PlanetMath: additive category

Proof. We shall prove the fact if the product

of objects

and

exists, then

is also their coproduct. The other direction is dual.

Suppose is the product of and , with morphisms

$\displaystyle \xymatrix@1{D\ar[r]^{\pi_A}&A}$ and $\displaystyle \qquad\xymatrix@1{D\ar[r]^{\pi_B}&B}.$

From these two morphisms, we construct two commutative diagrams

$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{&A \\ A \ar[ur]^1 \ar[dr]_0 \ar@{... ...r[dr]_1 \ar@{-->}[r]^{\beta} & D \ar[u]_{\pi_A} \ar[d]^{\pi_B}\\ &B} } \end{xy}$

where 0 and

are zero morphisms and identity morphisms on

and

, and $\alpha$ and $\beta$ are morphisms based on the definition of the product

Then it's not hard to see that is a coproduct of and with morphisms $\alpha$ and $\beta$ , for if $r:A\rightarrow C$ and $s:B\rightarrow C$ are two morphisms into an object , we can form two morphisms $r\pi_A$ and $s\pi_B$ , both from to . 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 is also the coproduct of and with morphisms $\alpha$ and $\beta$ . $\qedsymbol$

As a result of the above proposition, in an additive category, finite products and finite coproducts are synonymous. Given objects

, we denote $A\oplus B$ to be their product. We also call it the direct sum of

and

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 $\mathbb{Z}/p \mathbb{Z}$ and $\mathbb{Z}/q \mathbb{Z}$ exists in $\textbf{CyclGrp}$ only when

and

are coprime.