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Definition

A cocartesian monoidal dagger category is a monoidal dagger category $C$ with

• a morphism $i_A: hom(A,A \otimes B)$ for $A:C$ and $B:C$.
• a morphism $i_B: hom(B,A \otimes B)$ for $A:C$ and $B:C$.
• a morphism $d_{A \otimes B}: hom(A \otimes B,D)$ for an object $D:C$ and morphisms $d_A: hom(A,D)$ and $d_B: hom(B,D)$
• an identity $u_A: d_{A \otimes B} \circ p_A = d_A$ for an object $D:C$ and morphisms $d_A: hom(A,D)$ and $d_B: hom(B,D)$
• an identity $u_B: d_{A \otimes B} \circ p_B = d_B$ for an object $D:C$ and morphisms $d_A: hom(A,D)$ and $d_B: hom(B,D)$
• a morphism $a: hom(\Iota,A)$ for every object $A:C$

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

• Category theory
• dagger category