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\begin{document}

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\section*{Grothendieck group}

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

\hypertarget{homological_algebra}{}\paragraph*{{Homological algebra}}\label{homological_algebra}

[[!include homological algebra - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{OfCommutativeMonoids}{Of commutative monoids}\dotfill \pageref*{OfCommutativeMonoids} \linebreak
\noindent\hyperlink{OfStableCategories}{Of stable $\infty$-categories}\dotfill \pageref*{OfStableCategories} \linebreak
\noindent\hyperlink{of_an_abelian_category}{of an abelian category}\dotfill \pageref*{of_an_abelian_category} \linebreak
\noindent\hyperlink{of_a_quillen_exact_category}{of a Quillen exact category}\dotfill \pageref*{of_a_quillen_exact_category} \linebreak
\noindent\hyperlink{of_a_waldhausen_category}{of a Waldhausen category}\dotfill \pageref*{of_a_waldhausen_category} \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}

The term \emph{Grothendieck group} has two closely related meanings:

\begin{itemize}%
\item \hyperlink{OfCommutativeMonoids}{Grothendieck group of commutative monoids}


\item \hyperlink{OfStableCategories}{Grothendieck group of stable infinity-categories}



\end{itemize}
In its restricted sense the Grothedieck group of a commutative monoid (i.e. of a commutative [[semi-group]] with [[unitality|unit]]) $A$ is a specific presentation of its [[group completion]], given as a certain [[group]] structure on a [[quotient]] of the [[Cartesian product]] $A \times A$. This is such that applied to the additive monoid of [[natural numbers]] $\mathbb{N}$ it produces the additive group of [[integers]] $\mathbb{Z}$ represented by pairs of natural numbers $(n_+, n_-)$ subject to an [[equivalence relation]] which identifies them with their [[difference]] $n_+ - n_-$.

A vaguely similar procedure applied to [[decategorification|isomorphism classes]] of certain [[Quillen exact category|Quillen exact categories]] happens to compute a group that is called the [[algebraic K-theory]] of these categories. See [[K-theory]] for some general abstract nonsense behind this.

Notably, the Grothendieck group completion of the [[decategorification]] of the category of [[topological vector bundles]] on some ([[compact Hausdorff space|compact Hausdorff]]) [[topological space]] $X$ produces the group known as the [[topological K-theory]] of $X$.

Given this one speaks more generally of the ([[algebraic K-theory|algebraic]]) [[K-theory]] group of a suitable category (one presenting a [[stable (∞,1)-category]] in some way) as its \emph{Grothendieck group} .

In that sense, the Grothendieck group of a ($\infty$-)category $C$ with a notion of [[fibration sequence|cofibration sequence]]s is the [[decategorification]] set $K(C)$ equipped with a notion of addition that is encoded in these homotopy exact sequences.

We now first state the definition of ``Grothendieck group completion'' -- which is really just the \emph{free group completion of an abelian [[monoid]]} -- and then the definition of Grothendieck group in the sense of [[algebraic K-theory]]. Notice that a priori both concepts are entirely independent constructions on different entities. But in various special case both can be applied to specific objects so as to produce the same result.

\hypertarget{OfCommutativeMonoids}{}\subsection*{{Of commutative monoids}}\label{OfCommutativeMonoids}

Every [[abelian group]] is in particular a [[commutative monoid]]. The [[forgetful functor]] $U \colon Ab \to CMon$ from the [[category]] [[Ab]] to the category [[CMon]] has a [[left adjoint]] $F$ (the corresponding [[free functor]]), the \emph{[[group completion]]} functor

\begin{displaymath}
Ab
    \underoverset
      {\underset{U}{\longrightarrow}}
      {\overset{F}{\longleftarrow}}
      {\bot}
  CMon
\end{displaymath}
The \emph{[[Grothendieck group construction on commutative monoids]]} (def. \ref{GrothendieckGroupViaQuotientOfCartesianProduct} below) is an explict presentation of this [[group completion]] functor.

For more details see at \emph{[[Grothendieck group of a commutative monoid]]}.

\begin{remark}
\label{}\hypertarget{}{}
The idea of the free group on an abelian monoid is a very simple algebraic idea that, at least for a [[cancellative monoid]] (so that the unit is monic and one can reasonably use the term `completion') certainly predates [[Grothendieck]]. That the [[integers]] $\mathbb{Z}$ is the group completion of the [[natural numbers]] $\mathbb{N}$ goes back at least to [[Kronecker]].

