nLab compact closed dagger category

Definition

A compact closed dagger category is a symmetric monoidal dagger category $C$ with

• an object $A^*$ called the dual for every object $A:Ob(C)$

• for every object $A:Ob(C)$, a morphism $\iota^R_A: \Iota \to A \otimes A^*$ called the right unit

• for every object $A:Ob(C)$, a morphism $\iota^L_A: \Iota \to A^* \otimes A$ called the left unit

• for every object $A:Ob(C)$, a morphism $\eta^L_A: A \otimes A^* \to \Iota$ called the left counit

• for every object $A:Ob(C)$, a morphism $\eta^R_A: A^* \otimes A \to \Iota$ called the right counit

such that

• for all objects $A:Ob(C)$, ${\iota^R_A}^\dagger = \eta^L_A$

• for all objects $A:Ob(C)$, ${\iota^L_A}^\dagger = \eta^R_A$

• for all objects $A:Ob(C)$, $\iota^L_{A}A \circ A\eta^L_{A}$

• for all objects $A:Ob(C)$, $\iota^R_{A^*}A^* \circ A\eta^R_{A^*}$

• for all objects $A:Ob(C)$, $A\iota^R_{A} \circ \eta^R_{A}A$

• for all objects $A:Ob(C)$, $A^*\iota^R_{A^*} \circ \eta^R_{A^*}A^*$

Examples

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

See also

• dagger category

• monoidal dagger category

• symmetric monoidal dagger category

• compact closed category

References

• Chris Heunen, Andre Kornell, Axioms for the category of Hilbert spaces (arXiv:2109.07418)