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> Is there a category with binary biproducts but no zero morphism?
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I'm wondering if the definition of [biproducts](https://ncatlab.org/nlab/show/biproduct) 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.



Category with binary biproducts but no zero morphism
Karvonen has an [article on arXiv](https://arxiv.org/pdf/1801.06488.pdf) showing that if a category has all binary biproducts (using the alternative definition Tom mentioned) then it has zero morphisms (Corollary 3.3).


