Category Theory/Additive categories

Definition (biproduct):

Let ${\mathcal {C}}$ be a category, and let $(X_{\alpha })_{\alpha \in A}$ be a family of objects in ${\mathcal {C}}$. A biproduct of $(X_{\alpha })_{\alpha \in A}$ is an object of ${\mathcal {C}}$ that is usually denoted as

$\bigoplus _{\alpha \in A}X_{\alpha }$

and for which there exist arrows

$\iota _{\alpha }:X_{\alpha }\to \bigoplus _{\alpha \in A}X_{\alpha }$ and $\pi _{\alpha }:\bigoplus _{\alpha \in A}X_{\alpha }\to X_{\alpha }$

for all $\alpha \in A$ that have the following properties:

1. $\alpha \neq \beta \Rightarrow \pi _{\alpha }\circ \iota _{\beta }=0$ and $\forall \alpha \in A:\pi _{\alpha }\circ \iota _{\alpha }=\operatorname {Id} _{X_{\alpha }}$
2. $\bigoplus _{\alpha \in A}X_{\alpha }$, together with the morphisms $(\iota _{\alpha })_{\alpha \in A}$, constitutes a coproduct in the category ${\mathcal {C}}$
3. $\bigoplus _{\alpha \in A}X_{\alpha }$, together with the morphisms $(\pi _{\alpha })_{\alpha \in A}$, constitutes a product in the category ${\mathcal {C}}$

Definition (additive category):

An additive category is a category ${\mathcal {C}}$ that satisfies each of the following requirements:

1. Every morphism in ${\mathcal {C}}$ has a kernel and a cokernel
2. For every two objects $X,Y$ of ${\mathcal {C}}$, there exists a biproduct $X\oplus Y$
3. For every two objects $X,Y$ of ${\mathcal {C}}$, the assignment $(f,g)\mapsto h$, where $h:X\to Y$ is the morphism that arises from postcomposing the morphism $X{\overset {\Delta }{\to }}X\oplus X{\overset {f\times g}{\to }}Y\oplus Y$ (where $\Delta$ shall denote the diagonal) with the anti-diagonal $\nabla :Y\oplus Y\to Y$, turns $\operatorname {Hom} (X,Y)$ into an abelian group