sequential spectrum

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###Context###
#### Stable Homotopy theory
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[[!include stable homotopy theory - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

In [[stable homotopy theory]], a _sequential ([[prespectrum|pre-]])[[spectrum]]_ $E$ (also _Boardman spectrum_, after ([Boardman 65](#Boardman65))) is a sequence of [[pointed homotopy types]] ([[pointed topological spaces]], pointed [[simplicial sets]]) $E_n$, for $n \in \mathbb{N}$, together with maps $\Sigma E_n \to E_{n+1}$ from the [[reduced suspension]] of one into the next space in the sequence.

This is the original definition of _[[spectrum]]_ (or [[pre-spectrum]]), and still, the one predominantly meant to be default, as used in, say, the [[Brown representability theorem]]. But in view of many other definitions (all giving rise to equivalent [[stable homotopy theory]]) that involve systems of spaces indexed on more than just the integers (such as [[coordinate-free spectra]], [[excisive functors]], [[equivariant spectra]]) or that are of different flavor altogether (such as [[combinatorial spectra]]), one says _sequential spectrum_ for emphasis (e.g. [Schwede 12, def. 2.1](#Schwede12)).

## Definition

### In components

In what follows, [[sSet]] denotes the [[category]] of [[simplicial sets]] and $sSet^{\ast/}$ the category $\ast/sSet$ of [[pointed objects|pointed]] simplicial sets (the [[undercategory]] under the [[terminal object]] $\ast$), which may be thought of as a possible base of [[enriched category|enrichment]].

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

A _sequential_ [[pre-spectrum]] in [[simplicial sets]], is an $\mathbb{N}$-[[graded object|graded]] [[pointed object|pointed]] [[simplicial set]] $X_\bullet$ equipped with morphisms $\sigma_n \colon S^1 \wedge X_n \to X_{n+1}$ for all $n \in \mathbb{N}$, where $S^1 \coloneqq \Delta[1]/\partial\Delta[1]$ is the minimal simplicial [[circle]], and where $\wedge$ is the [[smash product]] of [[pointed objects]].

A [[homomorphism]] $f \colon X \to Y$ of sequential prespectra is a collection $f_\bullet \colon X_\bullet \to Y_\bullet$ of homomorphisms of [[pointed simplicial sets]], such that all [[diagrams]] of the form

$$
  \array{
    S^1 \wedge X_n &\stackrel{S^1 \wedge f_n}{\longrightarrow}& S^1 \wedge Y_n
    \\
    \downarrow^{\mathrlap{\sigma_n^X}} && \downarrow^{\mathrlap{\sigma_n^Y}}
    \\
    X_{n+1} &\stackrel{f_{n+1}}{\longrightarrow}& Y_{n+1}
  }
$$

[[commuting diagram|commute]].

This gives a [[category]] $SeqSpec(sSet)$ of sequential prespectra.

=--

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

For $X \in SeqSpec(sSet)$ and $K \in $ [[sSet]], hence $K_+ \in sSet^{\ast/}$ then $X \wedge K_+$ is the spectrum which is degreewise given by the [[smash product]] of [[pointed objects]]

$$
  (X \wedge K_+)_n \coloneqq (X_n \wedge K_+)
$$

And whose structure maps are given by

$$
  S^1 \wedge (X_n \wedge K_+) \simeq (S^1 \wedge X_n) \wedge K_+ \stackrel{\sigma_n \wedge K_+}{\longrightarrow} X_{n+1}\wedge K_+
  \,.
$$

=--

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

The category $SeqSpec$ of def. \ref{SequentialSpectra} becomes a [[simplicially enriched category]] (in fact an $sSet^{\ast/}$-[[enriched category]]) with [[hom objects]] $[X,Y]\in sSet$ given by

$$
  [X,Y]_n \coloneqq Hom_{SeqSpec(sSet)}(X\wedge \Delta[n]_+,Y)
  \,.
$$

=--

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

An _[[Omega-spectrum]]_ in the following is a sequential prespectrum $X$, def. \ref{SequentialSpectra}, such that after [[geometric realization]]/[[Kan fibrant replacement]] ${\vert -\vert}$ the smash$\dashv$pointed-hom [[adjuncts]]

$$
  {\vert X_n\vert} \stackrel{}{\longrightarrow} {\vert X^{n+1}\vert}^{{\vert S^1\vert}}
$$

of the structure maps ${\vert \sigma_n\vert}$ are [[weak homotopy equivalences]].

=--

### As diagram spectra
 {#AsDiagramSpectra}

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

Write $S^1_{std} \coloneqq \Delta[1]/\partial\Delta[1]\in sSet^{\ast/}$ for the standard minimal pointed simplicial [[1-sphere]].

Write

$$
  \iota \;\colon\; StdSpheres \longrightarrow sSet^{\ast/}_{fin}
$$

for the non-full $sSet^{\ast/}$-[[enriched category|enriched]] [[subcategory]] of pointed [[simplicial object|simplicial]] [[finite sets]], def. \ref{SimplicialSetsPointedAndFinite} whose

* [[objects]] are the [[smash product]] powers $S^n_{std} \coloneqq (S^1_{std})^{\wedge^n}$ (the standard minimal simplicial [[n-spheres]]);

* [[hom-objects]] are

  $$
    [S^{n}_{std}, S^{n+k}_{std}]_{StdSpheres}
    \coloneqq
    \left\{
      \array{
        \ast & for & k \lt 0
        \\
        im(S^{k}_{std} \stackrel{}{\to} [S^n_{std}, S^{n+k}_{std}]_{sSet^{\ast/}_{fin}})
        & otherwise
      }
    \right.
  $$

=--

([Lydakis 98, def. 4.2](#Lydakis98)), see also ([MMSS 00](#MMSS00)) [this example](Model+categories+of+diagram+spectra#SequentialSpectraAsFunctorsOnFreeSSequModules).

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

There is an $sSet^{\ast/}$-[[enriched functor]]

$$
  (-)^seq
  \;\colon\;
  [StdSpheres,sSet^{\ast/}]
    \longrightarrow
  SeqSpec(sSet)
$$

(from the category of $sSet^{\ast/}$-[[enriched presheaves|enriched copresheaves]] on the categories of standard simplicial spheres of def. \ref{CategoriesOfStandardSpheres} to the category of [[sequential prespectra]] in [[sSet]]) given on objects by sending $X \in [StdSpheres,sSet^{\ast/}]$ to the sequential prespectrum $X^{seq}$ with components

$$
  X^{seq}_n \coloneqq X(S^n_{std})
$$

and with structure maps

$$
  \frac{S^1_{std} \wedge X^{seq}_n \stackrel{\sigma_n}{\longrightarrow} X^{seq}_n}{S^1_{std} \longrightarrow [X^{seq}_n, X^{seq}_{n+1}]}
$$

given by

$$
  S^1_{std}
    \stackrel{\widetilde{id}}{\longrightarrow}
  [S^n_{std}, S^{n+1}_{std}]
    \stackrel{X_{S^n_{std}, S^{n+1}_{std}}}{\longrightarrow}
  [X^{seq}_n, X^{seq}_{n+1}]
  \,.
$$

This is an $sSet^{\ast/}$ [[enriched category theory|enriched]] [[equivalence of categories]].

=--

([Lydakis 98, prop. 4.3](#Lydakis98)), see also ([MMSS 00](#MMSS00))

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

Prop. \ref{SequentialSpectraAsDiagramSpectra} is a special case of a more general statement expressing structured spectra equivalently as enriched functors. Analogous statements hold for [[symmetric spectra]] and [[orthogonal spectra]]. See at _[[Model categories of diagram spectra]]_ [this lemma](Model+categories+of+diagram+spectra#SModulesAsEnrichedFunctors) and [this example](Model+categories+of+diagram+spectra#SequentialSpectraAsFunctorsOnFreeSSequModules).

=--

## Properties

### Limits and colimits

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

The category $SeqSpec(Top_{cq})$ of sequential spectra (def. \ref{SequentialSpectra}) has all [[limits]] and [[colimits]], and they are computed objectwise:

Given

$$
  X_\bullet
    \;\colon\;
  I
    \longrightarrow
  SeqSpec(Top_{cg})
$$

a [[diagram]] of sequential spectra, then:

1. its colimiting spectrum has component spaces the colimit of the component spaces formed in $Top_{cg}$ (via [this prop.](Introduction+to+Stable+homotopy+theory+--+P#DescriptionOfLimitsAndColimitsInTop) and [this corollary](Introduction+to+Stable+homotopy+theory+--+P#kTopIsCoreflectiveSubcategory)):

   $$
     (\underset{\longrightarrow}{\lim}_i X(i))_n
     \simeq
      \underset{\longrightarrow}{\lim}_i X(i)_n
      \,,
   $$

1. its limiting spectrum has component spaces the limit of the component spaces formed in $Top_{cg}$ (via [this prop.](Introduction+to+Stable+homotopy+theory+--+P#DescriptionOfLimitsAndColimitsInTop) and [this corollary](Introduction+to+Stable+homotopy+theory+--+P#kTopIsCoreflectiveSubcategory)):

   $$
     (\underset{\longleftarrow}{\lim}_i X(i))_n
     \simeq
      \underset{\longleftarrow}{\lim}_i X(i)_n
      \,;
   $$

Moreover:

1. the colimiting spectrum has structure maps in the sense of def. \ref{SequentialSpectra} given by

   $$
     S^1 \wedge (\underset{\longrightarrow}{\lim}_i X(i)_n)
     \simeq
     \underset{\longrightarrow}{\lim}_i ( S^1 \wedge X(i)_n )
     \overset{\underset{\longrightarrow}{\lim}_i \sigma_n^{X(i)}}{\longrightarrow}
     \underset{\longrightarrow}{\lim}_i X(i)_{n+1}
   $$

   where the first isomorphism exhibits that $S^1 \wedge(-)$ preserves all colimits, since it is a [[left adjoint]] by prop. \ref{SuspensionAndLoopAdjunctionInClassicalHomotopyTheory};

1. the limiting spectrum has adjunct structure maps in the sense of def. \ref{SequentialSpectrumViaAdjunctStructureMaps} given by

   $$
     \underset{\longleftarrow}{\lim}_i X(i)_n
      \overset{\underset{\longleftarrow}{\lim}_i \tilde \sigma_n^{X(i)}}{\longrightarrow}
     \underset{\longleftarrow}{\lim}_i Maps(S^1, X(i)_n)_\ast
     \simeq
     Maps(S^1, \underset{\longleftarrow}{\lim}_i X(i)_n)_\ast
   $$

   where the last isomorphism exhibits that $Maps(S^1,-)_\ast$ preserves all limits, since it is a [[right adjoint]] by prop. \ref{SuspensionAndLoopAdjunctionInClassicalHomotopyTheory}.

=--

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

That the limits and colimits exist and are computed objectwise follows via prop. \ref{SequentialSpectraAsDiagramSpectra} from the general statement for categories of topological functors ([prop.](Introduction+to+Stable+homotopy+theory+--+P#TopologicallyEnrichedCopresheavesHaveAllLimitsAndColimits)). But it is also immediate to directly check the [[universal property]].

=--

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

The [[coproduct]] of [[spectra]] $X, Y \in SeqSpec(Top_{cg})$, called the **wedge sum of spectra**

$$
  X \vee Y
    \coloneqq
  X \sqcup Y
$$

is componentwise the [[wedge sum]] of pointed topological spaces ([exmpl.](Introduction+to+Stable+homotopy+theory+--+P#WedgeSumAsCoproduct))

$$
  (X \vee Y)_n
  =
  X_n \vee Y_n
$$

with structure maps

$$
  \sigma_n^{X \vee Y}
    \;\colon\;
  S^1 \wedge (X \vee Y)
    \simeq
  S^1 \wedge X \,\vee\, S^1 \wedge Y
    \overset{(\sigma_n^X, \sigma_n^Y)}{\longrightarrow}
  X_{n+1} \vee Y_{n+1}
  \,.
$$

=--

### Tensoring and powering over pointed spaces

The following defines [[tensoring]] and [[powering]] of sequential spectra over [[pointed topological spaces]]/[[pointed simplicial sets]].

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

Let $X$ be a sequential spectrum and $K$ a [[pointed topological space]]/[[pointed simplicial set]]. Then

1. $X \wedge K$ is the sequential spectrum with

   * $(X \wedge K)_n \coloneqq X_n \wedge K$ ([[smash product]])

   * $\sigma_n^{X\wedge K} \coloneqq \sigma_n^{X} \wedge id_{K}$.

1. $X^K$ is the sequential spectrum with

   * $(X^K)_n \coloneqq (X_n)^K$ ([[pointed mapping space]])

   * $\sigma_n^{(X^k)} \colon S^1 \wedge X_n^K \to (S^1 \wedge X_n)^K \overset{(\sigma_n)^K}{\longrightarrow} (X_{n+1})^K$.

=--

### Model category structures
 {#ModelCategoryStructures}

There is a standard [[model structure on spectra]] for sequential spectra in [[Top]] the [[model structure on topological sequential spectra]] ([Kan 63](#Kan63), [MMSS 00](#MMSS00))  and in [[simplicial sets]], the [[Bousfield-Friedlander model structure]] ([Bousfield-Friedlander 78](#BousfieldFriedlander78)).

