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\begin{document}

%-------------------------------------------------------------------


\section*{additive category}

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

\hypertarget{enriched_category_theory}{}\paragraph*{{Enriched category theory}}\label{enriched_category_theory}

[[!include enriched category theory contents]]

\hypertarget{additive_and_abelian_categories}{}\paragraph*{{Additive and abelian categories}}\label{additive_and_abelian_categories}

[[!include additive and abelian categories - contents]]

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

[[!include homological algebra - contents]]

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

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

\begin{defn}
\label{AdditiveCategory}\hypertarget{AdditiveCategory}{}
An \textbf{additive category} is a [[category]] which is

\begin{enumerate}%
\item an [[Ab-enriched category]];

(sometimes called a [[pre-additive category]]--this means that each [[hom-set]] carries the structure of an [[abelian group]] and composition is [[bilinear map|bilinear]])


\item which admits [[finite limit|finite]] [[coproducts]]

(and hence, by prop. \ref{ProductsAreBiproducts} below, finite [[products]] which coincide with the coproducts, hence finite [[biproducts]]).



\end{enumerate}
The natural [[morphisms]] \emph{between} additive categories are [[additive functors]].

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
A [[pre-abelian category]] is an additive category which also has [[kernels]] and [[cokernels]]. Equivalently, it is an Ab-enriched category with all [[finite limits]] and finite [[colimits]]. An especially important sort of additive category is an [[abelian category]], which is a pre-abelian one satisfying the extra exactness property that all [[monomorphisms]] are [[kernels]] and all [[epimorphisms]] are [[cokernels]]. See at \emph{[[additive and abelian categories]]} for more.

\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
The [[Ab]]-enrichment of an additive category does not have to be given a priori. Every [[semiadditive category]] (a category with finite [[biproducts]]) is automatically [[enriched category|enriched]] over [[commutative monoids]] (as described at \emph{[[biproduct]]}), so an additive category may be defined as a category with finite biproducts whose [[hom-object|hom-monoids]] happen to be [[groups]]. (The requirement that the hom-monoids be groups can even be stated in elementary terms without discussing enrichment at all, but to do so is not very enlightening.) Note that the entire $Ab$-enriched structure follows automatically for [[abelian category|abelian categories]].

\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
Some authors use \textbf{additive category} to simply mean an Ab-enriched category, with no further assumptions. It can also be used to mean a $CMon$-enriched (commutative monoid enriched) category, with or without assumptions of products.

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

\begin{prop}
\label{ProductsAreBiproducts}\hypertarget{ProductsAreBiproducts}{}
In an [[Ab-enriched category]] (or even just a $CMon$-enriched category), a [[finite product|finite]] [[product]] is also a [[coproduct]], and dually.

This statement includes the zero-ary case: any [[terminal object]] is also an [[initial object]], hence a [[zero object]] (and dually), hence every additive category has a [[zero object]].

More precisely, for $\{X_i\}_{i \in I}$ a [[finite set]] of objects in an Ab-enriched category, the unique morphism

\begin{displaymath}
\underset{i \in I}{\coprod} X_i \longrightarrow \underset{j \in I}{\prod} X_j
\end{displaymath}
whose components are identities for $i = j$ and are [[zero morphism|zero]] otherwise is an [[isomorphism]].

\end{prop}
\begin{proof}
Consider first the nullary (i.e., zero-ary) case. Given a [[terminal object]] $\ast$, the unique morphism $id_\ast: \ast \to \ast$ is the zero morphism $0$ in its hom-object. For any object $A$, the zero morphism $0_A: \ast \to A$ must equal any morphism $f: \ast \to A$ on account of $f = f id_\ast = f 0 = 0_A$ where the last equation is by $CMon$-enrichment. Hence $\ast$ is initial. (N.B.: this argument applies more generally to categories [[enriched category|enriched]] in [[pointed sets]], and is self-[[formal duality|dual]].)

Consider now the case of binary (co-)products. Using [[zero morphisms]], in addition to its canonical [[projection]] maps $p_i \colon X_1 \times X_2 \to X_i$, any binary [[product]] also admits ``injection'' maps $X_i \to X_1 \times X_2$, and dually for the [[coproduct]]:

\begin{displaymath}
\itexarray{  
    X_1 && && X_2
    \\
    & \searrow^{\mathrlap{(id,0)}} && {}^{\mathllap{(0,id)}}\swarrow
    \\
    {}^{\mathllap{id_{X_1}}}\downarrow && X_1 \times X_2 && \downarrow^{\mathrlap{id_{X_2}}}
    \\
    & \swarrow_{\mathrlap{p_{X_1}}} && {}_{\mathllap{p_{X_2}}}\searrow
    \\
    X_1 && && X_2
  }
  \;\;\;\;\;\;\;\;\;\;\;\;\,,\;\;\;\;\;\;\;\;\;\;\;\;
  \itexarray{  
    X_1 && && X_2
    \\
    & \searrow^{\mathrlap{i_{X_1}}} && {}^{\mathllap{i_{X_2}}}\swarrow
    \\
    {}^{\mathllap{id_{X_1}}}\downarrow 
     && X_1 \sqcup X_2 && \downarrow^{\mathrlap{id_{X_2}}}
    \\
    & \swarrow_{\mathrlap{(id,0)}} && {}_{\mathllap{(0,id)}}\searrow
    \\
    X_1 && && X_2
  }
  \,.
\end{displaymath}
Observe some basic compatibility of the $Ab$-enrichment with the product:

First, for $(\alpha_1,\beta_1), (\alpha_2, \beta_2)\colon R \to X_1 \times X_2$ then

\begin{displaymath}
(\star)
  \;\;\;\;\;\;
  (\alpha_1,\beta_1) + (\alpha_2, \beta_2) 
    = 
  (\alpha_1+ \alpha_2 , \; \beta_1 + \beta_2)
\end{displaymath}
(using that the projections $p_1$ and $p_2$ are linear and by the universal property of the product).

Second, $(id,0) \circ p_1$ and $(0,id) \circ p_2$ are two projections on $X_1\times X_2$ whose sum is the identity:

\begin{displaymath}
(\star\star)
  \;\;\;\;\;\;
  (id, 0) \circ p_1
  +
  (0, id) \circ p_2
    =
  id_{X_1 \times X_2} 
  \,.
\end{displaymath}
(We may check this, via the [[Yoneda lemma]] on [[generalized elements]]: for $(\alpha, \beta) \colon R \to X_1\times X_2$ any morphism, then $(id,0)\circ p_1 \circ (\alpha,\beta) = (\alpha,0)$ and $(0,id)\circ p_2\circ (\alpha,\beta) = (0,\beta)$, so the statement follows with equation $(\star)$.)

Now observe that for $f_i \;\colon\; X_i \to Q$ any two morphisms, the sum

\begin{displaymath}
\phi 
    \;\coloneqq\; 
  f_1 \circ p_1 + f_2 \circ p_2
    \;\colon\;
  X_1 \times X_2
    \longrightarrow
  Q
\end{displaymath}
gives a morphism of [[cocones]]

\begin{displaymath}
\itexarray{  
    X_1 && && X_2
    \\
    & \searrow^{\mathrlap{(id,0)}} && {}^{\mathllap{(0,id)}}\swarrow
    \\
    {}^{\mathllap{id_{X_1}}}\downarrow && X_1 \times X_2 && \downarrow^{\mathrlap{id_{X_2}}}
    \\
    &  && 
    \\
    X_1 && \downarrow^{\mathrlap{\phi}} && X_2
    \\
    & {}_{\mathllap{f_1}}\searrow && \swarrow_{\mathrlap{f_2}}
    \\
    && Q
  }
  \,.
\end{displaymath}
Moreover, this is unique: suppose $\phi'$ is another morphism filling this diagram, then, by using equation $(\star \star)$, we get

\begin{displaymath}
\begin{aligned}
    (\phi-\phi')
    & = 
    (\phi - \phi') \circ id_{X_1 \times X_2}
    \\
    &= 
    (\phi - \phi') \circ ( (id_{X_1},0) \circ p_1 + (0,id_{X_2})\circ p_2 )
    \\
    & =
    \underset{ = 0}{\underbrace{(\phi - \phi') \circ (id_{X_1}, 0)}} \circ p_1
    + 
    \underset{ = 0}{\underbrace{(\phi - \phi') \circ (0, id_{X_2})}} \circ p_2
    \\
    & = 0
  \end{aligned}
\end{displaymath}
and hence $\phi = \phi'$. This means that $X_1\times X_2$ satisfies the [[universal property]] of a [[coproduct]].

By a [[formal dual|dual]] argument, the binary coproduct $X_1 \sqcup X_2$ is seen to also satisfy the universal property of the binary product. By [[induction]], this implies the statement for all finite (co-)products. (If a particular finite (co-)product exists but binary ones do not, one can adapt the above argument directly to that case.)

\end{proof}
\begin{remark}
\label{}\hypertarget{}{}
Such products which are also coproducts as in prop. \ref{ProductsAreBiproducts} are sometimes called \emph{[[biproducts]]} or \emph{[[direct sums]]}; they are [[absolute limit|absolute limits]] for [[Ab]]-[[enriched category|enrichment]].

\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
The coincidence of products with biproducts in prop. \ref{ProductsAreBiproducts} does \emph{not} extend to infinite products and coproducts.) In fact, an [[Ab-enriched category]] is [[Cauchy complete category|Cauchy complete]] just when it is additive and moreover its [[idempotents]] split.

