$-toposes]], see [this Remark](groupoid+objects+in+an+infinity1-topos+are+effective#InterpretationInTermsOfInfinityStacksWithAtlases) at *[[groupoid objects in an (∞,1)-topos are effective]]*.

{#AtlasForACategory} Yet more generally, the notion generalizes to [[2-topos theory]] and higher. Over the point this yields the notion of *[[category with an atlas]]* and *[[flagged categories]]* (depending on the truncation of the atlas) and relates to the notions of *flagged higher categories* ([Ayala & Francis 2018](category+with+an+atlas#AyalaFrancis18)).

## Related concepts

* [[category with an atlas]]

* [[n-types cover]]

## References

Review of the classical concept of atlases for geometric stacks:

* {#Leman10} [[Eugene Lerman]], Section 4.4 of: _Orbifolds as stacks?_, L'Enseign. Math. (2) 56 (2010), no. 3-4, 315--363 ([arXiv:0806.4160](https://arxiv.org/abs/0806.4160))

Formalization in [[cohesive homotopy theory]] and [[cohesive homotopy type theory|cohesive]]/[[modal homotopy type theory]]:

* {#KhavkineSchreiber17} [[Igor Khavkine]], [[Urs Schreiber]], _Synthetic geometry of differential equations_ ([arXiv:1701.06238](https://arxiv.org/abs/1701.06238))

* {#Wellen18} [[Felix Wellen]], _[[schreiber:thesis Wellen|Formalizing Cartan Geometry in Modal Homotopy Type Theory]]_ ([arXiv:1806.05966](https://arxiv.org/abs/1806.05966))

* {#SatiSchreiber20} [[Hisham Sati]], [[Urs Schreiber]], p. 27 of: *[[schreiber:Proper Orbifold Cohomology]]* ([arXiv:2008.01101](https://arxiv.org/abs/2008.01101))

[[!redirects atlases]]


Atle Selberg

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

## related $n$Lab entries

* [[Selberg trace formula]]

* [[Selberg zeta function]]

* [[Rankin-Selberg method]]

category: people

[[!redirects Selberg]]


atom

> This article is on the mathematical concept of atom as used in the theory of [[preorder|preorders]], and related mathematical notions. For small projective objects in categories see at [[atomic object]]. For still other uses, see [[atom (disambiguation)]].

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### Context
#### Compact objects
+-- {: .hide}
[[!include compact object - contents]]
=--
=--
=--

# Atoms
* table of contents
{: toc}

## Idea

An atom in a [[poset]] is a [[minimal element]] among those which are not actually the [[bottom element|minimum]].  Thus an atom is as small as possible without being nothing.  In an atomic poset, every element may be broken down (typically not uniquely) into atoms.

A related but slightly weaker concept is that of "tiny element", which has important generalizations in the context of enriched category theory.

## Definitions

Let $S$ be a [[poset]] (or [[proset]]) with a [[bottom element]] $\bot$.  Recall that an element of $S$ is __[[positive element|positive]]__ if it is not a bottom element.  An element $a$ of $S$ is __atomic__ if, given any element $p \leq a$, $p$ is positive iff $a \leq p$.  An __atom__ of $S$ is simply an atomic element of $S$.  Note that every atom must be positive (since $a \leq a$).

The atoms are precisely the [[minimal elements]] of the set of positive elements.  For a poset, $a$ is atomic iff every $p \leq a$ is positive iff $p = a$.  Using [[classical logic]] too, $a$ is atomic iff every $p \leq a$ satisfies $p = \bot$ [[xor]] $p = a$.

The p(r)oset $S$ is __atomic__ (or more commonly in the literature, **atomistic**; see remarks below) if every element is a [[supremum]] of atoms.  In this case, every element $x$ is a supremum of those atoms $a \leq x$.  Note that $\bot$ is a supremum of no atoms, and every atom is a supremum of itself, so the condition is really about the nontrivial nonatomic elements.

In [[constructive mathematics]], we require a more complicated definition of a [[positive element]], but the other definitions above remain correct (under the stated conditions), once we have that.  In [[predicative mathematics|predicative]] constructive mathematics, positivity cannot be defined at all, and $S$ must come equipped with a [[positivity predicate]] before we may consider its atoms.

### Remarks on terminology

There is some terminological variance in the literature to the notion of atomic poset as defined here. In particular, [Wikipedia](http://en.wikipedia.org/wiki/Atom_%28order_theory%29) defines an _atomic poset_ to be a poset in which every positive element has an atom below it, and refers to our stronger notion of atomic poset by the term "atomistic poset". Note well that the Wikipedia conventions seem to be the ones observed in most lattice-theoretic texts.

"Atomic" and "Atomistic" differ for the simple example of the [[divisor lattice]] for some number $n$. The atoms in this lattice are prime numbers while it may also contain [[semi-atom|semi-atoms]] which are powers of primes. This lattice is atomic because any object not the [[bottom]],  $1$, is divisible by a prime in the lattice. However it is not generally atomistic, but is instead uniquely semi-atomistic (every  object is the product of a unique set of semi-atoms with bottom corresponding to the empty set), which is one way of stating the [[fundamental theorem of arithmetic]], also known as the *unique factorization theorem*.

The two notions coincide in the case of complete Boolean algebras $B$. Indeed, suppose $B$ is atomic in the Wikipedia sense, and for any element $b \in B$, consider the relative complement

$$c = b \wedge \neg (\bigvee \{atoms\, a: a \leq b\})$$

To show $B$ is atomistic, it suffices to show $c = 0$. If not, then there is an atom $a'$ such that $a' \leq c$, which means both $a' \leq b$ and

$$a' = a' \wedge a' \leq a' \wedge \bigvee \{atoms\, a: a \leq b\} = 0$$

since $a' \leq \neg(\bigvee \{atoms\, a: a \leq b\})$. This is a contradiction.

Our (*pro tem*) decision to define the word "atomic" in the idiosyncratic nLab sense above is consistent with its use elsewhere in category theory; see the sections below on atomic objects and on categorification.

## Examples

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

In a [[power set]] the atoms are the [[singleton subsets]].  Every power set is atomic, and in fact every atomic [[complete boolean algebra]] is (up to [[isomorphism]]) a power set.

=--

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

In a [[lattice of subtoposes]] the atoms are the 2-valued [[Boolean toposes]]. See [this proposition](subtopos#BooleantoposesAreAtoms).

