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\begin{document}

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\section*{sequential spectrum}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{stable_homotopy_theory}{}\paragraph*{{Stable Homotopy theory}}\label{stable_homotopy_theory}

[[!include stable homotopy theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{in_components}{In components}\dotfill \pageref*{in_components} \linebreak
\noindent\hyperlink{AsDiagramSpectra}{As diagram spectra}\dotfill \pageref*{AsDiagramSpectra} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{limits_and_colimits}{Limits and colimits}\dotfill \pageref*{limits_and_colimits} \linebreak
\noindent\hyperlink{tensoring_and_powering_over_pointed_spaces}{Tensoring and powering over pointed spaces}\dotfill \pageref*{tensoring_and_powering_over_pointed_spaces} \linebreak
\noindent\hyperlink{ModelCategoryStructures}{Model category structures}\dotfill \pageref*{ModelCategoryStructures} \linebreak
\noindent\hyperlink{SuspensionAndLooping}{Suspension and looping}\dotfill \pageref*{SuspensionAndLooping} \linebreak
\noindent\hyperlink{relation_to_excisive_functors}{Relation to excisive functors}\dotfill \pageref*{relation_to_excisive_functors} \linebreak
\noindent\hyperlink{SmashProduct}{Smash product}\dotfill \pageref*{SmashProduct} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

In [[stable homotopy theory]], a \emph{sequential ([[prespectrum|pre-]])[[spectrum]]} $E$ (also \emph{Boardman spectrum}, after (\hyperlink{Boardman65}{Boardman 65})) 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 \emph{[[spectrum]]} (or pre-spectrum) and still the one predominently meant 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 \emph{sequential spectrum} for emphasis (e.g. \hyperlink{Schwede12}{Schwede 12, def. 2.1}).

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\hypertarget{in_components}{}\subsubsection*{{In components}}\label{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]].

\begin{defn}
\label{SequentialSpectra}\hypertarget{SequentialSpectra}{}
A \emph{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

\begin{displaymath}
\itexarray{
    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}
  }
\end{displaymath}
[[commuting diagram|commute]].

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

\end{defn}
\begin{example}
\label{SmashProductOfSpectrumWithSimplicialSet}\hypertarget{SmashProductOfSpectrumWithSimplicialSet}{}
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]]

\begin{displaymath}
(X \wedge K_+)_n \coloneqq (X_n \wedge K_+)
\end{displaymath}
and whose structure maps are given by

\begin{displaymath}
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_+
  \,.
\end{displaymath}
\end{example}
\begin{prop}
\label{SimplicialEnrichment}\hypertarget{SimplicialEnrichment}{}
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

\begin{displaymath}
[X,Y]_n \coloneqq Hom_{SeqSpec(sSet)}(X\wedge \Delta[n]_+,Y)
  \,.
\end{displaymath}
\end{prop}
\begin{defn}
\label{OmegaSpectrum}\hypertarget{OmegaSpectrum}{}
An \emph{[[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]]

\begin{displaymath}
{\vert X_n\vert} \stackrel{}{\longrightarrow} {\vert X^{n+1}\vert}^{{\vert S^1\vert}}
\end{displaymath}
of the structure maps ${\vert \sigma_n\vert}$ are [[weak homotopy equivalences]].

\end{defn}
\hypertarget{AsDiagramSpectra}{}\subsubsection*{{As diagram spectra}}\label{AsDiagramSpectra}

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

Write

\begin{displaymath}
\iota \;\colon\; StdSpheres \longrightarrow sSet^{\ast/}_{fin}
\end{displaymath}
for the non-full $sSet^{\ast/}$-[[enriched category|enriched]] [[subcategory]] of pointed [[simplicial object|simplicial]] [[finite sets]], def. \ref{SimplicialSetsPointedAndFinite} whose

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


\item [[hom-objects]] are

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


\end{itemize}
\end{defn}
(\hyperlink{Lydakis98}{Lydakis 98, def. 4.2}), see also (\hyperlink{MMSS00}{MMSS 00}) \href{Model+categories+of+diagram+spectra#SequentialSpectraAsFunctorsOnFreeSSequModules}{this example}.

