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\begin{document}

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\section*{orthogonal 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{orthogonal_spectra}{Orthogonal spectra}\dotfill \pageref*{orthogonal_spectra} \linebreak
\noindent\hyperlink{HomotopyGroups}{Homotopy groups and Weak homotopy equivalences}\dotfill \pageref*{HomotopyGroups} \linebreak
\noindent\hyperlink{SmashProduct}{Smash product}\dotfill \pageref*{SmashProduct} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{relation_to_the_jhomomorphism}{Relation to the J-homomorphism}\dotfill \pageref*{relation_to_the_jhomomorphism} \linebreak
\noindent\hyperlink{relation_to_the_cobordism_hypothesis}{Relation to the cobordism hypothesis}\dotfill \pageref*{relation_to_the_cobordism_hypothesis} \linebreak
\noindent\hyperlink{relation_to_global_equivariant_stable_homotopy_theory}{Relation to global equivariant stable homotopy theory}\dotfill \pageref*{relation_to_global_equivariant_stable_homotopy_theory} \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}

Orthogonal spectra are one version of [[highly structured spectra]] that support a [[symmetric monoidal smash product of spectra]]. An orthogonal spectrum is a sequence of [[pointed topological spaces]] $\{X_n\}_{n \in \mathbb{N}}$ equipped with maps $X_n \wedge S^1 \longrightarrow X_{n+1}$ from the [[suspension]] of one into the next, but such that the $n$th [[topological space]] is equipped with an [[action]] of the [[orthogonal group]] $O(n)$ and such that the induced structure maps. $X_n \wedge S^k \longrightarrow X_{n+k}$ are all $O(n)\times O(k)$-[[equivariance|equivariant]], hence are [[action]] [[homomorphisms]]. There is a natural [[homotopy theory]] of such orthogonal spectra and it is equivalent to the standard [[stable homotopy theory]] (\hyperlink{MMSS08}{MMSS 98}).

The [[category]] of \emph{orthogonal spectra} is a [[presentable (∞,1)-category|presentation]] of the [[symmetric monoidal (∞,1)-category]] [[stable (infinity,1)-category of spectra|of spectra]], with the special property that it implements the [[smash product of spectra]] such as to yield itself a [[symmetric monoidal category|symmetric]] [[monoidal model category|monoidal]] [[model category of spectra]]: the \emph{[[model structure on orthogonal spectra]]}.

This implies in particular that with respect to this [[symmetric smash product of spectra]] an [[E-∞ ring]] is presented simply as a plain [[commutative monoid]] [[internalization|in]] orthogonal spectra (``[[highly structured ring spectrum]]''). See at \emph{[[orthogonal ring spectrum]]}.

Other presentations sharing this property are \emph{[[symmetric spectra]]} and \emph{[[S-modules]]}. In contrast to symmetric spectra, orthogonal spectra need to consist of [[topological spaces]] instead of [[simplicial sets]].

One advantages of orthogonal spectra over symmetric spectra is that for them the naive definition of [[homotopy groups]] comes out homotopically correct, while for symmetric spectra an intransparent replacement is needed first (see \href{symmetric+spectrum#HomotopyGroups}{symmetric spectrum -- Homotopy groups}).

Another advantage is that orthogonal spectra support a similarly good model for [[equivariant stable homotopy theory]] with equivariance by [[compact Lie groups]], while [[symmetric spectra]] share this property only for equivariance under [[finite groups]].

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

\hypertarget{orthogonal_spectra}{}\subsubsection*{{Orthogonal spectra}}\label{orthogonal_spectra}

\begin{defn}
\label{OrthogonalSpectrum}\hypertarget{OrthogonalSpectrum}{}
An \emph{orthogonal spectrum} $X$ consists of for each $n \in \mathbb{N}$

\begin{enumerate}%
\item a sequence of [[pointed topological spaces]] $X_n$ (the \emph{$n$th level});


\item a base-point preserving [[continuous function|continuous]] [[action]] of the [[topological group|topological]] [[orthogonal group]] $O(n)$ on $X_n$;