The _strict_ Bousfield-Friedlander model structure (of which the actual stable version is the [[Bousfield localization of model categories|left Bousfield localization]] at the [[stable weak homotopy equivalences]]) is equivalently the [[projective model structure on enriched functors]] for the presentation of sequential spectra from prop. \ref{SequentialSpectraAsDiagramSpectra}:

$$
  SeqSpec(sSet)_{stable}
  \stackrel{\longleftarrow}{\overset{Bousf.\;loc}{\longrightarrow}}
  SeqSpec(sSet)_{strict}
  =
  [StdSpheres, Top^{\ast/}_{Quillen}]_{proj}
  \,.
$$

### Suspension and looping
 {#SuspensionAndLooping}

There are _three_ common constructions of looping and suspension of sequential spectra (with analogues for [[highly structured spectra]]). While they are not isomorphic, they are stably equivalent.

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

For $X$ a [[sequential spectrum]] and $k \in \mathbb{Z}$, the $k$-fold **shifted spectrum** of $X$ is the sequential spectrum denoted $X[k]$ given by

* $(X[k])_n \coloneqq \left\{ \array{X_{n+k} & for \; n+k \geq 0 \\ \ast & otherwise } \right. $;

* $\sigma_n^{X[k]} \coloneqq \left\{ \array{ \sigma^X_{n+k} & for \; n+k \geq 0 \\ 0 & otherwise} \right. $.

=--

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

For $X$ a sequential spectrum, then

1. the **real suspension** of $X$ is $X \wedge S^1$ according to def. \ref{TensoringAndPoweringOfSequentialSpectra};

1. the **real looping** of $X$ is $X^{S^1}$ according to def. \ref{TensoringAndPoweringOfSequentialSpectra}.

=--

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

For $X$ a sequential spectrum, then

1. the **fake suspension** of $X$ is the sequential spectrum $\Sigma X$ with

   1. $(\Sigma X)_n \coloneqq S^1 \wedge X_n$

   1. $\sigma_n^{\Sigma X} \coloneqq S^1 \wedge (\sigma_n)$.

1. the **fake looping** of $X$ is the sequential spectrum $\Omega X$ with

   1. $(\Omega X)_n \coloneqq (X_n)^{S^1}$;

   1. $\tilde \sigma_n^{\Omega X} \coloneqq (\sigma_n)^{S^1}$.

Here $\tilde \Sigma_n$ denotes the $(\Sigma\dashv \Omega)$-[[adjunct]] of $\sigma_n$.

=--

e.g. ([Jardine 15, section 10.4](#Jardine15)).

+-- {: .num_defn #ShiftingCommutesWithLoopingAndSuspensionOfSequentialSpectra}
###### Remark

The looping and suspension operations in def. \ref{SequentialSpectrumRealSuspension} and def. \ref{SequentialSpectrumFakeSuspension} commute with shifting, def. \ref{ShiftedSpectrum}. Therefore in expressions like $\Sigma (X[1])$ etc. we may omit the parenthesis.

=--

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

The canonical morphism

1. $\Sigma X \longrightarrow X[1]$ is given in degree $n$ by $\sigma_n^X$.

1. $X[-1] \longrightarrow \Omega X$ is given in degree $n$ by $\tilde \sigma^X_{n-1}$.

=--

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

The constructions from def. \ref{ShiftedSpectrum}, def. \ref{SequentialSpectrumRealSuspension} and def. \ref{SequentialSpectrumFakeSuspension} form pairs of [[adjoint functors]] $SeqSpec \to SeqSpec$ like so:

1. $(-)[1] \;\dashv\; (-)[-1] \;\dashv\; (-)[1] \;\dashv\; \cdots $;

1. $(-)\wedge S^1 \dashv (-)^{S^1}$;

1. $\Sigma \dashv \Omega$.

=--

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

The first is immediate from the definition.

The second is just degreewise the adjunction [[smash product]]$\dashv$[[pointed mapping space]] (discussed [here](pointed+object#ClosedMonoidalStructure)), since by definition the smash product and mapping spaces here do not interact non-trivially with the structure maps.

The third follows by applying the [[smash product]]$\dashv$[[pointed mapping space]]-adjunction isomorphism twice, like so:

Morphisms $f\colon \Sigma X \to Y$ are in components given by commuting diagrams of this form:

$$
  \array{
    S^1 \wedge S^1 \wedge X_{n}
      &\overset{S^1 \wedge f_{n}}{\longrightarrow}&
    S^1 \wedge Y_{n}
    \\
    {}^{\mathllap{S^1 \wedge \sigma_n^X}}\downarrow
     &&
    \downarrow^{\mathrlap{\sigma^Y_n}}
    \\
    S^1 \wedge X_{n+1} &\underset{f_{n+1}}{\longrightarrow}& Y_{n+1}
  }
  \,.
$$

Applying the adjunction isomorphism diagonally gives a bijection to diagrams of this form:

$$
  \array{
   S^1 \wedge X_n &\overset{f_n}{\longrightarrow}& Y_n
   \\
   {}^{\mathllap{\sigma^X_n}}\downarrow && \downarrow^{\mathrlap{\tilde \sigma^Y_n}}
   \\
   X_{n+1} &\underset{\tilde f_{n+1}}{\longrightarrow}& (Y_{n+1})^{S^1}
  }
  \,.
$$

Then applying the same isomorphism diagonally once more gives a further bijection to  commuting diagrams of this form:

$$
  \array{
    X_n &\overset{\tilde f_n}{\longrightarrow}& (Y_n)^{S^1}
    \\
    {}^{\mathllap{\tilde \sigma_n}}\downarrow
      &&
    \downarrow^{\mathrlap{(\tilde \sigma^Y_n)^{S^1}}}
    \\
    (X_{n+1})^{S^1}
    &\underset{(\tilde f_n)^{S^1}}{\longrightarrow}&
    \left((Y_{n+1})^{S^1}\right)^{S^1}
  }
  \,.
$$

This finally equivalently exhibits morphisms of the form

$$
  X \longrightarrow \Omega Y
  \,.
$$

=--

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

For $X$ a sequential spectrum, then $X[-1][1] = X$ while $X[1][-1]$ is $X$ with its 0-th component space set to the point. The [[adjunction unit]] $X \to X[1][-1]$ has components

$$
  \array{
     \vdots && \vdots
     \\
     X_2 &\overset{id}{\longrightarrow}& X_2
     \\
     X_1 &\overset{id}{\longrightarrow}& X_1
     \\
     \underbrace{X_0} &\overset{0}{\longrightarrow}& \underbrace{\;\ast\;}
     \\
     X &\overset{\eta}{\longrightarrow}& X[1][-1]
  }
  \,.
$$

=--

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

For $X$ a sequential spectrum, then (using remark \ref{ShiftingCommutesWithLoopingAndSuspensionOfSequentialSpectra} to suppress parenthesis)

1. the structure maps constitute a homomorphism

   $$
     \Sigma X[-1]
     \longrightarrow
     X
   $$

   and this is a stable equivalence.

1. the adjunct structure maps constitute a homomorphism

   $$
     X
     \longrightarrow
     \Omega X[1]
     \,.
   $$

   If $X$ is an [[Omega-spectrum]] (def. \ref{OmegaSpectrum}) then this is a weak equivalence in the strict model structure, hence in particular, a stable equivalence.

=--

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

The diagrams that need to commute for the structure maps to give a homomorphism as claimed are in degree 0 this one

$$
  \array{
    S^1 \wedge S^1 \wedge \ast &\overset{0}{\longrightarrow}& X_0
    \\
    {}^{\mathllap{S^1 \wedge 0}}\downarrow && \downarrow^{\mathrlap{\sigma_0}}
    \\
    S^1 \wedge X_0 &\underset{\sigma_0}{\longrightarrow}& X_1
  }
$$

and in degree $n \geq 1$ these:

$$
  \array{
     S^1 \wedge S^1 \wedge X_{n-1}
        &\overset{S^1 \wedge \sigma_{n-1}}{\longrightarrow}&
     X_n
     \\
     {}^{\mathllap{S^1 \wedge \sigma_{n-1}}}\downarrow
       &&
     \downarrow^{\mathrlap{\sigma_n}}
     \\
     S^1 \wedge X_{n}
       &\underset{\sigma_n}{\longrightarrow}&
     X_{n+1}
  }
  \,.
$$

But in all these cases, commutativity it trivially satisfied.

Now as in the proof of prop. \ref{AdjunctionsBetweenLoopingAndDeloopingForSeqSpec}, under applying the $(S^1\wedge (-)) \dashv (-)^{S^1}$-adjunction isomorphism twice, these diagrams are in bijection to diagrams for $n \geq 1$ of the form

$$
  \array{
    X_{n-1} &\overset{\tilde \sigma_{n-1}}{\longrightarrow}& (X_n)^{S^1}
    \\
    {}^{\mathllap{\tilde \sigma_{n-1}}}\downarrow
      &&
    \downarrow^{\mathrlap{\tilde \sigma_n}}
    \\
    (X_n)^{S^1}
      &\underset{(\tilde \sigma_n)^{S^1}}{\longrightarrow}&
    \left((X_n)^{S^1}\right)^{S^1}
  }
  \,.
$$

This gives the claimed morphism $X \to \Omega X[-1]$.

If $X$ is an [[Omega-spectrum]], then by definition, this last morphism is already a weak equivalence in the strict model structure, hence in particular a weak equivalence in the stable model structure.

From this, it follows that also the first morphism is a stable equivalence, because for every [[Omega-spectrum]] $Y$, then by the adjunctions in prop. \ref{AdjunctionsBetweenLoopingAndDeloopingForSeqSpec}

$$
  \array{
     [X, Y]_{strict} &\overset{}{\longrightarrow}& [\Sigma X[-1],Y]_{strict}
     \\
     {}^{\mathllap{id}}\downarrow && \downarrow^{\mathrlap{\simeq}}
     \\
     [X,Y]_{strict}  &\underset{\simeq}{\longrightarrow}& [X, \Omega Y[1]]_{strict}
  }
  \,.
$$

=--

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

For $X$, a sequential spectrum in simplicial sets. Then there are stable equivalences

$$
  X\wedge S^1 \longrightarrow \Sigma X \longrightarrow X[1]
$$

between the real suspension (def. \ref{SequentialSpectrumRealSuspension}), the fake suspension (def. \ref{SequentialSpectrumFakeSuspension}) and the shift by +1 (def. \ref{ShiftedSpectrum}) of $X$.

If each $X_n$ is a [[Kan complex]], then there are stable equivalences

$$
  X^{S^1} \longrightarrow \Omega X \longrightarrow X[-1]
$$

between the real looping (def. \ref{SequentialSpectrumRealSuspension}), the fake looping (def. \ref{SequentialSpectrumFakeSuspension}) and the shift by -1 (def. \ref{ShiftedSpectrum}) of $X$.

=--

([Jardine 15, corollary 10.54](#Jardine15))

### Relation to excisive functors

We discuss aspects of the equivalence of sequential spectra carrying the [[Bousfield-Friedlander model structure]] with [[excisive (infinity,1)-functors]], modeled as [[simplicial functors]] carrying a [[model structure for excisive functors]].

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

Write

* [[sSet]] for the [[category]] of [[simplicial sets]];

* $sSet^{\ast/}$ for the category of [[pointed simplicial sets]];

* $sSet_{fin}^{\ast/}\simeq s(FinSet)^{\ast/} \hookrightarrow sSet^{\ast/}$ for the [[full subcategory]] of [[pointed simplicial set|pointed simplicial]] [[finite sets]].

Write

$$
  sSet^{\ast/}
  \stackrel{\overset{(-)_+}{\longleftarrow}}{\underset{u}{\longrightarrow}}
  sSet
$$

for the [[free-forgetful adjunction]], where the [[left adjoint]] functor $(-)_+$ freely adjoins a base point.

Write

$$
  \wedge \colon sSet^{\ast/} \times sSet^{\ast/} \longrightarrow sSet^{\ast/}
$$

for the [[smash product]] of [[pointed object|pointed]] [[simplicial sets]], similarly for its restriction to $sSet_{fin}^{\ast}$:

$$
  X \wedge Y
    \coloneqq
  cofib\left(
    \;
    \left(\,
      (u(X),\ast) \sqcup (\ast, u(Y))
    \,\right)
      \longrightarrow
     u(X) \times u(Y)
    \;
  \right)
  \,.
$$

This gives $sSet^{\ast/}$ and $sSet^{\ast/}_{fin}$ the structure of a [[closed monoidal category]] and we write

$$
  [-,-]_\ast \;\colon\; (sSet^{\ast/})^{op} \times sSet^{\ast/} \longrightarrow sSet^{\ast/}
$$

for the corresponding [[internal hom]], the pointed [[function complex]] functor.

=--

We regard all the categories in def. \ref{SimplicialSetsPointedAndFinite} canonically as [[simplicially enriched categories]], and in fact regard $sSet^{\ast/}$ and $sSet^{\ast/}_{fin}$ as $sSet^{\ast/}$-[[enriched categories]].

