nLab braided monoidal dagger category

Contents

Contents

Definition

A braided monoidal dagger category is a monoidal dagger category (C,,Ι)(C, \otimes, \Iota) which is also a braided monoidal category, in that:

  • for objects AOb(C)A \in Ob(C) and BOb(C)B \in Ob(C), there is a natural unitary isomorphism? called the braiding of AA and BB
β A,B:AB BA\beta_{A,B}: A\otimes B \cong^\dagger B\otimes A
  • all objects DOb(C)D \in Ob(C), EOb(C)E \in Ob(C), and FOb(C)F \in Ob(C) satisfy the first hexagon identity
α E,F,Dβ D,EFα D,E,F=(idβ D,F)α E,D,F(β D,Eid)\alpha_{E,F,D} \circ \beta_{D, E \otimes F} \circ \alpha_{D,E,F} = (id \otimes \beta_{D,F}) \circ \alpha_{E,D,F} \circ (\beta_{D, E} \otimes id)
  • all objects DOb(C)D \in Ob(C), EOb(C)E \in Ob(C), and FOb(C)F \in Ob(C) satisfy the second hexagon identity
α F,D,E 1β DE,Fα D,E,F 1=(β D,Fid)α D,F,E 1(idβ E,F)\alpha_{F,D,E}^{-1} \circ \beta_{D \otimes E, F} \circ \alpha^{-1}_{D,E,F} = (\beta_{D, F} \otimes id) \circ \alpha^{-1}_{D,F,E} \circ (id \otimes \beta_{E, F})

Examples

See also

Last revised on May 16, 2022 at 00:50:18. See the history of this page for a list of all contributions to it.