nLab compact closed dagger category

Contents

Contents

Definition

A compact closed dagger category is a symmetric monoidal dagger category CC with

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

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

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

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

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

such that

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

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

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

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

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

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

Examples

See also

References

Last revised on May 4, 2022 at 02:06:52. See the history of this page for a list of all contributions to it.