# Homological Algebra/Defintion of abelian category

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**Definition (-enriched category)**:

An **-enriched category** is a category such that:

- , is an abelian group.
- , is bilinear.

**Definition (zero object)**:

A **zero object** is an object in an -enriched category that is both initial and terminal. We usually denote it by .

**Definition (biproduct)**:

Given an -enriched category , a **biproduct** of is a tuple such that:

We usually denote by .

**Definition (additive category)**:

An **additive category** is an -enriched category such that:

- There is a zero product in .
- Every has a biproduct.

**Definition ((co-)kernel)**:

Given in an -enriched category. A **(co-)kernel** of is a (co-)equalizer of and .

**Definition (abelian category)**:

An **abelian category** is an additive category where:

- Every morphism has a kernel and cokernel.
- Every monomorphism is a kernel and every epimorphism is a cokernel.

**Example**:

The category of all left -modules of a ring is an abelian category.

## Exercises

[edit | edit source]- Given in an -enriched category with zero object. Prove that iff factors through .

- Given a biproduct of and . Prove that is a coproduct of and and is a product of and .

- In an -enriched category with zero object, a kernel of can be equivalently be characterized as a pullback of along .