=--

## Properties

If $a$ is an atom in a [[lattice]] or more generally a [[meet]] [[semilattice]] and $b$ any other element then (using classical logic)
$$ a \wedge b \in \{a, \bot\} .$$
This is simply because $a \wedge b \leq a$, so equals either $a$ or $\bot$.

## Atoms and tiny elements

If $E$ is a poset or preorder, in other words a $\mathbf{2}$-[[enriched category]], an element $e \in E$ is _[[tiny object|tiny]]_ if the hom $E(e, -)\colon E \to \mathbf{2}$ preserves all [[supremum|sups]] that exist in $E$. It is arguable (from an [[nPOV]]) that the weaker concept of tiny element is more fundamental than the notion of atom; for example, as we will see below, replacing atoms by tiny elements permits one to generalize the characterization of power sets as complete atomic Boolean algebras.

+-- {: .num_prop}
###### Proposition
A tiny element in a Boolean algebra is precisely an atom.
=--

+-- {: .proof}
###### Proof
Let $a$ be an atom. Let $\{x_i\}$ be a collection of elements that admits a supremum such that $a \leq \bigvee_i x_i$. Then

$$a = a \wedge \bigvee_i x_i = \bigvee_i a \wedge x_i$$

(where the second equation holds since $a \wedge -$ is a left [[adjoint functor|adjoint]], because $B$ is a [[Heyting algebra]]). Since $a$ is positive, for some $i$ the element $a \wedge x_i$ is positive as well. Trivially it holds that $a \wedge x_i \leq a$; since $a$ is an atom, the inequality is an equality. Thus $a \leq x_i$ for some $i$, which is what we want.

If $a$ is not an atom, i.e., if $0 \lt b \lt a$ for some $b$, then

$$a = b \vee (a \wedge \neg b)$$

If $B(a, -)$ preserved the join on the right, then either $a \leq b$ which is evidently false, or $a \leq a \wedge \neg b$, i.e., $a \leq \neg b$, i.e., $b = a \wedge b \leq 0$, also evidently false. Thus $B(a, -)$ does not preserve suprema.
=--

Only one half of this proposition holds (an atom is a tiny element) if we replace the Boolean algebra $B$ by a general [[frame]]. (In fact, this direction even holds in impredicative constructive mathematics, if the frame is equipped with a [[positive element|positivity predicate]].) On the other hand, tiny elements need not be atoms (an easy example is the frame of down-sets of a [[poset]], where principal down-sets are atomic objects, but generally not atoms in the underlying poset of the frame).

Be this as it may, [Lawvere](http://www.acsu.buffalo.edu/~wlawvere/ToposMotion.pdf#page=6) has written, "In order to settle once and for all the various terminological differences, perhaps we can use a.t.o.m. as an abbreviation for 'amazing tiny object model'." This is Lawvere's 'objective' way of abbreviating "atomic object"; the word 'amazing' here is presumably chosen to evoke what Lawvere has called the "amazing right adjoint" to an exponential functor $(-)^D$, particularly in the case of [[synthetic differential geometry]] where such adjoints exist for [[infinitesimal object|infinitesimal objects]] $D$.

## Generalization and categorification

The result that an atomic complete Boolean algebra is isomorphic to a power set -- hence to a [[presheaf]] with values in the [[0-category]] $\mathbf{2} = (-1)Grpd$ of [[(-1)-groupoid|(-1)-groupoids]] -- may be generalized and categorified as follows.

Let $E$ be a $V$-category, where $V$ is a [[cosmos]] (a complete, cocomplete, symmetric monoidal closed category). We define an object $e$ of $E$ to be **tiny** or **atomic** if $E(e, -) \colon E \to V$ preserves any $V$-colimit that exists in $E$. (As usual, the appropriate notion of colimit in the enriched setting is [[weighted colimit]].)

In what follows, we suppose the full $V$-subcategory $Tiny(E)$ of atomic objects in $E$ is essentially small. The inclusion $i \colon Tiny(E) \hookrightarrow E$ induces a restricted Yoneda embedding

$$E \to V^{Tiny(E)^{op}}$$

sending an object $e$ to $E(i-, e)$. We say that $E$ is **[[atomic category|atomic]]** if $i \colon Tiny(E) \hookrightarrow E$ is $V$-[[dense functor|dense]], in other words if every object $e$ of $E$ is a canonical colimit of atomic objects below it, in the precise sense that the following enriched [[coend]] exists, and its canonical map to $e$,

$$\int^{a \in Tiny(E)} E(i a, e) \cdot i a \to e,$$

is an isomorphism.

If $E$ is a preorder, i.e., is $\mathbf{2}$-enriched where $\mathbf{2}$ is the category of $(-1)$-categories, the coend amounts to the supremum

$$\sup \{i a: i a \leq p\}$$

so that $E$ is atomic precisely if every element is the sup of the tiny elements below it.

+-- {: .num_thm}
###### Theorem
A small-cocomplete atomic preorder $E$ is equivalent to the free sup-lattice $2^{T^{op}}$ generated by the preorder $T = Tiny(E)$ of tiny elements. Conversely, every free sup-lattice $2^{T^{op}}$ is small-cocomplete and atomic, where $T$ is the poset of tiny elements.
=--

N.B. "Free sup-lattice" refers to a left adjoint of the forgetful functor $U \colon SupLat \to Preord$ from sup-lattices to preorders.

+-- {: .proof}
###### Proof
Since $E$ is cocomplete, and since $\mathbf{2}^{Tiny(E)^{op}}$ is the free sup-lattice or cocomplete preorder generated from $Tiny(E)$, the inclusion $i \colon Tiny(E) \to E$ extends uniquely to a sup-preserving map

$$L \colon \mathbf{2}^{Tiny(E)^{op}} \to E$$

which sends $X \colon Tiny(E)^{op} \to \mathbf{2}$ to

$$\int^{a \in Tiny(E)} X(a) \cdot i a = \sup \{i a: X(a) = 1\}.$$

This $L$ is left adjoint to the restricted Yoneda embedding $R \colon E \to \mathbf{2}^{Tiny(E)^{op}}$. The condition that $E$ is atomic says that for each $e \in E$, the value of the counit of $L \dashv R$ at $e$ is an isomorphism

$$\int^a E(i a, e) \cdot i a \cong e.$$

On the other hand, the value of the unit of $L \dashv R$ at an object $X$ is given by a string of isomorphisms

$$\array{
X & \stackrel{Yoneda}{\cong} & \int^a X(a) \cdot Tiny(E)(-, a) \\
 & \cong & \int^a X(a) \cdot E(i-, i a) \\
 & \cong & E(i-, \int^a X(a) \cdot i a)
}$$

where the last isomorphism obtains from the fact that $E(i a, -)$ preserves colimits if $a$ is tiny. Thus the unit is also an isomorphism.

