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\begin{document}

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\section*{biproduct}

\begin{quote}%
This entry is about [[coproducts]] coinciding with [[products]]. For the notion of biproduct in the sense of [[bicategory]] theory see at [[2-limit]]. See at \emph{[[bilimit]]} for general disambiguation.


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

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

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

\hypertarget{category_theory}{}\paragraph*{{Category theory}}\label{category_theory}

[[!include category theory - contents]]

\hypertarget{limits_and_colimits}{}\paragraph*{{Limits and colimits}}\label{limits_and_colimits}

[[!include infinity-limits - 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{SemiadditiveCategories}{Semiadditive categories}\dotfill \pageref*{SemiadditiveCategories} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{SemiadditivityAsStructureProperty}{Semiadditivity as structure/property}\dotfill \pageref*{SemiadditivityAsStructureProperty} \linebreak
\noindent\hyperlink{BiproductsImplyEnrichment}{Biproducts imply enrichment -- Relation to additive categories}\dotfill \pageref*{BiproductsImplyEnrichment} \linebreak
\noindent\hyperlink{biproducts_as_enriched_cauchy_colimits}{Biproducts as enriched Cauchy colimits}\dotfill \pageref*{biproducts_as_enriched_cauchy_colimits} \linebreak
\noindent\hyperlink{biproducts_from_duals}{Biproducts from duals}\dotfill \pageref*{biproducts_from_duals} \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}

A \emph{biproduct} in a [[category]] $\mathcal{C}$ is an operation that is both a [[product]] and a [[coproduct]], in a compatible way. Morphisms between finite biproducts are encoded in a [[matrix calculus]].

Finite biproducts are best known from [[additive category|additive categories]]. A category which has biproducts but is not necessarily [[enriched category|enriched]] in [[Ab]], hence not necessarily [[additive category|additive]], is called a \emph{semiadditive category}.

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

Let $\mathcal{C}$ be a [[category]] with [[zero morphisms]]; that is, $C$ is [[enriched category|enriched]] over [[pointed sets]] (for example, $C$ might have a [[zero object]]). For $c_1, c_2$ two objects in $C$, suppose a [[product]] $c_1 \times c_2$ and a [[coproduct]] $c_1 \sqcup c_2$ both exist.

\begin{defn}
\label{TheCanonicalComparisonMorphism}\hypertarget{TheCanonicalComparisonMorphism}{}
Write

\begin{displaymath}
r_{c_1,c_2} : c_1 \sqcup c_2 \to c_1 \times c_2
\end{displaymath}
for the [[morphism]] which is uniquely defined (via the [[universal property]] of [[coproduct]] and [[product]]) by the condition that

\begin{displaymath}
\left(
    c_i \to c_1 \sqcup c_2 \stackrel{r}{\to} c_1 \times c_2 \to c_j
  \right)
  =
  \left\{
    \itexarray{
      Id_{c_i} & if \; i = j
      \\
      0_{i,j} & if \; i \neq j
    }
  \right.
  \,
\end{displaymath}
where the last and first morphisms are the [[projections]] and [[co-projections]], respectively, and where $0_{i,j}$ is the [[zero morphism]] from $c_i$ to $c_j$. Thus $r_{c_1, c_2} = (Id_{c_1}, 0_{1,2}) \coprod (0_{2,1}, Id_{c_2})$, where $(f, g): d \to a \times b$ denotes the map induced by $f : d \to a$ and $g : d \to b$.

\end{defn}
\begin{defn}
\label{Biproduct}\hypertarget{Biproduct}{}
If the morphism $r_{c_1,c_2}$ in def. \ref{TheCanonicalComparisonMorphism}, is an [[isomorphism]], then the isomorphic objects $c_1 \times c_2$ and $c_1 \sqcup c_2$ are called [[generalized the|the]] \textbf{biproduct} of $c_1$ and $c_2$. This object is often denoted $c_1 \oplus c_2$, alluding to the [[direct sum]] (which is often an example).

