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\section*{spectrum}

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


\vspace{.5em} \hrule \vspace{.5em}
\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{connective_and_nonconnective_spectra_infinite_loop_spaces}{Connective and non-connective spectra; infinite loop spaces}\dotfill \pageref*{connective_and_nonconnective_spectra_infinite_loop_spaces} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{SequentialPreSpectra}{Sequential pre-spectra}\dotfill \pageref*{SequentialPreSpectra} \linebreak
\noindent\hyperlink{OmegaSpectrum}{$\Omega$-spectra}\dotfill \pageref*{OmegaSpectrum} \linebreak
\noindent\hyperlink{coordinatefree_spectra}{Coordinate-free spectra}\dotfill \pageref*{coordinatefree_spectra} \linebreak
\noindent\hyperlink{combinatorial_spectra}{Combinatorial spectra}\dotfill \pageref*{combinatorial_spectra} \linebreak
\noindent\hyperlink{general_context}{General context}\dotfill \pageref*{general_context} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{stabilization}{Stabilization}\dotfill \pageref*{stabilization} \linebreak
\noindent\hyperlink{symmetric_monoidal_structure}{Symmetric monoidal structure}\dotfill \pageref*{symmetric_monoidal_structure} \linebreak
\noindent\hyperlink{closed_structure}{Closed structure}\dotfill \pageref*{closed_structure} \linebreak
\noindent\hyperlink{model_category_structure}{Model category structure}\dotfill \pageref*{model_category_structure} \linebreak
\noindent\hyperlink{relation_to_symmetric_monoidal_groupoids}{Relation to symmetric monoidal ∞-groupoids}\dotfill \pageref*{relation_to_symmetric_monoidal_groupoids} \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}

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

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

\hypertarget{connective_and_nonconnective_spectra_infinite_loop_spaces}{}\subsubsection*{{Connective and non-connective spectra; infinite loop spaces}}\label{connective_and_nonconnective_spectra_infinite_loop_spaces}

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

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

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

\begin{displaymath}
\itexarray{
  Ch_+ &\stackrel{Dold-Kan \; nerve}{\to}&
  ConnectSp(\infty Grp) \subset \infty Grpd
  \\
  (\cdots A_2 \stackrel{\partial}{\to} A_1 \stackrel{\partial}{\to} A_0 \to 0 \to 0 \to \cdots)
  &\stackrel{}{\mapsto}&
  N(A_\bullet)
 }
\end{displaymath}
The [[homology]] groups of the chain complex correspond precisely to the [[homotopy groups]] of the corresponding [[topological space]] or [[∞-groupoid]].

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

\begin{displaymath}
\itexarray{
  Ch &\stackrel{Dold-Kan \; nerve}{\to}&
   Sp(\infty Grp) 
  \\
  (\cdots A_2 \stackrel{\partial}{\to} A_1 \stackrel{\partial}{\to} A_0 \stackrel{\partial}{\to} A_{-1} 
  \stackrel{\partial}{\to} A_{-2} \to \cdots)
  &\stackrel{}{\mapsto}&
  N(A_\bullet)
 }
\end{displaymath}
So it is the homologically nontrivial parts of the chain complexes in negative degree that corresponds to the non-connectiveness of a spectrum.

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

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

\hypertarget{SequentialPreSpectra}{}\subsubsection*{{Sequential pre-spectra}}\label{SequentialPreSpectra}

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

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

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

For details see \emph{[[Introduction to Stable homotopy theory -- 1-1|Introduction to stable homotopy theory -- 1.1 Sequential Spectra]]}.

\hypertarget{OmegaSpectrum}{}\subsubsection*{{$\Omega$-spectra}}\label{OmegaSpectrum}

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

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

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

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

\begin{displaymath}
F_n \coloneqq \lim_{m\to\infty}\Omega^m E_{n+m}
\end{displaymath}
(using that $\Omega$ commutes with the [[filtered colimits]]).

\hypertarget{coordinatefree_spectra}{}\subsubsection*{{Coordinate-free spectra}}\label{coordinatefree_spectra}

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

\begin{itemize}%
\item [[coordinate-free spectrum]].

\end{itemize}
See there for details.

\hypertarget{combinatorial_spectra}{}\subsubsection*{{Combinatorial spectra}}\label{combinatorial_spectra}

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

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

\hypertarget{general_context}{}\subsubsection*{{General context}}\label{general_context}

See [[spectrum object]].