For the converse: each representable object $T(-, t)$ of $\mathbf{2}^{T^{op}}$ is tiny, because the covariant functor $2^{T^{op}}(T(-, t), -)$, being the same as evaluation at $t$ by the Yoneda lemma, preserves colimits. Furthermore, every functor $X: T^{op} \to \mathbf{2}$ is a canonical colimit of representables, so that $2^{T^{op}}$ is atomic in addition to being cocomplete.
=--

+-- {: .num_cor}
###### Corollary
A complete atomic Boolean algebra $B$ is isomorphic to $2^T$, where $T$ is the discrete preorder of atoms of $B$.
=--

The argument given for the theorem above carries over without obstruction to the general enriched setting. In particular, replacing $\mathbf{2} = (-1)$-$Cat$ by its categorification $Set = 0$-$Cat$, we get the following result, first enunciated in Bunge's thesis.

+-- {: .num_thm}
###### Theorem (Bunge)
A category $E$ is equivalent to a [[presheaf topos]] (functors with values in the 1-category [[Set]] of [[0-groupoids]]) if and only if it is cocomplete and atomic as a $Set$-category. Representables $C(-, c)$ are (among the) atomic objects of $Set^{C^{op}}$, and generate the presheaf topos by closing under all small colimits.
=--

(The literal statement in Bunge's thesis is that a category is equivalent to a presheaf category $Set^{C^{op}}$ if and only if it is cocomplete, regular, and has a generating set of atomic objects, but this is trivially the same since presheaf toposes are of course [[regular category|regular]].)

## Related concepts

* [[compact object]]

* [[connected object]]

* [[Cauchy completion]]

[[!redirects atom]]
[[!redirects atoms]]
[[!redirects atomic element]]
[[!redirects atomic elements]]


atom (disambiguation)

_Atom_ may refer to one of the following:

* In mathematics, an element which is minimal among non-minimal elements (in a preorder or poset). Related notions may be found at [[atom]].

* In set theory, there are theories (for example [[ZFA]]) which admit [[urelements]], which are also called atoms.

* In [[physics]], a basic unit of [[matter]] that represents a chemical element or one of its isotopes, consisting of a central [[nucleus]] of [[protons]] and [[neutrons]] and a "cloud" of [[electrons]] surrounding it. See [[atom (physics)]].

* In [[William Lawvere]]'s work a characterization of [[infinitesimally thickened point]], an _[[atomic object]]_.

* In [[Georg Hegel]], _[[Science of Logic]]_ : [Atomism](Science+Of+Logic#Atomism)


atom (physics)

> This entry is about the concept in [[physics]]. For other notions of the same name see [[atom (disambiguation)]]

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### Context
#### Physics
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[[!include physicscontents]]
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#### Fields and quanta
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[[!include fields and quanta - table]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[physics]] an _atom_ is a [[bound state]] of a [[nucleus]] and a shell of [[electrons]], where the nucleus itself is a bound state of [[protons]] and [[neutrons]], which themselves are bound states of [[quarks]], which, finally, in the present [[standard model of particle physics]], are [[elementary particles]], as are the [[electrons]] in the shell of the atom.

In [[chemistry]] the different types of atoms that exist are also called _[[chemical elements]]_. Their [[bound states]] in turn are called _[[molecules]]_. The [[phenomenology]] of the [[elements]] that is captured by the [[periodic table of the elements]] is explained by the [[quantum physics]] of atoms. This transition area between [[quantum physics]] and [[chemistry]] is called _[[quantum chemistry]]_.

## Terminology

When the term _atom_ ("indivisible") for these physical objects was established, the inner structure of atoms was not known yet.  Similarly what today are called [[elementary particles|elementary]] (hence again: indivisible) [[particles]] are subject to speculation that they might be compound after all (e.g. in [[technicolor]] models, or [[string theory]]).

## Examples

* [[hydrogen atom]]

## Related concepts

* [[chemistry]], [[chemical element]]

* [[molecule]]

* [[particle]]

* [[periodic table of the elements]]

* [[atomtronics]]

## References

See also

* Wikipedia, _[Atom (physics)](http://en.wikipedia.org/wiki/Atom)_


atom physics

## Idea

The [[physics]] of [[atom (physics)|atoms]].

## References

* Wikipedia, _[Atom physics](http://en.wikipedia.org/wiki/Atomic_physics)_


atomic Boolean algebra

* table of contents
{: toc}

## Definition

Given an element $a$ of a [[Boolean algebra]] (or other [[poset]]) $A$, recall that $a$ is an **[[atom]]** in $A$ if $a$ is [[minimal element|minimal]] among non-trivial (non-[[bottom]]) elements of $A$.  That is, given any $b \in A$ such that $b \leq a$, either $b = 0$ or $b = a$.

$A$ is __atomic__ if we have $b = \bigvee_I a_i$ for every $b \in A$, where $\{a_i\}_I$ is some set of atoms in $A$.

If $A$ is complete we can write it: if for every $b \in A$, we have $b = \bigvee \mathcal{A}(b)$ where $\mathcal{A}(b)$ is the set of all the atoms $a$ in $A$ such that $a \leq b$. Or: for every $b \in A$, we have $b \le \bigvee \mathcal{A}(b)$.

## Properties

* A [[complete lattice|complete]] atomic Boolean algebra is necessarily a [[power set]]; see [[CABA]].

