[[!redirects semiadditive dagger categories]] #Contents# * table of contents {:toc} ## Definition ## A **semiadditive dagger category** is a [[cocartesian monoidal dagger category]] $(C, \oplus, 0, i_A, i_B, 0_A)$ with * an identity $m_A: i_A^\dagger \circ i_A = id_A$ for $A:C$ * an identity $m_B: i_B^\dagger \circ i_B = id_B$ for $B:C$ * an identity $i_b^\dagger \circ i_A = 0_B \circ 0_A^\dagger$ for $A:C$ and $B:C$. In a semiadditive dagger category, the coproduct is called a **biproduct** and the initial object is called a **zero object**. ## Examples ## (...) ## See also ## * [[Category theory]] * [[dagger category]] ## References ## * Martti Karvonen. Biproducts without pointedness ([abs:1801.06488](https://arxiv.org/abs/1801.06488)) * Chris Heunen and Martti Karvonen. Limits in dagger categories. Theory and Applications of Categories, 34(18):468–513, 2019. * Chris Heunen, Andre Kornell. Axioms for the category of Hilbert spaces ([arXiv:2109.07418](https://arxiv.org/abs/2109.07418))