\end{remark}
Conversely:

\begin{defn}
\label{SemiadditiveCategory}\hypertarget{SemiadditiveCategory}{}
A \textbf{[[semiadditive category]]} is a [[category]] that has all [[finite products]] which, moreover, are [[biproducts]] in that they coincide with finite [[coproducts]] as in def. \ref{ProductsAreBiproducts}.

\end{defn}
\begin{prop}
\label{SemiAdditivityInducesAbelianMonoidEnrichment}\hypertarget{SemiAdditivityInducesAbelianMonoidEnrichment}{}
In a [[semiadditive category]], def. \ref{SemiadditiveCategory}, the [[hom-sets]] acquire the structure of [[commutative monoids]] by defining the sum of two morphisms $f,g \;\colon\; X \longrightarrow Y$ to be

\begin{displaymath}
f + g 
    \;\coloneqq\;
  X \overset{\Delta_X}{\to} X \times X
  \simeq
  X \oplus X
  \overset{f \oplus g}{\longrightarrow}
  Y \oplus Y
  \simeq
  Y \sqcup Y
  \overset{\nabla_X}{\to}
  Y
  \,.
\end{displaymath}
With respect to this operation, [[composition]] is [[bilinear map|bilinear]].

\end{prop}
\begin{proof}
The [[associativity]] and commutativity of $+$ follows directly from the corresponding properties of $\oplus$. Bilinearity of composition follows from [[natural transformation|naturality]] of the [[diagonal]] $\Delta_X$ and [[codiagonal]] $\nabla_X$:

\begin{displaymath}
\itexarray{
    W 
      &\overset{\Delta_W}{\longrightarrow}& 
    W \times W
      &\overset{\simeq}{\longrightarrow}&
    W \oplus W
    \\
    \downarrow^{\mathrlap{e}}
      &&
    \downarrow^{\mathrlap{e \times e}} 
      && 
    \downarrow^{\mathrlap{e \oplus e}}
    \\
    X 
      &\overset{\Delta_X}{\to}& 
    X \times X
      &\simeq&
    X \oplus X
      &\overset{f \oplus g}{\longrightarrow}&
    Y \oplus Y
      &\simeq&
    Y \sqcup Y
      &\overset{\nabla_X}{\to}&
    Y
    \\
      &&
      &&
      &&
    \downarrow^{\mathrlap{h \oplus h}}
      &&
    \downarrow^{\mathrlap{h \sqcup h}}
      &&
    \downarrow^{\mathrlap{h}}
    \\
      && 
      &&
      &&
    Z \oplus Z
      &\simeq&
    Z \sqcup Z
      &\overset{\nabla_Z}{\to}&
    Z
  }
\end{displaymath}
\end{proof}
\begin{prop}
\label{SemiaddtiveStructureUnderlyingAdditiveInducesOriginalEnrichment}\hypertarget{SemiaddtiveStructureUnderlyingAdditiveInducesOriginalEnrichment}{}
Given an additive category according to def. \ref{AdditiveCategory}, then the enrichement in [[commutative monoids]] which is induced on it via prop. \ref{ProductsAreBiproducts} and prop. \ref{SemiAdditivityInducesAbelianMonoidEnrichment} from its underlying [[semiadditive category]] structure coincides with the original enrichment.

\end{prop}
\begin{proof}
By the proof of prop. \ref{ProductsAreBiproducts}, the [[codiagonal]] on any object in an additive category is the sum of the two projections:

\begin{displaymath}
\nabla_X \;\colon\; X \oplus X \overset{p_1 + p_2}{\longrightarrow} X
  \,.
\end{displaymath}
Therefore (checking on [[generalized elements]], as in the proof of prop. \ref{ProductsAreBiproducts}) for all morphisms $f,g \colon X \to Y$ we have [[commuting squares]] of the form

\begin{displaymath}
\itexarray{
    X &\overset{f+g}{\longrightarrow}& Y
    \\
    {}^{\mathllap{\Delta_X}}\downarrow && \uparrow^{\mathrlap{\nabla_Y =}}_{\mathrlap{p_1 + p_2}}
    \\
    X \oplus X 
      &\underset{f \oplus g}{\longrightarrow}& 
   Y\oplus Y
  }
  \,.
\end{displaymath}
\end{proof}
\begin{remark}
\label{}\hypertarget{}{}
Prop. \ref{SemiaddtiveStructureUnderlyingAdditiveInducesOriginalEnrichment} says that being an [[additive category]] is an extra [[property]] on a category, not extra [[structure]]. We may ask whether a given category is additive or not, without specifying with respect to which abelian group structure on the hom-sets.

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

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


\item [[abelian category]]


\item [[additive (∞,1)-category]]


\item [[triangulated category]]



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

Discussion of [[model category]] structures on additive categories is around def. 4.3 of

\begin{itemize}%
\item Apostolos Beligiannis, \emph{Homotopy theory of modules and Gorenstein rings}, Math. Scand. 89 (2001) (\href{http://users.uoi.gr/abeligia/mathscand.pdf}{pdf})

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



\end{document}