[[!redirects atomic Boolean algebra]]
[[!redirects atomic Boolean algebras]]


atomic category

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#### Category theory
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#Contents#
* table of contents
{:toc}

## Definition

An ordinary $Set$-enriched [[category]] $C$ is called **atomic** if it has a [[small set|small]] [[dense subcategory|dense]] [[full subcategory]] of [[atomic objects]], $Atom(C)$, so that every object $c$ of $C$ is a small colimit of the functor

$$Atom(C) \downarrow c \stackrel{proj}{\to} Atom(C) \stackrel{i}{\hookrightarrow} C.$$

More generally, for $V$ a [[cosmos]], a $V$-[[enriched category]] $C$ is _atomic_ if it admits a small $V$-dense full subcategory of atomic objects $Atom(C)$, such that every object $c$ is an enriched coend

$$\int^{a \in Atom(C)} C(i a, c) \cdot i a.$$

## Properties

### Relation to presheaf toposes

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

A category $E$ is equivalent to a [[presheaf topos]] (functors with values in the 1-category [[Set]] of [[0-groupoids]]) if and only if it is cocomplete and atomic.

=--

This is due to [[Marta Bunge]], who showed it is enough to have a regular cocomplete category with a [[generating set]] of atomic objects.

## Related concepts

* [[atom]]

* [[compact object]]

* [[Cauchy completion]]

[[!redirects atomic categories]]


atomic geometric morphism

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### Context
#### Topos Theory
+-- {: .hide}
[[!include topos theory - contents]]
=--
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#Contents#
* table of contents
{:toc}

## Definition

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

A [[geometric morphism]] $f : \mathcal{E} \stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*}{\to}} \mathcal{F}$ is called **atomic** if its [[inverse image]] $f^*$ is a [[logical functor]].

=--

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

A [[sheaf topos]] $\mathcal{E}$ is called **atomic** if its [[global section]] geometric morphism is atomic, or in other words, if the [[constant sheaf|constant sheaf functor]] $\Delta\colon Set\to\mathcal{E}$ is logical.

Generally, a topos over a [[base topos]] $\Gamma : \mathcal{E} \to \mathcal{S}$ is called an [[atomic topos]] if $\Gamma$ is atomic.

=--

+-- {: .num_note}
###### Note

As shown in prop. \ref{AtomicMeansLocallyConnected} below, every atomic morphism $f : \mathcal{E} \to \mathcal{S}$ is also a [[locally connected geometric morphism]]. The connected objects $A \in \mathcal{E}$, $f_! A \simeq *$ are called the **atoms** of $\mathcal{E}$.

=--

See ([Johnstone, p. 689](#Johnstone)).

## Properties

### General

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

Atomic morphisms are closed under composition.

=--

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

An atomic geometric morphism is also a [[locally connected geometric morphism]].

=--

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

By this proposition a logical morphism with a [[right adjoint]] has also a [[left adjoint]].

=--

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

If an atomic morphism is also a [[connected geometric morphism|connected]], then it is even [[hyperconnected geometric morphism|hyperconnected]].

=--

This appears as ([Johnstone, lemma 3.5.4](#Johnstone)).

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

A [[localic geometric morphism]] is atomic precisely if it is an [[etale geometric morphism]].

=--

This appears as ([Johnstone, lemma 3.5.4 (iii)](#Johnstone)).

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

Every [[étale geometric morphism]] is atomic.

=--

## Related concepts

 * [[atomic topos]]

 * [[atomic site]]

## References

* {#Johnstone}[[Peter Johnstone]], _[[Sketches of an Elephant]] vol. 2_ , Oxford UP 2002. (section C3.5, pp.684-695)

[[!redirects atomic geometric morphisms]]


atomic number

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#### Physics
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[[!include physicscontents]]
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#Contents#
* table of contents
{:toc}

## Idea

The _atomic number_ of a [[chemical element]] is the [[natural number|number]] of [[protons]] in any [[atomic nuclei]] of this element.

For instance [[carbon]] has atomic number 6.

The [[periodic table of chemical elements]] organizes them by their atomic number, together with other properties.

The corresponding [[natural number|number]] of [[neutrons]] on the nucleus tends to be close to the number of protons, hence to the atomic number of the element, but need not equal it and my differ considerably for heavy elements. These variants of neutron number for given atomic number specify the _[[isotopes]]_ of the [[chemical element]].

## References

See also

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

Speculative analogy to [[characteristic classes]] of [[complex surfaces]] (different from but inspired by [[Skyrmion]] [[model (in theoretical physics)|models]] for [[atomic nuclei]]):

* {#AtiyahManton16} [[Michael Atiyah]], [[Nicholas Manton]], _Complex Geometry of Nuclei and Atoms_ ([arXiv:1609.02816](https://arxiv.org/abs/1609.02816))

[[!redirects atomic numbers]]


atomic site

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### Context
#### Topos Theory
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[[!include topos theory - contents]]
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# Contents
* table of contents
{:toc}

## Idea
**Atomic sites** are [[sites]] $(\mathcal{C}, J_{at})$ equipped with the _atomic topology_ $J_{at}$. The corresponding [[sheaf toposes]] $Sh(\mathcal{C}, J_{at})$ are precisely the [[atomic topos|atomic Grothendieck toposes]].

## Definition

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

A [[site]] $(\mathcal{C}, J_{at})$ is called **atomic** if the [[covering sieve|covering]] [[sieves]] $S$ of $J_{at}$ are exactly the [[inhabited set|inhabited]] sieves $S\neq\emptyset$. A [[Grothendieck topology]] $J_{at}$ of this form is called _atomic_.
=--

## Example

Let $FinSet^{op}_{mono}$ be the opposite of the category $FinSet_{mono}$ with objects finite sets and monomorphisms. Then $(FinSet^{op}_{mono}, J_{at})$ is an atomic site and the corresponding sheaf topos $Sh(FinSet^{op}_{mono}, J_{at})$ is the [[Schanuel topos]].
That $J_{at}$ is indeed a [[Grothendieck topology]] is ensured by prop. \ref{atomic_ore}.

## Properties

+-- {: .num_prop #atomic_ore}
###### Proposition
Let $\mathcal{C}$ be a [[category]]. Then $\mathcal{C}$ can be made into an atomic site if and only if for any diagram
$$
\array{
  & & A \\
  & & \downarrow\\
  B & \to & C
}
$$
there is an object $D$ and arrows $D \to A, B$ such that the following diagram commutes:
$$
\array{
  D & \to & A \\
  \downarrow & & \downarrow\\
  B & \to & C
}
$$
=--

+-- {: .proof}
###### Proof
This is exactly what is needed for the pullback stability axiom to hold, and the other axioms are immediate.
=--

The condition occurring in the proposition is called the (right) [[Ore condition]]. It is a result by [[Peter Johnstone|P. T. Johnstone]] (1979) that $Set^{\mathcal{C}^{op}}$ is a [[De Morgan topos]] precisely if $\mathcal{C}$ satisfies the Ore condition. Whence we see that every [[atomic topos|atomic Grothendieck toposes]] is a ([[Boolean topos|Boolean]]) subtopos of a De Morgan [[presheaf topos]].