\item based-point preserving [[continuous functions]] $\sigma_n \colon X_n \wedge S^1 \longrightarrow X_{n+1}$ from the [[smash product]] with the [[1-sphere]] (the \emph{$n$th structure map})



\end{enumerate}
such that for all $n,k \in \mathbb{N}$ with $k \geq 1$

\begin{itemize}%
\item the [[continuous functions]] $\sigma^k \colon X_n \wedge S^k \longrightarrow X_{n+k}$ given as the [[compositions]]

\begin{displaymath}
\sigma^k \colon
  X_n \wedge S^k  
   \stackrel{\sigma_n \wedge S^{k-1}}{\longrightarrow}
  X_{n+1} \wedge S^{k-1}
   \stackrel{\sigma_{n-1} \wedge S^{k-2}}{\longrightarrow}
  X_{n+2} \wedge S^{k-2}
   \stackrel{\sigma_{n-2} \wedge S^{k-3}}{\longrightarrow}
   \cdots
   \stackrel{\sigma_{n+k-2} \wedge S^{1}}{\longrightarrow}
 X_{n+k-1} \wedge S^1
   \stackrel{\sigma_{n+k-1} }{\longrightarrow}
  X_{n+k}
\end{displaymath}
is $O(n) \times O(k)$-equivariant

(with respect to the $O(k)$-[[action]] on $S^k$ regarded as the [[representation sphere]] of the defining action on $\mathbb{R}^k$ and via the diagonal embedding $O(n)\times O(k) \hookrightarrow O(n+k)$).



\end{itemize}
A [[homomorphism]] $f \colon X \longrightarrow Y$ of orthogonal spectra is a sequence of $O(n)$-equivariant based continuous functions $f_n \colon X_n \longrightarrow Y_n$ [[commuting diagram|commuting]] with the structure maps

\begin{displaymath}
\itexarray{
    X_n \wedge S^1 & \stackrel{\sigma_n^X}{\longrightarrow} & X_{n+1}  
    \\
    \downarrow^{\mathrlap{f_n}} && \downarrow^{\mathrlap{f_{n+1}}}
    \\
    Y_n \wedge S^1 & \stackrel{\sigma_n^Y}{\longrightarrow} & Y_{n+1}      
  }
  \,.
\end{displaymath}
We write $OrthSpectra$ for the [[category]] of orthogonal spectra with homomorphisms between them.

\end{defn}
\hypertarget{HomotopyGroups}{}\subsubsection*{{Homotopy groups and Weak homotopy equivalences}}\label{HomotopyGroups}

\begin{defn}
\label{StabilizationMap}\hypertarget{StabilizationMap}{}
Given an orthogonal spectrum $X$, def. \ref{OrthogonalSpectrum}, then for $n,k \in \mathbb{N}$ the \emph{stabilization map} $\iota_{n,k}$ on [[homotopy groups]] $\pi_\bullet(X_\bullet)$ of the level spaces $X_\bullet$ is

\begin{displaymath}
\iota_{n,k}
    \;\colon\;
  \pi_{n+k} X_n
    \stackrel{(-)\wedge S^1}{\longrightarrow}
  \pi_{n+k+1}(X_n \wedge S^1)
    \stackrel{(\sigma_n)_\ast}{\longrightarrow}
  \pi_{n+k+1} X_{n+1}
  \,.
\end{displaymath}
\end{defn}
\begin{defn}
\label{StableHomotopyGroup}\hypertarget{StableHomotopyGroup}{}
Given an orthogonal spectrum $X$, def. \ref{OrthogonalSpectrum}, then for $k \in \mathbb{Z}$ its $k$th \emph{stable [[homotopy group]]} is the [[colimit]]

\begin{displaymath}
\pi_k X 
  \;\coloneqq\;
  \underset{\longrightarrow}{\lim}_n
  \pi_{n+k} X_n
\end{displaymath}
of the [[homotopy groups]] of the level spaces, taken with respect to the stabilization maps, def. \ref{StabilizationMap}.

