nLab semiadditive dagger category

Definition

A semiadditive dagger category is a cocartesian monoidal dagger category $(C, \oplus, 0, i_A, i_B, 0_A)$ such that

• for all objects $A \in Ob(C)$, $i_A^\dagger \circ i_A = id_A$ and $i_A^\dagger \circ i_A = id_A$
• for all objects $A \in Ob(C)$ and $B \in Ob(C)$, $i_B^\dagger \circ i_A = 0_B \circ 0_A^\dagger$

In a semiadditive dagger category, the coproduct is called a biproduct and the initial object is called a zero object.

Examples

• The dagger category Hilb of Hilbert spaces and continuous linear maps is a semiadditive dagger category.

• The dagger category Rel of sets and relations is a semiadditive dagger category.

See also

• dagger category
• monoidal dagger category
• cocartesian monoidal dagger category
• cartesian monoidal dagger category
• semiadditive dagger 2-poset

References

• Martti Karvonen. Biproducts without pointedness (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)