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. is also relevant as it specifies the key properties of all Abelian categories :
“The following statements are equivalent:
- is an Abelian category;
- has kernels, cokernels, finite products, finite coproducts, and is both normal and comormal;
- has pushouts and pullbacks and is both normal and conormal”.
- 2. Barry Mitchell. "Theory of Categories". Academic Press: New York and London, 1965, (Theorem 20.1 on p.33).