spectral super-scheme

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## Idea

The concept of _spectral super-scheme_ is supposed to be the refinement of the concept of [[super-scheme]] as one passes to [[spectral geometry]] in the sense of [[derived algebraic geometry]] over [[E-infinity rings]] ([[E-infinity geometry]]).

## Definition
 {#Definition}

The following is an argument for a good definition of spectral supergeometry. This was originally motivated from the observation in [Kapranov 13](#Kapranov13) and uses results due to [Rezk 09](#Rezk09) and [Sagave-Schlichtkrull 2011](#SagaveSchlichtkrull11).

Observe that

1. {#EInfinityGeometryAlreadyZGraded} [[E-∞ geometry|$E_\infty$-geometry]] is _already in itself_ a [[higher geometry|higher geometric]] version of $\mathbb{Z}$-graded supergeometry (in the sense discussed at _[[geometry of physics -- superalgebra]]_).

   $\,$

   At the level of [[homotopy groups]] this is the following basic fact:

   For $E$ a [[homotopy commutative ring spectrum]], its [[stable homotopy groups]] $\pi_\bullet(E)$ inherit the structure of a $\mathbb{Z}$-graded [[super-commutative ring]]  (according to [this](geometry+of+physics+--+superalgebra#SupercommutativeSuperalgebraZGraded)). See [this proposition](Introduction+to+Stable+homotopy+theory+--+1-2#HomotopyGroupsOfHomotopyCommutativeRingSpectrum) in the section *[1-2 Homotopy commutative ring spectra](Introduction+to+Stable+homotopy+theory+--+1-2#HomotopyRingSpectra)* of *[[Introduction to Stable homotopy theory]]*.

   $\,$

   But more is true: the $E_\infty$-analog of the [[integers]], $\mathbb{Z}$, is the [[sphere spectrum]], $\mathbb{S} \,\simeq\, \Sigma^\infty S^0$, and every [[E-infinity ring|$E_\infty$-ring]] $(E, \cdot)$ is canonically $\mathbb{S}$-graded, in that  ([Sagave-Schlichtkrull 2011, theorem 1.7-1.8](#SagaveSchlichtkrull11)):

   on [underlying](Introduction+to+Stable+homotopy+theory+--+1-1#SigmaInfinityOmegaInfinity) [[E-infinity spaces|$E_\infty$-spaces]] $E_0 \,\coloneqq\, \Omega^\infty(E)$, at least, realized as $\Omega^{\mathcal{J}}(E)$ in [SaSc11 (4.4)](#SagaveSchlichtkrull11), they are canonically equipped with an [[commutative monoid in a symmetric monoidal (infinity,1)-category|$E_\infty$-monoid]] [[homomorphism]]

   $$
     (E_0, \cdot)
       \xrightarrow{\;}
    (\mathbb{S}_0, +)
   $$

   to the *additive* $E_\infty$-space underlying the [[sphere spectrum]] ([traditionally](Bousfield-Friedlander+model+structure#Spectrification) denoted $Q S^0 $, which is notation for a construction that yields $\Omega^\infty \Sigma^\infty S^0$).

   The $E_\infty$-monoid homomorphisms of this form are the evident homotopy-theoretic generalization of morphisms of [[commutative monoids]] to the additive group $(\mathbb{Z}, +)$ of the [[integers]], and these are evidently equivalent to $\mathbb{Z}$-[[graded monoid|gradings]] on the [[domain]] monoid.

   $\,$

   So [[E-∞ geometry|$E_\infty$-geometry]] in itself is already a [[categorification|categorified]]/homotopified version of [[supergeometry]], but of $\mathbb{Z}$-graded supergeometry, not of the proper $\mathbb{Z}/2$-graded supergeometry.

   $\,$

   (That grading over the [[sphere spectrum]] is closely related to [[superalgebra]] had been highlighted in [Kapranov 2013](#Kapranov13), but the issue of the difference between homotopified $\mathbb{Z}$-grading compared to homotopified $\mathbb{Z}/2$-grading had been left open.)

   $\,$

1. {#SuperGeometryAsPeriodifiedZGradedGeometry} But ordinary $\mathbb{Z}/2$-graded [[supercommutative superalgebra]] is equivalently $\mathbb{Z}$-graded [[supercommutative superalgebra]] over the free even periodic $\mathbb{Z}$-graded supercommutative superalgebra ([this prop.](geometry+of+physics+--+superalgebra#ModulesOverRbeta)).

   $\,$

1. {#PassageToAlgebraOverPeriodicRingSpectra} In view of the [first point](#EInfinityGeometryAlreadyZGraded), the [second point](#SuperGeometryAsPeriodifiedZGradedGeometry) has an evident analog in [[E-∞ geometry]]:

   The higher/derived analog of an even periodic $\mathbb{Z}$-graded commutative algebra is an [[E-infinity algebra]] over an [[even cohomology theory|even]] [[periodic ring spectrum]].

   $\,$

   That [[E-infinity algebras]] over [[even cohomology theory|even]] [[periodic ring spectra]] are usefully regarded from the point of view of [[supercommutative superalgebra]] was highlighted in [Rezk 09, section 2](#Rezk09).

Hence it makes sense to say:

**Definition.** _Spectral/$E_\infty$ super-geometry_ is simply the [[E-∞ geometry|$E_\infty$-geometry]] over [[even cohomology theory|even]] [[periodic ring spectra]].

## Examples

### Spectral superpoint
  {#SpectralSuperpoint}

The ordinary [[superpoint]] over some [[field]] $k$ is the [[spectrum of a commutative ring]] of the graded [[symmetric algebra]] on a single odd generator ("graded [[ring of dual numbers]]")

$$
  \mathbb{A}_k^{0 \vert 1}
  \;\simeq\;
  Spec( \,Sym_k (k[1])\, )
$$

Accordingly, for $R$ an [[even cohomology theory|even]] [[periodic ring spectrum]], then the _spectral superpoint_ $R^{0 \vert 1}$ should be the [[spectral scheme]] given by the [[spectral symmetric algebra]] on the [[suspension spectrum]] of $R$:

$$
  \begin{aligned}
    R^{0 \vert 1}
     &\coloneqq
    Spec
    \left(
      Sym_R (\Sigma R)
    \right)
    \\
    & \simeq
    Spec\left(
    R
    \wedge
    \left(
       \underset{n \in \mathbb{N}}{\coprod}
        B \Sigma(n)^{\tau_n}
    \right)_+
    \right)
    \\
    & \simeq
    Spec\left(
    R \wedge
    Sym_{\mathbb{S}}(\Sigma \mathbb{S})
    \right)
  \end{aligned}
  \,.
$$

where on the right we have the [[Thom space]] of the [[vector bundle]] $\tau_n$ [[associated bundle|associated]] to the $\Sigma(n)$-[[universal principal bundle]] via the canonical [[action]] of $\Sigma(n)$ on $\mathbb{R}^n$ (see also at [symmetric group -- Classifying space and Thom space](permutation#ClassifyingSpaceAndThomSpace)).

## References

The proposal for spectral super-geometry [above](#Definition) invokes observations from

* {#Rezk09} [[Charles Rezk]], section 2 of _The congruence criterion for power operations in Morava E-theory_, Homology, Homotopy and Applications, Vol. 11 (2009), No. 2, pp.327-379 ([arXiv:0902.2499](https://arxiv.org/abs/0902.2499))

* {#SagaveSchlichtkrull11} [[Steffen Sagave]], [[Christian Schlichtkrull]], _Diagram spaces and symmetric spectra_, Advances in Mathematics, Volume 231, Issues 3–4, October–November 2012, Pages 2116–2193 ([arXiv:1103.2764](https://arxiv.org/abs/1103.2764))

The proposal [above](#Definition) was originally motivated from the discussion of the [[sphere spectrum]] in relation to [[super algebra]] highlighted in

* {#Kapranov13} [[Mikhail Kapranov]], _Categorification of supersymmetry and stable homotopy groups of spheres_, talk at _[Algebra, Combinatorics and Representation Theory: in memory of Andrei Zelevinsky (1953-2013)](https://web.archive.org/web/20130617191515/http://mathserver.neu.edu/~bwebster/ACRT/)_ (April 2013) [[[Webster-ACRTAbstracts2013.pdf:file]],  video:[YT](https://www.youtube.com/watch?v=WBsZrcYOrxI)]

  > **Abstract:**. The "minimal sign skeleton" necessary to formulate the [[signs in supergeometry|Koszul sign rule]] is a certain [[Picard category]], a [[symmetric monoidal category]] with all [[invertible object|objects]] and [[invertible morphism|morphisms invertible]]. It can be seen as the free Picard category generated by one object and corresponds, by [[Grothendieck]]'s dictionary, to the [[n-truncation|truncation]] of the [[sphere spectrum|spherical spectrum]] $S$ in degrees 0 and 1, so that $\{\pm 1\}$ appears as the [[first stable homotopy group of spheres]] $\pi_{n+1}(S^n)$. This suggest a "[[higher structures|higher]]" or [[categorification|categorified]] versions of [[super-algebra|super-mathematics]] which utilize deeper structure of [[sphere spectrum|$S$]]. The first concept on this path is that of a supersymmetric monoidal category which is [[categorification|categorified]] version of the concept of a [[supercommutative algebra]].

* {#Kapranov15} [[Mikhail Kapranov]], _Supergeometry in mathematics and physics_, in [[Gabriel Catren]], [[Mathieu Anel]], (eds.) _[[New Spaces for Mathematics and Physics]]_ ([arXiv:1512.07042](http://arxiv.org/abs/1512.07042))

* {#Kapranov15b} [[Mikhail Kapranov]], _Super-geometry_, talk at _[New Spaces for Mathematics & Physics](http://ercpqg-espace.sciencesconf.org/program)_, IHP Paris (Oct-Sept 2015) [[video recording](https://www.youtube.com/watch?v=bjsNwKYT8JE)]

A closely related suggestion later appears in:

* MathOverflow comment  [MO:a/40327](https://mathoverflow.net/a/403274/381), Sep 2021

[[!redirects spectral super-schemes]]

[[!redirects spectral supergeometry]]
[[!redirects E-∞ supergeometry]]
[[!redirects E-infinity supergeometry]]

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

[[!redirects spectral superpoint]]
[[!redirects spectral superpoints]]
[[!redirects spectral super point]]
[[!redirects spectral super points]]
[[!redirects spectral super-point]]
[[!redirects spectral super-points]]

[[!redirects spectral superscheme]]
[[!redirects spectral superschemes]]


spectral symmetric algebra

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## Idea

Given a [[ring spectrum]] $R$ and [[module spectrum]] $E$ over $R$, there is the [[free construction|free]] $R$-[[algebra spectrum]] induced by $E$, this is the _symmetric algebra_ spectrum $Sym_R E$, in direct analogy to the construction of [[symmetric algebras]] on [[vector spaces]] (e.g. [Khan 16](#Khan16)).

The underlying spectrum of the symmetric algebra is simply the [[direct sum]] ([[wedge sum]] of spectra) of all the $n$-fold [[smash products]] of $E$ over $R$ [[homotopy quotient|homotopy quotiented]] by the canonical [[∞-action]] of the [[symmetric group]] $\Sigma(n)$ (by [[permutation]] of factors):

$$
  Sym_R E
  \;\coloneqq\;
  \underset{n \in \mathbb{N}}{\vee}
  (E^{\wedge^n})/\Sigma(n)
  \,.
$$

In contrast to ordinary [[symmetric algebras]] on ordinary [[modules]] over ordinary [[rings]], this means that the symmetric algebra on $R$ itself, regarded as a [[module spectrum]] itself has interesting structure:

$$
  \begin{aligned}
    Sym_R R
    & \simeq
    \underset{n \in \mathbb{N}}{\vee}
     R / \Sigma(n)
    \\
    & \simeq
    \underset{n \in \mathbb{N}}{\vee} R \wedge (B \Sigma(n))_+
    \\
    & \simeq
    R \wedge
    \left(
      \underset{n \in \mathbb{N}}{\coprod}
      B \Sigma(n)
    \right)_+
  \end{aligned}
  \,,
$$

where $B \Sigma_n$ denotes the [[homotopy type]] of the [[classifying space]] of the [[symmetric group]] on $n$ elements (given for instance by the [[topological space]] $Emb(\{1, \cdots, n\}, \mathbb{R}^{\infty})/\Sigma(n)$).

In particular for $R = \mathbb{S}$ the [[sphere spectrum]], then  the _absolute [[spectral affine line]]_ $Sym_{\mathbb{S}}\mathbb{S}$ is nontrivial. This, and its structure of a [[Hopf ring spectrum]], is discussed in [Strickland-Turner 97](#StricklandTurner97).

More generally,

$$
  R\{x_1, \cdots, x_n\} \coloneqq Sym_R (R^{\vee^n})
$$

is like a [[polynomial ring|polynomial]] [[algebra spectrum]] over $R$.
Beware that there is a similar construction that is in general different, namely

$$
  R[x_1, \cdots, x_n] \coloneqq R \wedge \Sigma^\infty(\mathbb{N}_+)
$$

where on the right we have the "monoid $\infty$-algebra" of the [[natural numbers]], directly analogous to the [[∞-group ∞-ring]] construction. There is a canonical comparison homomorphisms

$$
  R\{x_1, \cdots, x_n\} \longrightarrow R[x_1, \cdots, x_n]
  \,.
$$

This is an [[equivalence]] if $R$ is of [[characteristic zero]] ([Khan 16, prop. 2.7.4](#Khan16)).

