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\section*{G-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{representation_theory}{}\paragraph*{{Representation theory}}\label{representation_theory}

[[!include representation theory - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{models}{Models}\dotfill \pageref*{models} \linebreak
\noindent\hyperlink{IndexedOnAllRepresentations}{Indexed on all representations}\dotfill \pageref*{IndexedOnAllRepresentations} \linebreak
\noindent\hyperlink{via_orthogonal_spectra_with_action}{Via orthogonal spectra with $G$-action}\dotfill \pageref*{via_orthogonal_spectra_with_action} \linebreak
\noindent\hyperlink{RelationToMackeyFunctors}{Via Mackey functors}\dotfill \pageref*{RelationToMackeyFunctors} \linebreak
\noindent\hyperlink{via_excisive_functors}{Via excisive functors}\dotfill \pageref*{via_excisive_functors} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{relation_to_spectra_with_action}{Relation to spectra with $G$-action}\dotfill \pageref*{relation_to_spectra_with_action} \linebreak
\noindent\hyperlink{relation_to_borelequivariant_spectra}{Relation to Borel-equivariant spectra}\dotfill \pageref*{relation_to_borelequivariant_spectra} \linebreak
\noindent\hyperlink{EquivariantWhitehead}{Equivariant stable Whitehead theorem}\dotfill \pageref*{EquivariantWhitehead} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

For $G$ a [[compact Lie group]] (or more generally a [[compact topological group]]) the concept of \emph{$G$-spectrum} (or \emph{$G$-equivariant spectrum}) is the generalization of that of [[spectrum]] as one passes from [[stable homotopy theory]] to [[equivariant stable homotopy theory]], or more generally, as $G$ is allow to vary, to [[global equivariant stable homotopy theory]].

Where the ordinary concept of [[spectrum]] is given in terms of [[looping and delooping]] of ordinary [[topological spaces]] by ordinary [[spheres]], a $G$-spectrum is instead given by looping and delooping of [[topological G-spaces]] with respect to [[representation spheres]] of $G$, namely [[one-point compactifications]] of linear $G$-[[representations]], for all representations appearing in a chosen ``[[G-universe]]''.

Such a [[G-universe]] is called \emph{complete} if it contains every [[irreducible representation]] of $G$, and the spectra modeled on such a complete $G$-universe are the \emph{genuine $G$-spectra}. At the other extreme, if the [[G-universe]] contains only the trivial representations, then the resulting spectra are the \emph{[[spectra with G-action]]}, also called \emph{[[naive G-spectra]]} for emphasis of the distinction to the previous case.

The genuine $G$-spectra are richer than \emph{[[spectra with G-action]]} and have better homotopy-theoretic properties. In particular the genuine [[equivariant cohomology]] theories which they [[Brown representability theorem|represent]] have [[suspension isomorphisms]] for suspension by all [[representation spheres]] and with respect to [[RO(G)-grading]].

When $G$ is the trivial group, a $G$-spectrum is also known as a [[coordinate-free spectrum]].

\hypertarget{models}{}\subsection*{{Models}}\label{models}

There are various equivalent ways to present genuine $G$-spectra.

\hypertarget{IndexedOnAllRepresentations}{}\subsubsection*{{Indexed on all representations}}\label{IndexedOnAllRepresentations}

(\hyperlink{May96}{May 96, chapter XII}, \hyperlink{GreenleesMay}{Greenlees-May, section 2})

Fix a [[G-universe]]. For $V$ any [[orthogonal group|orthogonal]] [[representation]] in the universe, write $S^V$ for its [[representation sphere]]. For $V \hookrightarrow W$ a subrepresentation, write $W-V$ for the [[orthogonal complement]] representation.

A \emph{$G$-prespectrum} $E$ is an assignment of a [[pointed topological space|pointed]] [[G-space]] $E_V$ to each $G$-[[representation]] $V$ (in the given [[G-universe]]), equipped for each subrepresentation $V \hookrightarrow W$ with a pointed $G$-equivariant [[continuous function]]

\begin{displaymath}
\sigma_{V,W}
  \;\colon\;
  S^{W-V} \wedge E_V \longrightarrow E_W
\end{displaymath}
such that

\begin{enumerate}%
\item $\Sigma_{V,V} = id$;


\item for any $Z \hookrightarrow V \hookrightarrow W$ we have [[commuting diagrams]]

\begin{displaymath}
\itexarray{
    S^{Z-W}\wedge S^{V-W} \wedge E_V
    &\stackrel{S^{Z-W}\wedge(\sigma_{V,W})}{\longrightarrow}& S^{Z-W}E_W
    \\
    \downarrow && \downarrow^{\mathrlap{\Sigma_{Z,W}}}
    \\
    S^{Z-V}E_V &\stackrel{\sigma_{V,Z}}{\longrightarrow}& E_Z
  }
\end{displaymath}


\end{enumerate}
Write

\begin{displaymath}
\tilde \sigma_{V,W} \;\colon\; E_V \longrightarrow \Omega^{W-V}E_W
\end{displaymath}
for the [[adjuncts]] of these structure maps.

