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\section*{K-theory}

\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{special_cases_and_models}{Special cases and models}\dotfill \pageref*{special_cases_and_models} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{related_entries}{Related entries}\dotfill \pageref*{related_entries} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

Given a [[stable (∞,1)-category]] $C$, its naive [[decategorification]] (its set of objects modulo the relation of [[equivalence]]) naturally inherits the structure of an abelian [[monoid]] from the [[biproducts]] in $C$. That is, we define $[a]+[b] = [a\oplus b]$. We can then further pass to the [[Grothendieck group]], adding formal additive inverses to get an abelian [[group]].

Note that such a biproduct sits in a [[split exact sequence|split]] (co)[[fibration sequence]] $a\to a\oplus b \to b$. Often we would like to obtain an analogous additivity result for \emph{arbitrary} (co)fibration sequences (a.k.a. [[distinguished triangles]]). This requires imposing the [[generators and relations|relation]] $[c] = [a] + [b]$ for any fibration sequence $a\to c\to b$, thereby passing to a quotient abelian group called $K_0(C)$, the \textbf{K-group} or [[Grothendieck group]] of $C$; see the latter entry for more details.

The ``K'' is chosen by Grothendieck for the German word \emph{Klasse} for ``class''. The K-group of $C$ is the group of equivalence classes of $C$: it is a group due to the existence of a notion of exact sequences in $C$.

K-theory starts with the study of these K-groups and their higher analogues $K_n(C)$, collectively denoted $K(C)$. Sometimes the K-groups themselves are called ``K-theory''. One would say for instance: ``$K(C)$ is the K-theory of $C$.''

More generally, there is a [[symmetric monoidal (infinity,1)-category|symmetric groupal ∞-groupoid]] $\mathbf{K}(C)$ -- i.e. a connective [[spectrum]] -- in between the [[decategorification]] from $C$ to $K(C)$, for which $K_0(C)$ is the set of [[simplicial homotopy group|connected components]]

\begin{displaymath}
C \mapsto \mathbf{K}(C) \to \pi_0 \mathbf{K}(C) = K_0(C)
  \,.
\end{displaymath}
and more generally $K_n(C) = \pi_n \mathbf{K}(C)$.

In nice cases this is the degree 0 part of a non-connective [[spectrum]] which is then the \textbf{K-theory spectrum} of $C$. This is also called the \textbf{Waldhausen K-theory} of $C$.

\hypertarget{special_cases_and_models}{}\subsection*{{Special cases and models}}\label{special_cases_and_models}

Much of the literature on K-theory discusses constructions that \emph{model} the above abstract setup in terms of [[model category|model categories]], or just their [[homotopy category|homotopy categories]], often of the [[derived category|derived catgeories]] type and then often expressed in terms of the [[abelian category]] or more generally [[Quillen exact category]] from which the derived category is derived.

Only a subset of the structure on a [[model category]] is necessary in order to conveniently extract the K-groups of the [[presentable (infinity,1)-category|presented]] [[stable (∞,1)-category]]. For that reason the axioms of a [[Waldhausen category]] have been devised to provide just the necessary convenient prerequisites to compute the K-groups of the [[(∞,1)-category]] [[presentable (infinity,1)-category|presented]] by the underlying [[homotopical category]].

\begin{itemize}%
\item In particular, the K-group associated to the [[stable (∞,1)-category]] $Ch^b(A)$ of \emph{bounded} [[chain complex]]es in an [[abelian category]] or [[exact category]] $A$ is often called the K-group of $A$ itself and just denoted

\begin{displaymath}
K(A) := K(Ch^b(A))
  \,.
\end{displaymath}
Most explicit constructions of K-theory spectra start with the data of an [[exact category]], such as notably Quillen's [[Q-construction]] and the [[Waldhausen S-construction]].


\item In particular if the exact category $A$ is that of [[vector bundle]]s on a [[topological space]] $X$

\begin{displaymath}
A = VectBund(X)
\end{displaymath}
the corresponding K-group is degree 0 [[topological K-theory]]. This was the original of the notion and the term K-theory.



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

Recall that given a [[(∞,1)-category]] $C$, we may regard it as a [[complete Segal space]] $C_{\bullet,\bullet}$, a [[bisimplicial object|bisimplicial set]]. For instance if $C$ is originally given as a [[quasicategory]] then

\begin{displaymath}
C_{\bullet,\bullet} : [n],[m] \mapsto
  Core(Func(\Delta^n,C))_{m}
  \,,
\end{displaymath}
where $Core(Func(\Delta^n,C))$ denotes the maximal [[Kan complex]] inside the [[(∞,1)-category of (∞,1)-functors]] from $\Delta^n$ to $C$.

\begin{defn}
\label{Gap}\hypertarget{Gap}{}
For $\mathcal{C}$ an [[(∞,1)-category]] and $n \in \mathbb{N}$, write $Gap(\Delta^n, \mathcal{C})$ for the full sub-$\infty$-category on $Func(Arr(\Delta^n),\mathcal{C} )$ on those objects $F$ for which

\begin{itemize}%
\item the [[diagonal]] $F(n,n)$ is inhabited by [[zero objects]], for all $n$;


\item all [[diagrams]] of the form

\begin{displaymath}
\itexarray{
    F(i,j) &\to& F(i,k)
    \\
    \downarrow && \downarrow
    \\
    F(j,j) &\to& F(j,k)
  }
\end{displaymath}
is an [[(∞,1)-pushout]].