Recall that the [[dense topology]] $J_d$ on a category $\mathcal{C}$ consists of all sieves $S\in J_d(C)$ with the property that given $f:D\to C$ there exists $g:E\to D$ such that $f\cdot g\in S$. The atomic topology is a special case of this:

+-- {: .num_prop #atomic_dense}
###### Proposition
Let $\mathcal{C}$ be a [[category]] satisfying the [[Ore condition]]. Then the atomic topology $J_{at}$ coincides with the dense topology $J_d$.
=--

+-- {: .proof}
###### Proof
For $\mathcal{C}=\emptyset$ the claim is trivial. So let $C\in\mathcal{C}$ be an object and $S$ a sieve on $C$.

Assume $S\in J_d(C)$, then for $id\colon C\to C$ there exists $g:E\to C$ with $id\cdot g\in S$ whence $S\in J_{at}(C)$.

Conversely, assume $S\in J_{at}(C)$ and let $f:D\to C$ be a morphism. Then there exists $g\in S$ by assumption and the diagram $D\overset{f}{\rightarrow} C \overset{g}{\leftarrow} E$ can be completed to a commutative square $f\cdot i = g\cdot h$ but $g\cdot h\in S$ since $g\in S$ and $S$ is a sieve. Whence $f \cdot i\in S$ and, accordingly, $S\in J_d(C)$.
=--

In other words, the atomic topology is just the [[dense topology]] on  categories satisfying the Ore condition. Since the corresponding sheaf toposes of the dense topology are just the double negation subtoposes of the corresponding presheaf topos we finally get:

+-- {: .num_prop}
###### Proposition
Atomic Grothendieck toposes i.e. toposes (equivalent to) $Sh(\mathcal{C}, J_{at})$ for $(\mathcal{C}, J_{at})$ an atomic site are precisely (the toposes  equivalent to) the [[double negation|double negation subtoposes]] $Sh_{\neg\neg}(Set^{\mathcal{C}^{op}})$ for a De Morgan presheaf topos $Set^{\mathcal{C}^{op}}$. $\qed$
=--

The sheaves of atomic sheaf toposes $Sh(\mathcal{C}, J_{at})$ are easy to describe:

+-- {: .num_prop #atomic_sheaf}
###### Proposition
Let $(\mathcal{C}, J_{at})$ be an atomic site. A presheaf $P\in Set^{\mathcal{C}^{op}}$ is a sheaf for $J_{at}$ iff for any morphism $f:D\to C$ and any $y\in P(D)$ , if $P(g)(y)=P(h)(y)$ for all diagrams
$$
E\overset{g}{\underset{h}{\rightrightarrows}} D\overset{f}{\to} C
$$
with $f\cdot g=f\cdot h$ , then $y=P(f)(x)$ for a unique $x\in P(C)$.
=--

For the proof see Mac Lane-Moerdijk ([1994](#MM94), pp.126f).

## The more general definition

Thus far we have presented the classical approach as presented in Mac Lane-Moerdijk ([1994](#MM94)) going back to Barr-Diaconescu ([1980](#BD80)) but it was observed by [[Olivia Caramello|O. Caramello]] ([2012](#Ca12)) that the atomic topology can in fact be defined on arbitrary categories not only on those satisfying the [[Ore condition]].

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

Let $\mathcal{C}$ be a category. The _atomic topology_ $J_{at}$ on $\mathcal{C}$ is the smallest [[Grothendieck topology]] containing all the nonempty sieves. A site of the form $(\mathcal{C}, J_{at})$ is called _atomic_.
=--

Note that $J_{at}$ is well defined as the intersection of all Grothendieck topologies with the property that all nonempty sieves cover. The following proposition justifies the terminology:

+-- {: .num_prop}
###### Proposition
Let $(\mathcal{C}, J_{at})$ be an atomic site. Then $Sh(\mathcal{C}, J_{at})$ is an [[atomic topos|atomic Grothendieck topos]].
=--

+-- {: .proof}
###### Proof
The main idea is to consider the full subcategory $\mathcal{C}'$ on those objects $U$ with $\emptyset\notin J_{at}(U)$ together with the [[dense sub-site|induced topology]] $J'_{at}=J_{at}|_{\mathcal{C}'}$. Then one shows that $\mathcal{C}'$ satisfies the Ore condition and concludes by the [[comparison lemma]] that $Sh(\mathcal{C}', J'_{at})\simeq Sh(\mathcal{C}, J_{at})$. For the details see Caramello ([2012](#Ca12), prop.1.4).
=--

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

Consider the category $\mathcal{C}$ on the 'walking co-span' $A\overset{f}{\rightarrow} C\overset{g}{\leftarrow} B$. $\mathcal{C}$  does not satisfy the [[Ore condition]]. The atomic topology $J_{at}$ is given by

$$
J_{at}(A)=\{\{id_A\},\emptyset\} \qquad J_{at}(B)=\{\{id_B\},\emptyset\}
$$
$$
J_{at}(C)=\{\{ id_C, f,g\},\{f \}, \{ g \},\{f,g\},\emptyset\} \quad .
$$

Here $\emptyset\in J_{at}(A)$ , respectively $\emptyset\in J_{at}(B)$, due to the stability axiom applied to $f^\ast(\{g\})=\emptyset$ , respectively to $g^\ast(\{f\})=\emptyset$ . Whereas $\emptyset\in J_{at}({C})$ by the transitivity axiom applied to $\{f\}\in J_{at}(C)$ and the sieve $\emptyset$ since $f^{\ast}(\emptyset)=\emptyset\in J_{at}(A)$.

