nLab semiadditive dagger category

Contents

Context

Category theory

Monoidal categories

monoidal categories

With braiding

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products

Semisimplicity

Morphisms

Internal monoids

Examples

Theorems

In higher category theory

Contents

Definition

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

  • for all objects AOb(C)A \in Ob(C), i A i A=id Ai_A^\dagger \circ i_A = id_A
  • for all objects AOb(C)A \in Ob(C) and BOb(C)B \in Ob(C), i B i A=0 B0 A i_B^\dagger \circ i_A = 0_B \circ 0_A^\dagger

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

Examples

See also

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)

Last revised on July 10, 2025 at 08:55:07. See the history of this page for a list of all contributions to it.