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\section*{Mackey functor}

\begin{quote}%
under construction


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

\hypertarget{homotopy_theory}{}\paragraph*{{Homotopy theory}}\label{homotopy_theory}

[[!include homotopy - contents]]

\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{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{dress_mackey_functors}{Dress' Mackey functors}\dotfill \pageref*{dress_mackey_functors} \linebreak
\noindent\hyperlink{EquivariantSpectra}{Equivariant spectra}\dotfill \pageref*{EquivariantSpectra} \linebreak
\noindent\hyperlink{Cohomology}{Cohomology with coefficients in a Mackey functor}\dotfill \pageref*{Cohomology} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{plain_mackey_functors}{Plain Mackey functors}\dotfill \pageref*{plain_mackey_functors} \linebreak
\noindent\hyperlink{spectral_mackey_functors}{Spectral Mackey functors}\dotfill \pageref*{spectral_mackey_functors} \linebreak
\noindent\hyperlink{in_equivariant_stable_homotopy_theory}{In equivariant stable homotopy theory}\dotfill \pageref*{in_equivariant_stable_homotopy_theory} \linebreak
\noindent\hyperlink{in_goodwillie_calculus}{In Goodwillie calculus}\dotfill \pageref*{in_goodwillie_calculus} \linebreak


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

Generally, the term \emph{Mackey functor} refers to an [[additive functor]] from a ([[subcategory]] of) a [[category of correspondences]] (in a [[disjunctive category]] $\mathcal{C}$) to possibly any other [[additive category]] which however usually is the ``base'' abelian category. More generally the term now refers to the fairly obvious [[homotopy theory|homotopy theoretic]] or [[higher category theory|higher categorical]] refinements of this concept.

Therefore the concept of Mackey functors is similar to that of [[sheaves with transfer]] and as such appears (implicitly) in the discussion of [[motives]] (explicitly e.g. in \hyperlink{KahnYamazaki11}{Kahn-Yamazaki 11, section 2}, where $\mathcal{C}$ is a category of suitable [[schemes]]).

Specifically, the concept was introduced and named as such in the context of [[representation theory]] (\hyperlink{Dress71}{Dress 71}) and [[equivariant homotopy theory]]/[[equivariant cohomology]] (\hyperlink{May96}{May 96}). Here the underlying [[category of correspondences]] is that in [[finite set|finite]] [[G-sets]], called the \emph{[[Burnside category]]}.

The [[equivariant homotopy groups]] $\pi_n(E)$ of a (genuine) [[G-spectrum]] $E$ organize into a Mackey functor on the [[Burnside category]] with values in [[abelian groups]]. This plays a key role in the [[equivariant Whitehead theorem]]. In fact, genunine [[G-spectra]] themselves are equivalent to Mackey [[(∞,1)-functors]] from the [[Burnside category]] to the [[(∞,1)-category of spectra]] (\hyperlink{GuillouMay11}{GuillouMay 11}, \hyperlink{Barwick14}{Barwick 14}).

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

We follow the modern account in (\hyperlink{Barwick14}{Barwick 14}).

Let $\mathcal{C}$ be a [[disjunctive (∞,1)-category]] and write $Corr_1(\mathcal{C})^\otimes$ for the [[(∞,1)-category of correspondences]] in $\mathcal{C}$, regarded as a [[symmetric monoidal (∞,1)-category]] with respect to its [[coproduct]] (which is a [[biproduct]] by disjunctiveness of $\mathcal{C}$).

Write $\mathcal{A} =$[[Spectra]]${}^{\oplus}$ for the [[(∞,1)-category of spectra]] regarded as a [[symmetric monoidal (∞,1)-category]] with respect to [[direct sum]]. More generally $\mathcal{A}$ could be any [[symmetric monoidal (∞,1)-category|symmetric monoidal]] [[stable (∞,1)-category]]

Then a (spectral) \emph{Mackey functor} on $\mathcal{C}$ is a [[monoidal (∞,1)-functor]] of the form

\begin{displaymath}
S \;\colon\; Corr_1(\mathcal{C})^\otimes \longrightarrow \mathcal{A}^{\oplus}
  \,.
\end{displaymath}
Notice that this means that $S$ is in particular

\begin{enumerate}%
\item a covariant [[(∞,1)-functor]] $(-)_\ast \colon\mathcal{C} \to \mathcal{A}$;


\item a contravariant [[(∞,1)-functor]], hence $(-)^\ast \colon\mathcal{C}^{op} \to \mathcal{A}$;


\item satisfying the [[Beck-Chevalley condition]].