{#OnSuspensionOfGroundRingInIntroduction} Similarly, if $E = \Sigma^n R \simeq R \wedge S^n$ is the $n$-fold [[suspension]] of $R$, regarded as an $R$-[[module spectrum]], then
$$
  \begin{aligned}
    Sym_R(\Sigma^n R)
    & \simeq
    R \wedge \left(
      \underset{k \in \mathbb{N}}{\coprod} S^{n k}/\Sigma(k)
    \right)_+
    \\
    & \simeq
    R \wedge \left(
      \underset{k \in \mathbb{N}}{\coprod}
      (B \Sigma(k))^{n \tau_k}
    \right)
  \end{aligned}
  \,,
$$
where on the right we have the [[Thom space]] of the [[vector bundle]] $\tau_n$ [[associated bundle|associated]] to the $\Sigma(n)$-[[universal principal bundle]] via the canonical [[action]] of $\Sigma(n)$ on $\mathbb{R}^n$ (see also at [symmetric group -- Classifying space and Thom space](permutation#ClassifyingSpaceAndThomSpace)).

The operation $Sym_R$ is of course functorial, and hence any choice of $R$-linear map $f_x \colon R \to E$ induces morphisms

$$
  R \wedge B \Sigma(n)_+ \simeq Sym^n_R(R) \longrightarrow Sym_R^n(E)
  \,.
$$

These correspond to [[power operation]] in [[generalized (Eilenberg-Steenrod) cohomology]] ([Rezk 10, slide 4](#Rezk10)).

## Examples

### Spectral affine lines

For $R$ a ([[connective spectrum|connective]]) [[E-∞ ring]], the

$$
  \mathbb{A}^1_R \coloneqq Spec( Sym_R(R) )
$$

is the [[spectral affine line]] over $R$.

### Spectral superpoints

By the discussion at _[[spectral super scheme]]_, it makes good sense to regard the concept of a [[spectral scheme]] over an [[even cohomology theory|even]] [[periodic ring spectrum]] $R$ as the lift to [[spectral geometry]] of the concept of ordinary [[super schemes]]. Under this identification the ordinary [[superpoint]], which is the [[spectrum of a commutative ring]] of the graded symmetric algebra on a single odd generator ("[[ring of dual numbers]]")

$$
  \mathbb{A}_k^{0 \vert 1}
  \;\simeq\;
  Spec( Sym_k (k[1]) )
$$

is given by the corresponding [[spectral scheme]] given by the spectral symmetric algebra on the [[suspension spectrum]] of $R$:

$$
  \begin{aligned}
    R^{0 \vert 1}
     &\coloneqq
    Spec
    \left(
      Sym_R (\Sigma R)
    \right)
    \\
    & \simeq
    Spec\left(
      R \wedge Sym_{\mathbb{S}}(\Sigma \mathbb{S})
    \right)
  \end{aligned}
$$

### Free $\mathbb{E}_\infty$-algebras

The free $\mathbb{E}_\infty$-algebra over $R$ on $n$ generators is the spectrum $Sym_R(R^{\vee n})$. This is typically denoted $R\{x_1, \ldots, x_n\}$. If $R$ is connective, $\pi_0(R\{x_1, \ldots, x_n\})$ can be identified with the polynomial algebra $(\pi_0 R)[x_1, \ldots, x_n]$. The spectrum $R\{x_1, \ldots, x_n\}$ satisfies the following universal property: for any other $\mathbb{E}_\infty$-$R$-algebra $T$,
$$
  Map_{Alg_R^{\mathbb{E}_\infty}}(R\{x_1, \ldots, x_n\}, T) \simeq (\Omega^\infty T)^{\times n}.
$$
See ([Lurie 2018, Notation B.1.1.2](#Lurie18))

## References

The symmetric algebra spectrum of the [[sphere spectrum]], and its structure as a [[Hopf ring spectrum]] is discussed in

* {#StricklandTurner97} [[Neil Strickland]], [[Paul Turner]], _Rational Morava $E$-theory and $D S^0$_, Topology Volume 36, Issue 1, January 1997, Pages 137-151 ([pdf](http://hopf.math.purdue.edu/Strickland-PTurner/rme.pdf))

Symmetric algebras in the context of [[power operation]] on [[generalized (Eilenberg-Steenrod) cohomology]] are discussed in

* {#Rezk 09} [[Charles Rezk]], _Power operations in Morava E-theory --  a survey_ (2009) ([pdf](http://www.math.uiuc.edu/~rezk/midwest-2009-power-ops.pdf))

* {#Rezk10} [[Charles Rezk]] _Power operations in Morava $E$-Theory_ (2010) ([pdf](http://www.math.uiuc.edu/~rezk/baltimore-2010-power-ops-handout.pdf))

* {#Rezk14} [[Charles Rezk]], _Isogenies, power operations, and homotopy theory_, article ([pdf](http://www.math.uiuc.edu/~rezk/rezk-icm-talk-posted.pdf)) and talk at ICM 2014 ([pdf](http://www.math.uiuc.edu/~rezk/rezk-icm-2014-slides.pdf))

See also

* [[Jacob Lurie]], examples 3.5.4 in _Elliptic Cohomology I_ ([pdf](http://www.math.harvard.edu/~lurie/papers/Elliptic-I.pdf))

* {#Lurie18} [[Jacob Lurie]], notation B.1.1.2 in _Spectral Algebraic Geometry_ ([pdf](http://www.math.harvard.edu/~lurie/papers/SAG-rootfile.pdf))

* {#Khan16} [[Adeel Khan]], sections 2.6 and 2.7 of _Brave new motivic homotopy theory I_ ([arXiv:1610.06871](https://arxiv.org/abs/1610.06871))

[[!redirects spectral symmetric algebra]]

[[!redirects symmetric algebra spectra]]

[[!redirects symmetric algebra spectrum]]

[[!redirects spectral polynomial algebra]]
[[!redirects spectral polynomial algebras]]

[[!redirects spectral polynomial ring]]
[[!redirects spectral polynomial rings]]


spectral theorem

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## Idea

The __spectral theorems__ form a cornerstone of [[functional analysis]]. They are a vast generalization to infinite-dimensional [[Hilbert spaces]] of a basic result in linear algebra: an $n \times n$ Hermitian matrix can be diagonalized or conjugated to a diagonal matrix with real entries along the diagonal.

There is a caveat, though: if we consider a separable Hilbert space $\mathcal{H}$ then we can choose a countable orthonormal (Hilbert) basis $\{e_n\}$ of $\mathcal{H}$, a linear operator $A$ then has a matrix representation in this basis just as in finite dimensional linear algebra. The spectral theorem does _not_ say that for every selfadjoint $A$ there is a basis so that $A$ has a diagonal matrix with respect to it. The situation that comes closest to the analogy of finite dimensions is the spectral theorem for compact operators.

There are several versions of the spectral theorem, or several spectral theorems, differing in the kind of operator considered (bounded or unbounded, selfadjoint or normal) and the phrasing of the statement (via spectral measures, multiplication operator norm), which is why this page does not consist of _one_ statement only. Plus there are several different strategies to prove e.g. the spectral theorem for bounded selfadjoint operators, so there is no _canonical_ way to prove it (none is particularly simple).

## Versions of the theorem

### Memory hook

This is a short description of a way to prove the spectral theorem for bounded, selfadjoint operators in the spectral measure form, it may serve as a memory hook:

Let $\mathcal{H}$ be a Hilbert space and A be an selfadjoint operator on $\mathcal{H}$. Then we can form the smallest [[von Neumann algebra]] $\mathcal{R}$ generated by both A and $\mathbb{1}$, the identity operator. This is an abelian algebra, that is, via the [[Gelfand representation]], the abelian version of the [[Gelfand-Naimark theorem]], isometric isomorph to the algebra of continuous complex valued functions of a compact Hausdorff space. The spectral integral of A now becomes the Lebesgue (or Riemann) integral of a continuous function.

### Selfadjoint operators

* theorem: there is a one to one correspondence of (bounded or unbounded) selfadjoint operators $A$ and [[spectral measure]]s E such that
$$
      A = \integral \lambda E(d\lambda)
$$
A is bounded iff E is bounded.

### Borel functional calculus

As stated on the [[spectral measure]] page, given a resolution of the identity $E$ one can make sense of $E(u)$ for any function $u$ that is Borel measurable. Since we have for a selfadjoint operator $A = E(u)$ for $u(\lambda) = \lambda$, one can define a bounded operator $u(A)$ for every _bounded_ Borel measurable function $u$.

### Compact operators

The special case of compact operators is mentioned here because it comes closest to the finite dimensional situation of diagonalizing Hermitian matrixes.

* theorem: let $A$ be a compact selfadjoint operator, then the spectrum of $A$ consists of eigenvalues only, and all eigenvalues $\neq 0$ have finite multiplicity. Therefore, the spectral theorem says in this case, that
$$
         A = \sum \lambda_k |x\rangle \langle x|
$$
where $\lambda_k$ is an eigenvalue (each eigenvalue appears n times if n is its multiplicity) and $x$ an eigenvector with eigenvalue $\lambda_k$.

The converse statement is of course true, too, given a sequence $(\lambda_n)$ in $\mathbb{C}$ convergent to $0$, the sum above (with arbitrary $x \in \mathcal{H}, \|x\| = 1$) defines a normal compact operator, and a selfadjoint compact operator iff all $\lambda_n$ are real.
It is possible to generalize the statement to compact operators on (not necessarily complete) normed spaces.

### Normal operators

(...) [[normal operator]] (...)

## Proofs

### ...

### By Lagrange multipliers

For the case of symmetric real matrices there is a
simple proof using [[Lagrange multiplier]] techniques.

See [Lagrange multiplier -- Applications -- To Spectral theory](http://ncatlab.org/nlab/show/Lagrange+multiplier#ApplicationToSpectralTheory).

## Related entries

* For [[unitary matrices]] the spectal theorem is equivalently [Cartan's theorem](maximal+torus#Properties) that every element in $U(n)$ is conjugate to an element in its [[maximal torus]].

[[!redirects spectral theorems]]


spectral theory

# Spectral theory
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{: toc}

## Idea

In [[functional analysis]] and [[operator algebra]], the point of a [[spectrum of an operator]] algebra should be thought as a common generalized eigenvalue of the corresponding family of operators. Thus the spectra of (operator) *algebras*, presenting the "points" of underlying space and spectra of (families of) *operators* have closely related origin in main examples.

__Spectral theory__ concerns generalizations of the theory of the set of eigenvalues of a linear operator on a finite-dimensional vector space, and [[spectrum of a linear operator]], not necessarily bounded, on an infinite-dimensional _complex_ vector space. Most of such generalizations occur in functional analysis (spectral theory of bounded Hermitean operator on a Hilbert space, spectral theory of unbounded operators, spectral theory of families of operators, spectral theory of complex Banach algebras). Spectral theory is in the basis of Gel'fand-Neumark theorem. The points of the Gel'fand spectrum of a commutative $C^*$-algebra have spectral origin in the sense of [[functional analysis]]; the structure of an operator algebra reflects this -- the characters encode related information, and in more general cases, the ideal and even module structure.

While spectra generalize eigenvalues, the generalized eigenmodes are studied in harmonic analysis, included noncommutative where irreducible representations generalize characters. It is traditional that by spectral theory in functional setup one subsumes only those "harmonic" issues which are closely  related to operator theoretic holomorphic calculus. As the operator is decomposed into an integral over a spectral measure, categories are often characterized by being generated by some special simple, smallest in some sense objects, which show some singular feature. Thus there are many spectra of categories.

## In different contexts

### Spectral theory of Banach algebras

This is the main case of spectral theory studied in 20th century functional analysis. ...

### Spectra in algebraic geometry

Grothendieck has defined a prime spectrum of commutative unital ring having in mind Gel'fand's spectrum of a commutative $C^*$-algebra. In noncommutative ring theory, this was followed by a number of spectral constructions from  lattices of ideals, including prime and primitive spectrum, Gabriel's spectrum of indecomposable injectives and so on; some of these constructions were equipped with a structure sheaf. These construction are typically not faithful unlike Grothendieck's prime spectrum in commutative case and Gel'fand's spectrum. To get more information one needs to consider an entire category of modules, not only the ideals of the ring. Rosenberg has constructed a  richer spectrum of an abelian category involving a structure stack which is in heart of his reconstruction theorem for commutative [[scheme]]s (not necessarily [[affine scheme|affine]]) and in the ring case it reduces to his earlier construction of the left spectrum of a noncommutative ring. Later he noticed that there is an abundance of spectral constructions which all pass via an intermediate step of a construction of a preorder category, or of its relative analogue, and then applying the [[spectral cookbook]] to this datum.

## Related notions

* [[spectral flow]] in geometry; related to study of [[index theorems]]

* Very recently, Gnang, Elgammal and Retakh proved some analogues of spectral theorems for tensors of higher rank, rather than for matrices.