\end{remark}
\begin{defn}
\label{GrothendieckGroupViaQuotientOfCartesianProduct}\hypertarget{GrothendieckGroupViaQuotientOfCartesianProduct}{}
\textbf{([[Grothendieck group of a commutative monoid]])}

Let $(A,+)$ be a [[commutative monoid]] (i.e. a commutative [[semi-group]]).

On the [[Cartesian product]] of underlying sets $A \times A$ (the set of ordered [[pairs]] of elements in $A$), consider the [[equivalence relation]]

\begin{displaymath}
\big(
    (a_+, a_-) \sim_1 (b_+, b_-)
  \big) 
    \;\Leftrightarrow\; 
  \left(
    \underset{k \in A}{\exists}
    \left( 
      a_+ + b_- + k = b_+ + a_- + k
    \right)
  \right)
\end{displaymath}
or equivalently the equivalence relation

\begin{displaymath}
\big(
    (a_+, a_-) \sim_2 (b_+, b_-)
  \big) 
    \;\Leftrightarrow\; 
  \left(
    \underset{k_1, k_2 \in A}{\exists}
    \left(
      (a_+ + k_1, a_- + k_1)
      =
      (b_+ + k_2, b_- + k_2)
    \right)
  \right)
  \,.
\end{displaymath}
Write

\begin{displaymath}
G(A) \coloneqq (A \times A)/\sim
\end{displaymath}
for the set of [[equivalence classes]] under this equivalence relation. This inherits a binary operation

\begin{displaymath}
+ \;\colon\; G(A) \times G(A) \longrightarrow G(A)
\end{displaymath}
by applying the addition in $A$ on representatives:

\begin{displaymath}
[a_+,a_-] + [b_+,b_-] \coloneqq [ a_+ + b_+ , a_- + b_- ]
  \,.
\end{displaymath}
This defines the structure of an [[abelian group]]

\begin{displaymath}
(G(A),+)
\end{displaymath}
and this is the \emph{Grothendieck group} of $A$.

This comes with a canonical [[homomorphism]] of [[semigroups]]

\begin{displaymath}
\itexarray{
    A &\overset{\phantom{A} \eta_A \phantom{A} }{\longrightarrow}& G(A)
    \\
    a &\overset{\phantom{AAA}}{\mapsto}& [a,0]
  }
  \,.
\end{displaymath}
\end{defn}
\hypertarget{OfStableCategories}{}\subsection*{{Of stable $\infty$-categories}}\label{OfStableCategories}

In this sense, a Grothendieck group is fundamentally something assigned to a [[stable (∞,1)-category]]. We start with the naive [[decategorification]] of $C$, i.e. the set of [[equivalence]] classes of objects, which inherits the structure of an abelian monoid. Then in addition to group-completing it as above, we add additional [[generators and relations|relations]] by the rule that for every [[fibration sequence]]

\begin{displaymath}
A \to X \to B
\end{displaymath}
in $C$, the equivalence classes $[A]$, $[B]$ and $[X]$ must satisfy

\begin{displaymath}
[X] = [A] + [B]
  \,.
\end{displaymath}
The result is also called the [[K-theory]] of $C$.

In particular, the additive inverse $-[A]$ of an element $[A]$ is the class of its [[loop space object]] $\Omega A$, or equivalently of its [[delooping]] $\mathbf{B} A$ called the \emph{suspension} $\Sigma A$, since by definition the sequences

\begin{displaymath}
\Omega A \to 0 \to A
\end{displaymath}
and

\begin{displaymath}
A \to 0 \to \Sigma A
\end{displaymath}
are [[fibration sequence]], so that

\begin{displaymath}
[A] + [\Omega A] = 0
\end{displaymath}
and

\begin{displaymath}
[A] + [\Sigma A] = 0
 \,.
\end{displaymath}
But there are many ways to \emph{model} a [[stable (∞,1)-category]] by an ordinary category. Essentially for each of these ways there is a seperate prescription for how to model the above general construction in terms of concrete 1-categorical constructions.

In particular from an

\begin{itemize}%
\item [[abelian category]]

\end{itemize}
and a

\begin{itemize}%
\item [[Quillen exact category]]

\end{itemize}
one obtains the corresponding [[category of chain complexes|categories of chain complexes]]. These are [[stable (infinity,1)-category|stable (∞,1)-categories]]. Below we list presciptions for how to compute the Grothendieck/K-theory groups of these in terms of the underlying 1-categories.

