On the book,categories for working mathematician, by S.Mac Lane, he gave a definition as follows: an additive category is an Ab-category which has a zero object 0 and a biproduct for each pair of its objects.However,the common definition is defined in the following way, A category C is called additive if satisfies the following conditions :

  1. Every Hom set is an abelian group;
  2. The composition of morphism satisfies two distribution laws;
  3. Exists zero object;
  4. The coproduct of finite objects exists.

By this definition, it's clear that additive category is defined on any abstract category, so why Mac Lane define it on an abelian groups category? Is it reasonable? There must be some reasons behind it . Just for simplicity or for some embedding theorem ? I am confused … Thanks in advance, any help will be greatly appreciated!

  • $\begingroup$ 1 and 2 are what it means for a category to be Ab-enriched, so the only difference is that 4 appears to be weaker than having biproducts, but you can easily show given 1-3 that the coproducts are also products (and thus biproducts). $\endgroup$ – Derek Elkins May 24 '17 at 4:16
  • $\begingroup$ @DerekElkins yeah,thank you! $\endgroup$ – Jiabin Du May 24 '17 at 8:50

The two definitions are the equivalent. Maybe even isomorphic.

In regard to the explanation of your doubt, note that the forgetful functor $\mathrm{Ab} \to \mathrm{Set}$ means that every $\mathrm{Ab}$-enriched category is canonically also a locally small category.

The bulk of the "common definition" you refer to is simply expressing the $\mathrm{Ab}$-category structure in terms of its underlying locally small category structure.


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