* _[[spectrum of an operator]]_, spectra of families, of a Banach algebra, _[[Gelfand spectrum]]_, prime (primitive etc.) spectrum of a commutative or noncommutative ring, _[[analytic spectrum]]_, _[[spectrum of an abelian category]]_, _[[spectrum of a triangulated category]]_, _[[Berkovich spectrum]]_, _[[Orlov spectrum]]_ etc.

* in algebraic geometry, especially in [[birational geometry]] and arithmetic algebraic geometry points are sometimes reconstructed by looking at [[valuation]]s on the field of rational functions.

## Literature

* [[Michael Reed]], [[Barry Simon]]: *Methods of Modern Mathematical Physics*, Academic Press (1978) Volume IV: *Analysis of Operators* $[$[ISBN:9780080570457](https://www.elsevier.com/books/iv-analysis-of-operators/reed/978-0-08-057045-7)$]$

* [[William Arveson]], *A Short Course on Spectral Theory*, Graduate Texts in Mathematics **209**, Springer (2002) $[$[doi:10.1007/b97227](https://link.springer.com/book/10.1007/b97227)$]$

* [[eom]]: [Spectral theory of linear operators](http://eom.springer.de/S/s086520.htm) (by Hazewinkel)

* wikipedia: [spectral theory](http://en.wikipedia.org/wiki/Spectral_theory)

* [[Tom Leinster]], _Spectra of operators and rings_, [nCafé](http://golem.ph.utexas.edu/category/2011/10/spectra_of_operators_and_rings.html)

* Gerald B. Folland, _A course in abstract harmonic analysis_, Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1995. x+276 pp. [gBooks](http://books.google.com/books?hl=en&lr=&id=0VwYZI1DypUC)

* Nelson Dunford, Jacob T Schwartz, _Linear operators, spectral theory, self adjoint operators in Hilbert space_ (Part 2) (1967, paperback 1988). Wiley.

* Jean Dieudonné, _Fundamentals of modern analysis_, vol. I, chapter XI: Elementary spectral theory, 1969

* Jean Dieudonné, _Sur la théorie spectrale_,  J. Math. Pures Appl. (9) 35 (1956), 175–187, MR0077894

* [[Joel Robbin]], [[Dietmar Salamon]], _The spectral flow and the Maslov index_, Bull. London Math. Soc. __27__ (1995), no. 1, 1–33, [MR96d:58021](http://www.ams.org/mathscinet-getitem?mr=96d:58021), [doi](http://dx.doi.org/10.1112/blms/27.1.1)
* Edinah K. Gnang, Ahmed Elgammal, Vladimir Retakh, _A generalized spectral theory for tensors_, [arxiv/1008.2923](http://arxiv.org/abs/1008.2923)

* [[A. L. Rosenberg]], _The spectrum of abelian categories and reconstructions of schemes_, in Rings, Hopf Algebras, and Brauer groups, Lectures Notes in Pure and Appl. Math. __197__, Marcel Dekker, New York, 257--274, 1998; MR99d:18011; and Max Planck Bonn preprint _Reconstruction of Schemes_, [MPIM1996-108](http://www.mpim-bonn.mpg.de/preprints/send?bid=3948) (1996)

* [[A. L. Rosenberg]], _Spectra of noncommutative spaces_, MPIM2003-110 [ps](http://www.mpim-bonn.mpg.de/preprints/send?bid=1946) [dvi](http://www.mpim-bonn.mpg.de/preprints/send?bid=1945) (2003);
_Underlying spaces of noncommutative schemes_, MPIM2003-111, [dvi](http://www.mpim-bonn.mpg.de/preprints/send?bid=1947), [ps](http://www.mpim-bonn.mpg.de/preprints/send?bid=1948)

* [[Valter Moretti]], _Spectral Theory and Quantum Mechanics: Mathematical Structure of Quantum Theories, Symmetries and introduction to the Algebraic Formulation_ 2nd Edition, Springer 2018 ISBN 978-3-319-70705-1

category: analysis, geometry, noncommutative geometry


spectral triple

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Noncommutative geometry
+--{: .hide}
[[!include noncommutative geometry - contents]]
=--
#### Functorial quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

### Algebraically

A _spectral triple_ ([Connes-Moscovici 95](#ConnesMoscovici95)) is [[operator algebra|operator algebraic]] data that mimics the geometric data provided by a [[smooth manifold|smooth]] [[Riemannian manifold]] $X$ with [[spin structure]] ([Connes 08](#Connes08)) and generalizes it to [[noncommutative geometry]]. It is effectively a [[Fredholm module]] with possibly unbounded [[Fredholm operator]] and refined by the specification of a [[dense subalgebra]] of the [[C-star-algebra]] of bounded operators on that module. As such, spectral triples have close ties to [[algebraic K-theory]] and so also to the [[physics]] described by these (see also at _[[spectral action]]_).

In a little more detail, a spectral triple consists of

1. a $\mathbb{Z}_2$-graded [[Hilbert space]] $\mathcal{H}$, to be thought of as the space of (square integrable) [[sections]] of the [[spinor bundle]] of $X$;

1. An [[associative algebra]] $A$ with a [[dense subalgebra|dense embedding]] $A \hookrightarrow B(H)$ into the [[C-star-algebra]] of [[bounded operators]] on $H$, to be thought of as the algebra of [[smooth functions]] on $X$;

   These two items encode the [[topology]] and [[smooth structure]].

1. A [[Fredholm operator]] $D$ acting on $\mathcal{H}$ and satisfying some conditions, to be thought of as the [[Dirac operator]] acting on the spinors.

   This item encodes the [[Riemannian metric]] and possibly a [[connection on a bundle|connection]].

(This is, or is a slight variant of, the concept of an unbounded [[Fredholm modules]] (e.g. [Carey-Philips 98](#CareyPhilips98)))

Below we discuss how one may think  of a spectral triple as being precisely the algebraic data of [[supersymmetric quantum mechanics]] defining the worldvolume [[QFT]] of the quantum [[relativistic particle|super particle]] propagating on a Riemannian target space (a [[sigma-model]].) Accordingly this is just the beginning of a pattern. One degree up a [[2-spectral triple]] is algebraic data encoding a Riemannian manifold with [[string structure]].

### As 1-dimensional FQFTs
 {#As1DimensionalFQFTs}

Here is an **unorthodox way** to state the idea of spectral triple in terms of [[FQFT]], which is in part just the reformulation of the [[quantum mechanics]] motivation that [[Alain Connes]] derived his definition from in the modern light of [[FQFT]], but which more concretely follows work by [[Maxim Kontsevich|Kontsevich]]-[[Yan Soibelman|Soibelman]], see  ([Soibelman 11](#Soibelman11)) and see the references at _[[2-spectral triple]]_.

> (but maybe eventually we should have a traditional idea section and move this here to a subsection on further interpretations)

Let $R Cob_{1|1}^{Feyn}$ be the [[cobordism category]] of [[Feynman graph]]s for the [[superparticle]] with a single type of interaction along the lines of [[(1,1)-dimensional Euclidean field theories and K-theory]]. So its morphisms are generated from $(1|1)$-dimensional super-Riemannian manifolds (i.e. super-intervals) and from a single interaction vertex

$$
  \array{
    \bullet
    \\
    & \searrow
    \\
    && \bullet & \to
    \\
    & \nearrow
    \\
    \bullet
  }
$$

subject to the obvious associativity condition.

Then a spectral triple $(A,H,D)$ is the data encoding a sufficiently nice smooth [[functor]]

$$
  Z_{(A,H,D)} : R Cob_{1|1}^{Feyn}  \to sVect
$$

to the [[category]] of [[super vector space]]s.

Here

* $A = Z_{(A,H,D)}(\bullet)_0$ is the even part of the [[super vector space]] assigned by the functor to the point, equipped with the structure of a [[algebra]] whose product is given by the image of the interaction vertex

  $$
    Z_{(A,H,D)}
    \left(
  \array{
    \bullet
    \\
    & \searrow
    \\
    && \bullet & \to
    \\
    & \nearrow
    \\
    \bullet
  }
    \right)
  $$

* $H$ is some completion of $Z_{(A,H,D)}(\bullet)$  to a [[super Hilbert space]]

* and $D \in End(H)$ is an odd self-adjoint operator on $H$, which gives the value of the functor on the super-interval $(t,\theta)$ by

  $$
    (t,\theta) \mapsto \exp( - t D^2 + \theta D )
    \,.
  $$

(For technical details that I am glossing over see the field theory link above).

So this is the [[quantum mechanics]] of a [[superparticle]]. In the simplest case this comes from a spinor particle propagating on a [[spin structure]] [[Riemannian manifold]] $X$in which case

* $H = L^2(S)$ is the space of square integrable spinor sections;

* $D$ is the [[Dirac operator]]

* $A = C^\infty(X)$ is the space of smooth functions on $X$.

One point of a spectral triple is to take the view of world-line [[quantum mechanics]] as basic and _characterize_ the spin Riemannian geometry of $X$ entirely by this algebraic data. In particular the [[Riemannian metric]] on $X$ is encoded in the [[operator spectrum]] of $D$, which is where the notion "spectral triple" gets its name from.

Then with all the ordinary geoemtry re-encoded algebraically this way, in terms of the 1-dimensional [[quantum field theory]] that _probes_ this geometry, one can then use the same formulas to interpret spectral triple geometrically that do _not_ come from an ordinary geometry as in the above example.

## Properties

### Metric dimension and KO-dimension

For a space described by a spectral triple there are several notions of [[dimension]] which all coincide when the space is a classical [[smooth manifold]] but which may differ on more general spectral spaces.

Notably there is the [[metric dimension]], which determines the growth of the [[eigenvalues]] of the [[Dirac operator]] ([Connes95, p. 6](#Connes95)).

And then there is the [[KO-dimension]].

## Related concepts

* [[spectral geometry]]

* [[hearing the shape of a drum]]

* [[KK-theory]]

* [[spectral action]]

* [[Connes-Lott-Chamseddine model]]

* [[higher category theory and physics]]: _[Spectral standard model and gravity](higher+category+theory+and+physics#SpecStandModAndGravity)_

* [[(1,1)-dimensional Euclidean field theories and K-theory]]

* [[2-spectral triple]]

## References

### General

The standard textbook is

* [[Alain Connes]], _Noncommutative Geometry_ , Academic Press (1994)

See also

* [[Alain Connes]], [[Matilde Marcolli]], chapter I, section 10 of _[[Noncommutative Geometry, Quantum Fields and Motives]]_

The terminology  of spectral triples was introduced in

* {#ConnesMoscovici95} [[Alain Connes]], [[Henri Moscovici]],  _The Local Index Formula in Noncommutative Geometry_, Geometry and Funct. Analysis 5 (1995) 174-243.

* {#Connes95} [[Alain Connes]], _Noncommutative geometry and reality_, J. Math. Phys. 36 (11), 1995 ([doi:10.1063/1.531241](https://doi.org/10.1063/1.531241))

and that of [[spectral action]] in

* [[Alain Connes]], _Gravity coupled with matter and foundation of non-commutative geometry_ ([arXiv:hep-th/9603053](http://arxiv.org/abs/hep-th/9603053))

The characterization of ordinary [[compact space|compact]] [[smooth manifolds]] with [[spin structure]] in terms of spectral triples is in

* {#Connes08} [[Alain Connes]], _On the spectral characterization of manifolds_ ([arXiv:0810.2088](http://arxiv.org/abs/0810.2088))

See also

* {#CareyPhilips98} [[Alan Carey]], John Phillips, _Fredholm modules and spectral flow_  J. Canadian Math. Soc. **50** (1998) 673-718. ([publisher](https://cms.math.ca/10.4153/CJM-1998-038-x))

* R. da Rocha, A. A. Tomaz, _Hearing the shape of inequivalent spin structures and exotic Dirac operators_ ([arXiv:2003.03619](https://arxiv.org/abs/2003.03619))

Traditionally spectral triples are discussed without specifying their [[homomorphisms]]. Proposals to remedy this such as to obtain a sensible [[category]] of spectral triples include the following

* {#BCL05} [[Paolo Bertozzini]], [[Roberto Conti]], Wicharn Lewkeeratiyutkul, _A Category of Spectral Triples and Discrete Groups with Length Function_, Osaka Journal of Mathematics, 43 n. 2, 327-350 (2006) ([arXiv:math/0502583](http://arxiv.org/abs/math/0502583))

* {#BCL08} [[Paolo Bertozzini]], [[Roberto Conti]], Wicharn Lewkeeratiyutkul, _Non-Commutative Geometry, Categories and Quantum Physics_, East-West Journal of Mathematics "Contributions in Mathematics and Applications II" Special Volume 2007, 213-259 (2008) ([arXiv:0801.2826](http://arxiv.org/abs/0801.2826))

(See also the pointers concerning the relation to [[KK-theory]] [below](#ReferencesRelationToKKTheory)).