The category that supports a [[model structure for excisive functors]] is the  $sSet^{\ast/}$-[[enriched functor category]]

$$
  [sSet^{\ast/}_{fin}, sSet^{\ast/}]
  \,.
$$

([Lydakis 98, example 3.8, def. 4.4](#Lydakis98))

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

The [[adjunction]]

$$
  (\iota_\ast \dashv \iota^\ast)
  \;\colon\;
  [sSet^{\ast/}_{fin}, sSet^{\ast/}]_{Ly}
    \stackrel{\overset{\iota_\ast}{\longleftarrow}}{\underset{\iota^\ast}{\longrightarrow}}
  [StdSpheres, sSet^{\ast/}]
    \underoverset{\simeq}{(-)^{seq}}{\longrightarrow}
  SeqSpec(sSet)_{BF}
$$

(given by restriction $\iota^\ast$ along the defining inclusion $\iota$ of def. \ref{CategoriesOfStandardSpheres} and by left [[Kan extension]] $\iota_\ast$ along $\iota$, and combined with the equivalence $(-)^{seq}$ of prop. \ref{SequentialSpectraAsDiagramSpectra}) is a  [[Quillen adjunction]] and in fact a [[Quillen equivalence]] between the [[Bousfield-Friedlander model structure]] on sequential prespectra and Lydakis' [[model structure for excisive functors]].

=--

([Lydakis 98, theorem 11.3](#Lydakis98)) For more details see at _[[model structure for excisive functors]]_. The analogous statement for spectra in $Top$ is in ([MMSS 00](#MMSS00)).

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

Prop. \ref{QuillenEquivalenceBetweenSequentialBFAndExcisiveFunctors} shows why
plain sequential spectra do not carry a [[symmetric smash product of spectra]]:

By [this remark](smash+product+of+spectra#WhySequentialSpectraHaveNoSymmetricSmashProduct) at _[[smash product of spectra]]_
the graded-commutativity implicit in the [[braiding]] of the [[smash product]] of [[n-spheres]] is not reflected after restricting from $(sSet^{\ast/}, \wedge)$ to the non-full subcategory $StdSpheres$.

=--

### Smash product
 {#SmashProduct}

The [[smash product of spectra]] realized on sequential spectra never has good properties before passage to the [[stable homotopy category]] or lift to better models (see [here](smash%20product+of+spectra#GradedCommutativity)), but it may still be defined in various ways:

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

For $X,Y$ two sequential spectra, def. \ref{SequentialSpectra}, their smash product $X \wedge Y$ is the sequential spectrum which in even degrees is given by the [[smash product]] fo the pointed component spaces of half that degree

$$
  (X\wedge Y)_{2n} \coloneqq X_n \wedge Y_n
$$

and in odd degree by

$$
  (X\wedge Y)_{2n+1} \coloneqq S^1 \wedge X_n \wedge Y_n
$$

with structure maps being in even degree, the identity

$$
  \sigma^{X \wedge Y}_{2 n} \colon S^1 \wedge (X \wedge Y)_{2n} = S^1 \wedge X_n \wedge Y_n = (X \wedge Y)_{2n+1}
$$

and in odd degree as the composite

$$
  \sigma^{X\wedge Y}_{2n+1}
    \colon
  S^1 \wedge (X \wedge Y)_{2n+1}
  \simeq
  S^1 \wedge S^1 \wedge X_n \wedge Y_n
  \simeq
  S^1 \wedge X_n \wedge S^1 \wedge Y_n
  \stackrel{\sigma_n^X \wedge \sigma^Y_n}{\longrightarrow}
  X_{n+1} \wedge Y_{n+1}
  \simeq
  (X\wedge Y)_{2n+2}
  \,.
$$

=--

([Lydakis 98, def. 10.20](#Lydakis98), [Lydakis 98b, def. 5.9](Gamma-space#Lydakis98), [MMSS 00, def. 11.6](#MMSS00))

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

Under the Quillen equivalence of prop.
\ref{QuillenEquivalenceBetweenSequentialBFAndExcisiveFunctors} the symmetric monoidal [[Day convolution product]] on pre-[[excisive functors]] as well as the [[symmetric monoidal smash product of spectra|symmetric monoidal smash product]] of [[orthogonal spectra]] is identified with the  [[smash product of spectra]] realized on sequential spectra via def. \ref{SmashProductByDoublingDegrees}.

=--

([Lydakis 98, theorem 12.5](#Lydakis98), [MMSS 00, prop. 11.9](#MMSS00))

## Related concepts

* [[symmetric spectrum]], [[orthogonal spectrum]]

* [[spectrum object]]

* in [[homotopy type theory]]: [[sequential spectrum type]]

* [[cospectrum]]

## References

* {#Kan63} [[Daniel Kan]], _Semisimplicial spectra_, Illinois J. Math. Volume 7, Issue 3 (1963), 463-478. ([euclid.ijm/1255644953](http://projecteuclid.org/euclid.ijm/1255644953))

* {#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

* {#Switzer75} [[Robert Switzer]], chapter 8 of _Algebraic Topology - Homotopy and Homology_, Die  Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, Vol. 212, Springer-Verlag, New York, N. Y., 1975.

* {#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://people.math.rochester.edu/faculty/doug/otherpapers/bousfield-friedlander.pdf))

 * {#Lydakis98} [[Manos Lydakis]], _Simplicial functors and stable homotopy theory_ Preprint, 1998 (Hopf archive [pdf](http://hopf.math.purdue.edu/Lydakis/s_functors.pdf), [[LydakisSimplicialFunctors.pdf:file]])

  (mostly on a [[model structure on excisive functors]] on [[simplicial sets]])

* {#MMSS00} [[Michael Mandell]], [[Peter May]], [[Stefan Schwede]], [[Brooke Shipley]], section 11 of _[[Model categories of diagram spectra]]_, Proceedings London Mathematical Society Volume 82, Issue 2, 2000 ([pdf](http://www.math.uchicago.edu/~may/PAPERS/mmssLMSDec30.pdf), [publisher](http://plms.oxfordjournals.org/content/82/2/441.short?rss=1&ssource=mfc))

* {#Schwede12} [[Stefan Schwede]], _Symmetric spectra_, 2012 ([pdf](http://www.math.uni-bonn.de/~schwede/SymSpec-v3.pdf))

  > (mostly on [[symmetric spectra]])

* {#Jardine15} [[John F. Jardine]], section 10 of: _[[Local homotopy theory]]_, 2016

  (with regards to [[sheaves of spectra]])

Symmetric spectra in more general [[model categories]] (using the [[Bousfield-Friedlander theorem]]) are discussed in

* {#Schwede97} [[Stefan Schwede]], section 3 of _Spectra in model categories and applications to the algebraic cotangent complex_, Journal of Pure and Applied Algebra 120 (1997) 104 ([pdf](http://www.math.uni-bonn.de/people/schwede/modelspec.pdf))

* {#Hovey} [[Mark Hovey]], _Spectra and symmetric spectra in general model categories_, Journal of Pure and Applied Algebra Volume 165, Issue 1, 23 November 2001, Pages 63–127 ([arXiv:math/0004051](https://arxiv.org/abs/math/0004051))

[[!redirects sequential spectra]]

[[!redirects sequential prespectrum]]
[[!redirects sequential prespectra]]

[[!redirects sequential pre-spectrum]]
[[!redirects sequential pre-spectra]]

[[!redirects Boardman spectrum]]
[[!redirects Boardman spectra]]


sequential spectrum type

+-- {: .rightHandSide}
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### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### Stable Homotopy theory
+--{: .hide}
[[!include stable homotopy theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

This entry is about the formulation of [[sequential spectra]] [[spectrum object|in]] [[homotopy type theory]], specifically of sequential spectra that need not be genuine [[stable homotopy types]] in the sense of [[Omega-spectra]] and thus may be regarded as "[[pre-spectra]]" (they become genuine spectra under [[spectrification]]).

## Definition ##

In [[homotopy type theory]], a **prespectrum** or **sequential spectrum** is a [[dependent sequence|dependent]] [[bi-infinite sequence]] of [[types]] $n:\mathbb{Z} \vdash E_n\;\mathrm{type}$ with a dependent [[bi-infinite sequence]] of [[terms]] $n \colon \mathbb{Z} \vdash *_n \colon E_n$ and a dependent bi-infinite sequence of [[functions]] $n \colon \mathbb{Z} \vdash e_n \colon E_n \to \Omega E_{n+1}$ which preserves the point, where $\Omega A$ is the [[loop space type]] of $A$.

In [[Coq]] pseudocode, this becomes

    Definition prespectrum :=
      {X \colon int -> Type &
       { pt \colon forall n, X n &
        { glue \colon forall n, X n -> pt (S n) == pt (S n) &
          forall n, glue n (pt n) == idpath (pt (S n)) }}}.

## See also ##

* [[pointed type]]

* [[Omega-spectrum type]]

* [[spectrum]]

* [[sequential spectrum]] (in contrast to an [[Omega-spectrum]])

* [[spectrification of a sequential spectrum type]]

## 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))

[[!redirects sequential spectrum types]]

[[!redirects prespectrum type]]
[[!redirects prespectrum types]]
[[!redirects pre-spectrum type]]
[[!redirects pre-spectrum types]]


sequential topological space

+-- {: .rightHandSide}
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### Context
#### Topology
+-- {: .hide}
[[!include topology - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

A [[topological space]] is _sequential_ if (in a certain sense) you can do topology in it using only [[sequences]] instead of more general [[nets]].

Sequential spaces are a kind of [[nice topological space]].

## Definition

A __sequential topological space__ is a [[topological space]] $X$ such that a subset $A$ of $X$ is closed if (hence iff) it contains all the limit points of all [[sequences]] of points of $A$---or equivalently, such that $A$ is open if (hence iff) any sequence converging to a point of $A$ must eventually be in $A$.

Equivalently, a topological space is sequential iff it is a [[quotient space]] (in $Top$) of a [[metric space]].

## Examples

* Every [[quotient object|quotient]] of a sequential space is sequential.  In particular, every [[CW complex]] is also a sequential space.  (Conversely, every sequential space is a quotient of a metrizable space, giving the alternative definition).

## Properties

* The category of sequential spaces is a [[coreflective subcategory]] of the category of all topological spaces.

* The category of sequential spaces is a [[reflective subcategory]] of the category of [[subsequential spaces]], much as $Top$ itself is a reflective subcategory of the category of all [[pseudotopological spaces]].

* The category of sequential spaces is [[cartesian closed category|cartesian closed]]. See also [[convenient category of topological spaces]].

* The category of sequential spaces embeds fully faithfully into the category of light [[condensed sets]] ([Clausen-Scholze 2023](#ClausenScholze_lect3), see timestamp 9:00)

## In constructive mathematics

Everything above assumes [[excluded middle]] (and probably at least [[countable choice]]).  Without that, it\'s hard to prove the existence of any nontrivial sequential spaces.

For example, to prove that the [[real line]] is sequential as a topological space, we must find, given a set $A$ and a point $x$ such that every sequence converging to $x$ is eventually in $A$, a [[positive real number]] $\delta$ such that ${]x - \delta, x + \delta[} \subseteq A$, and it\'s not clear how to construct that number from the data at hand.  (One might consider various specific sequences that converge to $x$, such as $(x + 1/n)_n$ and $(x - 1/n)_n$, and use them to find upper bounds on $\delta$; but no finite set of sequences will give an entire interval around $x$, and proving that an infinite set of sequences that does cover an entire interval has a uniform positive upper bound on $\delta$ is very tricky.)

The usual proof that the real line (or any first-countable topological space) is sequential uses [[excluded middle]] and [[countable choice]]:  Supposing that $A$ is not open, consider $x \in A$ such that $x \notin Int(A)$, pick for each $\delta = 1/n$ (or for each of the countably many basic neighbourhoods $U_n$ of $x$ in a general first-countable space) a point $y_n$ such that $d(x,y_n) \lt 1/n$ (or such that $y_n \in U_n$) but $y_n \notin A$ (which must exist since none of these balls/neighbourhoods are contained in $A$), note that $\lim_n y_n = x$, and get a contradiction.

For this reason, constructive analysis often requires the use of general [[nets]] (or [[filters]]) in situations where classical analysis can get by with sequences.  (It is trivially true, in any topological space, that a set $A$ is open if every *net* that converges to an element $x$ of $A$ belongs eventually to $A$, or equivalently that $A$ belongs to any filter that converges to $x$; you just use the [[neighbourhood filter]] of $x$.)

## Related countability properties

[[!include topology - countability axioms]]

## References

* R. Engelking, _General topology_

* {#ClausenScholze_lect3} [[Dustin Clausen]], [[Peter Scholze]], _Analytic Stacks_, Lecture 3 [YouTube](https://youtu.be/me1KNo3WJHE)

[[!redirects sequential topological space]]
[[!redirects sequential topological spaces]]
[[!redirects sequential space]]
[[!redirects sequential spaces]]


sequentially Cauchy-complete space

[[!redirects sequentially Cauchy complete]]
[[!redirects sequentially Cauchy-complete]]
[[!redirects sequentially Cauchy complete space]]
[[!redirects sequentially Cauchy complete spaces]]
[[!redirects sequentially Cauchy-complete spaces]]

+-- {: .rightHandSide}
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### Context
#### Analysis
+-- {: .hide}
[[!include analysis - contents]]
=--
#### Constructivism, Realizability, Computability
+-- {: .hide}
[[!include constructivism - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition ##

Let $S$ be a [[Cauchy space]] (such as a [[uniform space]] or a [[metric space]]). $S$ is **sequentially Cauchy-complete** if every [[Cauchy sequence]] in $S$ [[converges]].

