> [!definition] > > Let $\catc$ be a [[Category|category]], then $\catc$ is **additive** if: > 1. For any $E, F \in \catc$, $\mor{E, F}$ is an [[Commutative Group|abelian group]]. > 2. The law of composition is morphisms is bilinear, and there exists a zero object $0$, such that $\mor{0, E}$ and $\mor{E, 0}$ have precisely one element for each $E \in \catc$. > 3. Finite [[Product|products]] and finite [[Coproduct|coproducts]] exist in this category. > [!theorem] > > Let $\catc$ be an additive category, $E, F \in \catc$, and $f \in \mor{E, F}$. Then for any $X \in \catc$, $f$ induces a [[Group Homomorphism|group homomorphism]] > $ > \mor{X, E} \to \mor{X, F} \quad g \mapsto f \circ g > $ > and another morphism > $ > \mor{F, X} \to \mor{E, X} \quad g \mapsto g \circ f > $ > *Proof*. Since the composition law is bilinear, composition commutes with addition. Let $\catc$ be an additive category, $E, F \in \catc$, and $f \in \mor{X}$. A **kernel** of $f$ is an object $K \in \catc$ and a morphism $\varphi \in \mor{K, E}$ such that for all $X \in \catc$, $ \begin{CD} 0 @>>> \mor{X, K} @>{\varphi \circ}>> \mor{X, E} @>{f \circ }>> \mor{X, F} \end{CD} $ is [[Exact Sequence|exact]]. If the kernel exists, then it is unique up to isomorphism. *Proof*. Let $E_1, E_2$ with $\varphi_1 \in \mor{E_1, E}$ and $\varphi_2 \in \mor{E_2, E}$ be