{#OverJordanAlgebras} Spectral triples over [[Jordan algebras]]:

* [[Latham Boyle]], [[Shane Farnsworth]], _The standard model, the Pati-Salam model, and "Jordan geometry"_ ([arxiv:1910.11888](https://arxiv.org/abs/1910.11888))

* Fabien Besnard, [[Shane Farnsworth]], *Particle models from special Jordan backgrounds and spectral triples* $[$[arXiv:2206.07039](https://arxiv.org/abs/2206.07039)$]$

See also

* Ludwik Dabrowski, Andrzej Sitarz, _Multiwisted real spectral triples_ ([arXiv:1911.12873](https://arxiv.org/abs/1911.12873))

### Relation to quantum physics

A discussion of spectral triples as [[FQFT]] data encoding a [[representation]] of a category of 1-dimensional [[cobordisms]] with [[Riemannian manifold|Riemannian]] structure and vertices is in section 1.4 of

* {#Soibelman11} [[Yan Soibelman]], _Collapsing Conformal Field Theories, spaces with non-negative Ricci curvature and non-commutative geometry_, in [[Hisham Sati]], [[Urs Schreiber]] (eds.) _[[schreiber:Mathematical Foundations of Quantum Field and Perturbative String Theory]]_, Proceedings of Symposia in Pure Mathematics, AMS 2011

A brief indication of some of the central ideas going into this is at

* [[Urs Schreiber]],  _[Spectral triples and graph field theory](http://golem.ph.utexas.edu/category/2007/06/had_the_pleasure_of_talking.html)_

A general introduction to and discussion of spectral triples with an eye on [[quantum mechanics]], [[quantum field theory]] and [[string theory]] is in

* [[Jürg Fröhlich]], Oiver Grandjean, [[Andreas Recknagel]], _Supersymmetric quantum theory, non-commutative geometry, and gravitation_  Lecture Notes Les Houches (1995) ([arXiv:hep-th/9706132](http://arxiv.org/abs/hep-th/9706132))

In section 7.2 of this an outlokk on how to regard the [[string theory|string's]] worldvolume [[CFT]] as a [[2-spectral triple]] is given.

A detailed derivation for how spectral triples arise as point particle limits of [[vertex operator algebra]]s for 2d [[CFT]]s:

* [[Daniel Roggenkamp]], [[Katrin Wendland]], _Limits and Degenerations of Unitary Conformal Field Theories_ ([arXiv:hep-th/0308143](http://arxiv.org/abs/hep-th/0308143))

A summary of this is in

* [[Daniel Roggenkamp]], [[Katrin Wendland]], _Decoding the geometry of conformal field theories_ ([arXiv:0803.0657](http://arxiv.org/abs/0803.0657))

Also

* [[Sebastiano Carpi]], Robin Hillier, [[Yasuyuki Kawahigashi]], [[Roberto Longo]], _Spectral Triples and the Super-Virasoro Algebra_ Communications in Mathematical Physics, Volume 295, Number 1 / April 2010 ([Springer pdf](http://www.springerlink.com/content/316wx4r4x885172w/fulltext.pdf))

Discussion in the context of [[D-brane]] [[matrix models]] is in

* T. Asakawa, S. Sugimoto, S. Terashima, _D-branes, Matrix Theory and K-homology_, JHEP 0203 (2002) 034 ([arXiv:hep-th/0108085](http://arxiv.org/abs/hep-th/0108085))

### von Neumann spectral triples

One variation uses [[von Neumann algebras]] instead of [[C-star algebras]].

This goes back to ([Carey-Philips 98](#CareyPhilips98)) and

* Alan L Carey, John Phillips, Fyodor Sukochev, _On unbounded $p$-summable Fredholm modules_ ([arXiv:math/9908091](http://arxiv.org/abs/math/9908091))

See also

* M-T. Benameur, T. Fack,, _Type II non-commutative geometry. I. Dixmier trace in von Neumann algebras_, Advances in Mathematics 199: 29-87, 2006.

* [[Alain Connes]], [[Henri Moscovici]], _Type III and spectral triples_ ([arXiv:math/0609703](http://arxiv.org/abs/math/0609703))

### Relation to K-theory
 {#ReferencesRelationToKKTheory}

Relation to [[K-theory]] and [[KK-theory]] is discussed in

* ([pdf](http://www.math.sc.chula.ac.th/cimpa/sites/default/file/bangkoknotes.pdf))

* [[Bram Mesland]], _Spectral triples and KK-theory: A survey_ ([arXiv:1304.3802](http://arxiv.org/abs/1304.3802))

* [[Alan Carey]], John Philips, Adam Rennie, _Spectral triples: examples and index theory_, in Alan Carey (ed.) _Noncommutative geometry and physics, Renormalization, Motives, Index theory_ (2011)

### Examples

On the [[spectral triple|spectral]] [[Riemannian geometry]] of the [[fuzzy 2-sphere]]:

* [[Francesco D'Andrea]], [[Fedele Lizzi]], [[Joseph Várilly]], _Metric Properties of the Fuzzy Sphere_, Lett. Math. Phys.103 (2013), 183-205 ([arXiv:1209.0108](https://arxiv.org/abs/1209.0108))

[[!redirects spectral triples]]
[[!redirects graph field theory]]

[[!redirects unbounded Fredholm module]]
[[!redirects unbounded Fredholm modules]]


spectrification

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

#Contents#
* table of contents
{:toc}

## Idea

The [[full subcategory|inclusion]] of actual [[spectra]] (e.g. [[sequential spectrum|sequential]] [[Omega-spectra]] or [[excisive functors]] on $(\infty Grpd_{fin}^{\ast/})^{op}$) into [[pre-spectra]] (e.g. [[sequential spectrum|sequential]] pre-spectra or all functors on $(\infty Grpd_{fin}^{\ast/})^{op}$) has a ([[adjoint (infinity,1)-functor|infinity]]-)[[left adjoint]], "spectrification".

## Existence

### For sequential spectra

An original account is ([Lewis 86](#Lewis86)). The following is the approach due to ([Joyal 08](#Joyal08)), in the generality of [[parameterized spectra]] (which happens to make the analysis easier instead of harder).

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

Let $seq$ be the [[diagram]] category as follows:

$$
  seq
  \coloneqq
  \left\{
    \array{
    && \vdots && \vdots
    \\
    && \downarrow &&
    \\
    \cdots &\to& x_{n-1} &\stackrel{p_{n-1}}{\longrightarrow}& \ast
    \\
    &&{}^{\mathllap{p_{n-1}}}\downarrow &\swArrow& \downarrow^{\mathrlap{i_n}} & \searrow^{\mathrm{id}}
    \\
    &&\ast &\underset{i_n}{\longrightarrow}& x_n &\stackrel{p_n}{\longrightarrow}& \ast
    \\
    && &{}_{\mathllap{id}}\searrow& {}^{\mathllap{p_n}}\downarrow &\swArrow& \downarrow^{\mathrlap{i_{n+1}}}
    \\
    && && \ast &\stackrel{i_{n+1}}{\longrightarrow}& x_{n+1} &\to& \cdots
    \\
     && && && \downarrow
     \\
     && && && \vdots
    }
  \right\}_{n \in \mathbb{N}}
  \,.
$$

=--

([Joyal 08, section 35.5](#Joyal08))

Let $\mathbf{H}$ be an [[(∞,1)-topos]], for instance $\mathbf{H} = $ [[∞Grpd]] for purposes of traditional [[stable homotopy theory]].

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

An [[(∞,1)-functor]]

$$
  X_\bullet \;\colon\; seq \longrightarrow \mathbf{H}
$$

is equivalently

1. a choice of [[object]] $B \in \mathbf{H}$ (the image of $\ast \in seq$);

1. a sequence of objects $\{X_n\} \in \mathbf{H}_{/B}$ in the [[slice (∞,1)-topos]] over $B$;

1. a sequence of [[morphisms]] $X_n \longrightarrow \Omega_B X_{n+1}$ from $X_n$ into the [[loop space object]] of $X_{n+1}$ in the slice.

This is a [[prespectrum object]] in the [[slice (∞,1)-topos]] $\mathbf{H}_{/B}$.

A [[natural transformation]]
$f \;\colon \;X_\bullet \to Y_\bullet$ between two such functors with components

$$
  \left\{
  \array{
    X_n &\stackrel{f_n}{\longrightarrow}& Y_n
    \\
    \downarrow^{\mathrlap{p_n^X}} && \downarrow^{\mathrlap{p_n^Y}}
    \\
    B_1 &\stackrel{f_b}{\longrightarrow}& B_2
  }
  \right\}
$$

is equivalently a morphism of base objects $f_b \;\colon\; B_1 \longrightarrow B_2$ in $\mathbf{H}$ together with morphisms $X_n \longrightarrow f_b^\ast Y_n$ into the [[(∞,1)-pullback]] of the components of $Y_\bullet$  along $f_b$.

Therefore the [[(∞,1)-presheaf (∞,1)-topos]]

$$
  \mathbf{H}^{seq}
  \coloneqq
  Func(seq, \mathbf{H})
$$

is the [[codomain fibration]] of $\mathbf{H}$ with "fiberwise pre-stabilization".

A genuine [[spectrum object]] is a [[prespectrum object]] for which all the structure maps $X_n \stackrel{\simeq}{\longrightarrow} \Omega_B X_{n+1}$ are [[equivalence in an (∞,1)-category|equivalences]]. The [[full sub-(∞,1)-category]]

$$
  T \mathbf{H} \hookrightarrow \mathbf{H}^{seq}
$$

on the genuine [[spectrum objects]] is therefore the "fiberwise [[stabilization]]" of the self-indexing, hence the tangent $(\infty,1)$-category.

=--

+-- {: .num_lemma #SpectrificationLemma}
###### Lemma
**(spectrification is left exact reflective)**

The inclusion of [[spectrum objects]] into $\mathbf{H}^{seq}$  is [[left exact (infinity,1)-functor|left]] [[reflective sub-(infinity,1)-category|reflective]], hence it has a [[left adjoint]] [[(∞,1)-functor]] $L$ -- [[spectrification]] --  which preserves [[finite (∞,1)-limits]].

$$
  T \mathbf{H}
  \stackrel{\overset{L_{lex}}{\leftarrow}}{\hookrightarrow}
  \mathbf{H}^{seq}
  \,.
$$

=--

([Joyal 08, section 35.1](#Joyal08))

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

Forming degreewise [[loop space objects]] constitutes an [[(∞,1)-functor]] $\Omega \colon \mathbf{H}^{seq} \longrightarrow \mathbf{H}^{seq}$ and by definition of $seq$ this comes with a [[natural transformation]] out of the identity

$$
  \theta \;\colon\; id \longrightarrow \Omega
  \,.
$$

This in turn is compatible with $\Omega$ in that

$$
  \theta \circ \Omega \simeq \Omega \circ \theta
  \;\colon\;
  \rho \longrightarrow \rho \circ \rho = \rho^2
  \,.
$$

Consider then a sufficiently deep [[transfinite composition]] $\rho^{tf}$. By the [[small object argument]] available in the [[presentable (∞,1)-category]] $\mathbf{H}$ this stabilizes, and hence provides a [[reflective sub-(infinity,1)-category|reflection]] $L \;\colon\; \mathbf{H}^{seq} \longrightarrow T \mathbf{H}$.

Since [[transfinite composition]] is a [[filtered (∞,1)-colimit]] and since in an [[(∞,1)-topos]] these commute with [[finite (∞,1)-limits]], it follows that spectrum objects are an [[exact (∞,1)-functor|left exact]] [[reflective sub-(∞,1)-category]].

=--

See also at _[[tangent (∞,1)-topos]]_.

### For excisive functors

See at _[n-excisive functor -- n-Excisive approximation and reflection](n-excisive+functor#nExcisiveApproximation)_

## Construction

### For sequential spectra

For $E$ a [[sequential spectrum|sequential]] [[CW-spectrum|CW-]][[pre-spectrum]], its spectrification to an [[Omega-spectrum]] may be constructed

$$
  (L E)_n = \underset{\longrightarrow}{\lim}_k \Omega^k E_{n+k}
  \,.
$$

([Lewis-May-Steinberger 86, p. 3](#LewisMaySteinberger86), [Weibel 94, 10.9.6 and topology exercise 10.9.2](#Weibel94))

In the special case that $E = \Sigma^\infty X$ a [[suspension spectrum]], i.e. with $E_n = \Sigma^n X$, then  $(L E)_0$ is the [[free infinite loop space]] construction.

If the pre-spectrum $E$ is not a [[CW-spectrum]] then the construction of the spectrification is more involved ([Lewis 86](#Lewis86)).

For $E$ a [[sequential spectrum|sequential]] [[prespectrum]] in [[pointed object|pointed]] [[simplicial sets]] the spectrification may be constructed by ([Bousfield-Friedlander 78, section 2.3](#BousfieldFriedlander78))

$$
  (L E)_n = \underset{\longrightarrow}{\lim}_k Sing {\Omega^k {\vert E_{n+k} \vert}}
$$

(i.e. by the previous formula combined with [[geometric realization]]/[[Kan fibrant replacement]]).

Generally, for sequential spectra in any [[proper model category|proper]] [[pointed category|pointed]] [[simplicial model category]] which admits a [[small object argument]], spectrification is discussed in ([Schwede 97, section 2.1](#Schwede97)).