\end{defn}
\begin{defn}
\label{WeakHomotopyEquivalences}\hypertarget{WeakHomotopyEquivalences}{}
A homomorphism $f\colon X \longrightarrow Y$ of orthogonal spectra, def. \ref{OrthogonalSpectrum}, is a \emph{[[weak homotopy equivalence]]} if it induces [[isomorphisms]] (of [[abelian groups]])

\begin{displaymath}
\pi_\bullet(f) \;\colon\;  \pi_\bullet(X) \longrightarrow \pi_\bullet(Y)
\end{displaymath}
on all stable homotopy groups, def. \ref{StableHomotopyGroup}.

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
The [[simplicial localization]] of the category of orthogonal spectra, def. \ref{OrthogonalSpectrum}, at the weak homotopy equivalences, def. \ref{WeakHomotopyEquivalences}, is [[equivalence of (infinity,1)-categories|equivalent]] to the [[(infinity,1)-category of spectra]]:

\begin{displaymath}
L_{whe} OrthSpectra \simeq Spectra
  \,.
\end{displaymath}
See at \emph{[[model structure on orthogonal spectra]]}.

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

\begin{defn}
\label{SmashProduct}\hypertarget{SmashProduct}{}
Given two orthogonal spectra $X,Y\in OrthSpectra$, def. \ref{OrthogonalSpectrum}, their \emph{[[smash product of spectra]]} is the orthogonal spectrum

\begin{displaymath}
X \wedge Y \in OrthSpectrum
\end{displaymath}
whose $n$th level space is the [[coequalizer]]

\begin{displaymath}
\left(
    \underset{p+1+q = n}{\bigvee}
     O(n)_+  \underset{O(p)\times 1 \times O(q)}{\wedge} X_p \wedge S^1 \wedge X_q
  \right)
  \stackrel{\overset{}{\longrightarrow}}{\underset{}{\longrightarrow}}
  \left(
   \underset{p+q = n}{\bigvee}
    O(n)_+ \underset{O(p)\times O(q)}{\wedge} X_p \wedge X_q
   \right)
   \longrightarrow
   \left(X\wedge Y\right)_{n}
\end{displaymath}
of the two maps whose components are $\sigma_p^X \wedge Y_q$ and $X_p \wedge \sigma_q^Y \circ X_p \wedge braid_{S^1, Y_q}$, respectively, and whose structure maps are induced, under the coequalizer, by the component maps $X_p\wedge \sigma_q^Y$.

\end{defn}
\begin{prop}
\label{}\hypertarget{}{}
The [[smash product of spectra]] from def. \ref{SmashProduct} naturally extends to a [[functor]]

\begin{displaymath}
(-)\wedge (-)
  \;\colon\;
  OrthSpectra \times OrthSpectra
  \longrightarrow
  OrthSpectra
\end{displaymath}
which makes $OrthSpectra$ into a [[symmetric monoidal category]] with [[unit]] the orthogonal [[sphere spectrum]] $\mathbb{S}$, example \ref{OrthogonalSphereSpectrum}.

\end{prop}
\begin{defn}
\label{BilinearHomomorphisms}\hypertarget{BilinearHomomorphisms}{}
For $X,Y,Z \in OrthSpectrum$, def. \ref{OrthogonalSpectrum}, a \emph{[[bilinear map|bilinear]]-homomorphism}

\begin{displaymath}
b 
    \;\colon\;
  (X,Y)
   \longrightarrow
  Z
  \,,
\end{displaymath}
is a collection of, for each $p,q\in \mathbb{N}$, base-point preserving $O(p) \times O(q)$-equivariant [[continuous functions]]

\begin{displaymath}
b_{p,q}
   \;\colon\;
  X_p \wedge X_q 
    \longrightarrow
  Z_{p+q}
\end{displaymath}
(out of the [[smash product]] of [[pointed topological spaces]]) which are \emph{[[bilinear map|bilinear]]} in that the following [[diagrams]] [[commuting diagram|commutes]]:

\begin{displaymath}
\itexarray{
    X_p \wedge X_q \wedge S^1
     &\stackrel{b_{p,q} \wedge S^1}{\longrightarrow}&
    Z_{p+q} \wedge S^1
    \\
    \downarrow^{\mathrlap{X_p \wedge \sigma_q}} 
      && 
    \downarrow^{\mathrlap{\sigma_{p+q}}}
    \\
    X_p \wedge Y_{q+1}
     &\stackrel{b_{p,q+1}}{\longrightarrow}&
    Z_{p+q+1}
  }
  \;\;\;\;,\;\;\;\;\;
  \itexarray{
    X_p \wedge X_q \wedge S^1
     &\stackrel{b_{p,q} \wedge S^1}{\longrightarrow}&
    Z_{p+q} \wedge S^1
    \\
    \downarrow^{\mathrlap{X_p \wedge braid_{X_q, S^1}}}
     &&
    \downarrow^{\mathrlap{id}}
    \\
    X_p \wedge S^1 \wedge X_q 
     &&
    Z_{p+q} \wedge S^1
    \\
    \downarrow^{\mathrlap{\sigma_p \wedge Y_q}} 
      && 
    \downarrow^{\mathrlap{\sigma_{p+q}}}
    \\
    X_p \wedge Y_{q+1}
     &\stackrel{b_{p,q+1}}{\longrightarrow}&
    Z_{p+q+1}
  }
  \,.
\end{displaymath}
\end{defn}
\begin{prop}
\label{}\hypertarget{}{}
The smash product of orthogonal spectra $X \wedge Y$, def. \ref{SmashProduct}, is the [[universal construction|universal]] recipient in $OrthSpectra$ of bilinear maps, def. \ref{BilinearHomomorphisms}, out of $(X,Y)$.

\end{prop}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{example}
\label{OrthogonalSphereSpectrum}\hypertarget{OrthogonalSphereSpectrum}{}
The canonical incarnation of the [[sphere spectrum]] $\mathbb{S}$ as an orthogonal spectrum, def. \ref{OrthogonalSpectrum}, has $n$th level space

\begin{displaymath}
\mathbb{S}_n = S^n
\end{displaymath}
the [[representation sphere]] of the defining [[linear representation]] of $O(n)$ on $\mathbb{R}^n$, and as structure maps the canonical [[smash product]] [[isomorphisms]] ([[homeomorphisms]])

\begin{displaymath}
S^p \wedge S^1  \longrightarrow S^{p+1}
  \,.
\end{displaymath}
\end{example}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{relation_to_the_jhomomorphism}{}\subsubsection*{{Relation to the J-homomorphism}}\label{relation_to_the_jhomomorphism}

relation to the [[J-homomorphism]]:

see (\hyperlink{Schwede15}{Schwede 15, example 4.22})

\hypertarget{relation_to_the_cobordism_hypothesis}{}\subsubsection*{{Relation to the cobordism hypothesis}}\label{relation_to_the_cobordism_hypothesis}

\begin{quote}%
check


\end{quote}
A [[connective spectrum]] is equivalently a grouplike [[E-∞ space]], hence a [[Picard ∞-groupoid]]. As such it is an [[(∞,0)-category]] of [[fully dualizable objects]]. By the [[cobordism hypothesis]] this means that it is equipped with an $O(n)$-[[∞-action]] for all $n$, coming from the action $O(n)$ on the [[framed manifold|n-framings]] of the point in the [[(∞,n)-category of cobordisms]]. This $O(n)$-action is that which is encoded by the definition of orthogonal spectrum (\hyperlink{Lurie09}{Lurie 09, example 2.4.15}).

\hypertarget{relation_to_global_equivariant_stable_homotopy_theory}{}\subsubsection*{{Relation to global equivariant stable homotopy theory}}\label{relation_to_global_equivariant_stable_homotopy_theory}

Since orthogonal spectra are by definition equipped with orthogonal group [[actions]], they serve as models for [[equivariant homotopy theory]] ``for all groups at once'', called \emph{[[global stable homotopy theory]]}.