\begin{prop}
\label{SequentialSpectraAsDiagramSpectra}\hypertarget{SequentialSpectraAsDiagramSpectra}{}
There is an $sSet^{\ast/}$-[[enriched functor]]

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

\begin{displaymath}
X^{seq}_n \coloneqq X(S^n_{std})
\end{displaymath}
and with structure maps

\begin{displaymath}
\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}]}
\end{displaymath}
given by

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

\end{prop}
(\hyperlink{Lydakis98}{Lydakis 98, prop. 4.3}), see also (\hyperlink{MMSS00}{MMSS 00})

\begin{remark}
\label{}\hypertarget{}{}
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 \emph{[[Model categories of diagram spectra]]} \href{Model+categories+of+diagram+spectra#SModulesAsEnrichedFunctors}{this lemma} and \href{Model+categories+of+diagram+spectra#SequentialSpectraAsFunctorsOnFreeSSequModules}{this example}.

\end{remark}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{limits_and_colimits}{}\subsubsection*{{Limits and colimits}}\label{limits_and_colimits}

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

Given

\begin{displaymath}
X_\bullet
    \;\colon\;
  I
    \longrightarrow
  SeqSpec(Top_{cg})
\end{displaymath}
a [[diagram]] of sequential spectra, then:

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

\begin{displaymath}
(\underset{\longrightarrow}{\lim}_i X(i))_n
  \simeq
   \underset{\longrightarrow}{\lim}_i X(i)_n
   \,,
\end{displaymath}

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

\begin{displaymath}
(\underset{\longleftarrow}{\lim}_i X(i))_n
  \simeq
   \underset{\longleftarrow}{\lim}_i X(i)_n
   \,;
\end{displaymath}


\end{enumerate}
moreover:

\begin{enumerate}%
\item the colimiting spectrum has structure maps in the sense of def. \ref{SequentialSpectra} given by

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


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

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



\end{enumerate}
\end{prop}
\begin{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 (\href{Introduction+to+Stable+homotopy+theory+--+P#TopologicallyEnrichedCopresheavesHaveAllLimitsAndColimits}{prop.}). But it is also immediate to directly check the [[universal property]].

\end{proof}
\begin{example}
\label{WedgeSumOfSpectra}\hypertarget{WedgeSumOfSpectra}{}
The [[coproduct]] of [[spectra]] $X, Y \in SeqSpec(Top_{cg})$, called the \textbf{wedge sum of spectra}

\begin{displaymath}
X \vee Y
    \coloneqq
  X \sqcup Y
\end{displaymath}
is componentwise the [[wedge sum]] of pointed topological spaces (\href{Introduction+to+Stable+homotopy+theory+--+P#WedgeSumAsCoproduct}{exmpl.})

\begin{displaymath}
(X \vee Y)_n
  =
  X_n \vee Y_n
\end{displaymath}
with structure maps

\begin{displaymath}
\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}
  \,.
\end{displaymath}
\end{example}
\hypertarget{tensoring_and_powering_over_pointed_spaces}{}\subsubsection*{{Tensoring and powering over pointed spaces}}\label{tensoring_and_powering_over_pointed_spaces}

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

\begin{defn}
\label{TensoringAndPoweringOfSequentialSpectra}\hypertarget{TensoringAndPoweringOfSequentialSpectra}{}
Let $X$ be a sequential spectrum and $K$ a [[pointed topological space]]/[[pointed simplicial set]]. Then

\begin{enumerate}%
\item $X \wedge K$ is the sequential spectrum with

\begin{itemize}%
\item $(X \wedge K)_n \coloneqq X_n \wedge K$ ([[smash product]])


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



\end{itemize}

\item $X^K$ is the sequential spectrum with

\begin{itemize}%
\item $(X^K)_n \coloneqq (X_n)^K$ ([[pointed mapping space]])


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



\end{itemize}


\end{enumerate}
\end{defn}
\hypertarget{ModelCategoryStructures}{}\subsubsection*{{Model category structures}}\label{ModelCategoryStructures}