These spectrification functors on sequential prespectra satisfy the conditions of the [[Bousfield-Friedlander theorem]], and hence the [[left Bousfield localization]] of pre-spectra with degree-wise fibrations weak equivalences at the morphisms of prespectra that become weak equivalences under spectrification exists. This is the stable _[[Bousfield-Friedlander model structure]]_.

### For coordinate-free spectra

Similarly for a [[coordinate-free spectrum]] $E$, if all the structure maps are inclusions

$$
  E_V \hookrightarrow \Omega^{W-V}E_W
$$

then the spectrification is

$$
  (L E)_V = \underset{V\subset W}{\lim} \Omega^{W-V}E_V
  \,-
$$

([Elmendorf-Kriz-May 95, p. 7](#ElmendorfKrizMay95))

### For symmetric spectra

For [[symmetric spectra]], see ([Schwede 12, prop. 4.39](#Schwede12)).

## See also

* [[spectrification of a sequential spectrum type]]

## References

* {#BousfieldFriedlander78} [[Aldridge Bousfield]], [[Eric Friedlander]], _Homotopy theory of $\Gamma$-spaces, spectra, and bisimplicial sets_, Springer Lecture Notes in Math., Vol. 658, Springer, Berlin, 1978, pp. 80-130. ([pdf](https://www.math.rochester.edu/people/faculty/doug/otherpapers/bousfield-friedlander.pdf))

* {#LewisMaySteinberger86} [[L. Gaunce Lewis]], [[Peter May]], M. Steinberger, _Equivariant stable homotopy theory_, Springer Lecture Notes in Mathematics, 1986 ([pdf](http://www.math.uchicago.edu/~may/BOOKS/equi.pdf))

* {#Lewis86} [[L. Gaunce Lewis]], _Analysis of the passage from prespectra to spectra_, appendix in  [[L. Gaunce Lewis]], [[Peter May]], M. Steinberger, _Equivariant stable homotopy theory_, Springer Lecture Notes in Mathematics, 1986 ([pdf](http://www.math.uchicago.edu/~may/BOOKS/equi.pdf))

* {#Weibel94} [[Charles Weibel]], _[[An introduction to homological algebra]]_,  Cambridge Studies in Adv. Math. 38, CUP 1994

* {#ElmendorfKrizMay95} [[Anthony Elmendorf]], [[Igor Kriz]], [[Peter May]], _[[Modern foundations for stable homotopy theory]]_, in [[Ioan Mackenzie James]] (ed.), _[[Handbook of Algebraic Topology]]_, Amsterdam: North-Holland, 1995 pp. 213–253,  ([pdf](http://hopf.math.purdue.edu/Elmendorf-Kriz-May/modern_foundations.pdf))

* {#Schwede97} [[Stefan Schwede]], _Spectra in model categories and applications to the algebraic cotangent complex_, Journal of Pure and Applied Algebra 120 (1997) 77-104 ([pdf](http://www.math.uni-bonn.de/people/schwede/modelspec.pdf))

* {#Schwede12} [[Stefan Schwede]], around prop. 4.39 of _[[Symmetric spectra]]_ (2012)

* {#Joyal08} [[André Joyal]], _Notes on Logoi_, 2008 ([pdf](http://www.math.uchicago.edu/~may/IMA/JOYAL/Joyal.pdf))

* [[Cary Malkiewich]], _Some facts about $Q X$_ ([pdf](http://www.math.uiuc.edu/~cmalkiew/Q.pdf))

[[!redirects spectrifications]]

[[!redirects Omega-spectrification]]
[[!redirects Omega-spectrifications]]


spectrification of a sequential spectrum type

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

#Contents#
* table of contents
{:toc}

## Idea

In classical [[algebraic topology]], there is a **[[spectrification]]** functor which is [[left adjoint]] to the inclusion of [[spectra]] in [[prespectra]]. For instance, this is how a [[suspension spectrum]] is constructed: by spectrifying the prespectrum $X_n \coloneqq \Sigma^n A$. There is an analogue in [[homotopy type theory]] which constructs the spectrification of a [[sequential spectrum type]] (i.e. prespectrum types) into a [[Omega-spectrum type]] (i.e. spectrum types).

## Definition

The following [[higher inductive type]] should construct spectrification in [[homotopy type theory]] (though this has not yet been verified formally). (There are some abuses of notation below, which can be made precise using Coq typeclasses and implicit arguments.)

    Inductive spectrify (X : prespectrum) : nat -> Type :=
    | to_spectrify : forall n, X n -> spectrify X n
    | spectrify_glue : forall n, spectrify X n ->
        to_spectrify (S n) (pt (S n)) == to_spectrify (S n) (pt (S n))
    | to_spectrify_is_prespectrum_map : forall n (x : X n),
        spectrify_glue n (to_spectrify n x)
        == loop_functor (to_spectrify (S n)) (glue n x)
    | spectrify_glue_retraction : forall n
        (p : to_spectrify (S n) (pt (S n)) == to_spectrify (S n) (pt (S n))),
        spectrify X n
    | spectrify_glue_retraction_is_retraction : forall n (sx : spectrify X n),
        spectrify_glue_retraction n (spectrify_glue n sx) == sx
    | spectrify_glue_section : forall n
        (p : to_spectrify (S n) (pt (S n)) == to_spectrify (S n) (pt (S n))),
        spectrify X n
    | spectrify_glue_section_is_section : forall n
        (p : to_spectrify (S n) (pt (S n)) == to_spectrify (S n) (pt (S n))),
        spectrify_glue n (spectrify_glue_section n p) == p.

We can unravel this as follows, using more traditional notation.  Let $L X$ denote the spectrification being constructed.  The first constructor says that each $(L X)_n$ comes with a map from $X_n$, called $\ell_n$ say (denoted `to_spectrify n` above).  This induces a basepoint in each type $(L X)_n$, namely the image $\ell_n(*)$ of the basepoint of $X_n$.  The many occurrences of

    to_spectrify (S n) (pt (S n)) == to_spectrify (S n) (pt (S n))

simply refer to the based *loop space* of $\Omega_{\ell_{n+1}(*)} (L X)_{n+1}$ of $(L X)_{n+1}$ at this base point.

Thus, the second constructor `spectrify_glue` gives the structure maps $(L X)_n \to \Omega (L X)_{n+1}$ to make $L X$ into a prespectrum.  Similarly, the third constructor says that the maps $\ell_n\colon X_n \to (L X)_n$ commute with the structure maps up to a specified homotopy.

Since the basepoints of the types $(L X)_n$ are induced from those of each $X_n$, this automatically implies that the maps $(L X)_n \to \Omega (L X)_{n+1}$ are pointed maps (up to a specified homotopy) and that the $\ell_n$ commute with these pointings (up to a specified homotopy).  This makes $\ell$ into a map of prespectra.

Finally, the fourth through seventh constructors say that $L X$ is a spectrum, by giving [[h-isomorphism]] data: a retraction and a section for each glue map $(L X)_n \to \Omega (L X)_{n+1}$.  We could use adjoint equivalence data as we did for localization, but this approach avoids the presence of level-3 path constructors.  (We could have used h-iso data in localization too, thereby avoiding even level-2 constructors there.)  It is important, in general, to use a sort of equivalence data which forms an [[h-prop]]; otherwise we would be adding [[stuff, structure, property|structure]] rather than merely the property of such-and-such map being an equivalence.

## See also

* [[higher inductive type]]

* [[sequential spectrum type]]

* [[Omega-spectrum type]]

* [[spectrification]]

## References ##

* [[Univalent Foundations Project]], *[[HoTT book|Homotopy Type Theory – Univalent Foundations of Mathematics]]* (2013)

* [[Floris van Doorn]] (2018), _On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory_, ([arXiv:1808.10690](https://arxiv.org/abs/1808.10690), [web](http://florisvandoorn.com/papers/dissertation.pdf))


Spectrum

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=--
#### Stable Homotopy theory
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#Contents#
* table of contents
{:toc}

## Idea

By $Spectrum$ or $Spectra$ one may denote the [[stable (∞,1)-category]] of [[spectrum|spectra]].  It is also denoted $Sp$, or sometimes $Spec$ although that [[affine scheme|can be confusing]].

Its [[homotopy category of an (infinity,1)-category|homotopy category]] is the classical [[stable homotopy category]].

For more see the entry _[[stable (∞,1)-category of spectra]]_.

## Properties
 {#Properties}

[[!include finite homotopy limits of spectra -- section]]

### Other abstract characterizations

$Spectra$ is equivalently

* the free stable [[locally presentable (∞,1)-category]] on one [[compact object|compact]] generator, namely the [[sphere spectrum]].

* the [[stable (infinity,1)-category]] of [[quasicoherent infinity-stacks]] on [[Spec(S)]].

## Related concepts

* [[model structure on spectra]]

* [[PermCat]]

category: category

[[!redirects Spec]]
[[!redirects Sp]]
[[!redirects Spectra]]


spectrum

This entry is about the notion of spectrum in [[stable homotopy theory]]. For other uses of the term ''spectrum'' see [[spectrum - disambiguation]].

***

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## Idea

A [[topological space|topological]] spectrum is an [[object]] in the universal [[stable (∞,1)-category]] $Sp(Top) \simeq Sp(\infty Grpd)$ that stabilizes the [[(∞,1)-category]] [[Top]] or $\simeq$ [[∞-Grpd]] of [[topological spaces]] or [[∞-groupoids]] under the operations of forming [[loop space objects]] and [[reduced suspensions]]: the [[stable (∞,1)-category of spectra]].

More generally, one may consider [[spectrum objects]] in any [[presentable (∞,1)-category]].

### Connective and non-connective spectra; infinite loop spaces

As opposed to the [[homotopy groups]] of a [[topological space]], the [[homotopy groups of a spectrum]] are defined and may be non-trivial in _negative_ integer degree. This follows from the fact that the [[loop space]] operation is by construction invertible on spectra, which implies that for every spectrum $E$ these and all $n$, the $n$-fold looping $\Omega^n$ has [[stable homotopy groups]] given by $\pi_{k-n}(\Omega^n E) \simeq \pi_k(E)$.

Those spectra for which the [[homotopy groups of spectra]] in negative degree happen to vanish are called _[[connective spectra]]_. They are equivalent to [[infinite loop spaces]], i.e. grouplike [[E-∞ spaces]].

Connective spectra in the image of the [[nerve]] operation of the classical [[Dold-Kan correspondence]]: this identifies [[∞-groupoids]] that are not only connective spectra but even have a _strict_ symmetric monoidal [[group]] structure with non-negatively graded [[chain complex|chain complexes]] of abelian groups.

$$
 \array{
  Ch_+ &\stackrel{Dold-Kan \; nerve}{\to}&
  ConnectSp(\infty Grp) \subset \infty Grpd
  \\
  (\cdots A_2 \stackrel{\partial}{\to} A_1 \stackrel{\partial}{\to} A_0 \to 0 \to 0 \to \cdots)
  &\stackrel{}{\mapsto}&
  N(A_\bullet)
 }
$$

The [[homology]] groups of the chain complex correspond precisely to the [[homotopy groups]] of the corresponding [[topological space]] or [[∞-groupoid]].

The free stabilization of the [[(∞,1)-category]] of non-negatively graded chain complexes is simpy the [[stable (∞,1)-category]] of arbitrary chain complexes. There is a [[stable Dold-Kan correspondence]] (see at _[[module spectrum]]_ the section _[stable Dold-Kan correspondence](module+spectrum#StableDoldKanCorrespondence)_ ) that identifies these with special objects in $Sp(Top)$.

$$
 \array{
  Ch &\stackrel{Dold-Kan \; nerve}{\to}&
   Sp(\infty Grp)
  \\
  (\cdots A_2 \stackrel{\partial}{\to} A_1 \stackrel{\partial}{\to} A_0 \stackrel{\partial}{\to} A_{-1}
  \stackrel{\partial}{\to} A_{-2} \to \cdots)
  &\stackrel{}{\mapsto}&
  N(A_\bullet)
 }
$$

So it is the homologically nontrivial parts of the chain complexes in negative degree that corresponds to the non-connectiveness of a spectrum.

## Definition

There are many "models" for spectra, all of which present the same ([[stable homotopy theory|stable]]) [[homotopy theory]] (and in fact, nearly all of them are [[Quillen equivalence|Quillen equivalent]] [[model category|model categories]]).  For more details see at _[[Introduction to Stable Homotopy Theory]]_.

### Sequential pre-spectra
 {#SequentialPreSpectra}

A simple first definition is to define a [[pre-spectrum]] $\mathbf{E}$ to be a sequence of pointed spaces $(E_n)_{n\in\mathbb{N}}$ together with structure maps $\Sigma{}E_n\to{}E_{n+1}$ (where $\Sigma$ denotes the [[reduced suspension]]). See at _[[model structure on sequential spectra]]_.

There are various conditions that can be put on the spaces $E_n$ and the structure maps, for example if the spaces are CW-complexes and the structure maps are inclusions of subcomplexes, the spectrum is called a **[[CW-spectrum]]**.

Without any condition, this is just called a **spectrum**, or sometimes a **[[pre-spectrum]]**. In order to distinguish from various other richer definitions (such as [[coordinate-free spectra]], one also speaks of _[[sequential spectra]]_).

For details see _[[Introduction to Stable homotopy theory -- 1-1|Introduction to stable homotopy theory -- 1.1 Sequential Spectra]]_.

