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\section*{K-theory of a permutative category}

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

\hypertarget{cohomology}{}\paragraph*{{Cohomology}}\label{cohomology}

[[!include cohomology - 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{ViaTopologicalGroupCompletion}{Via topological group completion}\dotfill \pageref*{ViaTopologicalGroupCompletion} \linebreak
\noindent\hyperlink{via_gamma_spaces}{Via Gamma spaces}\dotfill \pageref*{via_gamma_spaces} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{monoidal_functoriality}{Monoidal functoriality}\dotfill \pageref*{monoidal_functoriality} \linebreak
\noindent\hyperlink{Examples}{Examples}\dotfill \pageref*{Examples} \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}

To a [[permutative category]] $C$ is naturally associated a [[Gamma-space]], hence a [[symmetric spectrum]]. The [[generalized (Eilenberg-Steenrod) cohomology]] theory [[Brown representability theorem|represented]] by this is called the \emph{([[algebraic K-theory|algebraic]]) [[K-theory]]} of (or represented by) $C$.

If the category is even a [[bipermutative category]] then the corresponding [[K-theory of a bipermutative category]] in addition has [[E-infinity ring]] structure, hence is a [[multiplicative cohomology theory]].

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

\hypertarget{ViaTopologicalGroupCompletion}{}\subsubsection*{{Via topological group completion}}\label{ViaTopologicalGroupCompletion}

For $\mathcal{C}$ a [[category]], write ${\vert \mathcal{C}\vert} \in$ [[classical model structure on simplicial sets|sSet]] $\simeq_{Qu}$ [[classical model structure on topological spaces|Top]] for its [[nerve]]/[[geometric realization]]. (Beware that this is often denoted $B C$, instead, but that notation clashes with that for [[delooping]], which we also need in the following.)

For $\mathcal{C}$ a [[permutative category]] its [[nerve]]/[[geometric realization]] $\vert \mathcal{C} \vert$ is naturally a [[topological monoid]] (\hyperlink{Quillen70}{Quillen 70} see e.g. \hyperlink{May13}{May 13, theorem 4.10}), hence admits a [[bar construction]]/[[classifying space]] $B {\vert \mathcal{C}\vert}$. The [[loop space]] of that

\begin{displaymath}
(\mathbb{K}\mathcal{C})_0
  \;\coloneqq\; 
  \Omega B {\vert \mathcal{C}\vert}
  \,,
\end{displaymath}
being an [[∞-group]], may be regarded as the [[homotopy theory|homotopy theoretic]] [[group completion]] of the [[topological monoid]] ${\vert \mathcal{C}\vert}$.

This is the degree-0 space in the \emph{algebraic K-theory [[spectrum]] $\mathbb{K}\mathcal{C}$ of the permutative category} $\mathcal{C}$ (see e.g. \hyperlink{May13}{May 13, def 4.11}).

By (\hyperlink{DwyerKan80}{Dwyer-Kan 80, prop. 3.7, prop. 9.2, remark 9.7}) the operation $\Omega B (-)$ is the [[derived functor]] of [[group completion]], so that this construction ought to be a model for the [[K-theory of a symmetric monoidal (∞,1)-category]].

In particular, under [[decategorification]], its [[group]] of [[connected components]] is the actual [[Grothendieck group]] $K(-)$ of the [[isomorphism classes]] of [[objects]] in $\mathcal{C}$:

\begin{displaymath}
\pi_0
  \left( 
    \mathbb{K} \mathcal{C}
  \right)
  \;\simeq\;
  K\left(
    \mathcal{C}/_\sim, \oplus 
  \right)
\end{displaymath}
(recalled e.g. in \hyperlink{BohmannOsorno14}{Bohmann-Osorno 14, p. 14}).

In particular for $R$ a [[topological ring]] one considers $C$ a [[skeletal category|skeleton]] of the [[groupoid]] of ([[finitely generated module|finitely generated]]) [[projective modules]] over $R$. Then the K-theory of $C$ is the [[algebraic K-theory]] of $R$ (e.g. \hyperlink{May13}{May 13, p. 25})

\begin{displaymath}
\mathbb{K}\left( R Mod^{fin}_{proj} \right)
  \;\simeq\;
  K R
  \,.
\end{displaymath}
\hypertarget{via_gamma_spaces}{}\subsubsection*{{Via Gamma spaces}}\label{via_gamma_spaces}

Write $FinSet^{*/}$ for the [[category]] of [[pointed objects|pointed]] [[finite sets]].

For $C$ a [[permutative category]], there is naturally a [[functor]]

\begin{displaymath}
\widebar {C}_{(-)}
  \;\colon\; 
  FinSet^{*/} \to Cat
\end{displaymath}
\begin{displaymath}
A \mapsto \widebar C_A
\end{displaymath}
such that (\ldots{}).