Accordingly the subcategory $\mathcal{C}'$ is empty and $Sh(\mathcal{C},J_{at})\simeq 1$ is degenerate. In particular,  $Sh(\mathcal{C},J_{at})$ is not equivalent to $Sh(\mathcal{C},J_d)\simeq Sh_{\neg\neg}(Set^{\mathcal{C}^{op}})\simeq Set\times Set$. So we see that the atomic topology on $\mathcal{C}$ is distinct from the [[dense topology]]. For completeness we describe the latter:

$$
J_{d}(A)=\{\{id_A\}\} \qquad J_{d}(B)=\{\{id_B\}\}
$$
$$
J_{d}(C)=\{\{id_C, f,g\},\{f,g\}\} \quad .
$$

Further details on $Set^{\mathcal{C}^{op}}$, the topos of hypergraphs, may be found at [[hypergraph]].
=--

In the example, we observed that dense and the atomic topology need not coincide for categories not satisfying the [[Ore condition]].
In fact more can be said here:

+-- {: .num_prop}
###### Proposition
Let $\mathcal{C}$ be a [[category]]. Then  $J_{d}\subseteq J_{at}$ in general, but $J_d=J_{at}$ precisely if $\mathcal{C}$ satisfies the [[Ore condition]].
=--

+-- {: .proof}
###### Proof
The proof of prop. \ref{atomic_dense} already showed that the sieves of the dense topology $J_d$ are never empty regardless of the Ore condition. From prop. \ref{atomic_ore} follows that the atomic topology $J_{at}$ will additionally contain empty sieves precisely if $\mathcal{C}$ does not satisfy the Ore condition.
=--

In particular, $Sh(\mathcal{C},J_{at})\subseteq Sh(\mathcal{C},J_{d})\simeq Sh_{\neg\neg}(Set^{\mathcal{C}^{op}})$ .

## Related entries

* [[dense topology]]

* [[atomic topos]]

* [[atomic geometric morphism]]

* [[Ore condition]]

* [[De Morgan topos]]

## Reference

* {#BD80}[[Michael Barr]], [[Radu Diaconescu]], _Atomic Toposes_ ,  JPAA **17** (1980) pp.1-24. ([pdf](http://www.math.mcgill.ca/barr/papers/atom.top.pdf))

* {#Ca12}[[Olivia Caramello]], _Atomic toposes and countable categoricity_ , Appl. Cat. Struc. **20** no. 4 (2012) pp.379-391. ([arXiv:0811.3547](http://arxiv.org/abs/0811.3547))

* {#MM94} [[Saunders Mac Lane]], [[Ieke Moerdijk]], _[[Sheaves in Geometry and Logic]]_ , Springer Heidelberg 1994. (pp.115, 126)

[[!redirects atomic topology]]


atomic spectrum

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> This entry is about the conept in [[physics]]. For other uses of the term see at _[[spectrum - disambiguation]]_.

## References

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

[[!redirects atomic spectra]]

[[!redirects spectroscopy]]


atomic topos

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#Contents#
* table of contents
{:toc}
## Idea
An atomic topos is a topos $\mathcal{E}$ where the global sections functor $\Gamma: \mathcal{E} \to Set$ is [[atomic geometric morphism|atomic]]. In the case where $\mathcal{E}$ is a Grothendieck topos, there are some (arguably more explicit) alternative characterizations of atomic toposes, as in Theorem \ref{EquivChar}, which make it clearer why they are called "atomic".

## Definition

Recall that a [[geometric morphism]] $f$ is called [[atomic geometric morphism|atomic]] if its [[inverse image functor]] $f^*$ is [[logical functor|logical]].

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

A [[topos]] over a [[base topos]] $\Gamma : \mathcal{E} \to \mathcal{S}$ is called an **atomic topos** if $\Gamma$ is [[atomic geometric morphism|atomic]]. Unless otherwise specified, the base topos will be taken to be $Set$.
=--

+-- {: .num_defn}
###### Definition
A non-zero object $A$ of a topos $\mathcal{E}$ is an [[atom]] if its only subobjects are $A$ and $0$.
=--

+-- {: .num_theorem #EquivChar}
###### Theorem
Let $\mathcal{E}$ be a [[Grothendieck topos]]. Then the following are equivalent:

1. $\mathcal{E}$ is an atomic topos.

1. $\mathcal{E}$ is the [[category of sheaves]] on an [[atomic site]].

1. The [[subobject lattice]] of every object of $\mathcal{E}$ is a [[complete lattice|complete]] [[atomic Boolean algebra]].

1. $\mathcal{E}$ has a [[small set|small]] [[generating set]] of atoms.

1. Every object of $\mathcal{E}$ can be written as a disjoint union of atoms.
=--

+-- {: .proof}
###### Proof
See [Johnstone, C3.5.8](#Johnstone) and [Barr-Diaconescu, Theorem A](#Barr-Diaconescu80).
=--

## Properties
+-- {: .num_prop}
###### Proposition
Let $\mathcal{E}$ be an atomic topos. Then $\mathcal{E}$ is Boolean.
=--

This appears as one direction of ([Johnstone, cor. C3.5.2](#Johnstone)).

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

If $\Gamma^*$ is logical then it preserves the isomorphism $* \coprod * \simeq \Omega$ characterizing a [[Boolean topos]].
=--

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

Let $\mathcal{E}$ be a [[Boolean topos|Boolean]] [[Grothendieck topos]] with [[point of a topos|enough points]]. Then $\mathcal{E}$ is an atomic topos.
=--

+-- {: .proof}
###### Proof
See ([Johnstone, cor. C3.5.2](#Johnstone))
=--

+-- {: .num_prop}
###### Proposition
Atomic Grothendieck toposes $Sh(\mathcal{C}, J_{at})$ for $(\mathcal{C}, J_{at})$ an [[atomic site]] are precisely the [[double negation|double negation subtoposes]] $Sh_{\neg\neg}(Set^{\mathcal{C}^{op}})$ for a [[De Morgan topos|De Morgan]] [[presheaf topos]] $Set^{\mathcal{C}^{op}}$.
=--
+-- {: .proof}
###### Proof
For the argument see at [[atomic site]].
=--

### Decomposition of atomic toposes

Atomic toposes decompose as [[disjoint unions]] of [[connected topos|connected]] atomic toposes. Connected atomic toposes with a [[point of a topos|point]] are the [[classifying toposes]] of [[localic groups]].

An example of a connected atomic topos without a [[point of a topos|point]] is given in ([Johnstone, example D3.4.14](#Johnstone})).

## Examples

*  A [[category of presheaves]] $Set^{\mathcal{C}^{op}}$ is atomic precisely iff $\mathcal{C}$ is a groupoid (cf. [Barr-Diaconescu (1980)](#Barr-Diaconescu80)).