### $\Omega$-spectra
 {#OmegaSpectrum}

If $\Omega$ denotes the [[loop space]] functor on the category of pointed spaces, we know that $\Sigma$ is left adjoint to $\Omega$.
In particular, given a spectrum $\mathbf{E}$, the structure maps can be transformed into maps $E_n\to\Omega{}E_{n+1}$.
If these maps are isomorphisms (depending on the situation it can be weak equivalences or homeomorphisms), then $\mathbf{E}$ is called an **$\Omega$-spectrum**.

The idea is that $E_0$ contains the information of $\mathbf{E}$ in dimensions $k\ge 0$, $E_1$ contains the information of $\mathbf{E}$ in $k\ge -1$ (but shifted up by one, so that it is modeled by the $\ge 0$ information in the space $E_1$), and so on.

$\Omega$-spectra are special cases of [[sequential spectrum|sequential]] pre-spectra as [above](#SequentialPreSpectra), and are in fact the [[fibrant objects]] for some [[model structure on spectra]].

Given any sequential pre-spectrum $\mathbf{E}$, it induces an equivalent $\Omega$-spectrum $\mathbf{F}$ (a fibrant replacement of $\mathbf{E}$, its _[[spectrification]]_) given by ([Lewis-May-Steinberger 86, p. 3](#LewisMaySteinberger86))

$$
  F_n \coloneqq \lim_{m\to\infty}\Omega^m E_{n+m}
$$

(using that $\Omega$ commutes with the [[filtered colimits]]).

### Coordinate-free spectra

A definition of spectrum consisting of spaces indexed by index sets less "coordinatized" than the integers is a

* [[coordinate-free spectrum]].

See there for details.

### Combinatorial spectra

There might be a type of categorical structure related to a spectrum in the same way that $\infty$-categories are related to $\infty$-groupoids.  In other words, it would contain $k$-cells for all integers $k$, not necessarily invertible.  Some people have called this conjectural object a **$Z$-category**.   "Connective" $Z$-categories could perhaps then be identified with stably monoidal $\infty$-categories.

One realization of this kind of idea is the notion of [[combinatorial spectrum]].

### General context

See [[spectrum object]].

## Examples

* [[suspension spectrum]]

* [[sphere spectrum]]

* [[Eilenberg-MacLane spectrum]]

* [[K-theory spectrum]]

* [[elliptic spectrum]]

* [[tmf]]

* [[Thom spectrum]], [[complex cobordism spectrum]]

## Properties

### Stabilization

In direct analogy to how topological spaces form the archetypical example, [[Top]], of an [[(∞,1)-category]], spectra form the archetypical example $Sp(Top)$ of a [[stable (∞,1)-category]]. In fact, there is a general procedure for turning any [[pointed category|pointed]] [[(∞,1)-category]] $C$ into a stable $(\infty,1)$-category $Sp(C)$, and doing this to the category $Top_*$ of [[pointed object|pointed]] spaces yields $Sp(Top)$.

### Symmetric monoidal structure

* [[smash product of spectra]]

* [[symmetric monoidal smash product of spectra]]

### Closed structure

* [[mapping spectrum]]

### Model category structure

* [[model structure on spectra]]

### Relation to symmetric monoidal ∞-groupoids

* [[stable homotopy hypothesis]]

## Related concepts

* [[homotopy group of a spectrum]]

* [[sheaf of spectra]], [[parametrized spectrum]]

  * [[smooth spectrum]]

* [[spectrum with G-action]], [[G-spectrum]]

* [[ring spectrum]], [[algebra spectrum]], [[module spectrum]]

* [[zero spectrum]]

* [[finite spectrum]]

* [[p-local spectrum]]

* [[Bousfield localization of spectra]]

* [[motivic spectrum]]

* [[Brown representability theorem]]

* [[stable homotopy hypothesis]]

[[!include k-monoidal table]]

## References

According to ([Adams 74, p. 131](#Adams74)) the notion of spectrum is due to

* {#Lima58} E. L. Lima, _Duality and Postnikov invariants_, Thesis, University of Chicago, Chicago 1958

> It is generally supposed that [[George Whitehead|G. W. Whitehead]] also had something to do with it, but the latter takes a modest attitude about that. ([Adams 74, p. 131](#Adams74))

Early notes include

* {#Boardman65} [[Michael Boardman]], _Stable homotopy theory_, mimeographed notes, University of Warwick, 1965 onward

* {#Adams74} [[Frank Adams]], Part III, section 2 _[[Stable homotopy and generalised homology]]_, 1974

See the references at _[[stable homotopy theory]]_.

Lecture notes include

* _[[Introduction to Stable homotopy theory -- 1-1|Introduction to stable homotopy theory -- 1.1 Sequential Spectra]]_

More modern developments are due to

* {#LewisMaySteinberger86} [[L. Gaunce Lewis]], [[Peter May]], M. Steinberger, _Equivariant stable homotopy theory_, Springer Lecture Notes in Mathematics, 1986 ([pdf](http://www.math.uchicago.edu/~may/BOOKS/equi.pdf))

The quick idea is surveyed for instance in

* [[Cary Malkiewich]], _The stable homotopy category_, 2014 ([pdf](http://math.stanford.edu/~carym/stable.pdf))

* [[Aaron Mazel-Gee]], _An introduction to spectra_ ([pdf](https://math.berkeley.edu/~aaron/writing/an-introduction-to-spectra.pdf))

The first review of stable homotopy theory with [[symmetric monoidal smash product of spectra]] is (in terms of [[S-modules]]) in

* {#ElmendorfKrizMay95} [[Anthony Elmendorf]], [[Igor Kriz]], [[Peter May]], _[[Modern foundations for stable homotopy theory]]_, in [[Ioan Mackenzie James]], _[[Handbook of Algebraic Topology]]_, Amsterdam: North-Holland, (1995) pp. 213–253,  ([pdf](http://hopf.math.purdue.edu/Elmendorf-Kriz-May/modern_foundations.pdf))

A comprehensive account of the symmetric model in terms of [[symmetric spectra]] is in

* [[Stefan Schwede]], _Symmetric spectra_ ([pdf](http://www.math.uni-bonn.de/~schwede/SymSpec-v3.pdf))

and in terms of [[orthogonal spectra]] in

* [[Stefan Schwede]], _[[Global homotopy theory]]_ (take $\mathcal{F} = \{1\}$, on p. 4, to be the trivial collection of groups, in order to specialize from [[global equivariant stable homotopy theory]] to plain stable homotopy theory).

See also

* [[Robert Thomason]], _Symmetric Monoidal Categories Model All Connective Spectra_ ([web](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.37.8193))

* [[Frank Adams]], _Infinite loop spaces_, Princeton University Press, 1978

* {#Ayoub} [[Joseph Ayoub]], _Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique, I_. Astérisque, Vol. 314 (2008). Société Mathématique de France. ([pdf](http://user.math.uzh.ch/ayoub/PDF-Files/THESE.PDF))

[[!redirects spectra]]


spectrum (geometry)

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[[!include higher geometry - contents]]
=--
=--
=--

> To be distinguished from [[spectrum]] in the sense of [[stable homotopy theory]], see at [[spectrum - disambiguation]].

#Contents#
* table of contents
{:toc}

## Idea

In [[geometry]], a __spectrum__ is a geometric [[space]] constructed from some algebraic, category-theoretic, analytic or similar data which typically do not have an obvious/manifest geometric meaning.

Examples are

+ the [[Gelfand spectrum]] of a commutative [[C*-algebra]],

* the spectrum of a commutative [[von Neumann algebra]] is an enhanced [[measurable space]],

* the Grothendieck [[spectrum of a commutative ring]],

* the formal spectrum of a complete noetherian [[commutative ring]],

* the [[primitive spectrum]] of a noncommutative (but unital) ring,

* the [[left spectrum]] of a noncommutative ring,

* the [[Rosenberg spectrum]] of an [[abelian category]],

* the [[Gabriel spectrum]] of indecomposable injectives,

* the [[Pierce spectrum]],

* the [[Berkovich spectrum]] in rigid [[analytic geometry]],

* the [[Diers spectrum]] recovers many of the above constructions,

* ...

One sometimes says "spectrum" also for the underlying sets of such geometric spectra.

A spectrum does not necessarily give a faithful representation of the original data. For example, the Gelfand spectrum of a $C^*$-algebra is sufficient for reconstructing the $C^*$-algebra if we restrict to commutative $C^*$-algebras (Gelfand--Neimark [[reconstruction]] theorem), but is not a sufficient invariant if we consider all $C^*$-algebras.

The word 'spectrum' in this setup originates from the fact that the spectrum of a commutative [[Banach algebra]] is a natural extension of the theory of a spectrum of a family of commuting self-adjoint operators, which is in turn the generalization of the spectral theory of one self-adjoint operator. See also [[spectrum of a Banach algebra]]. The spectrum of an operator corresponds in quantum (and classical!) mechanics to frequencies of vibrations and waves, hence in optics to color. Newton experimentally observed the spectrum from white light passing through a prism, and with a surprise he considered it at first as a ghost like object, hence he named it after a Latin word for ghost or spirit (see also [Wikipedia](http://en.wikipedia.org/wiki/Spectrum)).

## References

* [MO discussion](http://mathoverflow.net/a/20293/381) on the history of the idea of a space of ideals on which a ring is a ring of functions, both in analysis and in algebraic geometry

For a general-abstract perspective:

* [[Monique Hakim]], _Topos annelés et schémas relatifs_, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 64, Springer, Berlin, New York (1972) ([doi:10.1007/978-3-662-59155-0](https://www.springer.com/gp/book/9783540055730))

* [[Jacob Lurie]], _[[Structured Spaces]]_

also

* [[Cole's theory of spectrum]]

See also:

* [[Yves Diers]], _Une construction universelle des spectres, topologies spectrales et faisceaux structuraux_, Communication in Algebra Volume 12, Issue 17-18 (1984) ([doi:10.1080/00927878408823101](https://doi.org/10.1080/00927878408823101))

* {#Osmond2020a} [[Axel Osmond]], _On Diers theory of Spectrum I: Stable functors and right multi-adjoints_, ([arXiv:2012.00853](https://arxiv.org/abs/2012.00853))

* {#Osmond2020b} [[Axel Osmond]], _On Diers theory of Spectrum II: Geometries and dualities_, ([arXiv:2012.02167](https://arxiv.org/abs/2012.02167))

* [[Axel Osmond]], _The general construction of Spectra_ ([arXiv:2102.01259](https://arxiv.org/abs/2102.01259))


spectrum - disambiguation

There are several different (sometimes related) mathematical objects called ''spectrum'' or whose name has ''spectral'' as prefix:

* [[spectrum]] in [[stable homotopy theory]].

* [[spectrum (geometry)]] in [[geometry]], [[Cole's theory of spectrum]]

* including [[spectral theory]] such _[[spectrum of an operator]]_, spectra of families, of a Banach algebra, _[[Gelfand spectrum]]_, prime (primitive etc.) [[spectrum of a commutative ring]], [[spectrum of a ring]], _[[analytic spectrum]]_, [[Cole's theory of spectrum]], _[[spectrum of an abelian category]]_, _[[Orlov spectrum]]_,  _[[spectrum of a triangulated category]]_, _[[prime spectrum of a symmetric monoidal stable (∞,1)-category]]_, etc.

There is also a notion of

* [[spectral sequence]]

which has no particular relation to other notions of _spectrum_, except that [[filtered spectra]] give rise to [[spectral sequences of filtered spectra]].

In [[physics]]:

* [[atomic spectrum]]


spectrum object

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## Idea

Every [[(∞,1)-category]] $\mathcal{C}$ with [[finite (∞,1)-limits]] has a [[stabilization]] to a [[stable (∞,1)-category]] $Stab(\mathcal{C})$. This stabilization may be defined by abstract properties, but it may also be constructed explicitly as the category of _spectrum objects_ in $\mathcal{C}$.

In the special case that $\mathcal{C} = $ [[∞Grpd]] $\simeq L_{whe}$[[Top]], a spectrum object in $\mathcal{C}$ is a [[spectrum]] in the traditional sense. There is an evident generalization of the traditional notion of [[Omega-spectrum]] from [[Top]] to any $(\infty,1)$-category $\mathcal{C}$ with [[finite (∞,1)-limits]]: a spectrum object $X_\bullet$ is essentially a list of [[pointed objects]] $X_i$ together with [[equivalence in an (∞,1)-category|equivalences]] $X_i \to \Omega X_{i+1}$, from every object in the list to the [[loop space object]] of its successor.

