Abelian category

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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:

  • 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”.


  • 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).