Let $F:C \rightarrow D$ be a functor between two additive categories. The claim (p53, prop 40) is :

If $F$ preserves biproducts then $F$ preserves finite products.

For the proof the author wrote:

It suffices to prove that $F$ preserves zero object.

I do not get this argument. I thought there is nothing to prove and this follows by the characterization that for any product $(A\times B, \pi_1, \pi_2)$ we may extend this to a biproduct. $(A \times B, \pi_1, \pi_2, i_1, i_2)$.

So $F$ preserving biproduct implies preserving products (including initial and terminal objects)?

  • $\begingroup$ Reading the notes, it seems clear that the claim is: "If $F$ preserves binary biproducts then $F$ preserves finite products." Finite products includes nullary products (i.e. the terminal object), but preservation of the terminal object doesn't trivially follow from preservation of binary biproducts. It would if we talked about finite biproducts as the zero object would be the nullary biproduct. $\endgroup$ – Derek Elkins Oct 25 '18 at 19:02

Preserving biproducts implies preserving products of pairs. An induction argument, shows that preserving terminal object and product of pairs implies preserving finite products.


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