## Definition

### Via $\Omega$-spectrum objects

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

For $\mathcal{C}$ an [[(∞,1)-category]], a **prespectrum object**  of $\mathcal{C}$ is

* a $(\infty,1)$-functor $X : \mathbb{Z} \times \mathbb{Z} \to \mathcal{C}$

* such that for all integers $i \neq j$ we have $X(i,j) = 0$ a [[zero object]] of $\mathcal{C}$

Notice that this definition is highly redundant. The point is that writing $X[n] \coloneqq X(n,n)$ a spectrum object is for all $n \in \mathbb{Z}$ a (homotopy) [[commuting diagram]]

$$
  \array{
    X[n] &\to& 0
    \\
    \downarrow && \downarrow
    \\
    0 &\to& X[n+1]
  }
    \,.
$$

Recalling that in an [[(infinity,1)-category]] with [[zero object]]

* $\Omega X[n+1]$ denotes the pullback of such a diagram;

* $\Sigma X[n]$ denotes the pushout of such a diagram

this induces maps

$$
  \alpha_n : \Sigma X[n] \to X[n+1]
$$

$$
  \beta_{n+1} : X[n] \to \Omega X[n+1]
  \,.
$$

A prespectrum object is

* a **spectrum object** if $\beta_m$ is an equivalence for all $m \in \mathbb{Z}$ (a __spectrum below $n$__, if $\beta_m$ is an equivalence for all $m \leq n$);

* a **suspension spectrum** if $\alpha_m$ is an equivalence for all $m \in \mathbb{Z}$ (a __suspension spectrum above $n$__, if $\alpha_m$ is an equivalence for all $m \geq n$).

=--

([StabCat](#StabCat))

One writes

* $Sp(\mathcal{C})$ for the [[full sub-(∞,1)-category]] of $Fun(\mathbb{Z} \times \mathbb{Z},C)$ on spectrum objects in $C$;

* $Stab(\mathcal{C}) \coloneqq Sp(\mathcal{C}_*)$ -- the **stabilization of $C$** for the $(\infty,1)$-category of spectrum objects in the $(\infty,1)$-category $C_*$ of [[pointed object]]s of $\mathcal{C}$.

### Via excisive functors
 {#ViaExcisiveFunctors}

Write $\infty Grpd_{fin}$ for the [[(∞,1)-category]] of [[finite homotopy types]], hence those freely generated by [[finite (∞,1)-colimits]] from the point. Write $\infty Grpd_{fin}^{\ast/}$ for the [[pointed object|pointed]] finite homotopy types.

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

Let $\mathcal{C}$ be an [[(∞,1)-category]] with [[finite (∞,1)-limits]]. Then a **spectrum object** in $\mathcal{C}$ is a reduced (i.e. [[terminal object in an (infinity,1)-category|terminal object]]-preserving) [[excisive (∞,1)-functor]] of the form

$$
  \infty Grpd_{fin}^{\ast/} \longrightarrow \mathcal{C}
  \,.
$$

=--

([HigherAlg, def. 1.4.2.8 and around p. 823](#HigherAlg)).

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

This generalizes for instance to [[G-spectra]] ([Blumberg 05](#Blumberg05)).

=--

### In an ordinary category

One can define $\Phi$-symmetric $F$-spectra in a category $C$, where $\Phi$ is a [[graded monoid]] in the [[category]] of [[groups]] and $F : C \to C$ is a $\Phi$-symmetric [[endofunctor]] of $C$.  Here we follow [Ayoub](#Ayoub).

(One recovers the classical case described at [[spectrum]] by taking $C$ to be the [[category]] of [[pointed spaces]], $\Phi$ to be the trivial [[graded monoid]], and $F$ to be the [[suspension functor]].)

Let $\Phi$ be a [[graded monoid]] in the category of [[groups]].  Write $Seq(\Phi, C)$ for the category of $\Phi$-[[symmetric sequences]].  Let $F : C \to C$ be a $\Phi$-[[symmetric endofunctor]] of $C$.  (Usually $F$ will be the functor $T \otimes_C -$ induced by [[tensor product]] with some object $T$.)

A **$\Phi$-symmetric $F$-spectrum** in $C$ is a $\Phi$-[[symmetric sequence]] $(X_n)_{n \in \mathbf{N}}$ together with **assembly morphisms**
  $$ \gamma_n : F(X_n) \to X_{n+1} $$
such that the composite morphism
  $$ F^m(X_n) \to F^{m-1}(X_{n+1}) \to \cdots \to X_{m+n} $$
is $(\Phi_m \times \Phi_n)$-[[equivariant]].  (Note that $\Phi_m$ acts on $F^m$ by the definition of [[symmetric endofunctor]], and $\Phi_n$ acts on $X_n$ by the definition of [[symmetric sequence]].)  A morphism of $\Phi$-symmetric $T$-spectra $X = (X_n)_n \to Y = (Y_n)_n$ is a morphism of $\Phi$-[[symmetric sequences]] making the obvious [[diagrams]] [[commutative diagram|commute]].  We write $Spect^{\Phi}_F(C)$ for the category of $\Phi$-symmetric $F$-spectra in $C$.

When $\Phi = \Sigma$, the [[graded monoid]] of [[symmetric groups]], $\Sigma$-symmetric $F$-spectra are called simply **symmetric $F$-spectra**.  When $\Phi = 1$, $1$-symmetric $F$-spectra are called simply **nonsymmetric $F$-spectra**.  When the endofunctor $F$ is given by $T \otimes -$ for some object $T \in C$, $F$-spectra are called $T$-spectra.

When $C$ is a [[symmetric monoidal category]], there is an induced [[symmetric monoidal structure on spectrum objects]].

When $C$ is a sufficiently nice [[model category]], there are induced [[model structures on spectrum objects]].

## Properties

### As a model for stabilization

If $C$ is a pointed $(\infty,1)$-category with finite [[limit in quasi-categories|limits]], then $Sp(C)$ is a [[stable (infinity,1)-category]].

### Reflection into pre-spectrum objects

For $\mathcal{C}$ an [[(∞,1)-category]] with [[(∞,1)-pullbacks]] and [[(∞,1)-colimits]], then the inclusion of spectrum objects into prespectum objects
should be a [[exact (∞,1)-functor|left exact]] [[reflective sub-(∞,1)-category]] inclusion ([Joyal 08, section 35](#Joyal08)).

$$
  Spec(\mathcal{C})
  \stackrel{\overset{lex}{\leftarrow}}{\hookrightarrow}
  PreSpec(\mathcal{C})
  \,.
$$

This implies in particular that the [[tangent (∞,1)-category]] of an [[(∞,1)-topos]]
is itself again an [[(∞,1)-topos]] ([Joyal 08, section 35.5](#Joyal08)), see at _[tangent (∞,1)-category -- Tangent (∞,1)-topos](tangent+%28infinity%2C1%29-category#TangentTopos)_ .

## Examples

* For $C = Top$, $Stab(C)$ is the $(\infty,1)$-category version of the classical stable homotopy category of spaces: the [[stable (infinity,1)-category of spectra]].

* In the [[equivariant homotopy theory]] of [[G-spaces]] a spectrum object is a _[[spectrum with G-action]]_.

* see also at _[[motivic spectrum]]_

## Related concepts

[[!include k-monoidal table]]

* [[stable homotopy type]]

* [[cospectrum]]

## References

Discussion in terms of [[stable (infinity,1)-categories]] is in

* {#StabCat} [[Jacob Lurie]], section 8 of _[[Stable Infinity-Categories]]_

* {#HigherAlg} [[Jacob Lurie]], section 1.4.2 _[[Higher Algebra]]_

* {#Joyal08} [[André Joyal]], section 35  _Notes on Logoi_, 2008 ([pdf](http://www.math.uchicago.edu/~may/IMA/JOYAL/Joyal.pdf))

Discussion of [[model structure on spectra|model structures for spectrum objects]] includes

* {#Hovey00} [[Mark Hovey]], _Spectra and symmetric spectra in general model categories_ ([arXiv:0004051](http://arxiv.org/abs/math/0004051))

* [[Bjørn Ian Dundas]], [[Oliver Röndigs]], [[Paul Arne Østvær]], _Enriched functors and stable homotopy theory_,  Doc. Math., 8:409–488, 2003 ([EuDML](https://eudml.org/doc/123650))

A detailed treatment of the 1-categorical case is in the last chapter of

* {#Ayoub} [[Joseph Ayoub]], _Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique, I_. Astérisque, Vol. 314 (2008). Société Mathématique de France. ([pdf](http://user.math.uzh.ch/ayoub/PDF-Files/THESE.PDF))

Generalization to [[G-spectra]] is in

* {#Blumberg05} [[Andrew Blumberg]], _Continuous functors as a model for the equivariant stable homotopy category_ ([arXiv:math.AT/0505512](http://arxiv.org/abs/math.AT/0505512))

[[!redirects spectrum objects]]

[[!redirects prespectrum object]]
[[!redirects prespectrum objects]]

[[!redirects pre-spectrum object]]
[[!redirects pre-spectrum objects]]


spectrum of a commutative ring

> This entry is about the formal dual [[topological space]] of a [[commutative ring]]. For the very different notion of a similar name in [[higher algebra]] see at [[ring spectrum]]. For more see at _[[spectrum - disambiguation]]_.

#Contents#
* table of contents
{:toc}

## Idea

Given a [[commutative ring]] $R$, its _spectrum_ is the [[topological space]] $Spec(R)$ whose points are the [[prime ideals]] of $R$ and whose [[topology]] is the [[Zariski topology]] on these prime ideals. This topological case is also called the __prime spectrum__ of $R$.

However, usually by $Spec(R)$ one means more: the [[locally ringed space]] which is obtained by equipping the above topological space by a unique sheaf of rings $\mathcal{O}$ such that for every principal localization of commutative rings $R\to R[f^{-1}]$ we have $\mathcal{O}(Spec R[f^{-1}]) = R[f^{-1}]$ and the restrictions $\mathcal{O}(Spec R)\to\mathcal{O}(Spec R[f^{-1}])$ and $\mathcal{O}(Spec R[g^{-1}])\to\mathcal{O}(Spec R[f^{-1}])$ where $f$ divides $g$ are the corresponding localizations of rings. If the prime spectrum is taken with a structure of a locally ringed space then one may also say the affine spectrum (this terminology never used just for the underlying topological space).

One obtains a sheaf of rings whose stalks are local rings. Every locally ringed space isomorphic to an affine spectrum is said to be an [[affine scheme]].

## Related concepts

* [[projective spectrum]]

* [[spectrum (geometry)]]

  * [[prime spectrum]]

  * [[formal spectrum]]

* [[modules over a ring are equivalent to quasicoherent sheaves over its spectrum]]

* [[prime spectrum of a monoidal stable (∞,1)-category]]

* [[affine scheme]]

* [[spectral topological space]]

## References

* Wikipedia, _[Spectrum of a ring](http://en.wikipedia.org/wiki/Spectrum_of_a_ring)_

[[!redirects affine spectrum]]
[[!redirects spectrum of a ring]]

[[!redirects spectra of commutative rings]]

[[!redirects spectra of rings]]

[[!redirects prime spectrum of a commutative ring]]
[[!redirects prime spectra of commutative rings]]


spectrum of a graph

Spectral theory of graphs, Ihara zeta function, graph Laplacians and Green functions on graphs etc.

Related $n$Lab entries include [[Euler product]], [[graph theory]], [[Selberg zeta function]], [[spectral theory]], [[Laplacian]], [[determinant]]

* H. Bass, _The Ihara-Selberg zeta function of a tree lattice_, Internat. J. Math. __3__ (1992) 717–797 [doi](http://dx.doi.org/10.1142/S0129167X92000357)
* An Huang, Shing-Tung Yau, _Graph invariant from ideas in quantum field theory_, [arxiv/1409.5853](http://arxiv.org/abs/1409.5853)
* wikipedia [Ihara zeta function](http://en.wikipedia.org/wiki/Ihara_zeta_function)
* L. Chekhov, _Spectral problem on graphs and L-functions_, Russ. Math. Surv. 54:6, 1197-1232 (1999) [cond-mat/9911244](http://arxiv.org/abs/cond-mat/9911244) [doi](http://dx.doi.org/10.1070/RM1999v054n06ABEH000231)
* Hirobumi Mizuno, _The scattering matrix of a graph_, [pdf](http://www.emis.de/journals/EJC/Volume_15/PDF/v15i1r96.pdf)
category: combinatorics

[[!redirects spectra of graphs]]
[[!redirects Ihara zeta function]]


spectrum of a tensor triangulated category

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## Idea

The concept of [[tensor triangulated category]] may be thought of as a [[categorification]] of the concept of a [[ring]] in that the [[monoidal category]] structure refines the [[monoid]] structure of a ring, and the fact that it is a [[triangulated category]] and hence [[stable (infinity,1)-category|stable]] makes it the [[homotopy theory]] analog of a monoid in [[abelian groups]], hence of a ring.

Accordingly, one may consider the [[prime spectrum]] of a tensor triangulated category in direct analogy: the [[prime ideals]] are taken to be the [[thick subcategories]] which behave as prime ideals under the [[tensor product]], in the obvious way.  ([Thomason 97](#Thomason97), [Balmer 02](#Balmer02)).

In terms of modern [[stable (∞,1)-category]] theory this may be understood as being the shadow on [[homotopy categories]] of the  _[[prime spectrum of a symmetric monoidal stable (∞,1)-category]]_.

A typical application of the spectrum construction for tensor triangulated categories is to the [[derived categories]] of [[quasicoherent sheaves]] of [[algebraic varieties]]. If one restricts to [[perfect complexes]] then often the variety may be reconstructed from the spectrum of its category of coherent sheaves. More generally this reconstruction theory applies to [[geometric stacks]], see at _[[Tannaka duality for geometric stacks]]_.

