Additive Category

Definition

An additive category is a Category enriched over abelian groups, providing the structure needed for basic homological algebra.

A category $\mathcal{C}$ is additive if:

1. $\mathcal{C}$ has a Zero Object (an object that is both initial and terminal)
2. Every pair of objects has a product (equivalently, a coproduct)
3. Every hom-set $\mathcal{C}(A,B)$ has the structure of an abelian group
4. Composition of morphisms is bilinear with respect to the abelian group structures

Properties

• In an additive category, finite products and coproducts coincide (biproducts)
• The zero morphism $0:A\to B$ exists for any objects $A$ and $B$
• Direct sums of objects exist
• Addition of morphisms is well-defined and satisfies group axioms
• The composition distributes over addition: $f\circ (g+h)=f\circ g+f\circ h$

Examples

• Ab (Category): The category of abelian groups
• The category of modules over a Ring
• Any Abelian Category is additive
• The category of chain complexes
• The opposite category of an additive category is additive

Relationship to Abelian Categories

Additive categories provide the foundation for abelian categories. An abelian category is an additive category with additional properties (kernels, cokernels, and exactness conditions).

The sequence of generalizations is:

• Category → Additive Category → Abelian Category

Related Concepts

• Abelian Category: Additive categories with kernels and cokernels
• Category: The general notion
• Ab (Category): The prototypical example
• Zero Object: Required for additive structure
• Biproduct: Products and coproducts in additive categories
• Preadditive Category: Categories enriched over abelian groups without zero object