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\begin{document}

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\section*{Burnside category}

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

\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{Definition}{Definition}\dotfill \pageref*{Definition} \linebreak
\noindent\hyperlink{AsEnrichedInPermutativeCategories}{As enriched in permutative categories}\dotfill \pageref*{AsEnrichedInPermutativeCategories} \linebreak
\noindent\hyperlink{AsEnrichedInabelianGroups}{As enriched in abelian groups}\dotfill \pageref*{AsEnrichedInabelianGroups} \linebreak
\noindent\hyperlink{EnrichedInSpectra}{As enriched in spectra}\dotfill \pageref*{EnrichedInSpectra} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{relation_to_spectra}{Relation to $G$-Spectra}\dotfill \pageref*{relation_to_spectra} \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}

By the \emph{Burnside category} of a [[finite group]] $G$ one means either

\begin{enumerate}%
\item the [[category of correspondences]] in [[finite set|finite]] [[G-sets]] (Prop. \ref{PermutativeCategoryOfFiniteGSets} below)


\item or an abelianization of that

\begin{enumerate}%
\item either by [[group completion]] of its monoidal [[hom-groupoids]] to an [[additive category|additive]] [[1-category]] (Def. \ref{AdditiveBurnsideCategory} below);


\item or by forming the [[K-theory of a permutative category]] of the [[hom-groupoids]] to a [[(∞,1)-category of spectra|spectra]]-[[enriched (∞,1)-category]] (Def. \ref{SpectralBurnsideCategory} below).



\end{enumerate}


\end{enumerate}
The plain \emph{[[Burnside ring]]} is the [[endomorphism ring]] of the [[terminal object|terminal]] [[G-set]] in the [[additive category|additive]] version of the category (Example \ref{BurnsideRingIsEndomorphismRingInBurnsideCategory} below).

The [[functors]]/[[additive functors]]/[[enriched (∞,1)-functors]] from the Burnside category to, in particular, the [[category of abelian groups]]/[[(∞,1)-category of spectra]] are called \emph{[[Mackey functors]]}.

A [[functor]] out of the Burnside category is called a \emph{[[Mackey functor]]}, specifically if it is an [[Ab]]-[[enriched functor]] to [[Ab]] out of the [[additive category]]-version of the Burnside category. A [[spectra]]-[[enriched functor]] out of the spectral Burnside category to [[Spectra]] is called a \emph{[[spectral Mackey functor]]}, for emphasis. These spectral Mackey functors on the Burnside category are [[equivalence of (infinity,1)-categories|equivalent]] to [[genuine G-spectra]] in [[equivariant stable homotopy theory]].

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

There are various incarnations of the Burnside category as [[enriched categories]], in varying degree of sophistication of the enriching ``[[cosmos]]''. They are all induced from the canonical [[structure]] of [[permutative categories]] on the [[correspondences]] between two fixed [[finite set|finite]] [[G-sets]] (made explicit as Prop. \ref{PermutativeCategoryOfFiniteGSets} below). This yields the Burnside category as

\begin{enumerate}%
\item \emph{\hyperlink{AsEnrichedInPermutativeCategories}{enriched in permutative categories}};


\item \emph{\hyperlink{AsEnrichedInabelianGroups}{enriched in abelian groups}};


\item \emph{\hyperlink{EnrichedInSpectra}{enriched in spectra}}.



\end{enumerate}
\begin{prop}
\label{PermutativeCategoryOfFiniteGSets}\hypertarget{PermutativeCategoryOfFiniteGSets}{}
\textbf{(the [[permutative category]] of [[finite set|finite]] [[G-sets]])}

For $G$ be a [[finite group]], write $G FinSet_{sk}$ for the [[skeleton]] of the [[category]] of [[finite set|finite]] [[G-sets]].

Its [[objects]] may be identified with [[pairs]] $(n,\rho)$ consisting of a [[natural number]] $n$, reflecting the [[finite set]] $\{1,2, \cdots, n\}$, and a [[group homomorphism]] $\rho \;\colon\; G \longrightarrow S_n = Aut(\{1,2, \cdots, n\})$ from $G$ to the [[symmetric group]] on $n$ elements, reflecting the [[automorphism]] group of that finite set.

Its [[morphisms]] $(n_1,\rho_1) \overset{ f }{\longrightarrow} (n_2, \rho_2)$ are [[functions]] $f \;\colon\; \{1, 2, \cdots, n_1\} \longrightarrow \{1,2, \cdots, n_2\}$ that intertwine $\rho_1$ and $\rho_2$.