(\hyperlink{ElmendorfMandell06}{Elmendorf-Mandell, theorem 4.2})

Accordingly, postcomposition with the [[nerve]] $N : Cat \to sSet$ produces from $C$ a [[Gamma-space]] $N \widebar C$. To this corresponds a [[spectrum]]

\begin{displaymath}
K^{Seg} C \coloneqq \{N \widebar C_{S_\bullet^n}\}
  \,.
\end{displaymath}
This is the \emph{K-theory spectrum of $C$}.

(\hyperlink{ElmendorfMandell06}{Elmendorf-Mandell, def. 4.3})

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

\hypertarget{monoidal_functoriality}{}\subsubsection*{{Monoidal functoriality}}\label{monoidal_functoriality}

\begin{prop}
\label{MonoidalFunctoriality}\hypertarget{MonoidalFunctoriality}{}
The construction of K-theory spectra of permutative categories constitutes a [[multifunctor]]

\begin{displaymath}
\mathbb{K}
  \;\colon\;
  PermCat
    \longrightarrow
  Spectra
\end{displaymath}
between [[multicategories]] of [[permutative categories]] (under [[Deligne tensor product]]) and [[spectra]] (under [[smash product of spectra]]).

Hence for a multilinear functor of permutative categories

\begin{displaymath}
F 
    \;\colon\;
  \mathcal{A}_1 \times \cdots \times \mathcal{A}_n
  \longrightarrow
  \mathcal{B}
\end{displaymath}
there is a compatibly induced morphism of K-[[spectra]] out of the [[smash product of spectra|smash product]]

\begin{displaymath}
\mathbb{K}(\mathcal{A}_1) 
    \wedge
    \cdots
    \wedge
  \mathbb{K}(\mathcal{A}_n)
  \longrightarrow
  \mathbb{K}(\mathcal{B})
  \,.
\end{displaymath}
This implies that the construction further extends to a [[2-functor]] from the [[2-category]] $PermCat Cat$ of [[enriched categories]] over the [[multicategory]] of [[permutative categories]] to that of [[enriched categories]] of [[spectra]]:

\begin{displaymath}
\mathbb{K}_\bullet
  \;\colon\;
  PermCat Cat \longrightarrow Spectra Cat
\end{displaymath}
which applies $\mathbb{K}$ to each [[hom-object]].

\end{prop}
(\hyperlink{May80}{May 80, theorem 1.6, Theorem 2.1}, \hyperlink{ElmendorfMandell04}{Elmendorf-Mandell 04, theorem 1.1}, \hyperlink{Guillou10}{Guillou 10, Theorem 1.1}

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

\begin{example}
\label{OrdinaryAlgebraicKTheoryFromPermutativeCategoryOfProjectiveModules}\hypertarget{OrdinaryAlgebraicKTheoryFromPermutativeCategoryOfProjectiveModules}{}
\textbf{(ordinary [[algebraic K-theory]])}

For $R$ a [[commutative ring]], let $\mathcal{C} = R Mod_{pr}$ its [[category of modules|category of]] [[finitely generated module|finitely generated]] [[projective modules]] regarded as a [[permutative category]]. Then

\begin{displaymath}
K (R Mod^{pr}_{fin}) \;\simeq\; K R
\end{displaymath}
is the classical [[algebraic K-theory]] spectrum of the ring $R$.

\end{example}
(e.g. \hyperlink{ElmendorfMandell04}{Elmendorf-Mandell 04, p. 10}).

\begin{example}
\label{StableCohomotopyIsKTheoryOfFinSet}\hypertarget{StableCohomotopyIsKTheoryOfFinSet}{}
\textbf{([[stable cohomotopy]] is K-theory of [[FinSet]])}

Let $\mathcal{C} =$ [[FinSet]] be a [[skeleton]] of the category of [[finite sets]], regarded as a [[permutative category]]. Then

\begin{displaymath}
K(FinSet) \;\simeq\; \mathbb{S}
\end{displaymath}
is the [[sphere spectrum]], hence represents the [[cohomology theory]] called \emph{[[stable cohomotopy]]}.

\end{example}
(due to \hyperlink{Segal74}{Segal 74, Prop. 3.5}, see also \hyperlink{Priddy73}{Priddy 73})

\begin{remark}
\label{StableCohomotopyIsAlgebraicKTheoryOverFieldWithOneElement}\hypertarget{StableCohomotopyIsAlgebraicKTheoryOverFieldWithOneElement}{}
\textbf{([[stable cohomotopy]] as [[algebraic K-theory]] over the [[field with one element]])}

Since ([[pointed sets|pointed]]) [[finite sets]] may be regarded as the modules over the ``[[field with one element]]'' $\mathbb{F}_1$ (see \href{field+with+one+element#Modules}{there}),

\begin{displaymath}
\mathbb{F}_1 Mod
  \;=\;
  FinSet^{\ast/}
\end{displaymath}
one may read example \ref{StableCohomotopyIsKTheoryOfFinSet} in view of example \ref{OrdinaryAlgebraicKTheoryFromPermutativeCategoryOfProjectiveModules} as saying that [[stable cohomotopy]] is the algebraic K-theory of the [[field with one element]]:

\begin{displaymath}
K \mathbb{F}_1 \;=\; \mathbb{S}
  \,.
\end{displaymath}
This perspective is highlighted for instance in (\hyperlink{Deitmar06}{Deitmar 06, p. 2}, \hyperlink{Guillot06}{Guillot 06}).

