nLab monoidal dagger category

Definition

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

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

• for all objects $D \in Ob(C)$, $E \in Ob(C)$, $F\in Ob(C)$, a natural unitary isomorphism? called the associator of $D$, $E$, and $F$,

$$\alpha_{D, E, F}:(D\otimes E)\otimes F \cong^\dagger D\otimes(E\otimes F)$$

• for objects $A \in Ob(C)$, a natural unitary isomorphism called the left unitor of $A$,

$$\lambda_{A}: \Iota\otimes A \cong^\dagger A$$

• for objects $A \in Ob(C)$, a natural unitary isomorphism called the right unitor of $A$

$$\rho_{A}: A\otimes\Iota \cong^\dagger A$$

• all objects $A \in Ob(C)$ and $B \in Ob(C)$ satify the triangle identity:

$$\rho_A \otimes \Iota_B = (1_A \otimes \rho_B) \circ \alpha_{A, \Iota, B}$$

• all objects $D \in Ob(C)$, $E \in Ob(C)$, $F \in Ob(C)$, $G \in Ob(C)$ satisfy the pentagon identity:

$$\alpha_{D, E, F \otimes G} \circ \alpha_{D \otimes E, F, G} = (id_D \otimes \alpha_{E, F, G}) \circ \alpha_{D, E \otimes F, G} \circ (\alpha_{D, E, F} \otimes id_G)$$

Examples

• monoidal groupoid, 2-group

• braided monoidal dagger category, braided monoidal groupoid, braided 2-group

• symmetric monoidal dagger category, symmetric monoidal groupoid, symmetric 2-group

• cartesian monoidal dagger category

• cocartesian monoidal dagger category

• semiadditive dagger category

• rigid monoidal dagger category?

• compact closed dagger category?

See also

• dagger category
• dagger functor