Apart from the case of abelian categories, this requires some handle on the fibration sequences. A tool developed to handle exactly this for the purpose of computing Grothendieck/K-theory groups is the notion of a [[Waldhausen category]]. That provides the sufficient extra information to get a hand on the homotopy exact sequences.

\hypertarget{of_an_abelian_category}{}\subsubsection*{{of an abelian category}}\label{of_an_abelian_category}

Let $C$ be an [[abelian category]]. The \textbf{Grothendieck group} or [[algebraic K-theory]] group of $C$, denoted $K(C)$, is the [[abelian group]] generated by [[decategorification|isomorphism class]]es of objects of $C$, with relations of the form

\begin{displaymath}
[X] = [A] + [B]
\end{displaymath}
whenever there is a [[short exact sequence]]

\begin{displaymath}
0 \to A \to X \to B \to 0
\end{displaymath}
\hypertarget{of_a_quillen_exact_category}{}\subsubsection*{{of a Quillen exact category}}\label{of_a_quillen_exact_category}

An exact category $C$ in the present sense is a full [[subcategory]] of an [[abelian category]] $\hat C$ such that the collection of all sequences $0 \to A \to X \to B \to 0$ in $C$ that are [[exact sequence]]s in $\hat C$ has the property that for every [[exact sequence]] $A \to X \to B$ in $\hat C$ with $A$ and $B \in C$ also their ``sum'' $X$ is in $C$.

Given an exact category $C$ with the inherited notion of exact sequences this way, the definition of its Grothendieck group is as above.

\hypertarget{of_a_waldhausen_category}{}\subsubsection*{{of a Waldhausen category}}\label{of_a_waldhausen_category}

A [[Waldhausen category]] is a [[category with weak equivalences]] with an [[initial object]] -- called $0$ -- and equipped with the notion of auxiliary morphism called \emph{cofibrations}. These satisfy some axioms which are such that the ordinary 1-categorical [[cokernel]] of a cofibration $A \hookrightarrow X$, i.e. the ordinary [[pushout]]

\begin{displaymath}
\itexarray{
    A &\hookrightarrow& X
    \\
    \downarrow && \downarrow
    \\
    0 &\to &  B 
  }
\end{displaymath}
computes the desired [[homotopy colimit|homotopy]] [[pushout]]. (This is exactly dual to the reasoning by which one computes [[homotopy pullback]]s in a [[category of fibrant objects]]. See there for details.)

Therefore in a Waldhausen category a [[fibration sequence|cofibration sequence]] is a [[pushout]] sequence

\begin{displaymath}
A \hookrightarrow X \to B
\end{displaymath}
where the first morphism is a cofibration.

The Grothendieck/[[K-theory]]-group of the [[Waldhausen category]] $C$ is then, as before, on the [[decategorification]] $K(C)$ the abelian group structure given by

\begin{displaymath}
[X] = [A] + [B]
\end{displaymath}
for all cofibration sequences as above.

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

\begin{itemize}%
\item The Grothendieck group of the [[Quillen exact category]] of [[vector bundle]]s on a space $X$ is called the [[topological K-theory]] of $X$.

Notice that vector bundles do not form an [[abelian category]].


\item The Grothendieck group of the category of finite-dimensional complex-linear [[representations]] of a [[group]] is called its [[representation ring]].



\end{itemize}
These two examples illustrate a general fact: the Grothendieck group of a [[monoidal category|monoidal]] [[abelian category]] inherits a ring structure from the tensor product in this category, and thus becomes a ring, called the [[Grothendieck ring]]. See also the general discussion at [[decategorification]].

\begin{itemize}%
\item Every [[Quillen exact category]] $C$ is canonically equipped with the structure of a [[Waldhausen category]]. The two different prescriptions for forming the Grothendieck group $K(C)$ of $C$ do coincide.

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

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


\item [[K-theory]]



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

\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})

section 2: \emph{The Grothendieck group $K_0$} (\href{http://www.math.rutgers.edu/~weibel/Kbook/Kbook.II.pdf}{pdf})



\end{itemize}
See also

\begin{itemize}%
\item \href{http://online.itp.ucsb.edu/online/ktheory/wu/}{The Grothendieck Construction} (UCSB ITP Seminar)


\item Wikipedia, \emph{\href{https://en.wikipedia.org/wiki/Grothendieck_group}{Grothendieck group}}



\end{itemize}
[[!redirects Grothendieck groups]]



\end{document}