Note that this is usually just called *Cauchy complete space* in some areas of mathematics.

## Examples ##

* The [[Dedekind real numbers]] are sequentially Cauchy complete.

## See also ##

* [[Cauchy space]]
* [[complete space]]
* [[modulated Cauchy complete space]]

## References ##

* Auke B. Booij, Analysis in univalent type theory ([pdf](https://etheses.bham.ac.uk/id/eprint/10411/7/Booij2020PhD.pdf))
* Univalent Foundations Project, [[HoTT book|Homotopy Type Theory – Univalent Foundations of Mathematics]] (2013)


sequentially compact metric spaces are equivalently compact metric spaces

+-- {: .rightHandSide}
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### Context
#### Analysis
+-- {: .hide}
[[!include analysis - contents]]
=--
#### Topology
+-- {: .hide}
[[!include topology - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

For general [[topological spaces]] the condition of being [[compact topological space|compact]] neither implies nor is implied by being [[sequentially compact]]. However for [[metric spaces]] the two conditions happen to be equivalent

## Statement

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

Using [[excluded middle]] and [[countable choice]], then:

If $(X,d)$ is a [[metric space]], regarded as a [[topological space]] via its [[metric topology]], then the following are equivalent:

1. $(X,d)$ is a [[compact topological space]].

1. $(X,d)$ is a [[sequentially compact topological space]].

=--

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

Assume first that $(X,d)$ is a [[compact topological space]]. Let $(x_k)_{k \in \mathbb{N}}$ be a [[sequence]] in $X$. We need to show that it has a sub-sequence which [[convergence|converges]].

Consider the [[topological closures]] of the sub-sequences that omit the first $n$ elements of the sequence

$$
  F_n
   \;\coloneqq\;
  Cl(\left\{
    x_k \,\vert\, k \geq n
  \right\})
$$

and write

$$
  U_n \coloneqq X \backslash F_n
$$

for their [[open subset|open]] [[complements]].

Assume now that the [[intersection]] of all the $F_n$ were [[empty set|empty]]

$$
  (\star)
  \phantom{AA}
  \underset{n \in \mathbb{N}}{\cap} F_n \;= \; \emptyset
$$

or equivalently that the [[union]] of all the $U_n$ were all of $X$

$$
  \underset{n \in \mathbb{N}}{\cup} U_n
  \;=\;
  X
  \,,
$$

hence that $\{U_n \to X\}_{n \in \mathbb{N}}$ were an [[open cover]]. By the assumption that $X$ is compact, this would imply that there is a [[finite set|finite]] [[subset]] $\{i_1 \lt i_2 \lt  \cdots \lt i_k\} \subset \mathbb{N}$ with

$$
  \begin{aligned}
    X & = U_{i_1} \cup U_{i_2} \cup \cdots \cup U_{i_k}
    \\
    & = U_{i_k}
  \end{aligned}
  \,.
$$

This in turn would mean that $F_{i_k} = \empty$, which contradicts the construction of $F_{i_k}$. Hence we have a [[proof by contradiction]] that assumption $(\ast)$ is wrong, and hence that there must exist an element

$$
  x \in \underset{n \in \mathbb{N}}{\cap} F_n
  \,.
$$

By definition of [[topological closure]] this means that for all $n$ the [[open ball]] $B^\circ_x(1/(n+1))$ around $x$ of [[radius]] $1/(n+1)$ must intersect the $n$th of the above subsequence:

$$
  B^\circ_x(1/(n+1))
  \,\cap\,
  \{x_k \,\vert\, k \geq n \}
  \;\neq\;
  \emptyset
  \,.
$$

Picking one point $(x'_n)$ in the $n$th such intersection for all $n$ hence defines a sub-sequence, which converges to $x$.

This proves that compact implies sequentially compact for metric spaces.

For the converse, assume now that $(X,d)$ is sequentially compact. Let $\{U_i \to X\}_{i \in I}$ be an [[open cover]] of $X$. We need to show that there exists a finite sub-cover.

Now by the [[Lebesgue number lemma]], there exists a positive real number $\delta \gt 0$ such that for each $x \in X$ there is $i_x \in I$ such that $B^\circ_x(\delta) \subset U_{i_x}$.
Moreover, since [[sequentially compact metric spaces are totally bounded]], there exists then  a [[finite set]] $S \subset X$ such that

$$
   X
     \;=\;
   \underset{s \in S}{\cup}
   B^\circ_s(\delta)
   \,.
$$

Therefore $\{U_{i_s} \to X\}_{s \in S}$ is a finite sub-cover as required.

=--

## Remarks

+-- {: .num_defn #weakerAssumptions}
###### Remark

In the proof of prop. \ref{sequentiallyCompactMetricSpacesAreEquivCompact} the implication that a compact topological space is sequentially compact requires less of $(X,d)$ than being a metric space. Actually, the proof works for any [[first-countable space]] that is a [[countably compact space]], i. e. any countable open cover admits a finite sub-cover. Hence [[countably compact metric spaces are equivalently compact metric spaces]].

=--

## Related statements

* [[compact spaces equivalently have converging subnets]]

* [[countably compact metric spaces are equivalently compact metric spaces]]

* [[sequentially compact metric spaces are totally bounded]]

* [[Lebesgue number lemma]]

* [[continuous metric space valued function on compact metric space is uniformly continuous]]


sequentially compact metric spaces are totally bounded

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### Context
#### Analysis
+-- {: .hide}
[[!include analysis - contents]]
=--
#### Topology
+-- {: .hide}
[[!include topology - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Statement

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

Using [[excluded middle]] and [[dependent choice]] then:

Let $(X,d)$ be a [[metric space]] which is [[sequentially compact topological space|sequentially compact]]. Then it is [[totally bounded metric space]].

=--

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

Assume that $(X,d)$ were not totally bounded. This would mean that there existed a [[positive number|positive]] [[real number]] $\epsilon \gt 0$ such that for every [[finite set|finite]] [[subset]] $S \subset X$ we had that $X$ is not the [[union]] of the [[open balls]] of [[radius]] $\epsilon$ around the elements of this finite subset

$$
  X \neq \underset{s \in S}{\cup} B^\circ_s(\epsilon)
  \,.
$$

This would mean that we could [[induction|inductively]] choose elements

* $x_0 \in X$

* $x_1 \in X\backslash B^\circ_{x_0}(\epsilon)$

* $x_{n+1} \in X \backslash \left( \underoverset{i = 0}{n}{\cup}  B^\circ_{x_i}(\epsilon)\right)$

This constituted a [[sequence]] $(x_i)$, and by assumption of sequential compactness this would have a [[convergence|converging]] sub-sequence. But this would then be a contradiction, since by the above assumption that $X$ is not totally bounded we have $d(x_j, x_k) \gt \epsilon$ for all $j \neq k$. Hence we have a [[proof by contradiction]] that $(X,d)$ in fact is totally bounded.

=--

## Related pages

* [[Lebesgue number lemma]]

* [[sequentially compact metric spaces are equivalently compact metric spaces]]

* [[countably compact metric spaces are equivalently compact metric spaces]]


sequentially compact topological space

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

#Contents#
* table of contents
{:toc}

## Idea

A [[topological space]] is called _sequentially compact_ if every [[sequence]] of points in that space has a sub-sequence which [[convergence|converges]]. In general this concept neither implies nor is implied by that of actual [[compact space|compactness]], but for some types of [[topological spaces]], such as [[metric spaces]], it is equivalent.

[[compact space|Compactness]] is an extremely useful concept in [[topology]].  The basic idea is that a [[topological space]] is [[compact space|compact]] if it isn't "fuzzy around the edges".

Whilst one can study a topological space by itself, it is often useful to probe it with known spaces.  A common choice for topological spaces, and in particular metric spaces, is to use the natural numbers $\mathbb{N}$, and the [[one-point compactification]] $\mathbb{N} \cup \{*\}$ of the natural numbers.  This is more traditionally known as studying the topology using sequences and convergent sequences.

Thus one can ask, "Can I detect compactness using probes from $\mathbb{N}$, and $\mathbb{N} \cup \{*\}$?".  The short answer to this is "No", but that just reveals that the question was too restrictive.  Rather, one should ask "What does compactness look like if all I'm allowed to use are probes from $\mathbb{N}$ and $\mathbb{N} \cup \{*\}$?".  The answer to that question is "sequential compactness".

Thus **sequential compactness** is what [[compact space|compactness]] looks like if all one has to test it are sequences.

## Definition

+-- {: .num_defn #seqcpt}
###### Definition ######
A topological space is **sequentially compact** if every [[sequence]] in it has a [[convergence|convergent]] [[subsequence]].
=--

## Properties

The following is a list of properties of and pertaining to sequentially compact spaces.

1. For a [[metric space]], the notions of sequential compactness and compactness coincide. See at _[[sequentially compact metric spaces are equivalently compact metric spaces]]_.

2. The [[Eberlein-Smulian theorem|Eberlein–Šmulian theorem]] states that in a [[Banach space]], for a subset with regard to the [[weak topology]], compactness and sequentially compactness are both equivalent to the weaker notion of [[countably compact space|countable compactness]].

3. A [[countable set|countable]] product of sequentially compact spaces is again sequentially compact.

+-- {: .proof}
###### Proof
Let $\{X_j\}$ be a countable family of sequentially compact spaces.  Let
$$\pi_k: \prod_j X_j \to X_k$$
denote the $k^{th}$ projection, and let $(x(n))$ be a sequence in $\prod X_j$. We build a sequence of nested subsequences $S_j = x_j(n)$ as follows. For $j = 1$, choose a subsequence $S_1$ of the given sequence $(x_n)$ such that $x_1(1) = x(1)$ and such that $\pi_1(S_1)$ converges to a point $y_1 \in X_1$. Then, supposing that $S_j$ has been constructed, choose $S_{j+1}$ to be a subsequence of $S_j$ that starts the same way as $S_j$ for the first $j$ steps,

$$x_{j+1}(1) = x_j(1), \ldots, x_{j+1}(j) = x_j(j),$$

and such that $\pi_{j+1}(S_{j+1})$ converges to $y_{j+1} \in X_{j+1}$. By the [[axiom of dependent choice]], we may suppose these choices have been made for all $j$, and then we form the diagonal sequence $S$ given by $x_j(j)$. This is a subsequence of each $S_j$, and so $\pi_j(S)$ converges to $y_j$ for every $j$, hence $S$ converges to the point $(y_j) \in \prod_j X_j$, and we have produced the desired convergent subsequence.
=--

   This shows that the example of a compact space that is not sequentially compact is about as simple as can be.

4. The theorem that a continuous bijection from a compact space to a Hausdorff space is a homeomorphism has a counterpart for sequentially compact spaces.

   +-- {: .num_theorem #SeqCptReg}
   ###### Theorem
   Let $\mathcal{T}_1$ and $\mathcal{T}_2$ be two topologies on a set $X$ such that:

   1. $\mathcal{T}_1 \supseteq \mathcal{T}_2$ (equivalently, the identity map on $X$ is continuous as a map $(X,\mathcal{T}_1) \to (X, \mathcal{T}_2)$)
   2. $\mathcal{T}_1$ is sequentially compact
   3. $\mathcal{T}_2$ is [[completely regular space|completely regular]] and singleton sets are $G_\delta$-[[G-delta set|sets]],

   then $\mathcal{T}_1 = \mathcal{T}_2$.
   =--

   +-- {: .proof}
   ###### Proof
   Let $V \subseteq X$ be such that $V \notin \mathcal{T}_2$.  Then it must be non-empty and there must be a point $v \in V$ such that $V$ is not a neighbourhood of $v$.  As $\mathcal{T}_2$ is completely regular and singleton sets are $G_\delta$ sets, there is a continuous function $g \colon (X, \mathcal{T}_2) \to \mathcal{R}$ such that $g^{-1}(0) = \{v\}$.  Since $V$ is not a neighbourhood of $v$, for each $n \in \mathbb{N}$, the set $g^{-1}(-\frac1n, \frac1n)$ is not wholly contained in $V$.  Thus for each $n$ there is a point $x_n \in X$ such that $x_n \notin V$ and $|g(x_n)| \lt \frac1n$.  As $\mathcal{T}_1$ is sequentially compact, this sequence has a $\mathcal{T}_1$-convergent subsequence, say $(x_{n_k})$ converging to $y$.  Since $g(x_n) \to 0$, $g(x_{n_k}) \to 0$ and thus $g(y) = 0$.  Thus $y = v$ and so $(x_{n_k}) \to v$ in $\mathcal{T}_1$.  As $x_{n_k}
 \notin V$ for all $n_k$, and $v \in V$, it must be the case that $V$ is not a $\mathcal{T}_1$-neighbourhood of $v$.  Hence $V \notin \mathcal{T}_1$.  Thus $\mathcal{T}_1 \subseteq \mathcal{T}_2$, whence they are equal.
   =--

5. The [[image]] of a sequentially compact space $X$ under a continuous map $f: X \to Y$ is also sequentially compact. For suppose $y_n$ is a sequence in $f(X)$, say $y_n = f(x_n)$. Then $x_n$ has a convergent subsequence $x_{n_j}$, converging to $x$ say, and by continuity $y_{n_j} = f(x_{n_j})$ converges to $f(x)$.