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

The \emph{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}:

\begin{displaymath}
SeqSpec(sSet)_{stable}
  \stackrel{\longleftarrow}{\overset{Bousf.\;loc}{\longrightarrow}}
  SeqSpec(sSet)_{strict}
  =
  [StdSpheres, Top^{\ast/}_{Quillen}]_{proj}
  \,.
\end{displaymath}
\hypertarget{SuspensionAndLooping}{}\subsubsection*{{Suspension and looping}}\label{SuspensionAndLooping}

There are \emph{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.

\begin{defn}
\label{ShiftedSpectrum}\hypertarget{ShiftedSpectrum}{}
For $X$ a [[sequential spectrum]] and $k \in \mathbb{Z}$, the $k$-fold \textbf{shifted spectrum} of $X$ is the sequential spectrum denoted $X[k]$ given by

\begin{itemize}%
\item $(X[k])_n \coloneqq \left\{ \itexarray{X_{n+k} & for \; n+k \geq 0 \\ \ast & otherwise } \right.$;


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



\end{itemize}
\end{defn}
\begin{defn}
\label{SequentialSpectrumRealSuspension}\hypertarget{SequentialSpectrumRealSuspension}{}
For $X$ a sequential spectrum, then

\begin{enumerate}%
\item the \textbf{real suspension} of $X$ is $X \wedge S^1$ according to def. \ref{TensoringAndPoweringOfSequentialSpectra};


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



\end{enumerate}
\end{defn}
\begin{defn}
\label{SequentialSpectrumFakeSuspension}\hypertarget{SequentialSpectrumFakeSuspension}{}
For $X$ a sequential spectrum, then

\begin{enumerate}%
\item the \textbf{fake suspension} of $X$ is the sequential spectrum $\Sigma X$ with

\begin{enumerate}%
\item $(\Sigma X)_n \coloneqq S^1 \wedge X_n$


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



\end{enumerate}

\item the \textbf{fake looping} of $X$ is the sequential spectrum $\Omega X$ with

\begin{enumerate}%
\item $(\Omega X)_n \coloneqq (X_n)^{S^1}$;


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



\end{enumerate}


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

\end{defn}
e.g. (\hyperlink{Jardine15}{Jardine 15, section 10.4}).

\begin{defn}
\label{ShiftingCommutesWithLoopingAndSuspensionOfSequentialSpectra}\hypertarget{ShiftingCommutesWithLoopingAndSuspensionOfSequentialSpectra}{}
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.

\end{defn}
\begin{defn}
\label{}\hypertarget{}{}
The canonical morphism

\begin{enumerate}%
\item $\Sigma X \longrightarrow X[1]$ is given in degree $n$ by $\sigma_n^X$.


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



\end{enumerate}
\end{defn}
\begin{prop}
\label{AdjunctionsBetweenLoopingAndDeloopingForSeqSpec}\hypertarget{AdjunctionsBetweenLoopingAndDeloopingForSeqSpec}{}
The constructions from def. \ref{ShiftedSpectrum}, def. \ref{SequentialSpectrumRealSuspension} and def. \ref{SequentialSpectrumFakeSuspension} form pairs of [[adjoint functors]] $SeqSpec \to SeqSpec$ like so:

\begin{enumerate}%
\item $(-)[1] \;\dashv\; (-)[-1] \;\dashv\; (-)[1] \;\dashv\; \cdots$;


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


\item $\Sigma \dashv \Omega$.



\end{enumerate}
\end{prop}
\begin{proof}
The first is immediate from the definition.

The second is just degreewise the adjunction [[smash product]]$\dashv$[[pointed mapping space]] (discussed \href{pointed+object#ClosedMonoidalStructure}{here}), 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:

\begin{displaymath}
\itexarray{  
    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}
  }
  \,.
\end{displaymath}
Applying the adjunction isomorphism diagonally gives a bijection to diagrams of this form:

\begin{displaymath}
\itexarray{
   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}
  }
  \,.
\end{displaymath}
Then applying the same isomorphism diagonally once more gives a further bijection to commuting diagrams of this form:

\begin{displaymath}
\itexarray{
    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}
  }
  \,.
\end{displaymath}
This finally equivalently exhibits morphisms of the form

\begin{displaymath}
X \longrightarrow \Omega Y
  \,.
\end{displaymath}
\end{proof}
\begin{example}
\label{}\hypertarget{}{}
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

\begin{displaymath}
\itexarray{
     \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]
  }
  \,.
\end{displaymath}
\end{example}
\begin{prop}
\label{}\hypertarget{}{}
For $X$ a sequential spectrum, then (using remark \ref{ShiftingCommutesWithLoopingAndSuspensionOfSequentialSpectra} to suppress parenthesis)

\begin{enumerate}%
\item the structure maps constitute a homomorphism

\begin{displaymath}
\Sigma X[-1]
  \longrightarrow
  X
\end{displaymath}
and this is a stable equivalence.