\end{itemize}
\end{defn}
\begin{defn}
\label{}\hypertarget{}{}
Let $C$ be a [[stable (∞,1)-category]]. Then its \textbf{Waldhausen K-theory}

\begin{displaymath}
\mathbf{K}(C) := \underset{\rightarrow}{\lim}_n Core(Gap(C^{\Delta^n}))
\end{displaymath}
is the [[geometric realization of simplicial topological spaces|geometric realization]] of/[[homotopy colimit]] of the degreewise [[core]] of the $Gap$, def. \ref{Gap}, of the corresponding [[complete Segal space]] (as a simplicial diagram of $\infty$-groupoids).

\end{defn}
This is \href{http://arxiv.org/PS_cache/math/pdf/0608/0608228v5.pdf#page=43}{remark 11.4} in [[stable (infinity,1)-category|StCat]]. See also \hyperlink{BGT}{Blumberg-Gepner-Tabuada, section 7}.

This construction is also conjectured in the last section of Toen-Vezzosi's \emph{A remark on K-theory} .

\begin{remark}
\label{}\hypertarget{}{}
In the case that $C$ is the [[simplicial localization]] of a [[Waldhausen category]] $\bar C$ the explicit way to obtain this is the [[Waldhausen S-construction]].

\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
It should be true that with this definition we have an isomorphism of groups

\begin{displaymath}
K(C) \simeq \pi_0 \mathbf{K}(C)
  \,.
\end{displaymath}
\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
This Waldhausen/hocolim-construction gives the \emph{connective} K-theory, taking values in [[connective spectra]]. The [[universal construction|universal]] completion to functor that sends [[homotopy cofibers]] of [[stable (infinity,1)-categories]] to homotopy cofibers of spectra is the corresponding unconnective $\mathbb{K}$-functor.

\end{remark}
There is a universal characterization of the construction of the $\mathbb{K}$-theory spectrum of a stable $(\infty,1)$-category $A$:

there is an $(\infty,1)$-functor

\begin{displaymath}
U : (\infty,1)StabCat \to N
\end{displaymath}
to a stable $(\infty,1)$-category which is universal with the property that it respects colimits and exact sequences in a suitable way. Given any stable $(\infty,1)$-category $A$, its (connective or non-connective, depending on details) algebraic K-theory spectrum is the hom-object

\begin{displaymath}
K(A) \simeq Hom(U(Sp), U(A))
  \,,
\end{displaymath}
where $Sp$ denotes the stable $(\infty,1)$-category of compact [[spectra]]. (\hyperlink{BGT}{BGT})

\hypertarget{related_entries}{}\subsection*{{Related entries}}\label{related_entries}

\begin{itemize}%
\item [[Grothendieck group]]


\item [[K-theory spectrum]]


\item [[topological K-theory]]

\begin{itemize}%
\item [[fiber integration in K-theory]]

\end{itemize}

\item [[Tate K-theory]]


\item [[K-orientation]]


\item [[algebraic K-theory]]

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


\item [[K-theory of a bipermutative category]]


\item [[differential algebraic K-theory]]


\item [[noncommutative motive]]



\end{itemize}

\item [[Waldhausen K-theory of a dg-category]]


\item [[groupoid K-theory]]


\item [[operator K-theory]]

\begin{itemize}%
\item [[equivariant operator K-theory]]


\item [[KK-theory]]


\item [[E-theory]]



\end{itemize}

\item [[equivariant K-theory]]


\item [[Karoubi K-theory]]


\item [[Morava K-theory]]


\item [[L-theory]]


\item [[twisted K-theory]]


\item [[differential K-theory]]


\item [[K-theory and physics]]



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

It was in

\begin{itemize}%
\item [[Bertrand Toen]], Gabriele Vezzosi, \emph{A remark on K-theory and $S$-categories} (\href{http://arxiv.org/PS_cache/math/pdf/0210/0210125v2.pdf}{arXiv}).

\end{itemize}
that it was proven that the the [[Waldhausen S-construction]] of the [[K-theory spectrum]] depends precisely on the [[simplicial localization]] of the [[Waldhausen category]], i.e. of the [[(∞,1)-category]] that it presents.

In view of this remark 11.4 in

\begin{itemize}%
\item [[Jacob Lurie]], [[stable (infinity,1)-category|Stable ∞-Categories]] .

\end{itemize}
interprets the construction of the K-theory spectrum as a natural operation of [[stable (infinity,1)-category|stable (∞,1)-categories]], as described above.

The universal property of the $(\infty,1)$-categorical definition is studied in

\begin{itemize}%
\item [[Andrew Blumberg]], [[David Gepner]], [[Goncalo Tabuada]], \emph{A universal characterization of higher algebraic K-theory} (\href{http://arxiv.org/abs/1001.2282}{arXiv:1001.2282}).

\end{itemize}
The standard constructions of K-theory spectra from [[Quillen exact categories]] are discussed in detail in chapter 1 of

\begin{itemize}%
\item [[Eric Friedlander]], [[Daniel Grayson]], \emph{Handbook of K-theory}, Springer Verlag .

\end{itemize}
A useful introduction to the definition and computation of K-groups (with a little on K-spectra) is

\begin{itemize}%
\item Charles Weibel, \emph{The K-book: An introduction to algebraic K-theory} (\href{http://www.math.rutgers.edu/~weibel/Kbook.html}{web})

\end{itemize}
[[!redirects Waldhausen K-theory]]



\end{document}