If $r_{c_1,c_2}$ is an isomorphism for all objects $c_1, c_2 \in \mathcal{C}$ and hence a [[natural isomorphism]]

\begin{displaymath}
r \;\colon\; (-)\coprod (-) \stackrel{\simeq}{\longrightarrow} (-) \times (-)
\end{displaymath}
then $\mathcal{C}$ is called a \hyperlink{SemiadditiveCategories}{semiadditive category}.

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
Definition \ref{Biproduct} has a straightforward generalization to biproducts of any number of objects (although this requires extra structure on the category in [[constructive mathematics]] if the set indexing these objects might not have [[decidable equality]]).

A [[zero object]] is the biproduct of no objects.

\end{remark}
\hypertarget{SemiadditiveCategories}{}\subsection*{{Semiadditive categories}}\label{SemiadditiveCategories}

A category $C$ with all finite biproducts is called a \textbf{semiadditive category}. More precisely, this means that $C$ has all finite products and coproducts, that the unique map $0\to 1$ is an isomorphism (hence $C$ has a zero object), and that the canonical maps $c_1 \sqcup c_2 \to c_1 \times c_2$ defined above are isomorphisms.

Amusingly, for $C$ to be semiadditive, it actually suffices to assume that $C$ has finite products and coproducts and that there exists \emph{any} [[natural transformation|natural]] family of isomorphisms $c_1 \sqcup c_2 \cong c_1 \times c_2$ --- not necessarily the canonical maps constructed above. A proof can be found in (\hyperlink{Lack09}{Lack 09}).

An [[additive category]], although normally defined through the theory of [[enriched categories]], may also be understood as a semiadditive category with an [[extra property]], as explained below at \emph{\hyperlink{BiproductsImplyEnrichment}{Properties -- Biproducts imply enrichment}}.

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

\hypertarget{SemiadditivityAsStructureProperty}{}\subsubsection*{{Semiadditivity as structure/property}}\label{SemiadditivityAsStructureProperty}

Given a category $\mathcal{C}$ with [[zero morphisms]], one may imagine equipping it with the [[structure]] of a chosen [[natural isomorphism]]

\begin{displaymath}
\psi_{(-),(-)} : (-)\coprod (-) \stackrel{\simeq}{\longrightarrow} (-)\times(-)
  \,.
\end{displaymath}
\begin{prop}
\label{}\hypertarget{}{}
(\hyperlink{Lack09}{Lack 09, proof of theorem 5}). If a [[category]] $\mathcal{C}$ with finite [[coproducts]] and [[products]] carries any [[natural isomorphism]] $\psi_{(-),(-)}$ from [[coproducts]] to [[products]], then

\begin{displaymath}
\array {
  c_1\coprod c_2 & \overset{\psi_{c_1, 0} + \psi_{0, c_2}}\rightarrow & c_1 \coprod c_2 \\
    & \searrow^{r_{c_1, c_2}}           & \downarrow^{\psi_{c_1, c_2}} \\
    &                        & c_1 \times c_2
}
\end{displaymath}
commutes for any two object $c_1$ and $c_2$.

\end{prop}
Hence $r_{c_1, c_2}$ is an isomorphism so that $\mathcal{C}$ is semi-additive. See [[non-canonical isomorphism]] for more.

\hypertarget{BiproductsImplyEnrichment}{}\subsubsection*{{Biproducts imply enrichment -- Relation to additive categories}}\label{BiproductsImplyEnrichment}

A semiadditive category is automatically [[enriched category|enriched]] over the [[monoidal category]] of [[commutative monoids]] with the usual [[tensor product]], as follows.