* Another example of an atomic Grothendieck topos is the [[Schanuel topos]]. More generally, any [[category of G-sets]] is an atomic Grothendieck topos.

## Related concepts

* [[atomic geometric morphism]]

* [[atomic site]]

* [[connected topos]]

## References

* {#Barr-Diaconescu80}[[Michael Barr]], [[Radu Diaconescu]], *Atomic Toposes*, J. Pure Appl. Algebra **17** (1980) 1-24 [doi:10.1016/0022-4049(80)90020-1, [pdf](http://www.math.mcgill.ca/barr/papers/atom.top.pdf), [[BarrDiaconescu-Atomictoposes.pdf:file]]]

* [[Olivia Caramello]], _Atomic toposes and countable categoricity_ , Appl. Cat. Struc. **20** no. 4 (2012) pp.379-391. ([arXiv:0811.3547](http://arxiv.org/abs/0811.3547))

* {#Johnstone}[[Peter Johnstone]], _[[Sketches of an Elephant]] vol. 2_ , Oxford UP 2002. (section C3.5, pp.684-695)

[[!redirects Atomic topos]]
[[!redirects atomic toposes]]
[[!redirects atomic topoi]]


atomtronics

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#### Quantum systems
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[[!include quantum systems -- contents]]
=--
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#Contents#
* table of contents
{:toc}

## Idea

(...)

## Related concepts

* [[atom (physics)]]

* [[quantum circuit]]

* [[quantum information theory]], [[quantum computation]]

## References

* [[Luigi Amico]], [[Malcolm G. Boshier]], *Atomtronics* [[arXiv:1511.07215](https://arxiv.org/abs/1511.07215)]

* [[Luigi Amico]], [[Malcolm G. Boshier]], [[Juan Polo Gomez]] et al.: *Roadmap on Atomtronics: State of the art and perspective*, AVS Quantum Sci. **3** 039201 (2021) [[arXiv:2008.04439](https://arxiv.org/abs/2008.04439), [doi:10.1116/5.0026178](https://doi.org/10.1116/5.0026178)]

* [[Luigi Amico]], Dana Anderson, [[Malcolm Boshier]], Jean-Philippe Brantut, Leong-Chuan Kwek, Anna Minguzzi, Wolf von Klitzing: *Colloquium: Atomtronic circuits: From many-body physics to quantum technologies*, Rev. Mod. Phys. **94** (2022) 041001 [[doi:10.1103/RevModPhys.94.041001](https://doi.org/10.1103/RevModPhys.94.041001)]

See also:

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


atoroidal 3-manifold

#Contents#
* table of contents
{:toc}

## Idea

A [[3-manifold]] that does not contain an essential [[torus]].

## Related concepts

* [[virtually fibered conjecture]]

## References

* Wikipedia, _[Atoroidal 3-manifold](http://en.wikipedia.org/wiki/Atoroidal)_

[[!redirects atoroidal 3-manifold]]


Atsushi Hosaka

* [webpage](https://www.phys.sci.osaka-u.ac.jp/en/research_groups/group/12-0_quark/hosaka/index.html)

## Selected writings

Equivalence between [[hidden local symmetry]]- and [[massive Yang-Mills theory]]-description of [[Skyrmion]] [[quantum hadrodynamics]]:

* [[Atsushi Hosaka]], H. Toki, [[Wolfram Weise]], _Skyrme Solitons With Vector Mesons: Equivalence of the Massive Yang-Mills and Hidden Local Symmetry Scheme, 1988, Z. Phys. A332 (1989) 97-102 ([spire:24079](http://inspirehep.net/record/24079))

Combination of the [[omega-meson]]-stabilized [[Skyrme model]] with the [[bag model for quark confinement|bag model]] for [[nucleons]]:

* [[Atsushi Hosaka]], _Omega stabilized chiral bag model with a surface $\omega q q$ coupling_, Nuclear Physics A Volume 546, Issue 3, 31 (1992) Pages 493-508 (doi:10.1016/0375-9474(92)90544-T)

On [[kaon]]-[[K*-meson]]-[[photon]] [[interaction]] via the [[WZW term]]:

* S. Ozaki, H. Nagahiro, [[Atsushi Hosaka]], Equations (3) and (9) in: _Magnetic interaction induced by the anomaly in kaon-photoproductions_, Physics Letters B Volume 665, Issue 4, 24 July 2008, Pages 178-181 ([arXiv:0710.5581](https://arxiv.org/abs/0710.5581), [doi:10.1016/j.physletb.2008.06.020](https://doi.org/10.1016/j.physletb.2008.06.020))

On [[heavy baryons]] in [[holographic QCD]] coming from higher [[KK-compactification|KK-modes]]:

* Daisuke Fujii, [[Atsushi Hosaka]], _Heavy baryons in holographic QCD with higher dimensional degrees of freedom_ ([arXiv:2003.13415](https://arxiv.org/abs/2003.13415))

category: people


Atsushi Matsuo

* [personal page](https://www.ms.u-tokyo.ac.jp/~matsuo/)

* [institute page](https://www.s.u-tokyo.ac.jp/en/people/matsuo_atsushi/)

## Selected writings

On the [hypergeometric integral construction](Knizhnik-Zamolodchikov+equation#BraidRepresentationsViaTwisteddRCohomologyOfConfigurationSpaces) of solutions to the [[Knizhnik-Zamolodchikov equation]]:

* [[Etsuro Date]], [[Michio Jimbo]], [[Atsushi Matsuo]], [[Tetsuji Miwa]], *Hypergeometric-type integrals and the $\mathfrak{sl}(2,\mathbb{C})$-Knizhnik-Zamolodchikov equation*,  International Journal of Modern Physics B **04** 05 (1990) 1049-1057   $[$[doi:10.1142/S0217979290000528](https://doi.org/10.1142/S0217979290000528)$]$

* [[Atsushi Matsuo]], *An application of Aomoto-Gelfand hypergeometric functions to the $SU(n)$ Knizhnik-Zamolodchikov equation*, Communications in Mathematical Physics **134** (1990) 65–77 $[$[doi:10.1007/BF02102089](https://doi.org/10.1007/BF02102089)$]$

category: people


Atsushi Tomoda

* [webpage](http://ton.prosou.nu/official/)

## Related $n$Lab entries

* [[splitting principle]]

* [[twisted vector bundle]], [[gerbe module]]

category: people


Attempt to Introduce the Concept of Negative Quantities into Philosophy

[[!redirects empty 154]]
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# Contents
* table of contents
{:toc}

## Idea

**"Versuch, den Begriff der negativen Grössen in die Weltweisheit einzuführen""** (1763) is a short article of [[Immanuel Kant]] concerning the philosophical foundations of the [[integer|integers]], in particular the _negative_ integers.