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

\begin{itemize}%
\item [[K-theory of a bipermutative category]]


\item [[algebraic K-theory of a symmetric monoidal (∞,1)-category]]



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

\begin{itemize}%
\item [[Daniel Quillen]], \emph{Cohomology of groups}, Proceedings of the international congress of mathematics 1970


\item [[Daniel Quillen]], \emph{On the group completion of a simplicial monoid}


\item [[Stewart Priddy]], \emph{Transfer, symmetric groups, and stable homotopy theory}, in \emph{Higher K-Theories}, Springer, Berlin, Heidelberg, 1973. 244-255 (\href{https://link.springer.com/content/pdf/10.1007/BFb0067060.pdf}{pdf})


\item [[Graeme Segal]], \emph{Catgeories and cohomology theories}, Topology vol 13 (1974) (\href{http://ncatlab.org/nlab/files/SegalCategoriesAndCohomologyTheories.pdf}{pdf})


\item [[Peter May]], \emph{The spectra associated to permutative categories}, Topology 17 (1978) (\href{http://www.math.uchicago.edu/~may/PAPERS/23.pdf}{pdf})


\item [[Peter May]], \emph{$E_\infty$-Spaces, group completions, and permutative categories}, in [[Graeme Segal]] (ed.) \emph{New Developments in Topology}, Cambirdge University Press 2013 (\href{http://www.math.uchicago.edu/~may/PAPERS/13.pdf}{pdf}, \href{https://doi.org/10.1017/CBO9780511662607.008}{doi:10.1017/CBO9780511662607.008})


\item [[Peter May]], \emph{Pairings of categories and spectra}, J. Pure Appl. Algebra, 19:299–346, 1980


\item [[William Dwyer]], [[Daniel Kan]], \emph{Simplicial localization of categories}, Journal of pure and applied algebra 17 (1980) 267-284


\item [[Anthony Elmendorf]], [[Michael Mandell]], \emph{Permutative categories as a model of connective stable homotopy}, in: [[Birgit Richter]] (ed.) \emph{Structured Ring spectra}, Cambridge University Press (2004)


\item [[Anthony Elmendorf]], [[Michael Mandell]], \emph{Rings, modules and algebras in infinite loop space theory}, Adv. in Math. 205 (2006), no. 1, 163-228 (\href{https://arxiv.org/abs/math/0403403}{arXiv:math/0403403})


\item [[Anthony Elmendorf]], [[Michael Mandell]], \emph{Permutative categories, multicategories, and algebraic K-theory}, Algebraic \& Geometric Topology 9 (2009) 2391-2441 (\href{http://arxiv.org/abs/0710.0082}{arXiv:0710.0082}, \href{https://projecteuclid.org/euclid.agt/1513797088}{euclid:1513797088})


\item [[Bertrand Guillou]], \emph{Strictification of categories weakly enriched in symmetric monoidal categories}, Theory Appl. Categ., 24:No. 20, 564–579, 2010.



\end{itemize}
The interpretation of [[stable cohomotopy]] as the algebraic K-theory over the [[field with one element]] is adopted in

\begin{itemize}%
\item [[Anton Deitmar]], \emph{Remarks on zeta functions and K-theory over $\mathbb{F}_1$} (\href{https://arxiv.org/abs/math/0605429}{arXiv:math/0605429})


\item [[Pierre Guillot]], \emph{Adams operations in cohomotopy} (\href{https://arxiv.org/abs/math/0612327}{arXiv:0612327})



\end{itemize}
The generalization of K-theory of permutative categories to [[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}
Generalization to [[equivariant stable homotopy theory]] and [[G-spectra]] is discussed in

\begin{itemize}%
\item [[Bert Guillou]], [[Peter May]], around section 4.4 of \emph{Permutative $G$-categories in equivariant infinite loop space theory} (\href{http://arxiv.org/abs/1207.3459}{arXiv:1207.3459})

\end{itemize}
[[!redirects K-theory of permutative categories]]

[[!redirects algebraic K-theory of a permutative category]] [[!redirects algebraic K-theory of permutative categories]]

[[!redirects K-theory spectrum of a permutative category]] [[!redirects K-theory spectra of permutative categories]]



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