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

\begin{itemize}%
\item [[free orthogonal spectrum]]


\item [[orthogonal ring spectrum]]



\end{itemize}
[[model structure on spectra]], [[symmetric monoidal smash product of spectra]]

\begin{itemize}%
\item [[symmetric spectrum]], [[model structure on symmetric spectra]]


\item \textbf{orthogonal spectrum}, [[model structure on orthogonal spectra]]


\item [[S-module]], [[model structure on S-modules]]


\item [[global equivariant stable homotopy theory]]



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

Reviews include

\begin{itemize}%
\item Knut Berg, \emph{Orthogonal spectra} (\href{http://folk.uio.no/rognes/theses/orthogonal.pdf}{pdf})


\item [[Cary Malkiewich]], section 2.3 of \emph{The stable homotopy category}, 2014 (\href{http://math.stanford.edu/~carym/stable.pdf}{pdf})



\end{itemize}
Lecture notes include

\begin{itemize}%
\item [[Stefan Schwede]], chapter I, section 7.1 of \emph{[[Symmetric spectra]]} (2012)


\item [[Stefan Schwede]], section 1 of \emph{[[Lectures on Equivariant Stable Homotopy Theory]]}, 2014 (\href{http://www.math.uni-bonn.de/people/schwede/equivariant.pdf}{pdf})



\end{itemize}
and (\hyperlink{Schwede15}{Schwede 15}) (take throughout $\mathcal{F} = \{1\}$ there to be the trivial family to restrict to the non-equivariant case).

Orthogonal spectra were introduced around

\begin{itemize}%
\item [[Michael Mandell]], [[Peter May]], [[Stefan Schwede]], [[Brooke Shipley]], \emph{Diagram spaces, diagram spectra, and FSP's}, 1998 (\href{http://www.math.uiuc.edu/K-theory/0319/}{KTheory:0319})

\end{itemize}
and their [[homotopy theory]] and [[Quillen equivalences]] of [[model categories of spectra]] were discussed in

\begin{itemize}%
\item [[Michael Mandell]], [[Peter May]], \emph{Orthogonal spectra and $S$-modules} (\href{http://www.math.uiuc.edu/K-theory/0318/}{K-theory:0318}, \href{http://www.math.uchicago.edu/~may/PAPERS/mmLMSDec30.pdf}{pdf})


\item [[Michael Mandell]], [[Peter May]], [[Stefan Schwede]], [[Brooke Shipley]], \emph{[[Model categories of diagram spectra]]}, 1998 (\href{http://www.math.uiuc.edu/K-theory/0320/}{KTheory:0320})



\end{itemize}
and for [[equivariant spectra]] in

\begin{itemize}%
\item [[Michael Mandell]], [[Peter May]], \emph{Equivariant orthogonal spectra and $S$-modules}, 2000 \href{http://www.math.uiuc.edu/K-theory/0408/}{KTheory:0408}

\end{itemize}
and for [[operads]] enriched over orthogonal spectra in

\begin{itemize}%
\item [[Tore Kro]], \emph{Model structure on operads in orthogonal spectra}, Homology Homotopy Appl. Volume 9, Number 2 (2007), 397-412.(\href{http://projecteuclid.org/euclid.hha/1201127343}{Euclid})

\end{itemize}
and in the context of [[equivariant stable homotopy theory]] in (\hyperlink{Schwede14}{Schwede 14}) and in

\begin{itemize}%
\item [[Michael Mandell]], [[Peter May]], \emph{Equivariant orthogonal spectra and S-modules}. Preprint, April 29, 2000, (\href{http://www.math.uiuc.edu/K-theory/0408/}{KTheory:0408})

\end{itemize}
and in [[global equivariant stable homotopy theory]]:

\begin{itemize}%
\item [[Stefan Schwede]], \emph{[[Global homotopy theory]]}, 2015 (\href{http://www.math.uni-bonn.de/people/schwede/global.pdf}{pdf})

\end{itemize}
See also

\begin{itemize}%
\item [[Jacob Lurie]], \emph{[[On the Classification of Topological Field Theories]]}, Current Developments in Mathematics Volume 2008 (2009), 129-280 (\href{http://arxiv.org/abs/0905.0465}{arXiv:0905.0465})

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



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