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Commutative Diagram Proof Isomorphism Category Additive Finite Boolean Valued Function Abelian Group Biconditional Categorical Direct Product Cyclic Group Matrix Operations Kernel Of A Morphism Categorical Direct Sum Subcategory Polynomial Ring Over Field Is Euclidean Domain Preadditive Category

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Additive Category

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

$\mathcal{C}$ is a preadditive category, and
for every pair of objects in $\mathcal{C}$ , their product exists.

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

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 $\beta$ . $\qedsymbol$

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

$\begin{pmatrix} \pi_A \\ \pi_B \end{pmatrix}\begin{pmatrix} \alpha & \beta \end{pmatrix}= \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$ 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 , 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.

1	Category
1	Isomorphism
1	Commutative Diagram
1	Proof
1	Boolean Valued Function
1	Finite
1	Additive
1	Biconditional
1	Abelian Group
2	Categorical Direct Product
2	Cyclic Group
4	Subcategory
4	Kernel Of A Morphism
4	Categorical Direct Sum
4	Matrix Operations
10	Preadditive Category
10	Polynomial Ring Over Field Is Euclidean Domain