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

\begin{itemize}%
\item [[suspension spectrum]]


\item [[sphere spectrum]]


\item [[Eilenberg-MacLane spectrum]]


\item [[K-theory spectrum]]


\item [[elliptic spectrum]]


\item [[tmf]]


\item [[Thom spectrum]], [[complex cobordism spectrum]]



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

\hypertarget{stabilization}{}\subsubsection*{{Stabilization}}\label{stabilization}

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

\hypertarget{symmetric_monoidal_structure}{}\subsubsection*{{Symmetric monoidal structure}}\label{symmetric_monoidal_structure}

\begin{itemize}%
\item [[smash product of spectra]]


\item [[symmetric monoidal smash product of spectra]]



\end{itemize}
\hypertarget{closed_structure}{}\subsubsection*{{Closed structure}}\label{closed_structure}

\begin{itemize}%
\item [[mapping spectrum]]

\end{itemize}
\hypertarget{model_category_structure}{}\subsubsection*{{Model category structure}}\label{model_category_structure}

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

\end{itemize}
\hypertarget{relation_to_symmetric_monoidal_groupoids}{}\subsubsection*{{Relation to symmetric monoidal ∞-groupoids}}\label{relation_to_symmetric_monoidal_groupoids}

\begin{itemize}%
\item [[stable homotopy hypothesis]]

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

\begin{itemize}%
\item [[homotopy group of a spectrum]]


\item [[sheaf of spectra]], [[parametrized spectrum]]

\begin{itemize}%
\item [[smooth spectrum]]

\end{itemize}

\item [[spectrum with G-action]], [[G-spectrum]]


\item [[ring spectrum]], [[algebra spectrum]], [[module spectrum]]


\item [[zero spectrum]]


\item [[finite spectrum]]


\item [[p-local spectrum]]


\item [[Bousfield localization of spectra]]


\item [[motivic spectrum]]


\item [[Brown representability theorem]]


\item [[stable homotopy hypothesis]]



\end{itemize}
[[!include k-monoidal table]]

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

According to (\hyperlink{Adams74}{Adams 74, p. 131}) the notion of spectrum is due to

\begin{itemize}%
\item E. L. Lima, \emph{Duality and Postnikov invariants}, Thesis, University of Chicago, Chicago 1958

\end{itemize}
\begin{quote}%
It is generally supposed that [[George Whitehead|G. W. Whitehead]] also had something to do with it, but the latter takes a modest attitude about that. (\hyperlink{Adams74}{Adams 74, p. 131})


\end{quote}
Early notes include

\begin{itemize}%
\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



\end{itemize}
See the references at \emph{[[stable homotopy theory]]}.

Lecture notes include

\begin{itemize}%
\item \emph{[[Introduction to Stable homotopy theory -- 1-1|Introduction to stable homotopy theory -- 1.1 Sequential Spectra]]}

\end{itemize}
More modern developments are due to

\begin{itemize}%
\item [[L. Gaunce Lewis]], [[Peter May]], M. Steinberger, \emph{Equivariant stable homotopy theory}, Springer Lecture Notes in Mathematics, 1986 (\href{http://www.math.uchicago.edu/~may/BOOKS/equi.pdf}{pdf})

\end{itemize}
The quick idea is surveyed for instance in

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


\item [[Aaron Mazel-Gee]], \emph{An introduction to spectra} (\href{https://math.berkeley.edu/~aaron/writing/an-introduction-to-spectra.pdf}{pdf})



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

\begin{itemize}%
\item [[Anthony Elmendorf]], [[Igor Kriz]], [[Peter May]], \emph{[[Modern foundations for stable homotopy theory]]}, in [[Ioan Mackenzie James]], \emph{[[Handbook of Algebraic Topology]]}, Amsterdam: North-Holland, (1995) pp. 213--253, (\href{http://hopf.math.purdue.edu/Elmendorf-Kriz-May/modern_foundations.pdf}{pdf})

\end{itemize}
A comprehensive account of the symmetric model in terms of [[symmetric spectra]] is in

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

\end{itemize}
and in terms of [[orthogonal spectra]] in

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

\end{itemize}
See also

\begin{itemize}%
\item [[Robert Thomason]], \emph{Symmetric Monoidal Categories Model All Connective Spectra} (\href{http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.37.8193}{web})


\item [[Frank Adams]], \emph{Infinite loop spaces}, Princeton University Press, 1978


\item [[Joseph Ayoub]], \emph{Les six op\'e{}rations de Grothendieck et le formalisme des cycles \'e{}vanescents dans le monde motivique, I}. Ast\'e{}risque, Vol. 314 (2008). Soci\'e{}t\'e{} Math\'e{}matique de France. (\href{http://user.math.uzh.ch/ayoub/PDF-Files/THESE.PDF}{pdf})



\end{itemize}
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