### Relationship to Compactness

[[compact space|Compactness]] does not imply sequentially compactness, nor does sequentially compactness imply compactness, without further assumptions, see at _[Examples and counter-examples](#Examples)_ below.

In [[metric spaces]] for example both notions coincide, see at [[sequentially compact metric spaces are equivalently compact metric spaces]]. (This is a consequence of the [[Lebesgue number lemma]] and the fact that  [[sequentially compact metric spaces are totally bounded]].)

This is _not_ a contradiction to the statement that compact is equivalent to every [[net]] having a convergent subnet: Given a sequence in a compact space, its convergent _subnet_ need not be a _subsequence_ (see [[net]] for a definition of subnet).

## Examples and counter-examples
 {#Examples}

A [[metric space]] is sequentially compact precisely if it is [[compact topological space|compact]]. See at _[[sequentially compact metric spaces are equivalently compact metric spaces]]_.

In general neither of these two properties implies the other:

* Examples of a sequentially compact spaces which are not compact are given in in ([Buskes-Rooij 97, example 13.5](#BuskesRooij97), [Patty 08, chapter 4, example 13](#Patty08), [Vermeeren 10, prop. 18](#Vermeeren10)). The [[long line]] is such an example.

* An example of a [[compact topological space]] which is not sequentially compact is given in ([Steen-Seebach 70, item 105](#SteenSeebach70)), recalled at [Vermeeren 10, prop. 17](#Vermeeren10). See at [compact space -- Compact spaces which are not sequentially compact](compact+space#CompactSpacesWhichAreNotSequentiallyCompact).

+-- {: .num_example #ACompactSpaceNotSequentiallyCompact}
###### Example
**(a compact space which is not sequentially compact)**

Consider the following uncountable power of the [[discrete space]] $2$, namely the [[product topological space]] (with its [[Tychonoff topology]])

$$
  X \coloneqq \underset{f: \mathbb{N} \to 2}{\prod} Disc(\{0,1\})
$$

of copies of the discrete space on two elements, indexed by functions $f: \mathbb{N} \to \{0, 1\}$. Since $Disc(\{0,1\})$ is a [[finite topological space|finite]] [[discrete topological space]] it is clearly compact. Therefore the [[Tychonoff theorem]] says that also $X$ is compact.

Let $(x_n)_{n: \mathbb{N}}$ be the sequence in $X$ given by the [[duality|double-dual embedding]]

$$\mathbb{N} \to 2^{2^\mathbb{N}},$$

i.e., define $x_n$ to have coordinate at $f: \mathbb{N} \to 2$ given by $(x_n)_f = f(n)$. We claim this sequence has no subsequence that converges in $X$, so that $X$ is not sequentially compact. We will argue by contradiction. (This [[refutation by contradiction]], refuting sequential compactness, is [[constructive logic|constructively valid]]. The argument above that affirms compactness, via the Tychonoff theorem, is not constructive.)

Suppose instead some subsequence $(x_{(n_k)})_{k \in \mathbb{N}}$ converges to some $x \in X$. Choose any $f: \mathbb{N} \to 2$ that is not eventually constant on the subsequence $(n_k)_{k: \mathbb{N}}$; for example, define $f: \mathbb{N} \to 2$ by $f(n_k) = k\; mod\; 2$, else $f(n) = 0$ if $n$ does not appear in the subsequence. Consider the open set $U_f = \{x_f\} \times \prod_{g: g \neq f} \{0, 1\}$, which is an open neighborhood of $x$. In order to have $x_{n_k} \in U_f$ for all $k \geq k_0$, we'd have to have $f(n_k) = x_f$ for all $k \geq k_0$, in other words $f$ would be eventually constant on the subsequence $n_k$. Contradiction.

=--

## Related concepts

* [[compact topological space]]

* [[countably compact topological space]]

* [[paracompact topological space]]

* [[locally compact topological space]]

* [[compactly generated topological space]]

## References

* {#BuskesRooij97} Buskes, van Rooij, _Topological Spaces: From  Distance to Neighbourhood_, Springer 1997

* {#SteenSeebach70} Lynn Steen, J. Arthur Seebach, _Counterexamples in Topology_, Springer-Verlag, New York (1970) 2nd edition,  (1978), Reprinted by Dover Publications, New York, 1995

* {#Patty08} Wayne Patty, _Foundations of Topology_, Jones and Bartlett Publishers (2008)

* {#Vermeeren10} [[Stijn Vermeeren]], _Sequences and nets in topology_, 2010 ([pdf](http://stijnvermeeren.be/download/mathematics/nets.pdf))

See also

* Wikipedia, _[Sequentially compact space](https://en.wikipedia.org/wiki/Sequentially_compact_space)_

[[!redirects sequentially compact space]]
[[!redirects sequentially compact spaces]]

[[!redirects sequentially compact]]
[[!redirects sequential compactness]]

[[!redirects sequentially compact metric space]]
[[!redirects sequentially compact metric spaces]]


Serap Gürer

* [webpage](http://cv.gsu.edu.tr/en/CV/serap-gurer)

## Selected writings

On [[manifolds with boundaries and corners]] as [[diffeological spaces]], and on [[differential forms]] on these:

* {#GurerIZ19} [[Serap Gürer]],  [[Patrick Iglesias-Zemmour]], _Differential forms on manifolds with boundary and corners_, Indagationes Mathematicae, Volume 30, Issue 5, September 2019, Pages 920-929 ([doi:10.1016/j.indag.2019.07.004](https://doi.org/10.1016/j.indag.2019.07.004))

On [[orbifolds]] as [[stratified space|stratified]] [[diffeological spaces]]:

* [[Serap Gürer]], [[Patrick Iglesias-Zemmour]], *Orbifolds as stratified diffeologies*, Differential Geometry and its Applications **86** (2023) 101969 [[doi:10.1016/j.difgeo.2022.101969](https://doi.org/10.1016/j.difgeo.2022.101969)]

## Related $n$Lab entries

* [[diffeological space]]

* [[manifold with boundary]], [[manifold with corners]]

category: people

[[!redirects Serap Gurer]]
[[!redirects Serap Guerer]]


Serge Haroche

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

## Selected writings

On [[quantum physics]]:

* [[Serge Haroche]], [[Jean-Michel Raimond]], *Exploring the Quantum: Atoms, Cavities, and Photons*, Oxford University Press (2006) [[doi:10.1093/acprof:oso/9780198509141.001.0001](https://doi.org/10.1093/acprof:oso/9780198509141.001.0001)]

category: people


Serge Lang

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

category: people


Serge Ochanine

* [webpage](https://math.as.uky.edu/users/ochanin)

## Selected writings

Introducing [[elliptic genera]]:

* {#Ochanine87} [[Serge Ochanine]], _Sur les genres multiplicatifs definis par des integrales elliptiques_, Topology, Vol. 26, No. 2, 1987 ([pdf](http://www.math.sciences.univ-nantes.fr/~hossein/GdT-Elliptique/Mult-genre-Ochanine.pdf))

A quick review of [[elliptic genera]]:

* {#Ochanine09} [[Serge Ochanine]], _What is... an elliptic genus_?, Notices of the AMS, volume 56, number 6  (2009) ([pdf](http://www.ams.org/notices/200906/rtx090600720p.pdf))

On (the failure of) the [[Conner-Floyd isomorphism]] for [[MSU]]$\to$[[KO]]:

* {#Ochanine87} [[Serge Ochanine]], _Modules de SU-bordisme. Applications_, Bulletin de la Société Mathématique de France, Tome 115 (1987) , pp. 257-289 ([doi:10.24033/bsmf.2078]( https://doi.org/10.24033/bsmf.2078))

category: people


Serge Tabachnikov

## Selected writings

On [[topological complexity]] of [[real projective space]] and connection with their [[immersion]]  into [[cartesian space]]:

* {#FarberTabachnikovYuzvinsky02} [[Michael Farber]], [[Serge Tabachnikov]], [[Sergey Yuzvinsky]], _Topological robotics: motion planning in projective spaces_ (2002), [arXiv:math/0210018](https://arxiv.org/abs/math/0210018);

category: people


Sergei Chmutov

* [webpage](https://math.osu.edu/people/chmutov.1)

## Selected writings

On [[Lie algebra weight systems]] for the [[special linear Lie algebra]] $\mathfrak{sl}(2)$:

* [[Sergei Chmutov]], [[Alexander Varchenko]], _Remarks on the Vassiliev knot invariants coming from $\mathfrak{sl}(2)$_, Topology 36 (1), 153-178, 1997 (doi:10.1016/0040-9383(95)00071-2)

On [[Vassiliev invariants]] ([[chord diagrams]], [[Jacobi diagrams]], [[weight systems]], [[knot invariants]] etc.)

* {#ChmutovDuzhinMostovoy11} [[Sergei Chmutov]], [[Sergei Duzhin]], [[Jacob Mostovoy]], _Introduction to Vassiliev knot invariants_, Cambridge University Press, 2012 ([arxiv:1103.5628](http://arxiv.org/abs/1103.5628), [doi:10.1017/CBO9781139107846](https://doi.org/10.1017/CBO9781139107846))

category: people


Sergei Dubovsky

* [Institute page](https://as.nyu.edu/faculty/sergei-dubovsky.html)

## Selected writings

On the [[confinement]]/[[mass gap problem]] in [[holographic QCD]]

* {#Dubovsky21} [[Sergei Dubovsky]], *Comments on (mostly long) QCD strings*, talk at [Strings 2021](https://www.ictp-saifr.org/strings2021/) ([pdf](https://www.ictp-saifr.org/wp-content/uploads/2021/06/Sergei.pdf), [video](https://youtu.be/B9NI2LE7d68?t=1183))

category: people


Sergei Duzhin

* [webpage](http://www.pdmi.ras.ru/~duzhin/)

## Selected writings

On [[Vassiliev invariants]] ([[chord diagrams]], [[Jacobi diagrams]], [[weight systems]], [[knot invariants]] etc.)

* {#ChmutovDuzhinMostovoy11} [[Sergei Chmutov]], [[Sergei Duzhin]], [[Jacob Mostovoy]], _Introduction to Vassiliev knot invariants_, Cambridge University Press, 2012 ([arxiv:1103.5628](http://arxiv.org/abs/1103.5628), [doi:10.1017/CBO9781139107846](https://doi.org/10.1017/CBO9781139107846))

List of literature on [[Vassiliev invariants]]:

* [[Dror Bar-Natan]], [[Sergei Duzhin]], _[Bibliography of Vassiliev Invariants](http://www.pdmi.ras.ru/~duzhin/VasBib/Long)_

category: people


Sergei Gelfand

* [Mathematics Genealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=41991)

## Selected writings

On [[homological algebra]] and [[homotopical algebra]] (via [[triangulated categories]] and including the [[model structure on dgc-algebras]] for [[rational homotopy theory]]):

* [[Sergei Gelfand]], [[Yuri Manin]], _[[Methods of homological algebra]]_,   transl. from the 1988 Russian (Nauka Publ.) original, Springer 1996. xviii+372 pp. 2nd corrected ed. 2002 ([doi:10.1007/978-3-662-12492-5](https://doi.org/10.1007/978-3-662-12492-5))

## Related $n$Lab entries ##

* [[model structure on dg-algebras]]

[[!redirects Sergei Gel'fand]]
[[!redirects Sergei Gel'fand]]
[[!redirects Сергей Гельфанд]]

category: people


Sergei Gukov

* [website](http://www.physics.ucsb.edu/~gukov)

## Selected writings

On [[M-theory on 8-manifolds]] and [[conifold transition]]:

* [[Sergei Gukov]], [[James Sparks]], [[David Tong]], *Conifold Transitions and Five-Brane Condensation in M-Theory on $Spin(7)$ Manifolds*, Class. Quant. Grav. **20** (2003) 665-706 [[arXiv:hep-th/0207244](https://arxiv.org/abs/hep-th/0207244)]

On [[moduli stabilization]] in [[heterotic string theory]]:

* [[Sergei Gukov]], [[Shamit Kachru]], Xiao Liu, [[Liam McAllister]], _Heterotic Moduli Stabilization with Fractional Chern-Simons Invariants_,  Phys.Rev.D **69** (2004) 086008 [[arXiv:hep-th/0310159](http://arxiv.org/abs/hep-th/0310159)]

On [[topological M-theory]]:

* [[Robbert Dijkgraaf]], [[Sergei Gukov]], [[Andrew Neitzke]], [[Cumrun Vafa]], *Topological M-theory as Unification of Form Theories of Gravity*, Adv. Theor. Math. Phys. **9** (2005) 603-665 [[arXiv:hep-th/0411073](http://arxiv.org/abs/hep-th/0411073), [doi:10.4310/ATMP.2005.v9.n4.a5](https://dx.doi.org/10.4310/ATMP.2005.v9.n4.a5)]

On [[perturbative quantum field theory|perturbative]] [[Chern-Simons theory with complex gauge group]]:

* {#DGLZ09} [[Tudor Dimofte]], [[Sergei Gukov]], [[Jonatan Lenells]], [[Don Zagier]], _Exact Results for Perturbative Chern-Simons Theory with Complex Gauge Group_, Commun. Num. Theor. Phys. **3** (2009) 363-443 [[arXiv:0903.2472](http://arxiv.org/abs/0903.2472), [doi:10.4310/CNTP.2009.v3.n2.a4](https://dx.doi.org/10.4310/CNTP.2009.v3.n2.a4)]

On [[topological quantum field theory]]:

* [[Sergei Gukov]], [[Anton Kapustin]], _[[Topological Quantum Field Theory, Nonlocal Operators, and Gapped Phases of Gauge Theories]]_ ([arXiv:1307.4793](http://arxiv.org/abs/1307.4793))

On [[KK-compactification]] of [[D=6 N=(2,0) SCFT]] on [[4-manifolds]] to [[2d CFTs]]:

* [[Abhijit Gadde]], [[Sergei Gukov]], [[Pavel Putrov]], *Fivebranes and 4-manifolds*, in: *Arbeitstagung Bonn 2013*, Progress in Mathematics **319**, Birkhäuser (2016) [[arXiv:1306.4320](https://arxiv.org/abs/1306.4320), [doi:10.1007/978-3-319-43648-7_7](https://doi.org/10.1007/978-3-319-43648-7_7)]

* [[Mykola Dedushenko]], [[Sergei Gukov]], [[Pavel Putrov]], *Vertex algebras and 4-manifold invariants*, Chapter 11 in: *Geometry and Physics: Volume I*  (2018) 249–318 [[arXiv:1705.01645](https://arxiv.org/abs/1705.01645), [doi:10.1093/oso/9780198802013.003.0011](https://doi.org/10.1093/oso/9780198802013.003.0011)]

* [[Boris Feigin]], [[Sergei Gukov]], $VOA[M_4]$, J. Math. Phys. **61** 012302 (2020) [[arXiv:1806.02470](https://arxiv.org/abs/1806.02470), [doi:10.1063/1.5100059](https://doi.org/10.1063/1.5100059)]

and in relation to the [[M5-brane elliptic genus]]:

* {#GukovPeiPutrovVafa18} [[Sergei Gukov]], [[Du Pei]], [[Pavel Putrov]], [[Cumrun Vafa]], *4-manifolds and topological modular forms*,  J. High Energ. Phys. **2021** 84 (2021) [[arXiv:1811.07884](https://arxiv.org/abs/1811.07884), doi:10.1007/JHEP05(2021)084, [spire:1704312](https://inspirehep.net/literature/1704312)]

On the [[3d-3d correspondence]]:

* Sungbong Chun, [[Sergei Gukov]], Sunghyuk Park, Nikita Sopenko, _3d-3d correspondence for mapping tori_ ([arxiv:1911.08456](https://arxiv.org/abs/1911.08456))

On [[Rozansky-Witten theory]]:

* [[Sergei Gukov]], Po-Shen Hsin, Hiraku Nakajima, Sunghyuk Park, Du Pei, Nikita Sopenko, _Rozansky-Witten geometry of Coulomb branches and logarithmic knot invariants_ ([arXiv:2005.05347](https://arxiv.org/abs/2005.05347))

On [[machine learning]] the detection of [[unknots]]:

* [[Sergei Gukov]], [[James Halverson]], [[Fabian Ruehle]], [[Piotr Sułkowski]], *Learning to Unknot*, Mach. Learn.: Sci. Technol. **2** (2022) 025035 [[arXiv:2010.16263](https://arxiv.org/abs/2010.16263), [doi:10.1088/2632-2153/abe91f](https://iopscience.iop.org/article/10.1088/2632-2153/abe91f)]

* [[Sergei Gukov]], *Machine learning and hard problems in topology*, talk at *[QFT and Cobordism](https://nyuad.nyu.edu/en/events/2023/march/quantum-field-theories-and-cobordisms.html)*, [[CQTS]] (Mar 2023) [[web](https://ncatlab.org/nlab/show/Center+for+Quantum+and+Topological+Systems#GukovMar2023)]

category: people

[[!redirects Sergey Gukov]]


Sergei K. Lando


Sergei Kerov

* [personal page](http://www.pdmi.ras.ru/~kerov/)

## Selected writings

On the asymptotic number of [[standard Young tableaux]] (dimension of [[Specht modules]]) with respect to the [[Plancherel measure]]:

* {#VershikKerov85} [[Anatoly Vershik]], [[Sergei Kerov]], *Asymptotic of the largest and the typical dimensions of irreducible representations of a symmetric group*, Functional Analysis and Its Applications volume 19, pages 21–31 (1985) ([doi:10.1007/BF01086021](https://doi.org/10.1007/BF01086021))

On [[character of a linear representation|characters]] on the [[symmetric group]] depending only on [[Cayley distance]]:

* {#GnedinGorinKerov11} [[Alexander Gnedin]], [[Vadim Gorin]], [[Sergei Kerov]], *Block characters of the symmetric groups*, Journal of Algebraic Combinatorics, 38, no. 1 (2013), 79-101 ([arXiv:1108.5044](https://arxiv.org/abs/1108.5044), [doi:10.1007/s10801-012-0394-9](https://doi.org/10.1007/s10801-012-0394-9))

category: people


Sergei Ketov

* [webpage](http://kiso.phys.se.tmu.ac.jp/~ketov/ketov.htm)

## Selected writings

On embedding the [[Starobinsky model of cosmic inflation]] into [[supergravity]]:

* Yermek Aldabergenov, Shuntaro Aoki, [[Sergei Ketov]], _Minimal Starobinsky supergravity coupled to dilaton-axion superfield_ ([arXiv:2001.09574](https://arxiv.org/abs/2001.09574))

and into [[string theory]]/[[higher curvature corrections]]:

* [[Sergei Ketov]], Ekaterina O. Pozdeeva, Sergey Yu. Vernov, *On the superstring-inspired quantum correction to the Starobinsky model of inflation* [[arXiv:2211.01546](https://arxiv.org/abs/2211.01546)]

## Related entries

* [[Starobinsky model of cosmic inflation]]

category: people


Sergei Merkulov

_Sergei Merkulov_ is a mathematician working on [[operads]], [[homotopy algebras]], [[deformation theory]], [[quantization]] and related questions.

* [institute page](https://wwwfr.uni.lu/recherche/fstm/dmath/people/sergei_merkulov)

## Selected writings

(...)

category: people

[[!redirects S. A. Merkulov]]
[[!redirects S. Merkulov]]
[[!redirects Sergei A. Merkulov]]


Sergei Novikov

Sergei Novikov (Russian: Серге́й Петро́вич Но́виков) is a Russian mathematician, Fields medalist (1970).

* Wikipedia entry

His father was also a mathematician [[Pyotr Sergeyevich Novikov]], with major results in mathematical logics.

## Selected writings

On [[cobordism theory]] ([[MO]], [[MSO]], [[MU]], [[MSU]] and [[MSp]]):

* {#Novikov62} [[Sergei Novikov]], _Homotopy properties of Thom complexes_, Mat. Sbornik 57 (1962), no. 4, 407–442, 407–442 ([pdf](http://www.mi-ras.ru/~snovikov/6.pdf), [[NovikovThomComplexes.pdf:file]])

On the [[Adams-Novikov spectral sequence]]:

* {#Novikov67} [[Sergei Novikov]], _The methods of algebraic topology from the viewpoint of cobordism theories_, Izv. Akad. Nauk. SSSR. Ser. Mat. 31 (1967), 855–951 (Russian). ([[Novikov67.pdf:file]])

On [[formal groups]] and [[Adams operations]]:

* [[Victor M. Buchstaber]], [[Sergei P. Novikov]], _Formal groups, power systems and Adams operators_, Math. USSR-Sb.13 (1971) 80-116 [[doi:10.1070/SM1971v013n01ABEH001030](https://iopscience.iop.org/article/10.1070/SM1971v013n01ABEH001030)]

On [[differential topology]] and [[differential geometry]]:

* B. A. Dubrovin, [[Sergei Novikov]], A. T. Fomenko, _Modern Geometry — Methods and Applications: Part II: The Geometry and Topology of Manifolds_, Graduate Texts in Mathematics 104, Springer-Verlag New York, 1985

On [[differential topology]] and [[algebraic topology]]:

* [[Sergei Novikov]], _Topology I -- General survey_, in: Encyclopedia of Mathematical Sciences Vol. 12, Springer 1986 ([doi:10.1007/978-3-662-10579-5](https://link.springer.com/book/10.1007/978-3-662-10579-5), [pdf](https://web.math.rochester.edu/people/faculty/doug/otherpapers/novikovsurv.pdf))

## Related entries

* [[Adams-Novikov spectral sequence]]

* [[Novikov conjecture]]

* [[Landweber-Novikov theorem]]

basic [[differential topology]]:

* [[degree of a continuous function]]

* [[Hopf degree theorem]]

* [[Poincaré–Hopf theorem]]

* [[Brouwer fixed point theorem]]

* [[multivalued group]]

[[!redirects Sergei P. Novikov]]
[[!redirects S. P. Novikov]]

category: people


Sergei O. Ivanov

* [Institute page](https://math-cs.spbu.ru/en/people/sergei-ivanov/)

## Selcted writings

On [[rationalization]] of possibly non-[[nilpotent homotopy types]] (such as via [[deck transformation|deck]]-[[Borel-equivariant rational homotopy theory]]):

* {#Ivanov21} [[Sergei O. Ivanov]], *An overview of rationalization theories of non-simply connected spaces and non-nilpotent groups* [[arXiv:2111.10694](https://arxiv.org/abs/2111.10694)]

category: people

[[!redirects Sergei Ivanov]]


Sergei Winitzki

* [personal page](https://sites.google.com/site/winitzki)

* [leanpub page](https://leanpub.com/u/winitzki)

* [GoogleScholar page](https://scholar.google.com/citations?user=Q9U40gUAAAAJ&hl=en)

## Selected writings

On [[functional programming]] (in [[Scala]]) such as with [[monads]] modelling [[computational effects]]:

* {#Winitzki22} [[Sergei Winitzki]], *The Science of Functional Programming -- A tutorial, with examples in Scala* (2022) [[leanpub:sofp](https://leanpub.com/sofp), [github:sofp](https://github.com/winitzki/sofp)]

## Related pages

* [[monad transformer]]

category: people


Sergey Antonyan

* [institute page](http://www.fciencias.unam.mx/directorio/32855)

* [MathematicsGenealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=127937)

## Selected writings

On the [[slice theorem]]:

* [[Sergey Antonyan]], *Characterizing slices for proper actions of locally compact groups*, Topology and its Applications Volume 239, 15 April 2018, Pages 152-159 ([arXiv:1702.08093](https://arxiv.org/abs/1702.08093), [doi:10.1016/j.topol.2018.02.026](https://doi.org/10.1016/j.topol.2018.02.026))

On [[proper actions]]:

* {#Antonyan02} [[Sergey Antonyan]], *Universal proper G-spaces*, Topology and its Applications Volume 117, Issue 1, 15 January 2002, Pages 23-43 (doi:10.1016/S0166-8641(00)00115-2)

category: people


Sergey Arkhipov

* institute page

## Selected writings

On [[homotopy limits]]/[[homotopy colimits]] via [[homotopy ends]]/[[homotopy coends]] and [[homotopy weighted colimit|homotopy weighted (co)limits]]:

* {#ArkhipovOrsted18} [[Sergey Arkhipov]], [[Sebastian Ørsted]], *Homotopy (co)limits via homotopy (co)ends in general combinatorial model categories* ([arXiv:1807.03266](https://arxiv.org/abs/1807.03266))

category: people


Sergey Cherkis

* [webpage](https://www.math.arizona.edu/people/cherkis)

## Selected writings

On the [[M2-brane 3-algebra]] and the [[BLG model]]:

* {#CherkisSaemann08} [[Sergey Cherkis]], [[Christian Saemann]], _Multiple M2-branes and Generalized 3-Lie algebras_, Phys. Rev. D78:066019, 2008 ([arXiv:0807.0808](https://arxiv.org/abs/0807.0808))

category: people


Sergey Fomin

A Russian-American mathematician. His main works are in combinatorics and algebra, often motivated by mathematical physics, representation theory and other applications. With A. Zelevinsky, he introduced [[cluster algebra]]s.

* [wikipedia](https://en.wikipedia.org/wiki/Sergey_Fomin)

## Selected works

* [[Sergey Fomin]], [[Andrei Zelevinsky]], _Cluster algebras. I. Foundations_, J. Amer. Math. Soc. 15 (2002), no. 2, 497--529, [math.RT/0104151 ](https://arxiv.org/abs/math/0104151); _Cluster algebras. II. Finite type classifications._ Invent. Math. 154 (2003), no. 1, 63--121 -- classification of cluster algebras of finite type via root systems [math.RA/0208229](https://arxiv.org/abs/math/0208229), _Cluster algebras, IV: Coefficients_, Compos. Math. __143__ (2007), 112--164, [MR2295199](https://www.ams.org/mathscinet-getitem?mr=2295199) [doi](https://doi.org/10.1112/S0010437X06002521)
* [[Sergey Fomin]], [[Andrei Zelevinsky]], _[[Y-system]]s and generalized associahedra_, Ann. Math. __158__ 977-1018 (2003) [doi](https://doi.org/10.4007/annals.2003.158.977)

category: people


Sergey Frolov

> This page is about S. Frolov working in [[high energy physics]]/[[string theory]]. There is also *[[Sergey M. Frolov]]* working on [[condensed matter theory]]/[[Majorana zero modes]].

