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# additive category

Let $\mathcal{C}$ be a category. Then $\mathcal{C}$ is an *additive category* if

1. $\mathcal{C}$ is a preadditive category, and

2.

Proposition. In a preadditive category, coproduct of two objects exists iff their product exists. Furthermore, they are isomorphic.

###### 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.

Suppose $D$ is the product of $A$ and $B$, with morphisms

$\xymatrix@1{D\ar[r]^{{\pi_{A}}}&A}\qquad\mbox{ and }\qquad\xymatrix@1{D\ar[r]^% {{\pi_{B}}}&B}.$ |

From these two morphisms, we construct two commutative diagrams

$\xymatrix{&A\\ A\ar[ur]^{1}\ar[dr]_{0}\ar@{-->}[r]^{{\alpha}}&D\ar[u]_{{\pi_{A}}}\ar[d]^{{\pi% _{B}}}\\ &B}\qquad\mbox{ and }\qquad\xymatrix{&A\\ B\ar[ur]^{0}\ar[dr]_{1}\ar@{-->}[r]^{{\beta}}&D\ar[u]_{{\pi_{A}}}\ar[d]^{{\pi_% {B}}}\\ &B}$ |

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 $D$.

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 $\beta$. ∎

An easy way to remember the relationships among the various morphisms in the above proof are the following two matrix products:

$\begin{pmatrix}\pi_{A}\\ \pi_{B}\end{pmatrix}\begin{pmatrix}\alpha&\beta\end{pmatrix}=\begin{pmatrix}1&% 0\\ 0&1\end{pmatrix}\qquad\mbox{ and }\qquad\begin{pmatrix}r&s\end{pmatrix}\begin{% pmatrix}\pi_{A}\\ \pi_{B}\end{pmatrix}=f$.

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 *direct sum* of $A$ and $B$.

Many preadditive categories are also examples of additive categories. The category 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 $\mathbb{Z}/p\mathbb{Z}$ and $\mathbb{Z}/q\mathbb{Z}$ exists in CyclGrp only when $p$ and $q$ are coprime.

## Mathematics Subject Classification

18E05*no label found*

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