TITLE: Category with binary biproducts but no zero morphism
QUESTION [5 upvotes]: Is there a category with binary biproducts but no zero morphism?

I'm wondering if the definition of biproducts as objects that are simultaneously products and coproducts that obey some identities on the projections/injections is 'different' than the definition involving a zero object. Whenever a zero object exists the definitions are trivially equivalent, and an empty biproduct is a zero object, so we would need a category with binary biproducts but no nullary ones. Any assistance is appreciated.

REPLY [2 votes]: Karvonen has an article on arXiv showing that if a category has all binary biproducts (using the alternative definition Tom mentioned) then it has zero morphisms (Corollary 3.3).