***

* [institute page](https://qft.physik.hu-berlin.de/frolov)

## Selected writings

On [[multi-trace operators]]:

* [[Gleb Arutyunov]], [[Sergey Frolov]], A. C. Petkou, _Perturbative and instanton corrections to the OPE of CPOs in $\mathcal{N}=4$ $SYM_4$_, Nucl. Phys. B602:238-260, 2001; Erratum-ibid. B609:540, 2001 ([arXiv:hep-th/0010137](https://arxiv.org/abs/hep-th/0010137))

On [[single trace operators]]/[[BMN operators]] in [[D=4 N=4 super Yang-Mills theory]] identified as [[integrable system|integrable]] [[spin chains]] with respect to the [[dilatation opetator]], and the correspondence of their spectrum with the [[classical field theory|classical]] [[Green-Schwarz superstring]] on [[anti de Sitter spacetime|AdS5]] under the [[AdS/CFT correspondence]]:

* {#BeisertFrolovStaudacherTseytlin03} [[Niklas Beisert]], [[Sergey Frolov]], [[Matthias Staudacher]], [[Arkady Tseytlin]], _Precision Spectroscopy of AdS/CFT_, JHEP 0310:037, 2003 ([arXiv:hep-th/0308117](https://arxiv.org/abs/hep-th/0308117))

## Related $n$Lab entries

* [[Green-Schwarz superstring]]

category: people


Sergey Gorchinskiy

* [MathNet page](https://www.mathnet.ru/php/person.phtml?option_lang=eng&personid=18422)

* [GoogleScholar page](https://scholar.google.com/citations?user=XrFGgm4AAAAJ&hl=en)

## Selected writings

On [[symmetric powers]] in [[higher algebra|higher]] [[homotopical algebra]] (and on [[monoidal localization|monoidal]] [[Bousfield localization of model categories]]):

* {#GorchinskiyGuletskii16} [[Sergey Gorchinskiy]], [[Vladimir Guletskii]], *Symmetric powers in abstract homotopy categories*, Adv. Math., **292** (2016) 707-754 [[arXiv:0907.0730](https://arxiv.org/abs/0907.0730), [doi:10.1016/j.aim.2016.01.011](https://doi.org/10.1016/j.aim.2016.01.011)]

category: people


Sergey Grigorian

* [webpage](http://portal.utpa.edu/utpa_main/daa_home/cosm_home/math_home/grigorian)

category: people


Sergey M. Frolov

> This page is about Sergey M. Frolov working in [[condensed matter theory]]/[[Majorana zero modes]]. There is also *[[Sergey Frolov]]* working in [[high energy physics]]/[[string theory]].

***

* [personal page](https://espressospin.org/about/)

## Selected writings

On (non-)detection of [[Majorana zero modes]] in [[experiment]]:

* {#Frolov21} [[Sergey Frolov]]. *So, You Think You Discovered a New State of Matter?*, Physics **14** 68 (2021) [[physics.aps:v14/68](https://physics.aps.org/articles/v14/68)]

* [[Sergey M. Frolov]], *Quantum computing’s reproducibility crisis: Majorana fermions*, Nature **592** (2021) 350-352 [[doi:10.1038/d41586-021-00954-8](https://doi.org/10.1038/d41586-021-00954-8)]

* {#FrolovMourik22} [[Sergey M. Frolov]], [[Vincent Mourik]], *We cannot believe we overlooked these Majorana discoveries* [[arXiv:2203.17060](https://arxiv.org/abs/2203.17060), [doi:10.5281/zenodo.6364928](https://doi.org/10.5281/zenodo.6364928), conclusion on: [p. 7](https://arxiv.org/ftp/arxiv/papers/2203/2203.17060.pdf#page=7)]

* {#Frolov22} [[Sergey M. Frolov]], [[Vincent Mourik]], *Majorana Fireside Podcast, Episode 1: The Microsoft TGP paper live review* [[video](https://www.youtube.com/watch?v=RnYghkDaHH0), conclusion at: [1:01:31](https://www.youtube.com/watch?v=RnYghkDaHH0&t=3691s)]

* {#Frolov22} [[Sergey M. Frolov]], *Superconductors and semiconductors, nanowires and majorana, research and integrity* [[video](https://www.youtube.com/watch?v=FR7J4yH67GA), general caution: [55:34](https://youtu.be/FR7J4yH67GA?t=3334), concrete criticism: [1:01:41](https://youtu.be/FR7J4yH67GA?t=3701)]

* [[Sergey M. Frolov]], *How do we discover MAJORANA PARTICLES in NANOWIRES?*, talk via [ Instituto de Física, Universidade de São Paulo](http://portal.if.usp.br/ifusp/) (May 2021) [[video](https://www.youtube.com/watch?v=7btnPqmSh_0)]

* [[Sergey M. Frolov]], *[The Fallen Angel Particle](https://espressospin.org/2022/11/17/the-fallen-angel-particle/)* (Nov 2022)

category: people


Sergey Mironov

* [InSpire page](https://inspirehep.net/authors/1065019)

## Selected writings

On [[topological quantum computation]] with [[Chern-Simons theory]]:

* [[Dmitry Melnikov]], [[Andrei Mironov]], [[Sergey Mironov]], [[Alexei Morozov]], [[Andrey Morozov]], _Towards topological quantum computer_, Nucl. Phys. B926 (2018) 491-508 ([arXiv:1703.00431](https://arxiv.org/abs/1703.00431), [doi:10.1016/j.nuclphysb.2017.11.016](https://doi.org/10.1016/j.nuclphysb.2017.11.016))

On universality of [[braid representation|braid]] [[quantum gates]] of [[su(2)-anyons]] at higher [[Chern-Simons level|level]] $k$ (and also for $SU(N)$-anyons with $N \gt 2$):

* [[Nikita Kolganov]], [[Sergey Mironov]], [[Andrey Morozov]], *Large $k$ topological quantum computer*, Nuclear Physics B
**987** (2023) 116072 [[arXiv:2105.03980](https://arxiv.org/abs/2105.03980), [doi:10.1016/j.nuclphysb.2023.116072](https://doi.org/10.1016/j.nuclphysb.2023.116072)]

category: people


Sergey Oblezin

Sergey Oblezin is an EPSRC research fellow at Nottingham University.

* [webpage](http://www.nottingham.ac.uk/mathematics/people/sergey.oblezin)

* EPSRC grant _[Topological field theories, Baxter operators and the Langlands programme](http://gow.epsrc.ac.uk/NGBOViewGrant.aspx?GrantRef=EP/L000865/1)_

Oblezin has been working on a  [[local Langlands correspondence|local]] [[archimedean field|Archimedean]] [[geometric Langlands duality|Langlands duality]] for [[Whittaker functions]] realized as [[mirror symmetry]] of a suitable [[A-model]] and [[B-model]]:

* {#GerasimovLebedevOblezin09} [[Anton Gerasimov]], Dimitri Lebedev, [[Sergey Oblezin]],

  _Archimedean L-factors and Topological Field Theories I_ ([arXiv:0906.1065](http://arxiv.org/abs/0906.1065))

  _Archimedean L-factors and Topological Field Theories II_ ([arXiv:0909.2016](http://arxiv.org/abs/0909.2016))

  _Parabolic Whittaker Functions and Topological Field Theories I_ ([arXiv:1002.2622](http://arxiv.org/abs/1002.2622))

category: people


Sergey Slavnov

* [webpage](http://www.math.cornell.edu/m/People/PhD/2005Aug/slavnov)

## Related entries

* [[quantum logic]], [[linear logic]], [[Weinstein symplectic category]]

## Selected writings

On [[quantum logic]] as multiplicative [[linear logic]] with [[categorical semantics]] in non-cartesian [[monoidal categories]] such as [[cobordism categories]]:

* {#Slavnov05} [[Sergey Slavnov]], *From proof-nets to bordisms: the geometric meaning of multiplicative connectives*, Mathematical Structures in Computer Science **15** 06 (2005) 1151-1178 [[doi:10.1017/S0960129505004974](https://doi.org/10.1017/S0960129505004974)]

* [[Sergey Slavnov]], *Geometrical semantics for linear logic (multiplicative fragment)*, Theoretical Computer Science **357** 1-3 (2006) 215-229 [[doi:10.1016/j.tcs.2006.03.020](https://doi.org/10.1016/j.tcs.2006.03.020)]

* [[Sergey Slavnov]], _Coherent phase spaces. Semiclassical semantics_, Annals of Pure and Applied Logic
Volume 131, Issues 1–3, January 2005, Pages 177-225, [doi:10.1016/j.apal.2004.05.003](https://doi.org/10.1016/j.apal.2004.05.003).

* [[Sergey Slavnov]], *On Banach spaces of sequences and free linear logic exponential modality*, Math. Struct. Comp. Sci. 29 (2019) 215-242 [[arXiv:1509.03853](https://arxiv.org/abs/1509.03853)]

category: people


Sergey Solodukhin

* [webpage](http://www.theorie.physik.uni-muenchen.de/cosmology/members/former_mem/solodukhin/index.html)

## related $n$Lab entries

* [[AdS-CFT correspondence]]

* [[Fefferman-Graham ambient construction]]

category: people


Sergey Yuzvinsky

* [personal page](https://pages.uoregon.edu/yuz/)

* [Institute page](https://math.uoregon.edu/profile/yuz)

## Selected writings

On [[local systems]] and their [[twisted de Rham cohomology]]/[[Dolbeault cohomology]]:

* [[Anatoly Libgober]], [[Sergey Yuzvinsky]], *Cohomology of local systems*, Advanced Studies in Pure Mathematics **27**, Mathematics Society of Japan (2000) 169-184 [[pdf](http://homepages.math.uic.edu/~libgober/otherpapers/export/2000sergeytokyo.pdf), [doi:10.2969/aspm/02710169](https://doi.org/10.2969/aspm/02710169)]

On [[topological complexity]] of [[real projective space]] and connection with their [[immersion]]  into [[cartesian space]]:

* {#FarberTabachnikovYuzvinsky02} [[Michael Farber]], [[Serge Tabachnikov]], [[Sergey Yuzvinsky]], _Topological robotics: motion planning in projective spaces_ (2002), [arXiv:math/0210018](https://arxiv.org/abs/math/0210018);

category: people


Sergio Albeverio

* [webpage](http://wt.iam.uni-bonn.de/albeverio/home/)

category: people


Sergio Benvenuti

* [InSpire page](https://inspirehep.net/authors/1039500?ui-citation-summary=true)

## Selected writings

On [[5-brane webs]] and [[D=4 N=2 super Yang-Mills theory]]:

* [[Francesco Benini]], [[Sergio Benvenuti]], [[Yuji Tachikawa]], _Webs of five-branes and $\mathcal{N}=2$ superconformal field theories_, JHEP 0909:052 (2009) [[arXiv:0906.0359](https://arxiv.org/abs/0906.0359)]

category: people


Sergio Cecotti

* [inspire page](https://inspirehep.net/authors/1014205)

## Selected writings

On [[supergravity]] with [[higher curvature corrections]] and the [[Starobinsky model of cosmic inflation]]:

* {#Cecotti87} [[Sergio Cecotti]], *Higher derivative supergravity is equivalent to standard supergravity coupled to matter*, Phys. Lett. B **190** 86 (1987) [doi:10.1016/0370-2693(87)90844-6]

On the [[quantum cohomology]] induced by [[supersymmetry|supersymmetric]] [[CP^N sigma-model|$\mathbb{C}P^N$ sigma-models]]:

* [[Sergio Cecotti]], [[Cumrun Vafa]], *Exact Results for Supersymmetric Sigma Models*, Phys. Rev. Lett. **68** (1992) 903-906 [[arXiv:hep-th/9111016](https://arxiv.org/abs/hep-th/9111016), [doi:10.1103/PhysRevLett.68.903](https://doi.org/10.1103/PhysRevLett.68.903)]

* {#CecottiVafa93} [[Sergio Cecotti]], [[Cumrun Vafa]], *On Classification of $N=2$ Supersymmetric Theories*, Comm. Math. Phys. **158** (1993) 569-644 [[arXiv:hep-th/9211097](https://arxiv.org/abs/hep-th/9211097), [doi:10.1007/BF02096804](https://doi.org/10.1007/BF02096804)]

On how the [[second quantization]] of the [[B-model]] yields [[Kodeira-Spencer gravity]]/[[BCOV theory]]:

* {#BCOV93} [[Michael Bershadsky]], [[Sergio Cecotti]], [[Hirosi Ooguri]], [[Cumrun Vafa]], *Kodaira-Spencer Theory of Gravity and Exact Results for Quantum String Amplitudes*, Commun. Math. Phys. **165** (1994) 311-428 [[arXiv:hep-th/9309140](http://arxiv.org/abs/hep-th/9309140), [doi:10.1007/BF02099774](https://doi.org/10.1007/BF02099774)]

