nLab cocartesian monoidal dagger category

Definition

A cocartesian monoidal dagger category is a monoidal dagger category $(C, +, 0)$ with

• a morphism $i_A \in Hom_C(A,A + B)$ for $A \in Ob(C)$ and $B \in Ob(C)$.
• a morphism $i_B \in Hom_C(B,A + B)$ for $A \in Ob(C)$ and $B \in Ob(C)$.
• a morphism $d_{A + B} \in Hom_C(A + B,D)$ for an object $D \in Ob(C)$ and morphisms $d_A \in Hom_C(A,D)$ and $d_B \in Hom_C(B,D)$
• a morphism $0_A \in Hom_C(0,A)$ for every object $A \in C$

such that

• for every object $D \in Ob(C)$ and morphisms $d_A \in Hom_C(A,D)$ and $d_B \in Hom_C(B,D)$, $d_{A + B} \circ i_A = d_A$
• for every object $D \in Ob(C)$ and morphisms $d_A \in Hom_C(A,D)$ and $d_B \in Hom_C(B,D)$, $d_{A + B} \circ i_B = d_B$
• for every object $A \in Ob(C)$ and $B \in Ob(C)$ and morphism $f \in Hom_C(A,B)$, $f \circ 0_A = 0_B$.

In a cocartesian monoidal dagger category, the tensor product is called a coproduct and the tensor unit is called an initial object.

Examples

• semiadditive dagger category

See also

• dagger category
• monoidal dagger category