In some good cases such reconstruction works even when forgetting the monoidal structure ([Bondal-Orlov 03](#BondalOrlov03)):
The reconstruction theorems of nice classes of schemes from the abelian categories of (quasi)coherent sheaves have versions from weaker data of the corresponding derived categories, viewed as [[triangulated category|triangulated]] or [[enhanced triangulated categories]] (see [[triangulated categories of sheaves]]). This is important for study of deformations, [[homological mirror symmetry]] and noncommutative generalizations.

## Related concepts

* [[spectrum of an abelian category]]

* [[prime spectrum of a symmetric monoidal stable (∞,1)-category]]

## References

An important theorem of reconstruction using the triangulated category (but not the monoidal structure) is the [[Bondal-Orlov reconstruction theorem]]:

* {#BondalOrlov03} [[Aleksei Bondal]], [[Dmitri Orlov]], _Reconstruction of a variety from the derived category and groups of autoequivalences_, Compositio Mathematica __125__ (03), 327-344 [doi](http://dx.doi.org/10.1023/A:1002470302976), [pdf](http://www.mi.ras.ru/~orlov/papers/Compositio2001.pdf)

Several spectra for triangulated categories were systematically studied by Rosenberg:

* [[A. L. Rosenberg]], _Geometry of Noncommutative 'Spaces' and Schemes_, [pdf](http://www.mpim-bonn.mpg.de/preblob/5148)

Along the lines of the general [[spectral cookbook]], these spectra are obtained considering certain preorders on objects of the triangulated category. The preorders can be easily modified to respect additional structure, say a monoidal product, if there.

In this vein, Thomason, Balmer, Garkusha and others considered instead tensor triangulated categories and corresponding [[spectral theory]], as well as stronger reconstruction theorems of schemes from derived categories of coherent sheaves (or, in some cases, perfect complexes) taking into account the monoidal structures:

* {#Thomason97} [[R. W. Thomason]], _The classification of triangulated subcategories_, Compositio Math. __105__(1):1–27, 1997 [MR98b:18017](http://www.ams.org/mathscinet-getitem?mr=1436741) [doi](http://dx.doi.org/10.1023/A:1017932514274)

* {#Balmer02} [[Paul Balmer]], _Presheaves of triangulated categories and reconstruction of schemes_, Mathematische Annalen __324__:3 (2002), 557-580 [dvi](https://www.math.ucla.edu/~balmer/Pubfile/Reconstr.dvi), [pdf](https://www.math.ucla.edu/~balmer/Pubfile/Reconstr.pdf) [ps](https://www.math.ucla.edu/~balmer/Pubfile/Reconstr.ps); _The spectrum of prime ideals in tensor triangulated categories_, J. Reine Angew. Math. __588__:149–168, 2005 [](http://www.math.ucla.edu/~balmer/Pubfile/Spectrum.dvi) [pdf](http://www.math.ucla.edu/~balmer/Pubfile/Spectrum.pdf) [ps](http://www.math.ucla.edu/~balmer/Pubfile/Spectrum.ps); _Spectra, spectra, spectra - Tensor triangular spectra versus Zariski spectra of endomorphism rings_, Alg. and Geom. Topology __10__:3 (2010) 1521-1563 [dvi](http://www.math.ucla.edu/~balmer/Pubfile/SSS.dvi) [pdf](http://www.math.ucla.edu/~balmer/Pubfile/SSS.pdf) [ps](http://www.math.ucla.edu/~balmer/Pubfile/SSS.ps)
* [[Grigory Garkusha]], M. Prest, _Reconstructing projective schemes from Serre subcategories_, J. Algebra __319__(3) (2008), 1132-1153, [pdf](http://www.swansea.ac.uk/media/garkpr44.pdf); _Triangulated categories and the Ziegler spectrum_, Algebras Repr. Theory 8 (2005), 499-523 [pdf](http://www.swansea.ac.uk/media/gark-pr2.pdf)

* {#Balmer04} [[Paul Balmer]], _The spectrum of prime ideals in tensor triangulated categories_. J. Reine Angew. Math., 588:149–168, 2005 ([arXiv:math/0409360](http://arxiv.org/abs/math/0409360))

* {#Balmer10} [[Paul Balmer]], _Spectra, spectra, spectra—tensor triangular spectra versus Zariski spectra of endomorphism rings_, Algebr. Geom. Topol., 10(3):1521–1563, 2010 ([pdf](http://www.math.ucla.edu/~balmer/research/Pubfile/SSS.pdf))

* [[Greg Stevenson]], _Tensor actions and locally complete intersection_ PhD thesis 2011 ([pdf](http://www.math.uni-bielefeld.de/~gstevens/Stevenson_thesis.pdf))

This line has been enhanced by exploring the theory of [[locales]] (for the case of quasicompact quasiseparated (i.e. coherent) schemes and derived category of [[perfect complexes]]) in

* [[Joachim Kock]], Wolfgang Pitsch, _Hochster duality in derived categories and point-free reconstruction of schemes_, [arxiv/1305.1503](http://arxiv.org/abs/1305.1503)

Further discussion in the context of [[2-algebraic geometry]] is in

* {#Brandenburg14} [[Martin Brandenburg]], _Tensor categorical foundations of algebraic geometry_ ([arXiv:1410.1716](http://arxiv.org/abs/1410.1716))

For the [[equivariant stable homotopy theory|equivariant stable homotopy category]]:

* [[Paul Balmer]], [[Beren Sanders]], _The spectrum of the equivariant stable homotopy category of a finite group_ ([arXiv:1508.03969](http://arxiv.org/abs/1508.03969))

In relation to [[Drinfeld centers]]:

* Kent B. Vashaw, _Balmer spectra and Drinfeld centers_ ([arXiv:2010.11287](https://arxiv.org/abs/2010.11287))

[[!redirects spectra of tensor triangulated categories]]

[[!redirects spectrum of a triangulated category]]
[[!redirects spectra of triangulated categories]]

category: geometry, noncommutative geometry, triangulated categories


spectrum of an abelian category

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## Idea

The __spectrum of an abelian category__ (or Rosenberg's spectrum) is a set attached to a svelte (equivalent to small) [[abelian category]] generalizing the left spectrum of a noncommutative ring. It can be equipped with natural Zariski topology and a stack of [[local abelian category|local abelian categories]], similar to the picture of locally ringed spaces. It has been invented by A. L. Rosenberg in 1980-s.

## Definition

Let $A$ be a svelte abelian category.
One first defines a preorder $\succ$ on the class of nonzero objects in $A$ by setting $M\succ N$ if $N$ is a [[subquotient]] of the direct sum of finitely many copies of $M$. In other words, $M\succ N$ if there exists a positive number $k$ and a subobject $U$ of a direct sum $\oplus_{i = 1}^k M$ of $k$ copies of $M$ and an epimorphism from $U$ to $N$.

As a set, spectrum $\mathbf{Spec}\,A$ of an abelian category $A$, can be described as a set of equivalence classes of [[topologizing subcategories]] $[M]$ generated by objects $M$ in $A$ such that $L\succ M$ for any nonzero subobject $L$ of $M$, equipped with a specialization preorder.

If $A = Qcoh(X)$ where $X$ is the abelian category of quasicoherent sheaves of $\mathcal{O}_X$ modules on a [[quasiseparated]] [[quasicompact]] scheme, then the scheme $X$ can be reconstructed up to isomorphisms in two steps. First one constructs the spectrum $\mathbf{Spec} Qcoh(X)$ as a topological space with a stack of local abelian categories. Then one does fibrewise the construction of a [[center of an abelian category]], that is replacing each fibre (which is an abelian category) by the commutative ring of endotransformations of the identity functor and sheafifies.
The Rosenberg's version of the [[Gabriel-Rosenberg reconstruction]] of commutative schemes states that the result is isomorphic to the original scheme. One can generalize to arbitrary schemes possessing a cover by affines whose inclusions have a direct image functor, but then one needs to use a different spectrum.

## Related concepts

* [[spectrum of a tensor triangulated category]]

## References

* [[Alexander Rosenberg]], _Geometry of Noncommutative 'Spaces' and Schemes_, MPIM 2011-68, [pdf](http://www.mpim-bonn.mpg.de/preblob/5148)

* [[A. L. Rosenberg]], _Underlying spaces of noncommutative spaces_, [dvi](http://www.mpim-bonn.mpg.de/preblob/1947), [ps](http://www.mpim-bonn.mpg.de/preblob/1948)

* [[A. L. Rosenberg]], _Topics in noncommutative algebraic geometry, homological algebra and K-theory_, preprint MPIM Bonn 2008-57 [pdf](http://www.mpim-bonn.mpg.de/preblob/3589)

* [[A. L. Rosenberg]], _The spectrum of abelian categories and reconstructions of schemes_, in Rings, Hopf Algebras, and Brauer groups, Lectures Notes in Pure and Appl. Math. __197__, Marcel Dekker, New York, 257--274, 1998; MR99d:18011; and Max Planck Bonn preprint _Reconstruction of Schemes_, MPIM1996-108, [pdf](http://www.mpim-bonn.mpg.de/preblob/3948) (1996).

* [[A. L. Rosenberg]], _Spectra of noncommutative spaces_, MPIM2003-110 [ps](http://www.mpim-bonn.mpg.de/preblob/1946) [dvi](http://www.mpim-bonn.mpg.de/preblob/1945) (2003)

* [[A. L. Rosenberg]], _Noncommutative schemes_, Compos. Math. __112__ (1998) 93--125 ([doi](http://dx.doi.org/10.1023/A:1000479824211))

* [[A. L. Rosenberg]], _The left spectrum, the Levitzki radical, and noncommutative schemes_, Proc. Nat. Acad. Sci. U.S.A. __87__  (1990),  no. 21, 8583--8586.

* [[A. L. Rosenberg]], _Noncommutative local algebra_,  Geom. Funct. Anal. __4__  (1994),  no. 5, 545--585.

* [[A. L.  Rosenberg]], _Noncommutative algebraic geometry and representations of quantized algebras_, MIA __330__, Kluwer Academic Publishers Group, Dordrecht, 1995. xii+315 pp. ISBN: 0-7923-3575-9

category: algebraic geometry, noncommutative geometry


spectrum of an operator

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## Idea

For a [[linear operator]] $A$ on a [[finite-dimensional vector space|finite-dimensional]] [[complex vector space]] $V$, the spectrum of $A$ is simply the [[subset]] of the [[complex numbers]] consisting of the [[eigenvalues]] of $A$.

In the case that $V$ is an infinite-dimensional complex [[separable Hilbert space]], the ([[normal eigenvalue|normal]]) eigenvalues only form the *discrete spectrum*.  Instead, the full spectrum is the set of all $\lambda \in \mathbb{C}$ for which the [[resolvent]] $(A-\lambda I)^{-1}$ is not a [[bounded operator]].

In other words, the spectrum is the [[complement]] of the subset of complex numbers for which the resolvent is a [[bounded operator]].

## Properties

If $A$ is a [[bounded linear operator]] on a complex separable Hilbert space, then the spectrum $\sigma(A)$ is a [[compact subset]] of $\mathbb{C}$.

The set $\sigma_d(A)$ of ordinary [[normal eigenvalues|normal]] [[eigenvalues]] of $A$ is a subset of $\sigma(A)$ called the *discrete spectrum* of $A$. In particular case when $A$ is a [[compact operator]] the spectrum consists of the discrete spectrum only, except for possible addition of point $0$.

## Related concepts

* [[spectral theory]]

* [[min-entropy]]

## References

See also

* Wikipedia, *Spectrum (functional_analysis)*

[[!redirects operator spectrum]]
[[!redirects spectrum of operator]]
[[!redirects spectrum of a linear operator]]
[[!redirects spectrum of linear operator]]

[[!redirects discrete spectrum]]
[[!redirects discrete spectra]]


spectrum with G-action

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## Idea

For $G$ a [[topological group]], a _spectrum with $G$-action_ is a [[spectrum]] equipped with an [[action]] of $G$. In other words, it is a functor from $BG$ to the [[(infinity,1)-category]] of spectra. The [[stable homotopy theory]] of spectra with $G$-action is part of the subject of [[equivariant stable homotopy theory]].

Spectra with $G$-action are sometimes called "doubly naive" $G$-spectra. They are even more naive than [[naive G-spectra]], and can be identified with the full subcategory of [[G-spectra]] consisting of "Borel-complete" $G$-spectra (see Prop. 6.17 in [MNN](#MNN) or Thm. II.2.7 in [NS](#NikolausScholze)).

## See also

* [[naive G-spectra]], [[G-spectra]]

* [[G-spaces]]

* [[equivariant stable homotopy theory]]

## References

* {#MNN} [[Akhil Mathew]], [[Niko Naumann]], [[Justin Noel]], _Nilpotence and descent in equivariant stable homotopy theory_, Adv. Math. 305 (2017), 994--1084, [arXiv:1507.06869](https://arxiv.org/abs/1507.06869).

* {#NikolausScholze} [[Thomas Nikolaus]], [[Peter Scholze]], _On topological cyclic homology_, [arXiv:1707.01799](https://arxiv.org/abs/1707.01799).

[[!redirects spectra with G-action]]