nLab monoidal dagger category

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Definition

A monoidal dagger category is a dagger category CC with a dagger functor ()():(C×C)C(-)\otimes(-): (C \times C) \to C from the product dagger category of CC to CC itself called the tensor product and an object ΙOb(C)\Iota \in Ob(C) called the tensor unit, such that

  • for all morphisms fHom C(D,E)f \in Hom_C(D,E) and gHom C(F,G)g \in Hom_C(F,G), (fg) =f g (f \otimes g)^\dagger = f^\dagger \otimes g^\dagger

  • for all objects DOb(C)D \in Ob(C), EOb(C)E \in Ob(C), FOb(C)F\in Ob(C), a natural unitary isomorphism? called the associator of DD, EE, and FF,

alpha D,E,F:(DE)F D(EF)alpha_{D, E, F}:(D\otimes E)\otimes F \cong^\dagger D\otimes(E\otimes F)
  • for objects AOb(C)A \in Ob(C), a natural unitary isomorphism called the left unitor of AA,
λ A:ΙA A\lambda_{A}: \Iota\otimes A \cong^\dagger A
  • for objects AOb(C)A \in Ob(C), a natural unitary isomorphism called the right unitor of AA
ρ A:AΙ A\rho_{A}: A\otimes\Iota \cong^\dagger A
  • all objects AOb(C)A \in Ob(C) and BOb(C)B \in Ob(C) satify the triangle identity:
ρ AΙ B=(1 Aρ B)α A,Ι,B\rho_A \otimes \Iota_B = (1_A \otimes \rho_B) \circ \alpha_{A, \Iota, B}
  • all objects DOb(C)D \in Ob(C), EOb(C)E \in Ob(C), FOb(C)F \in Ob(C), GOb(C)G \in Ob(C) satisfy the pentagon identity:
α D,E,FGα DE,F,G=(id Dα E,F,G)α D,EF,G(α D,E,Fid G)\alpha_{D, E, F \otimes G} \circ \alpha_{D \otimes E, F, G} = (id_D \otimes \alpha_{E, F, G}) \circ \alpha_{D, E \otimes F, G} \circ (\alpha_{D, E, F} \otimes id_G)

Examples

See also

Last revised on May 16, 2022 at 00:48:39. See the history of this page for a list of all contributions to it.