| Version 2 |
Version 1 |
| Let $\mathcal{C}$ be a category. Then $\mathcal{C}$ is an \emph{additive category} if |
Let $\mathcal{C}$ be a category. Then $\mathcal{C}$ is an \emph{additive category} if |
| \begin{enumerate} |
\begin{enumerate} |
| \item $\mathcal{C}$ is a preadditive category, and |
\item $\mathcal{C}$ is a preadditive category, and |
| \item for every pair of objects $A,B$ in $\mathcal{C}$, their \PMlinkname{product}{CategoricalDirectProduct} exists. |
\item for every pair of objects $A,B$ in $\mathcal{C}$, their \PMlinkname{product}{CategoricalDirectProduct} exists. |
| \end{enumerate} |
\end{enumerate} |
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| \textbf{Proposition}. In a preadditive category, coproduct of two objects exists iff their product exists. Furthermore, they are isomorphic. |
\textbf{Proposition}. In a preadditive category, coproduct of two objects exists iff their product exists. Furthermore, they are isomorphic. |
| \begin{proof}We shall prove the fact if the product $D$ of objects $A$ and $B$ exists, then $D$ is also their coproduct. The other direction is dual. |
\begin{proof}We shall prove the fact if the product $D$ of objects $A$ and $B$ exists, then $D$ is also their coproduct. The other direction is dual. |
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| Suppose $D$ is the product of $A$ and $B$, with morphisms |
Suppose $D$ is the product of $A$ and $B$, with morphisms |
| $$\xymatrix@1{D\ar[r]^{\pi_A}&A}\qquad\mbox{ and }\qquad\xymatrix@1{D\ar[r]^{\pi_B}&B}.$$ |
$$\xymatrix@1{D\ar[r]^{\pi_A}&A}\qquad\mbox{ and }\qquad\xymatrix@1{D\ar[r]^{\pi_B}&B}.$$ |
| From these two morphisms, we construct two commutative diagrams |
From these two morphisms, we construct two commutative diagrams |
| $$\xymatrix{&A \\ A \ar[ur]^1 \ar[dr]_0 \ar@{-->}[r]^{\alpha} & D |
$$\xymatrix{&A \\ A \ar[ur]^1 \ar[dr]_0 \ar@{-->}[r]^{\alpha} & D |
| \ar[u]_{\pi_A} \ar[d]^{\pi_B}\\ &B} |
\ar[u]_{\pi_A} \ar[d]^{\pi_B}\\ &B} |
| \qquad\mbox{ and }\qquad |
\qquad\mbox{ and }\qquad |
| \xymatrix{&A \\ B \ar[ur]^0 \ar[dr]_1 \ar@{-->}[r]^{\beta} & D |
\xymatrix{&A \\ B \ar[ur]^0 \ar[dr]_1 \ar@{-->}[r]^{\beta} & D |
| \ar[u]_{\pi_A} \ar[d]^{\pi_B}\\ &B}$$ |
\ar[u]_{\pi_A} \ar[d]^{\pi_B}\\ &B}$$ |
| where $0$ and $1$ are zero morphisms and identity morphisms on $A$ and $B$, and $\alpha$ and $\beta$ are morphisms based on the definition of the product $D$. |
where $0$ and $1$ are zero morphisms and identity morphisms on $A$ and $B$, and $\alpha$ and $\beta$ are morphisms based on the definition of the product $D$. |
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| Then it's not hard to see that $D$ is a coproduct of $A$ and $B$ with morphisms $\alpha$ and $\beta$, for if $r:A\rightarrow C$ and $s:B\rightarrow C$ are two morphisms into an object $C$, we can form two morphisms $r\pi_A$ and $s\pi_B$, both from $D$ to $C$. Since $\operatorname{hom}(D,C)$ is an abelian group, these two can then be added to form $f:=r\pi_A+s\pi_B$. Then $f\alpha=(r\pi_A+s\pi_B)\alpha=r$, and similarly $f\beta=s$. This shows that $D$ is also the coproduct of $A$ and $B$ with morphisms $\alpha$ and $\beta$. |
Then it's not hard to see that $D$ is a coproduct of $A$ and $B$ with morphisms $\alpha$ and $\beta$, for if $r:A\rightarrow C$ and $s:B\rightarrow C$ are two morphisms into an object $C$, we can form two morphisms $r\pi_A$ and $s\pi_B$, both from $D$ to $C$. Since $\operatorname{hom}(D,C)$ is an abelian group, these two can then be added to form $f:=r\pi_A+s\pi_B$. Then $f\alpha=(r\pi_A+s\pi_B)\alpha=r$, and similarly $f\beta=s$. This shows that $D$ is also the coproduct of $A$ and $B$ with morphisms $\alpha$ and $\beta$. |
| \end{proof} |
\end{proof} |
| An easy way to remember the relationships among the various morphisms in the above proof are the following two matrix products: |
An easy way to remember the relationships among the various morphisms in the above proof are the following two matrix products: |
| \begin{center} |
\begin{center} |
| $ |
$ |
| \begin{pmatrix} |
\begin{pmatrix} |
| \pi_A \\ |
\pi_A \\ |
| \pi_B |
\pi_B |
| \end{pmatrix} |
\end{pmatrix} |
| \begin{pmatrix} |
\begin{pmatrix} |
| \alpha & \beta |
\alpha & \beta |
| \end{pmatrix} |
\end{pmatrix} |
| = |
= |
| \begin{pmatrix} |
\begin{pmatrix} |
| 1 & 0 \\ |
1 & 0 \\ |
| 0 & 1 |
0 & 1 |
| \end{pmatrix} |
\end{pmatrix} |
| \qquad |
\qquad |
| \mbox{ and } |
\mbox{ and } |
| \qquad |
\qquad |
| \begin{pmatrix} |
\begin{pmatrix} |
| r & s |
r & s |
| \end{pmatrix} |
\end{pmatrix} |
| \begin{pmatrix} |
\begin{pmatrix} |
| \pi_A \\ |
\pi_A \\ |
| \pi_B |
\pi_B |
| \end{pmatrix} |
\end{pmatrix} |
| =f$. |
=f$. |
| \end{center} |
\end{center} |
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| As a result of the above proposition, in an additive category, finite products and finite coproducts are synonymous. Given objects $A,B$, we denote $A\oplus B$ to be their product. We also call it the \emph{direct sum} of $A$ and $B$. |
As a result of the above proposition, in an additive category, finite products and finite coproducts are synonymous. Given objects $A,B$, we denote $A\oplus B$ to be their product. We also call it the \emph{direct sum} of $A$ and $B$. |
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Many preadditive categories are also examples of additive categories. The category $\textbf{CyclGrp}$ of cyclic groups as the subcategory of the category of abelian groups is an example of a preadditive category that is not additive, for the product of two cyclic groups $Z_p$ and $Z_q$ where $p$ and $q$ are not coprime does not exist in $\textbf{CyclGrp}$.
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Many preadditive categories are also examples of additive categories. The category $\textbf{CyclGrp}$ of cyclic groups as the subcategory of the category of abelian groups is an example of a preadditive category that is not additive, for the product of two cyclic groups $Z_p$ and $Z_q$ where $p$ and $q$ are coprime does not exist in $\textbf{CyclGrp}$.
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