A $G$-prespectrum is called (at least in (\hyperlink{May96}{May 96, chapter XII}))

\begin{itemize}%
\item a \emph{$G$-$\Omega$-spectrum} if all the $\sigma_{V,W}$ are [[weak homotopy equivalences]];


\item a \emph{$G$-spectrum} if all the $\sigma_{V,W}$ are [[homeomorphisms]].



\end{itemize}
\hypertarget{via_orthogonal_spectra_with_action}{}\subsubsection*{{Via orthogonal spectra with $G$-action}}\label{via_orthogonal_spectra_with_action}

While the definition of \hyperlink{IndexedOnAllRepresentations}{spectra indexed on all representations} manifestly relates to the [[suspension isomorphism]] for smashing with [[representation spheres]] and shifting in [[RO(G)-grading]], the information encoded in the objects in this definition has much redundancy. A ``smaller'' definition of genuine $G$-spectra is given by [[orthogonal spectra]] equipped with $G$-action (\hyperlink{Mandellmay04}{Mandell-May 04}, \hyperlink{Schwede15}{Schwede 15}).

\hypertarget{RelationToMackeyFunctors}{}\subsubsection*{{Via Mackey functors}}\label{RelationToMackeyFunctors}

For $G$ a [[finite group]] then genuine $G$-spectra are equivalent to \emph{[[Mackey functors]]} on the category of [[finite set|finite]] [[G-sets]].

(\hyperlink{GuillouMay11}{Guillou-May 11, theorem 0.1}, \hyperlink{Barwick14}{Barwick 14, below example B.6}).

\hypertarget{via_excisive_functors}{}\subsubsection*{{Via excisive functors}}\label{via_excisive_functors}

Characterization of $G$-spectra \href{}{via excisive functors} on [[G-spaces]] is in (\hyperlink{Blumberg05}{Blumberg 05}).

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

\hypertarget{relation_to_spectra_with_action}{}\subsubsection*{{Relation to spectra with $G$-action}}\label{relation_to_spectra_with_action}

(e.g. \hyperlink{Carlsson92}{Carlsson 92, p. 14}, \hyperlink{GreenleesMay}{GreenleesMay, p.16})

\hypertarget{relation_to_borelequivariant_spectra}{}\subsubsection*{{Relation to Borel-equivariant spectra}}\label{relation_to_borelequivariant_spectra}

In the general context of ([[global equivariant stable homotopy theory|global]]) [[equivariant stable homotopy theory]], [[Borel equivariant cohomology|Borel-equivariant]] spectra are those which are \emph{right induced} from plain [[spectra]], hence which are in the [[essential image]] of the [[right adjoint]] to the [[forgetful functor]] from [[equivariant spectra]] to plain [[spectra]].

(\hyperlink{Schwede18}{Schwede 18, Example 4.5.19})

\hypertarget{EquivariantWhitehead}{}\subsubsection*{{Equivariant stable Whitehead theorem}}\label{EquivariantWhitehead}

The equivariant version of the stable [[Whitehead theorem]] holds:

a map of $G$-spectra $f \colon E \longrightarrow F$ is a [[weak equivalence]] (e.g. an $RO(G)$-degree-wise [[weak homotopy equivalence]] of [[topological G-spaces]] in the \hyperlink{IndexedOnAllRepresentations}{model via indexing on all representations}) precisely it if induces [[isomorphisms]] $\pi_\bullet(f) \colon \pi_\bullet(E) \longrightarrow \pi_{\bullet}(F)$ on all [[equivariant homotopy group]] [[Mackey functors]].

(\hyperlink{GreenleesMay}{Greenlees-May, theorem 2.3})

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

\begin{itemize}%
\item [[equivariant stable homotopy category]]


\item [[RO(G)-grading]]


\item [[fixed point spectrum]]


\item [[geometric fixed points]]



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

\begin{itemize}%
\item [[equivariant sphere spectrum]]

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

Good lecture notes are

\begin{itemize}%
\item [[Andrew Blumberg]], \emph{Equivariant homotopy theory}, 2017 (\href{https://www.ma.utexas.edu/users/a.debray/lecture_notes/m392c_EHT_notes.pdf}{pdf}, \href{https://github.com/adebray/equivariant_homotopy_theory}{GitHub})

\end{itemize}
The concept of genuine $G$-spectra is due to

\begin{itemize}%
\item [[L. Gaunce Lewis, Jr.]], [[Peter May]], M. Steinberger, and J. E. McClure, \emph{Equivariant stable homotopy theory}, Lecture Notes in Mathematics, vol. 1213, Springer-Verlag, Berlin, 1986, With contributions by J. E. McClure. MR 866482 (88e:55002)

