# Abelian category

An Abelian category is defined as follows according to Barry Mitchell (1965).

Definition: "An Abelian category is an exact, additive category with finite products."

The following theorem from ref.[2] is also relevant as it specifies the key properties of all Abelian categories[1] :

“The following statements are equivalent:

• ${\displaystyle Ab}$ is an Abelian category;
• ${\displaystyle Ab}$ has kernels, cokernels, finite products, finite coproducts, and is both normal and comormal;
• ${\displaystyle Ab}$ has pushouts and pullbacks and is both normal and conormal”.

## References

• http://planetphysics.us/encyclopedia/AbelianCategory2.html The Fundamental Abelian Category Theorem
• 2. Barry Mitchell. "Theory of Categories". Academic Press: New York and London, 1965, (Theorem 20.1 on p.33).