Given two morphisms $f, g: a \to b$ in $C$, let their sum $f + g: a \to b$ be

\begin{displaymath}
a \to a \times a \cong a \oplus a \overset{f \oplus g}{\to} b \oplus b \cong b \sqcup b \to b .
\end{displaymath}
One proves that $+$ is associative and commutative. Of course, the zero morphism $0: a \to b$ is the usual [[zero morphism]] given by the zero object:

\begin{displaymath}
a \to 1 \cong 0 \to b .
\end{displaymath}
One proves that $0$ is the neutral element for $+$ and that this matches the $0$ morphism that we began with in the definition. Note that in addition to a zero object, this construction actually only requires biproducts of an object with itself, i.e. biproducts of the form $a\oplus a$ rather than the more general $a\oplus b$.

If additionally every morphism $f: a \to b$ has an inverse $-f: a \to b$, then $C$ is enriched over the category $Ab$ of [[abelian groups]] and is therefore (precisely) an \textbf{[[additive category]]}.

If, on the other hand, the addition of morphisms is idempotent ($f+f=f$), then $C$ is enriched over the category $SLat$ of [[semilattices]] (and is therefore a kind of [[2-poset]]).

\hypertarget{biproducts_as_enriched_cauchy_colimits}{}\subsubsection*{{Biproducts as enriched Cauchy colimits}}\label{biproducts_as_enriched_cauchy_colimits}

Conversely, if $C$ is already known to be enriched over abelian monoids, then a binary biproduct may be defined purely diagrammatically as an object $c_1\oplus c_2$ together with injections $n_i:c_i\to c_1\oplus c_2$ and projections $p_i:c_1\oplus c_2 \to c_i$ such that $p_j n_i = \delta_{i j}$ (the [[Kronecker delta]]) and $n_1 p_1 + n_2 p_2 = 1_{c_1\oplus c_2}$. It is easy to check that makes $c_1\oplus c_2$ a biproduct, and that any binary biproduct must be of this form. Similarly, an object $z$ of such a category is a zero object precisely when $1_z= 0_z$, its identity is equal to the zero morphism. It follows that functors enriched over abelian monoids must automatically preserve finite biproducts, so that finite biproducts are a type of [[Cauchy colimit]]. Moreover, any product \emph{or} coproduct in a category enriched over abelian monoids is actually a biproduct.

For categories enriched over [[suplattices]], this extends to all small biproducts, with the condition $n_1 p_1 + n_2 p_2 = 1_{c_1\oplus c_2}$ replaced by $\bigvee_{i} n_i p_i = 1_{\bigoplus_i c_i}$. In particular, the category of suplattices has all small biproducts.

\hypertarget{biproducts_from_duals}{}\subsubsection*{{Biproducts from duals}}\label{biproducts_from_duals}

The existence of [[dual objects]] tends to imply (semi)additivity; see (\hyperlink{Houston06}{Houston 08}, \href{http://mathoverflow.net/questions/14402/semiadditivity-and-dualizability-of-2/21307}{MO discussion}).

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

Categories with biproducts include:

\begin{itemize}%
\item [[K(n)-local stable homotopy theory]] (\hyperlink{HopkinsLurie14}{Hopkins-Lurie 14})

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

\begin{itemize}%
\item [[bilimit]]


\item [[semiadditive (∞,1)-category]]



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

\begin{itemize}%
\item [[Stephen Lack]], \emph{Non-canonical isomorphisms}, (\href{http://arxiv.org/abs/0912.2126}{arXiv:0912.2126}).


\item [[Robin Houston]], \emph{Finite Products are Biproducts in a Compact Closed Category}, Journal of Pure and Applied Algebra, Volume 212, Issue 2, February 2008, Pages 394-400 (\href{http://arxiv.org/abs/math/0604542}{arXiv:math/0604542})


\item [[Michael Hopkins]], [[Jacob Lurie]], \emph{[[Ambidexterity in K(n)-Local Stable Homotopy Theory]]} (2014)



\end{itemize}
A related discussion is archived at $n$\href{http://nforum.mathforge.org/discussion/4966/zero-morphism-and-additive-categories/?Focus=39983#Comment_39983}{Forum}.

[[!redirects biproducts]] [[!redirects semiadditive category]] [[!redirects semiadditive categories]] [[!redirects semi-additive category]] [[!redirects semi-additive categories]]



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