\end{enumerate}
(More generally one may specify suitably chosen sub-$(\infty,1)$-categories $\mathcal{C}^\dagger, \mathcal{C}_\dagger \subset \mathcal{C}$ and restrict $Corr_1$ to [[correspondences]] whose left leg is in $\mathcal{C}_\dagger$ and whose right leg is in $\mathcal{C}^\dagger$ (\hyperlink{Barwick14}{Barwick 14, section 5}).)

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

\hypertarget{dress_mackey_functors}{}\subsubsection*{{Dress' Mackey functors}}\label{dress_mackey_functors}

For $\mathcal{A}$ taken to be (the [[derived category]]) of an [[abelian category]] (or better: postcomposed with a [[homological functor]] ) this definition reduces (\hyperlink{Barwick14}{Barwick 14}) to that of Mackey functors as originally defined in (\hyperlink{Dress71}{Dress 71}).

\hypertarget{EquivariantSpectra}{}\subsubsection*{{Equivariant spectra}}\label{EquivariantSpectra}

Let $G$ be a [[finite group]]. Let $\mathcal{C}= G Set$ be its category of [[G-sets]]. Then $Corr_1(\mathcal{C})$ is essentially what is called the [[Burnside category]] of $G$ (possibly after abelianizing/stabilizing the hom-spaces suitably, but as (\hyperlink{Barwick14}{Barwick 14}) highlights, this is unnecessary when one is mapping out of this into something abelian/stable, as is the case here).

For $G$ finite, Mackey functors on $\mathcal{C}$ are equivalent to genuine [[G-spectra]] (\hyperlink{GuillouMay11}{Guillou-May 11, theorem 0.1}, \hyperlink{Barwick14}{Barwick 14, below example B.6}) (Notice that this equivalence does not in general hold if $G$ is not a finite group.)

(\ldots{})

For $E$ a [[genuine G-spectrum]], the corresponding spectral Mackey functor is given by the [[fixed point spectra]] of $E$

\begin{displaymath}
G/H \mapsto E(G/H) =  [\Sigma^\infty_+ G/H, E]^G \simeq E^H
  \,,
\end{displaymath}
where on the right we have the $G$-equivariant [[mapping spectrum]] from the [[equivariant suspension spectrum]] of the [[transitive action|transitive]] [[G-set]] $G/H$ to $E$.

(e.g. (\hyperlink{GuillouMay11}{Guillou-May 11, remark 2.5}), see also (\hyperlink{Schwede15}{Schwede 15, p. 16}) for restriction and section 4 culminating on p. 37 for transfer and compatibility).

Further, the corresponding abelian-group valued Mackey functor is

\begin{displaymath}
\pi_n(E) \colon G/H \mapsto [G/H_+\wedge S^n, X]_G
  \,,
\end{displaymath}
where now on the right we have just the [[homotopy classes]] of maps, i.e. the morphisms in the [[equivariant stable homotopy category]] (e.g. \hyperlink{GreenleesMay95}{Greenlees-May 95, p. 43})

(\ldots{})

\hypertarget{Cohomology}{}\subsection*{{Cohomology with coefficients in a Mackey functor}}\label{Cohomology}

We discuss [[cohomology]] of [[topological G-spaces]] with [[coefficients]] in a Mackey functor, following notation and conventions as in (\hyperlink{May96}{May 96, sections IX, X}). See also (\hyperlink{GreenleesMay95}{Greenlees-May 95, p. 9}).

\begin{defn}
\label{CohomologyOfGSpace}\hypertarget{CohomologyOfGSpace}{}
For $X$ a [[pointed object|pointed]] [[G-CW complex]], define the [[chain complex]] $C_\bullet(X)$ of Mackey functors to be given by the stable [[equivariant homotopy groups]] of the quotient spaces $X^{\bullet}/X^{\bullet-1}$:

\begin{displaymath}
C_n(X) \coloneqq \pi_n(X^n/X^{n-1})
  \,,
\end{displaymath}
Then for $A$ any Mackey functor, the ordinary cohomology of $X$ with [[coefficients]] in $A$ is the [[cochain cohomology]] of the complex of homs of Mackey functors $C_n(X) \to A$:

\begin{displaymath}
H_G^n(X,A) \coloneqq H^n( Hom(C_\bullet(X), A) )
  \,.
\end{displaymath}
More generally, for $V$ a G-[[representation]], the $(n-V)$-[[RO(G)-grading|RO(G)-graded]] cohomology of $X$ with coefficients in $A$ is

\begin{displaymath}
H_G^{n-V}(X,A) = H_G^n(S^V \wedge X,A)
  \,.
\end{displaymath}
\end{defn}
(\hyperlink{May96}{May 96, section X.4 def. 4.1, def. 4.2})

\begin{remark}
\label{}\hypertarget{}{}
The corresponding [[reduced cohomology]] $\tilde H^n(-,A)$ is represented by maps into the [[Eilenberg-MacLane space|Eilenberg-MacLane G-space]]:

\begin{displaymath}
\tilde H^n(X,A) \simeq [X,K(A,n)]_G
  \,.
\end{displaymath}
\end{remark}
(\hyperlink{GreenleesMay95}{Greenlees-May 95})

\begin{remark}
\label{}\hypertarget{}{}
For this kind of cohomology, there is equivariant [[Serre spectral sequence]] (\hyperlink{Kronholm10}{Kronholm 10}).

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

\begin{itemize}%
\item [[Tambara functor]]

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

\hypertarget{plain_mackey_functors}{}\subsubsection*{{Plain Mackey functors}}\label{plain_mackey_functors}

The original article is

\begin{itemize}%
\item A. W. M. Dress, \emph{Notes on the theory of representations of finite groups. Part I: The Burnside ring of a finite group and some AGN-applications, Bielefeld}, 1971,

\end{itemize}
Reviews and surveys include

\begin{itemize}%
\item [[Andrew Blumberg]], section 3.2 of \emph{The Burnside category}, 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})


\item [[John Greenlees]], [[Peter May]], \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})


\item [[Peter May]], section IX.4 of \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. With contributions by M. Cole, G. Comezana, S. Costenoble, A. D. Elmenddorf, J. P. C. Greenlees, L. G. Lewis, Jr., R. J. Piacenza, G. Triantafillou, and S. Waner. (\href{http://www.math.rochester.edu/u/faculty/doug/otherpapers/alaska1.pdf}{pdf})


\item [[Stefan Schwede]], around p. 16 of \emph{[[Lectures on Equivariant Stable Homotopy Theory]]}, 2015 (\href{http://www.math.uni-bonn.de/people/schwede/equivariant.pdf}{pdf})


\item Peter Webb, \emph{A Guide to Mackey Functors} (\href{http://www.math.rochester.edu/people/faculty/doug/otherpapers/WebbMF.pdf}{pdf})


\item [[Michael Hill]], [[Michael Hopkins]], [[Douglas Ravenel]], section 4 of \emph{The Arf-Kervaire problem in algebraic topology: Sketch of the proof} ([[HHRKervaire.pdf:file]])

(with an eye towards application to the [[Arf-Kervaire invariant problem]])


\item [[Megan Shulman]], chapter 2 of \emph{Equivariant local coefficients and the $RO(G)$-graded cohomology of classifying spaces} (\href{http://arxiv.org/abs/1405.1770}{arXiv:1405.1770})



\end{itemize}
See also

\begin{itemize}%
\item [[Tammo tom Dieck]], \emph{Transformation groups}, Studies in Mathematics, vol. 8, Walter de Gruyter, Berlin, New York, 1987, x + 311 pp.,


\item Serge Bouc, chapter 1 of \emph{Green Functors and G-sets}, LNM 1671 (1997; paperback 2008) \href{http://dx.doi.org/10.1007/BFb0095821}{doi:10.1007/BFb0095821}


\item [[Tammo tom Dieck]], Equivariant homology and Mackey functors, Mathematische Annalen \textbf{206}, no.1, pp. 67--78, 1973 \href{http://dx.doi.org/10.1007/BF01431529}{doi:10.1007/BF01431529}


\item [[John Greenlees]], [[Peter May]], appendix A of \emph{Generalized Tate cohomology}, Mem. Amer. Math. Soc. 113 (1995) no 543 (\href{http://www.math.rochester.edu/people/faculty/doug/otherpapers/GM-Tate-543.pdf}{pdf})