\item the adjunct structure maps constitute a homomorphism

\begin{displaymath}
X
  \longrightarrow
  \Omega X[1]
  \,.
\end{displaymath}
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.



\end{enumerate}
\end{prop}
\begin{proof}
The diagrams that need to commute for the structure maps to give a homomorphism as claimed are in degree 0 this one

\begin{displaymath}
\itexarray{
    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
  }
\end{displaymath}
and in degree $n \geq 1$ these:

\begin{displaymath}
\itexarray{
     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}
  }
  \,.
\end{displaymath}
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

\begin{displaymath}
\itexarray{
    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}
  }
  \,.
\end{displaymath}
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}

\begin{displaymath}
\itexarray{
     [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}
  }
  \,.
\end{displaymath}
\end{proof}
\begin{prop}
\label{}\hypertarget{}{}
For $X$ a sequential spectrum in simplicial sets. Then there are stable equivalences

\begin{displaymath}
X\wedge S^1 \longrightarrow \Sigma X \longrightarrow X[1]
\end{displaymath}
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

\begin{displaymath}
X^{S^1} \longrightarrow \Omega X \longrightarrow X[-1]
\end{displaymath}
between the real looping (def. \ref{SequentialSpectrumRealSuspension}), the fake looping (def. \ref{SequentialSpectrumFakeSuspension}) and the shift by -1 (def. \ref{ShiftedSpectrum}) of $X$.

\end{prop}
(\hyperlink{Jardine15}{Jardine 15, corollary 10.54})

\hypertarget{relation_to_excisive_functors}{}\subsubsection*{{Relation to excisive functors}}\label{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]].

\begin{defn}
\label{SimplicialSetsPointedAndFinite}\hypertarget{SimplicialSetsPointedAndFinite}{}
Write

\begin{itemize}%
\item [[sSet]] for the [[category]] of [[simplicial sets]];


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


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



\end{itemize}
Write

\begin{displaymath}
sSet^{\ast/}
  \stackrel{\overset{(-)_+}{\longleftarrow}}{\underset{u}{\longrightarrow}}
  sSet
\end{displaymath}
for the [[free-forgetful adjunction]], where the [[left adjoint]] functor $(-)_+$ freely adjoins a base point.

Write

\begin{displaymath}
\wedge \colon sSet^{\ast/} \times sSet^{\ast/} \longrightarrow sSet^{\ast/}
\end{displaymath}
for the [[smash product]] of [[pointed object|pointed]] [[simplicial sets]], similarly for its restriction to $sSet_{fin}^{\ast}$:

\begin{displaymath}
X \wedge Y 
    \coloneqq 
  cofib\left(
    \; 
    \left(\,
      (u(X),\ast) \sqcup (\ast, u(Y)) 
    \,\right) 
      \longrightarrow 
     u(X) \times u(Y) 
    \;
  \right)
  \,.
\end{displaymath}
This gives $sSet^{\ast/}$ and $sSet^{\ast/}_{fin}$ the structure of a [[closed monoidal category]] and we write

\begin{displaymath}
[-,-]_\ast \;\colon\; (sSet^{\ast/})^{op} \times sSet^{\ast/} \longrightarrow sSet^{\ast/}
\end{displaymath}
for the corresponding [[internal hom]], the pointed [[function complex]] functor.

\end{defn}
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]]

\begin{displaymath}
[sSet^{\ast/}_{fin}, sSet^{\ast/}]
  \,.
\end{displaymath}
(\hyperlink{Lydakis98}{Lydakis 98, example 3.8, def. 4.4})

\begin{prop}
\label{QuillenEquivalenceBetweenSequentialBFAndExcisiveFunctors}\hypertarget{QuillenEquivalenceBetweenSequentialBFAndExcisiveFunctors}{}
The [[adjunction]]

\begin{displaymath}
(\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}
\end{displaymath}
(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]].

