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 C C with a dagger functor ( − ) ⊗ ( − ) : ( C × C ) → C (-)\otimes(-): (C \times C) \to C from the product dagger category of C C to C C itself called the tensor product and an object Ι ∈ Ob ( C ) \Iota \in Ob(C) called the tensor unit , such that
for all morphisms f ∈ Hom C ( D , E ) f \in Hom_C(D,E) and g ∈ Hom C ( F , G ) g \in Hom_C(F,G) , ( f ⊗ g ) † = f † ⊗ g † (f \otimes g)^\dagger = f^\dagger \otimes g^\dagger
for all objects D ∈ Ob ( C ) D \in Ob(C) , E ∈ Ob ( C ) E \in Ob(C) , F ∈ Ob ( C ) F\in Ob(C) , a natural unitary isomorphism? called the associator of D D , E E , and F F ,
α D , E , F : ( D ⊗ E ) ⊗ F ≅ † D ⊗ ( E ⊗ F ) \alpha_{D, E, F}:(D\otimes E)\otimes F \cong^\dagger D\otimes(E\otimes F)
for objects A ∈ Ob ( C ) A \in Ob(C) , a natural unitary isomorphism called the left unitor of A A ,
λ A : Ι ⊗ A ≅ † A \lambda_{A}: \Iota\otimes A \cong^\dagger A
for objects A ∈ Ob ( C ) A \in Ob(C) , a natural unitary isomorphism called the right unitor of A A
ρ A : A ⊗ Ι ≅ † A \rho_{A}: A\otimes\Iota \cong^\dagger A
all objects A ∈ Ob ( C ) A \in Ob(C) and B ∈ Ob ( C ) B \in Ob(C) satify the triangle identity :
ρ A ⊗ Ι B = ( 1 A ⊗ ρ B ) ∘ α A , Ι , B \rho_A \otimes \Iota_B = (1_A \otimes \rho_B) \circ \alpha_{A, \Iota, B}
all objects D ∈ Ob ( C ) D \in Ob(C) , E ∈ Ob ( C ) E \in Ob(C) , F ∈ Ob ( C ) F \in Ob(C) , G ∈ Ob ( C ) G \in Ob(C) satisfy the pentagon identity :
α D , E , F ⊗ G ∘ α D ⊗ E , F , G = ( id D ⊗ α E , F , G ) ∘ α D , E ⊗ F , G ∘ ( α D , E , F ⊗ id 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.
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