The [[coproduct]] $\sqcup$ of [[G-sets]] ([[disjoint union]]) makes this [[skeleton]] a [[permutative category]] $\big( G FinSet_{sk}, \sqcup \big)$.

In the same way its [[slice category]] over any object $X \in G FinSet_{sk}$ becomes a [[permutative category]] $\big( G FinSet_{sk}/_{X}, \sqcup \big)$ under disjoint union of [[domains]].

Similarly, the [[Cartesian product]] of finite $G$-sets can be restricted to these [[skeleta]] to produce [[bipermutative categories]] $\big( G FinSet_{sk}/_{X}, \sqcup, \times \big)$.

Finally, restriction to [[isomorphisms]] (passage to [[cores]]) yields the bipermutative [[groupoids]] $\big( Core(G FinSet_{sk})/_{X}, \sqcup, \times \big)$.

\end{prop}
(e.g. \hyperlink{GuillouMay11}{Guillou-May 11, Def. 1.3}, \hyperlink{BohmannOsorno14}{Bohmann-Osorno 14, Def. 1.4})

\hypertarget{AsEnrichedInPermutativeCategories}{}\subsubsection*{{As enriched in permutative categories}}\label{AsEnrichedInPermutativeCategories}

\begin{defn}
\label{AdditiveBurnsideCategory}\hypertarget{AdditiveBurnsideCategory}{}
\textbf{([[PermCat]]-[[enriched category|enriched]] [[Burnside category]])}

The [[(2,1)-category of correspondences]] $Corr(G FinSet)$ is [[equivalence of (infinity,1)-categories|equivalent]] to the [[(2,1)-category]] whose [[objects]] are the [[finite set|finite]] [[G-sets]] and whose [[hom-categories]] are the [[permutative category|permutative]] [[cores]] of [[skeleta]] of [[slice categories]] of [[G-sets]] from Def. \ref{PermutativeCategoryOfFiniteGSets}, over the [[Cartesian product]] of [[source]] and [[domain]] [[G-sets]]:

\begin{displaymath}
Corr(G FinSet)\big( S_1, S_2  \big)
  \;\simeq\;
  Core(G FinSet_{sk}/_{S_1 \times S_2})
  \,.
\end{displaymath}
This locally skeletal [[(2,1)-category]] is the \emph{[[PermCat]]-[[enriched category|enriched]] Burnside category} $G Burn_{pc}$\_

It may be regarded as an [[enriched category]] over the [[multicategory]] [[PermCat]] of [[permutative categories]] (a $\mathbf{PC}$-cateory in the sense of \hyperlink{Guillout10}{Guillout 10}).

\end{defn}
(\hyperlink{GuillouMay11}{Guillou-May 11, Def. 1.6}, \hyperlink{BohmannOsorno14}{Bohmann-Osorno 14, Def. 7.1, 7.2})

\hypertarget{AsEnrichedInabelianGroups}{}\subsubsection*{{As enriched in abelian groups}}\label{AsEnrichedInabelianGroups}

\begin{defn}
\label{AdditiveBurnsideCategory}\hypertarget{AdditiveBurnsideCategory}{}
\textbf{(additive Burnside category)}

The \emph{additive Burnside category} $G Burn_{ad}$ of $G$ is the [[additive category]] obtained from the [[PermCat]]-[[enriched category|enriched]] Burnside category $G Burn_{pc}$ (Def. \ref{AdditiveBurnsideCategory}) under replacing each [[hom-object|hom]]-[[permutative category]] $Core(G FinSet_{sk}/_{S_1 \times S_2})$ with its [[Grothendieck group]], hence with the [[abelian group]] which is the [[group completion]]

\begin{displaymath}
K \;\colon\; PermCat \overset{\text{iso classes}}{\longrightarrow} CMon \overset{K}{\longrightarrow} Ab
\end{displaymath}
of the [[commutative monoid]] of [[isomorphism classes]] of [[objects]] in $Core(G FinSet_{sk}/_{S_1 \times S_2})$, under [[disjoint union]]:

\begin{displaymath}
G Burn_{ab}
  \;\coloneqq\;
  K_{\bullet} G Burn_{pc}  
  \,.
\end{displaymath}
\end{defn}
\begin{example}
\label{BurnsideRingIsEndomorphismRingInBurnsideCategory}\hypertarget{BurnsideRingIsEndomorphismRingInBurnsideCategory}{}
\textbf{([[Burnside ring]] is [[endomorphism ring]] of additive Burnside category)}

