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.

Also see

• Equivalence of Definitions of Additive Category
• Definition:Abelian Category
• Finite Product in Preadditive Category is Biproduct
• Finite Coproduct in Preadditive Category is Biproduct

Sources

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