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According to this article in Wikipedia the following first three axioms in the definition of of an abelian category imply that hom-sets are enriched over the monoidal category of Abelian groups:

  • existence of a zero object
  • existence of binary products and binary coproducts
  • existence of all kernels and cokernels

(see this image for a saved copy of the relevant section)

My intuition is that it should follow from a diagram like this:

$$A\to A\otimes A \to B \oplus B \to B$$

where the first map is the diagonal map, the second is the product of the two maps one wishes to sum, i.e. $\phi\times \psi$ and the third map is the sum. But how could I use the first and third axioms to prove the existence of the diagonal and sum maps?

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marked as duplicate by Zhen Lin, Did, Yiorgos S. Smyrlis, Claude Leibovici, user127.0.0.1 Feb 23 '14 at 18:35

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up vote 2 down vote accepted

Well, the diagonal morphism exists by the universal property i.e. definition of the product, and the "sum morphism" exists by the very definition of a coproduct.

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Hi Martin! How do you get the sum morphism from the definition of a coproduct? The coproduct is defined more generally than just as a "direct sum" of the same object multiple times, so I don't see where the "sum morphism" is built in by the very definition of a coproduct. Also, why do we need kernels and cokernels to show that it is enriched over the category of abelian groups if they don't show up in any part of my construction of a sum of two morphisms? – Rodrigo Feb 23 '14 at 21:18
$id_B : B \to B$ and $id_B : B \to B$ give $(id_B,id_B) : B \oplus B \to B$ (univ.prop. of coproduct). – Martin Brandenburg Feb 23 '14 at 21:53

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