## Overview

> Diese kleine Schrift ist eine der tiefsinnigsten und lichtvollsten, die nicht bloß Kant geschrieben, sondern welche die philosophische Literatur überhaupt aufzuweisen hat. Man tut Kant nicht unrecht, wenn man behauptet, daß sie ihm wie ein Meteor entschlüpft und selbst nicht wieder zu Gesicht gekommen ist.[^trans]  Karl Rosenkranz ([1840, p.118](#Rosenkranz40))

[^trans]: _'This small text is one of the deepest and most lightfull not only in the writings of Kant but in the whole philosophical literature. One does no injustice to Kant if one says that it snapped out of him like a meteor and was never seen again by him.'_

Starting with a discussion of the meaning of the minus sign in negative numbers and as a designator of the subtraction operation, Kant sets out to distinguish between (predicate) negation in logic and substraction in arithmetic. He is thereby led to argue for a sharp distinction between _logical_ and _real opposition_: whereas the former leads to logical contradictions, the latter does not.

Since he conceives logical contradictions as the simultaneous predication of a predicate $p$ and its negative $\neg p$ to the same subject $S$, a contradiction destroys the possibility of the thing represented by the subject.

In contrast, in a real opposition, e.g. two forces acting in opposite directions on the same point mass, merely the effects of the opposites are cancelled without affecting their reality, i.e. the cause is still effective but its effect are covered by the effects of its opposite.

As result, a gulf opens between the realm of logic with its analytic modes of reasoning and the realm of reality throwing into crisis the rationalist thought of the Leibniz-Wolffian school that maintained that in principle all truth was analytical and that denied any difference in principle between empirical and logical truths.

Hence the distinction between logical and real oppositions gave Kant strong incentive to reconceptualize the relation between logical and empirical propositions finally leading to the distinction between the formal and transcendental logic in his [[Critique of Pure Reason]] of 1781.

On more general grounds, Kant also recalibrates in the article the relation between philosophy and mathematics, urging the former to modesty e.g. when it comes to deny the existence of [[infinitesimal|infinitesimals]] on metaphysical grounds.

## Repercussions

The concept of real opposition preserving (the reality of) the opposites positively in its result proved to be decisive not only for Kantian but also for the post Kantian philosophy since, as already pointed out by [[Karl Rosenkranz|K. Rosenkranz]] surrounding the discussions around the [[Science of Logic|Hegelian logic]] in the 1840-50s, it can be viewed as a germ of Hegel's concept of contradiction and [[Aufhebung|sublation]] in the _'Wissenschaft der Logik'_. In fact, the section on 'contradiction' in the second volume even has a discussion of arithmetic and negative numbers (cf. [Wolff 2010](#Wolff10)).

## Related entries

* [[Immanuel Kant]]

* [[Karl Rosenkranz]]

* [[Hermann Grassmann]]

* [[transcendental ideal]]

* [[infinite judgement]]

* [[Aufhebung]]

* [[negation]]

* [[Ausdehnungslehre]]

* [[construction in philosophy]]

## References

* {#Giovanelli15} M. Giovanelli, *Trendelenburg and the Concept of Negation in Post-Kantian Philosophy*, to appear in Munk (ed.), *Proceedings of the Amsterdam 2010 Colloquium: Natur des Denkens und das Denken der Natur: Spinoza, Trendelenburg und H. Cohen*. ([draft](https://www.dropbox.com/s/mxm4db8id7je924/Giovanelli%2C%20Marco%20-%20Trendenleburg%20and%20the%20Concept%20of%20Negation%20in%20Post-Kantian%20Philosophy%20Proceedings%20of%20the%20Connference.pdf?dl=0))

* {#Kaestner58} Abraham Gotthelf Kästner, _Anfangsgründe der Arithmetik, Algebra, Geometrie, ebenen und sphärischen Trigonometrie, und Perspectiv_, Göttingen 1758. ([gdz](http://gdz.sub.uni-goettingen.de/dms/load/toc/?PID=PPN812429885))

* Srećko Kovač, _In what sense is Kantian principle of contradiction non-classical?_, Logic and Logical Philosophy **17** (2008) pp.251-274. ([link](http://dx.doi.org/10.12775/LLP.2008.013))

* {#Rosenkranz40} [[Karl Rosenkranz]], _Geschichte der Kant'schen Schule_, Akademie-Verlag Berlin 1987[1840].

* {#Wolff10} Michael Wolff, _Der Begriff des Widerspruchs - Eine Studie zur Dialektik Kants und Hegels_, Frankfurt UP ²2010.

[[!redirects Attempt to introduce the concept of negative quantities into philosophy]]
[[!redirects concept of negative quantities]]
[[!redirects Concept of Negative Quantities]]


Attila Szabo

* Wikipedia entry

## Selected writings

On [[quantum chemistry]]:

* [[Attila Szabo]], [[Neil S. Ostlund]], *Modern Quantum Chemistry -- Introduction to Advanced Electronic Structure Theory*, Macmillan (1982), McGraw-Hill (1989), Dover (1996) $[$[pdf](https://chemistlibrary.files.wordpress.com/2015/02/modern-quantum-chemistry.pdf)$]$

## Related $n$Lab entries

* [[Slater determinant]]

category: people


Atul Sharma

* [inspire page](https://inspirehep.net/authors/1737875)

## Selected writings

On [[celestial holography]] via [[topological strings]] on [[twistor space]]:

* [[Kevin Costello]], [[Natalie M. Paquette]], [[Atul Sharma]], *Top-down holography in an asymptotically flat spacetime* [[arXiv:2208.14233](https://arxiv.org/abs/2208.14233)]

* [[Kevin Costello]], [[Natalie M. Paquette]], [[Atul Sharma]], *Burns space and holography* [[arXiv:2306.00940](https://arxiv.org/abs/2306.00940)]

category: people