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

\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}
also (developed for the [[Arf-Kervaire invariant problem]])

\begin{itemize}%
\item [[Michael Hill]], [[Michael Hopkins]], [[Doug Ravenel]], \emph{The Arf-Kervaire invariant problem in algebraic topology: introduction}, Current developments in mathematics, 2009, Int. Press, Somerville, MA, 2010, pp. 23--57. MR 2757358

\emph{On the non-existence of elements of Kervaire invariant one}, (\href{http://arxiv.org/abs/0908.3724}{arXiv:0908.3724})

\emph{The Arf-Kervaire problem in algebraic topology: sketch of the proof}, Current developments in mathematics, 2010, Int. Press, Somerville, MA, 2011, pp. 1--43



\end{itemize}
Surveys and introductions include

\begin{itemize}%
\item [[Gunnar Carlsson]], \emph{A survey of equivariant stable homotopy theory},Topology, Vol 31, No. 1, pp. 1-27, 1992 (\href{http://www.maths.ed.ac.uk/~aar/papers/carlsson1.pdf}{pdf})


\item [[John Greenlees]], [[Peter May]], section 2 of \emph{[[Equivariant stable homotopy theory]]}, in I.M. James (ed.), \emph{[[Handbook of Algebraic Topology]]} , pp. 279-325. 1995. (\href{http://www.math.uchicago.edu/~may/PAPERS/Newthird.pdf}{pdf})



\end{itemize}
Lecture notes include

\begin{itemize}%
\item [[Peter May]], \emph{Equivariant homotopy and cohomology theory} CBMS Regional Conference Series in Mathematics, vol. 91, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1996. (\href{http://www.math.uchicago.edu/~may/BOOKS/alaska.pdf}{pdf})


\item [[Stefan Schwede]], \emph{[[Lectures on Equivariant Stable Homotopy Theory]]}



\end{itemize}
See also

\begin{itemize}%
\item [[Stefan Schwede]], \emph{[[Global homotopy theory]]} 2018 (\href{https://arxiv.org/abs/1802.09382}{arXiv:1802.09382})

\end{itemize}
Relation to [[Mackey functors]]:

\begin{itemize}%
\item [[Bert Guillou]], [[Peter May]], \emph{Models of $G$-spectra as presheaves of spectra, (\href{http://arxiv.org/abs/1110.3571}{arXiv:1110.3571})}

\emph{Permutative $G$-categories in equivariant infinite loop space theory (\href{http://arxiv.org/abs/1207.3459}{arXiv:1207.3459})}


\item [[Denis Nardin]], section 2.6 of \emph{Stability and distributivity over orbital ∞-categories}, 2012 (\href{https://www.math.univ-paris13.fr/~nardin/thesis.pdf}{pdf})


\item [[Clark Barwick]], \emph{Spectral Mackey functors and equivariant algebraic K-theory (I)}, Adv. Math., 304:646–727 (\href{http://arxiv.org/abs/1404.0108}{arXiv:1404.0108})



\end{itemize}
For more references see at \emph{[[equivariant stable homotopy theory]]} and at \emph{[[Mackey functor]]}

Characterization via [[excisive functors]] is in

\begin{itemize}%
\item [[Andrew Blumberg]], \emph{Continuous functors as a model for the equivariant stable homotopy category} (\href{http://arxiv.org/abs/math.AT/0505512}{arXiv:math.AT/0505512})

\end{itemize}
In the case of a [[cyclic group]] of prime order, genuine $G$-spectra admit a simple model which amounts to specifying a spectrum $E$, a $G$-action on $E$, a genuine fixed point spectrum $E^G$, and a diagram $E_{hG} \to E^G \to E^{hG}$. See Example 3.29 in:

\begin{itemize}%
\item [[Saul Glasman]], \emph{Stratified categories, geometric fixed points and a generalized Arone-Ching theorem}, \href{https://arxiv.org/abs/1507.01976}{arXiv:1507.01976}.

\end{itemize}
A perspective on the category of genuine G-spectra as a lax limit over those of ``naive'' G-spectra is given in

\begin{itemize}%
\item [[David Ayala]], [[Aaron Mazel-Gee]], [[Nick Rozenblyum]], \emph{A naive approach to genuine $G$-spectra and cyclotomic spectra} (\href{https://arxiv.org/abs/1710.06416}{arXiv:1710.06416})

\end{itemize}
[[!redirects G-spectra]]

[[!redirects genuine G-spectrum]] [[!redirects genuine G-spectra]] [[!redirects G-equivariant spectrum]] [[!redirects G-equivariant spectra]] [[!redirects equivariant spectrum]] [[!redirects equivariant spectra]]

[[!redirects genuine equivariant spectrum]] [[!redirects genuine equivariant spectra]]



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