\item D. Tambara, \emph{The Drinfeld center of the category of Mackey functors}, J. Algebra \textbf{319}, 10, pp. 4018-4101 (2008) \href{http://dx.doi.org/10.1016/j.jalgebra.2008.02.011}{doi:10.1016/j.jalgebra.2008.02.011}


\item Elango Panchadcharam, \emph{Categories of Mackey Functors}, PhD thesis, Macquarie Univ. 2006


\item [[William Kronholm]], \emph{The $RO(G)$-graded Serre spectral sequence}, Homology Homotopy Appl. Volume 12, Number 1 (2010), 75-92. (\href{http://www.swarthmore.edu/NatSci/wkronho1/serre.pdf}{pdf}, \href{https://projecteuclid.org/euclid.hha/1296223823}{Euclid})



\end{itemize}
Relation of Mackey functors to [[sheaves with transfer]] in the theory of [[motives]]:

\begin{itemize}%
\item [[Bruno Kahn]], Takao Yamazaki, \emph{Voevodsky's motives and Weil reciprocity}, Duke Mathematical Journal 162, 14 (2013) 2751-2796 (\href{http://arxiv.org/abs/1108.2764}{arXiv:1108.2764})

\end{itemize}
[[categorification|Categorification]] to Mackey [[2-functors]] is discussed found in

\begin{itemize}%
\item [[Paul Balmer]], [[Ivo Dell'Ambrogio]], \emph{Mackey 2-functors and Mackey 2-motives}, (\href{https://arxiv.org/abs/1808.04902}{arXiv:1808.04902})

\end{itemize}
\hypertarget{spectral_mackey_functors}{}\subsubsection*{{Spectral Mackey functors}}\label{spectral_mackey_functors}

The general concept of spectral Mackey functors

\begin{itemize}%
\item [[Clark Barwick]], \emph{Spectral Mackey functors and equivariant algebraic K-theory (I)}, Adv. Math., 304:646–727, 2017 (\href{https://doi.org/10.1016/j.aim.2016.08.043}{doi:10.1016/j.aim.2016.08.043}, \href{http://arxiv.org/abs/1404.0108}{arXiv:1404.0108})


\item [[Clark Barwick]], [[Saul Glasman]], Jay Shah, \emph{Spectral Mackey functors and equivariant algebraic K-theory (II)} (\href{http://arxiv.org/abs/1505.03098}{arXiv:1505.03098})


\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}, \href{http://www-users.math.umn.edu/~sglasman/strattalk.pdf}{talk notes pdf})



\end{itemize}
\hypertarget{in_equivariant_stable_homotopy_theory}{}\paragraph*{{In equivariant stable homotopy theory}}\label{in_equivariant_stable_homotopy_theory}

The construction of [[equivariant stable homotopy theory]] in terms of spectral Mackey functors is originally due to

\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})}


\item [[Bert Guillou]], [[Peter May]], \emph{Permutative $G$-categories in equivariant infinite loop space theory}, Algebr. Geom. Topol. 17 (2017) 3259-3339 (\href{http://arxiv.org/abs/1207.3459}{arXiv:1207.3459})


\item [[Denis Nardin]], section 2.6 and appendix A of \emph{Stability and distributivity over orbital ∞-categories}, 2012 (\href{http://hdl.handle.net/1721.1/112895}{hdl.handle.net/1721.1/112895}, \href{https://www.math.univ-paris13.fr/~nardin/thesis.pdf}{pdf})



\end{itemize}
The generalization of [[K-theory of permutative categories]] to spectral Mackey functors is discussed in

\begin{itemize}%
\item [[Anna Marie Bohmann]], [[Angélica Osorno]], \emph{Constructing equivariant spectra via categorical Mackey functors}, Algebraic \& Geometric Topology 15.1 (2015): 537-563 (\href{http://arxiv.org/abs/1405.6126}{arXiv:1405.6126})

\end{itemize}
\hypertarget{in_goodwillie_calculus}{}\paragraph*{{In Goodwillie calculus}}\label{in_goodwillie_calculus}

Discussion of [[Goodwillie calculus]] via [[spectral Mackey functors]]

\begin{itemize}%
\item [[Saul Glasman]], \emph{Goodwillie calculus and Mackey functors} (\href{https://arxiv.org/abs/1610.03127}{arXiv:1610.03127})

\end{itemize}
[[!redirects Mackey functors]]

[[!redirects spectral Mackey functor]] [[!redirects spectral Mackey functors]]



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