[[!redirects cocartesian monoidal dagger categories]] #Contents# * table of contents {:toc} ## Definition ## A **cocartesian monoidal dagger category** is a [[monoidal dagger category]] $(C, +, 0)$ with * a morphism $i_A: hom(A,A + B)$ for $A:C$ and $B:C$. * a morphism $i_B: hom(B,A + B)$ for $A:C$ and $B:C$. * a morphism $d_{A + B}: hom(A + B,D)$ for an object $D:C$ and morphisms $d_A: hom(A,D)$ and $d_B: hom(B,D)$ * an identity $u_A: d_{A + B} \circ i_A = d_A$ for an object $D:C$ and morphisms $d_A: hom(A,D)$ and $d_B: hom(B,D)$ * an identity $u_B: d_{A + B} \circ i_B = d_B$ for an object $D:C$ and morphisms $d_A: hom(A,D)$ and $d_B: hom(B,D)$ * a morphism $0_a: hom(0,A)$ for every object $A:C$ * an identity $u_0: f \circ 0_A = 0_B$ for for $A:C$ and $B:C$ and $f:hom(A,B)$. 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 ## * [[Category theory]] * [[dagger category]]