\end{prop}
(\hyperlink{Lydakis98}{Lydakis 98, theorem 11.3}) For more details see at \emph{[[model structure for excisive functors]]}. The analogous statement for spectra in $Top$ is in (\hyperlink{MMSS00}{MMSS 00}).

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

By \href{smash+product+of+spectra#WhySequentialSpectraHaveNoSymmetricSmashProduct}{this remark} at \emph{[[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 $StdSpheres$.

\end{remark}
\hypertarget{SmashProduct}{}\subsubsection*{{Smash product}}\label{SmashProduct}

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

\begin{defn}
\label{SmashProductByDoublingDegrees}\hypertarget{SmashProductByDoublingDegrees}{}
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

\begin{displaymath}
(X\wedge Y)_{2n} \coloneqq X_n \wedge Y_n
\end{displaymath}
and in odd degree by

\begin{displaymath}
(X\wedge Y)_{2n+1} \coloneqq S^1 \wedge X_n \wedge Y_n
\end{displaymath}
with structure maps being in even degree the identity

\begin{displaymath}
\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}
\end{displaymath}
and in odd degree as the composite

\begin{displaymath}
\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}
  \,.
\end{displaymath}
\end{defn}
(\hyperlink{Lydakis98}{Lydakis 98, def. 10.20}, \href{Gamma-space#Lydakis98}{Lydakis 98b, def. 5.9}, \hyperlink{MMSS00}{MMSS 00, def. 11.6})

\begin{prop}
\label{SmashProductCompatibleWithTheOneOnSpectra}\hypertarget{SmashProductCompatibleWithTheOneOnSpectra}{}
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}.

\end{prop}
(\hyperlink{Lydakis98}{Lydakis 98, theorem 12.5}, \hyperlink{MMSS00}{MMSS 00, prop. 11.9})

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[symmetric spectrum]], [[orthogonal spectrum]]


\item [[spectrum object]]


\item [[cospectrum]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item [[Daniel Kan]], \emph{Semisimplicial spectra}, Illinois J. Math. Volume 7, Issue 3 (1963), 463-478. (\href{http://projecteuclid.org/euclid.ijm/1255644953}{Euclid})


\item [[Michael Boardman]], \emph{Stable homotopy theory}, mimeographed notes, University of Warwick, 1965 onward


\item [[Frank Adams]], Part III, section 2 \emph{[[Stable homotopy and generalised homology]]}, 1974


\item [[Robert Switzer]], chapter 8 of \emph{Algebraic Topology - Homotopy and Homology}, Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, Vol. 212, Springer-Verlag, New York, N. Y., 1975.


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


\item Lydakis, \emph{Simplicial functors and stable homotopy theory} Preprint, available via Hopf archive, 1998 (\href{http://hopf.math.purdue.edu/Lydakis/s_functors.pdf}{pdf})


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


\item [[Stefan Schwede]], \emph{Symmetric spectra}, 2012 (\href{http://www.math.uni-bonn.de/~schwede/SymSpec-v3.pdf}{pdf})


\item [[John F. Jardine]], section 10 of \emph{[[Local homotopy theory]]}, 2016



\end{itemize}
Symmetric spectra in more general [[model categories]] (using the [[Bousfield-Friedlander theorem]]) are discussed in

\begin{itemize}%
\item [[Stefan Schwede]], section 3 of \emph{Spectra in model categories and applications to the algebraic cotangent complex}, Journal of Pure and Applied Algebra 120 (1997) 104 (\href{http://www.math.uni-bonn.de/people/schwede/modelspec.pdf}{pdf})


\item [[Mark Hovey]], \emph{Spectra and symmetric spectra in general model categories}, Journal of Pure and Applied Algebra Volume 165, Issue 1, 23 November 2001, Pages 63--127 (\href{https://arxiv.org/abs/math/0004051}{arXiv:math/0004051})



\end{itemize}
[[!redirects sequential spectra]]

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

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

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



\end{document}