nLab monoidal 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 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,

α 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 July 10, 2025 at 08:51:37. See the history of this page for a list of all contributions to it.