The [[endomorphism ring]] of the [[terminal object|terminal]] [[G-set]] (the [[point]] $\ast$ equipped with the, necessarily, [[trivial action]]) in the additive Burnside category (Def. \ref{AdditiveBurnsideCategory}) is the \emph{[[Burnside ring]]} $A(G)$:

\begin{displaymath}
End_{G Burn_{ad}}(\ast, \ast)
  \;\simeq\;
  A(G)
\end{displaymath}
\end{example}
\hypertarget{EnrichedInSpectra}{}\subsubsection*{{As enriched in spectra}}\label{EnrichedInSpectra}

\begin{defn}
\label{SpectralBurnsideCategory}\hypertarget{SpectralBurnsideCategory}{}
(\textbf{[[Spectra]]-[[enriched category|enriched]] [[Burnside category]])}

Since construction of [[K-theory spectra of permutative categories]] applies [[hom-object]]-wise to [[PermCat]]-[[enriched categories]] (\href{multifunctor#MonoidalFunctoriality}{this Prop.})

\begin{displaymath}
\mathbb{K}_\bullet
  \;\colon\;
  PermCat Cat \longrightarrow Spectra Cat
\end{displaymath}
the image of the [[PermCat]]-[[enriched category|enriched]] [[Burnside category]] $G \mathcal{E}$ from Def. \ref{AdditiveBurnsideCategory} under forming [[hom-object]]-wise the [[K-theory spectra of permutative categories]] yields a [[Spectra]]-[[enriched category]]

\begin{displaymath}
G Burn_{sp}
  \;\coloneqq\;
  \mathbb{K}_\bullet G Burn_{pc}  
  \,.
\end{displaymath}
This is called the \emph{spectral Burnside category}.

\end{defn}
(\hyperlink{GuillouMay11}{Guillou-May 11, Def. 1.12}, \hyperlink{BohmannOsorno14}{Bohmann-Osorno 14, Def. 7.3})

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

\hypertarget{relation_to_spectra}{}\subsubsection*{{Relation to $G$-Spectra}}\label{relation_to_spectra}

\begin{prop}
\label{GSpectraAreSpectralMackeyFunctors}\hypertarget{GSpectraAreSpectralMackeyFunctors}{}
\textbf{([[G-spectra]] are spectral presheaves on the spectral Burnside category)}

There is a [[zig-zag]] of [[Quillen equivalences]]

\begin{displaymath}
[ G Burn_{sp}, Spectra ]
  \;\simeq_{Qu}^{zigzag}\;
  G Spectra
\end{displaymath}
between the [[model structure on functors|model category of]] [[Spectra]]-[[enriched presheaves]] over the [[spectral Burnside category]] from Def. \ref{SpectralBurnsideCategory} (``[[spectral Mackey functors]]'') and that of [[genuine G-spectra]].

This equivalence is such that the [[spectral Mackey functor]] corresponding to a fibrant [[G-spectrum]] $E$ assigns to the [[transitive action|transitive]] [[G-set]] $G/H$ the [[fixed point spectrum]] $E^H$:

\begin{displaymath}
G/H 
    \;\mapsto\; 
  E(G/H)
    \;\coloneqq\;
  [\Sigma^\infty_G G/H, E]^G
    \;\simeq\;
  E^H
  \,.
\end{displaymath}
\end{prop}
(\hyperlink{GuillouMay11}{GuillouMay 11, theorem 1.13, corollary 1.14, remark 2.5}).

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

\begin{itemize}%
\item [[Burnside ring]]


\item [[Mackey functor]], [[equivariant stable homotopy theory]]


\item [[disjunctive (∞,1)-category]]



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

Lecture notes include:

\begin{itemize}%
\item [[Andrew Blumberg]], section 3.1 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})

\end{itemize}
The spectrally-enriched version and its role in the equivalent description of [[G-spectra]] via [[spectral Mackey functors]] is 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})}

\end{itemize}
A beautified review is given in

\begin{itemize}%
\item [[Anna Marie Bohmann]], [[Angélica Osorno]], section 7 of \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}
making use of

\begin{itemize}%
\item [[Bertrand Guillou]], \emph{Strictification of categories weakly enriched in symmetric monoidal categories}, Theory Appl. Categ., 24:No. 20, 564–579, 2010 (\href{http://www.tac.mta.ca/tac//volumes/24/20/24-20abs.html}{TAC})

\end{itemize}
[[!redirects Burnside categories]]

[[!redirects spectral Burnside category]] [[!redirects spectral Burnside categories]]

[[!redirects additive Burnside category]] [[!redirects additive Burnside categories]]



\end{document}