# Definition:Additive Category

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## Definition

Let $A$ be a preadditive category.

### Definition 1: binary biproducts

The category $A$ is an **additive category** if and only if:

- it has a zero object
- it has all binary biproducts.

### Definition 2: binary products

The category $A$ is an **additive category** if and only if:

- it has a zero object
- it has all binary products.

### Definition 3: binary coproducts

The category $A$ is an **additive category** if and only if:

- it has a zero object
- it has all binary coproducts.

### Definition 4: finite biproducts

The category $A$ is an **additive category** if and only if it has all finite biproducts.

### Definition 5: finite products

The category $A$ is an **additive category** if and only if it has all finite products.

### Definition 6: finite coproducts

The category $A$ is an **additive category** if and only if it has all finite coproducts.