On [[wall crossing]] and [[topological string theory]]:

* [[Sergio Cecotti]], [[Cumrun Vafa]], *BPS wall crossing and topological strings*  [[arXiv/0910.2615](http://arxiv.org/abs/0910.2615)]

* [[Sergio Cecotti]], _Trieste lectures on wall-crossing invariants_ (2010) [[pdf](http://people.sissa.it/~cecotti/ictptext.pdf)&rbrack

category: people


Sergio Doplicher

__Sergio Doplicher__ is a [[mathematical physicist]] from Rome working in the context of [[AQFT]].

* [web](http://www.mat.uniroma1.it/people/show.php?id_utente=57)

## related $n$Lab entries

* [[Doplicher-Roberts reconstruction theorem]]

* [[DHR superselection theory]]

[[!redirects S. Doplicher]]
[[!redirects Doplicher]]

category: people


Sergio Ferrara

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

One of the co-discoverers of [[supergravity]] ([[4d supergravity]]):

## Selected writings

Introducing [[super Yang-Mills theory]]:

* {#FerraraZumino74} [[Sergio Ferrara]], [[Bruno Zumino]], _Supergauge invariant Yang-Mills theories_, Nuclear Physics B Volume 79, Issue 3, 25 September 1974, Pages 413-421 (doi:10.1016/0550-3213(74)90559-8)

Introducing [[4d supergravity]]:

* {#FreedmanNieuwenhuizenFerrara76} [[Daniel Freedman]], [[Peter van Nieuwenhuizen]], [[Sergio Ferrara]], _Progress toward a theory of supergravity_, Phys. Rev. D13 (1976) 3214 ([doi.org/10.1103/PhysRevD.13.3214](https://doi.org/10.1103/PhysRevD.13.3214))

Formulation of [[D=11 supergravity]] in [[superspace]] (via [[supergeometry]]):

* {#CremmerFerrara80} [[Eugene Cremmer]], [[Sergio Ferrara]], *Formulation of Eleven-Dimensional Supergravity in Superspace*, Phys.Lett. B **91** (1980) 61 [doi:10.1016/0370-2693(80)90662-0]

On [[D=6 supergravity]]:

* [[Sergio Ferrara]], [[Fabio Riccioni]], [[Augusto Sagnotti]], _Tensor and Vector Multiplets in Six-Dimensional Supergravity_, Nucl. Phys. B519 (1998) 115-140 ([arXiv:hep-th/9711059](https://arxiv.org/abs/hep-th/9711059))

Discussion of [[multi-trace operators]] in [[super Yang-Mills theory]] and of their [[AdS-CFT duality|AdS-CFT dual]] [[gravity]]/[[string theory]] incarnation:

* [[Laura Andrianopoli]], [[Sergio Ferrara]], _On short and long $SU(2,2/4)$ multiplets in the AdS/CFT correspondence_, Lett. Math.Phys. 48 (1999) 145-161 ([arXiv:hep-th/9812067](https://arxiv.org/abs/hep-th/9812067))

## Related entries

* [[supergravity]]

* [[Starobinsky model of cosmic inflation]]

* [[singleton representation]]

category: people


Sergiu Klainerman

* [webpage](https://web.math.princeton.edu/~seri/homepage/)

## Writings

lecture notes on [[functional analysis]]:

* _Lecture notes in analysis_, 2011 ([pdf](https://web.math.princeton.edu/~seri/homepage/courses/Analysis2011.pdf))

## Related $n$Lab entries

* [[functional analysis]]

* [[distribution]]

  * [[compactly supported distribution]]

  * [[support of a distribution]]

  * [[tensor product of distributions]]

  * [[Fourier transform of distributions]]

  * [[fundamental solution]]

* [[wave equation]]

* [[Minkowski spacetime]]

category: people


series

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Analysis
+-- {: .hide}
[[!include analysis - contents]]
=--
#### Algebra
+-- {: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

# Infinite series
* table of contents
{: toc}

## Idea

In [[analysis]] a *series* is a formal precursor to a various notions of a [[sum]] of an infinite sequence of [[numbers]].

## Definition

An (ordinary) __series__ $\sum_{n=0}^\infty a_n$ whose members are the elements $a_n$ in a given additive [[group]] or [[semigroup]] is an ordered pair of a [[sequence]] $(a_n)_{n=0}^\infty$ and a sequence $(b_k)_k$ of its partial [[sums]] $b_k = \sum_{n=0}^k a_k$.

### As an operator

For any group or semigroup $G$, we can [[inductive definition|inductively define]] the finite sum $\mathbb{N}$-[[action]]

$$\sum_{n=0}^{(-)} (-)(n) : (\mathbb{N} \to G) \times \mathbb{N} \to G$$

as

$$\sum_{n=0}^{0} a(n) = a(0)$$

$$\sum_{n=0}^{k+1} a(n) = a(k+1) + \sum_{n=0}^{k} a(n)$$

for all $k:\mathbb{N}$ and $a:\mathbb{N} \to G$, such that [[currying]] the action results in the infinite series [[operator]]

$$\sum_{n=0}^\infty (-)(n) : (\mathbb{N} \to G) \to (\mathbb{N} \to G)$$

over the [[function algebra|function $G$-module]] $\mathbb{N} \to G$.

## Internalisation

In a [[cartesian closed category]] $C$, let $(G,+)$ be an additive [[magma object]], let $(N,0,s)$ be a [[natural numbers object]] in $C$, and let $a:N\to G$ be an __infinite sequence object__ of elements in $G$. Then there exists an infinite sequence object $b:N\to G$ called the __left partial sum infinite sequence object__ of $a$ inductively defined by $b(0) = a(0)$ and $b(s(n)) = b(n) + a(s(n))$, and an infinite sequence object $c:N\to G$ called the __right partial sum infinite sequence object__ of $a$ inductively defined by $c(0) = a(0)$ and $c(s(n)) = a(s(n)) + c(n)$. The element $b(n)$ is called the __left $n$-th partial sum__ of the infinite sequence $a$, and the element $c(n)$ is called the __right $n$-th partial sum__ of the infinite sequence $a$. A __left series object__ is an infinite sequence object with its left partial sum infinite sequence object, and a __right series object__ is an infinite sequence object with its right partial sum infinite sequence object. If the magma object is [[commutative]] and [[associative]], then the left and right partial sum infinite sequence objects of $a$ are equal and just called a __partial sum infinite sequence object__ $d = \sum_{n:N} a(n)$ of $a$, where the element $d(n)$ is the __$n$-th partial sum__. Then s __series object__ is an infinite sequence object with its partial sum infinite sequence object,

## Sum of a series

The most straightforward notion of the **sum** of a series is the [[limit of a sequence|limit]] of its sequence of partial sums, if this sequence converges, relative to some [[topological space|topology]] on the space where the members of the sequence belong to.  A series that does not converge in this sense is called [[divergent series|divergent]]; sometimes these can also be "summed" by fancier techniques.

## Examples

* [[geometric series]]

* [[Mercator series]]

* [[Taylor series]]

* The formal [[e]], $e = \sum_{n=0}^\infty \frac{1}{n!}$, where $\frac{1}{(-)!}:\mathbb{N}\to \mathbb{Q}$

## Related concepts

* [[power series]]

* [[asymptotic series]]

* [[infinite product]]

* [[divergent series]]

* [[partial sum]]

* [[Riemann series theorem]]

[[!redirects series]]
[[!redirects infinite series]]

[[!redirects infinite sum]]
[[!redirects infinite sums]]


Serre automorphism

see [here](cobordism+hypothesis#ExamplesOfTheCanonicalOnActions)

([Lurie 09, example 2.4.14](#Lurie09), [DSPS 13, section 2.3](#DSPS13))

## Refrences

* {#Lurie09} [[Jacob Lurie]], _[[On the Classification of Topological Field Theories]]_, Current Developments in Mathematics Volume 2008 (2009), 129-280 ([arXiv:0905.0465](http://arxiv.org/abs/0905.0465))

* {#DSPS13} [[Chris Douglas]], [[Chris Schommer-Pries]], [[Noah Snyder]], _Dualizable tensor categories_ ([arXiv:1312.7188](http://arxiv.org/abs/1312.7188))

[[!redirects Serre automorphisms]]


Serre duality

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

## Contents
* table of contents
{:toc}

## Idea

Serre duality in [[complex analytic geometry]] is the duality induced by the [[Hodge star operator]] on the [[Dolbeault complex]]. This generalizes to suitable non-singular [[projective algebraic varieties]] over other base rings.

## Statement

### In complex analytic geometry

Let $X$ be a [[Hermitian manifold]] of complex [[dimension]] $dim_{\mathbb{C}}(\Sigma) = n$. Its [[Riemannian metric]] induces a [[Hodge star operator]] which acts on the pieces in the [[Dolbeault complex]] as

$$
  \star \colon \Omega^{p,q}(X)\to \Omega^{n-q, n-p}(X)
$$

(see at _[Hodge star operator -- On a Kähler manifold](Hodge%20star%20operator#OnAKahlerManifold)_).

Moreover, [[complex conjugation]] gives $\mathbb{C}$-[[antilinear functions]]

$$
  \overline{(-)} \colon \Omega^{p,q}(X)\to \Omega^{q,p}(X)
  \,.
$$

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

Write

$$
  \bar \star \;\coloneqq\; \overline{(-)} \circ \star = \star \circ \overline{(-)}
  \;\colon\;
  \Omega^{p,q}(X)\to \Omega^{n-p, n-q}(X)
$$

for the [[composition|composite]] [[antilinear function]].

=--

e.g. ([Huybrechts 04, def. 4.1.6](#Huybrechts04))

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

By the [basic properties of the Hodge star](Hodge+star+operator#BasicProperties) it follows that restricted to $\Omega^{p,q}(X)$

$$
  \bar \star \circ \bar \star = (-1)^{p+q}
  \,.
$$

=--

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

For $X$ a [[compact topological space|compact]] [[Hermitian manifold]], define a [[bilinear form]]

$$
  \int_X (-) \wedge \bar \star (-)
  \;\colon\;
  \Omega^{p,q}(X) \otimes \Omega^{n-p,n-q}(X)
   \longrightarrow
   \mathbb{C}
$$

as the [[integration of differential forms]]

$$
  (\alpha,\beta) \mapsto \int_X \alpha \wedge \bar \star \beta
$$

of the wedge product of $\alpha$ with the image of $\beta$ under the complex conjugated Hodge star operator of def. \ref{ComplexConjugatedHodgeStar}.

=--

+-- {: .num_prop #StatementForTrivialBundleOverHermitianManifold}
###### Proposition
**(Serre duality)**

The pairing of def. \ref{ThePairing} induces a non-degenerate sesquilinear (i.e. hermitian) form  on [[Dolbeault cohomology]]

$$
  \int_X (-) \wedge \bar \star (-)
  \;\colon\;
  \H^{p,q}(X) \otimes H^{n-p,n-q}(X)
   \longrightarrow
   \mathbb{C}
$$

=--

e.g. ([Huybrechts 04, prop 4.1.15](#Huybrechts04))

## Properties

### Relation to Poincaré duality

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

For $\Sigma$ a [[compact topological space|compact]] [[Kähler manifold]] the [[Hodge theorem]] gives an [[isomorphism]]

$$
  H^k(X, \mathbb{C})
  \simeq
  \underset{p+q = k}{\oplus} H^{p,q}(X)
$$

between the [[ordinary cohomology]] of the underlying [[topological space]] with [[coefficients]] in the [[complex numbers]], and the [[direct sum]] of all the [[Dolbeault cohomology]] groups in the same total degree.

Therefore for $\Sigma$ of complex [[dimension]] $dim_{\mathbb{C}}(\Sigma)= n$ then Serre duality in the form of prop. \ref{StatementForTrivialBundleOverHermitianManifold} induces an [[isomorphism]] in [[ordinary cohomology]] of the form

$$
  H^k(X, \mathbb{C})
  \stackrel{\simeq}{\longrightarrow}
  H^{2n-k}(X, \mathbb{C})
  \,.
$$

=--

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

The isomorphism in remark \ref{SerreDualityActionOnComplexordinaryCohomologyOfKaehlerManifold}
coincides with [[Poincaré duality]].

=--

## Related concepts

* [[complex conjugation]], [[Hodge star operator]]

## References

* Wikipedia, _[Serre duality](http://en.wikipedia.org/wiki/Serre_duality)_

Discussion in [[complex analytic geometry]] ([[Hermitian manifolds]]) includes

* {#Huybrechts04} [[Daniel Huybrechts]] around prop. 4.1.15 of _Complex geometry - an introduction_. Springer (2004). Universitext. 309 pages. ([pdf](http://www.math.uh.edu/~shanyuji/2012/Geometry/Huybrechts.pdf))

* R.O. Wells, _Differential Analysis on Compact Manifolds_, Second Edition,
Springer, 1980. 14

and review with emphasis on the case of [[Kähler manifolds]] includes

* Julien Meyer, _Hodge theory on Kähler manifolds_ ([pdf](http://homepages.ulb.ac.be/~jmeyer/Templates/hodgetheory.pdf))