[[!redirects cartesian monoidal dagger categories]] #Contents# * table of contents {:toc} ## Definition ## A **cartesian monoidal dagger category** is a [[monoidal dagger category]] $C$ with * a morphism $p_A: hom(A \otimes B,A)$ for $A:C$ and $B:C$. * a morphism $p_B: hom(A \otimes B,B)$ for $A:C$ and $B:C$. * a morphism $d_{A \otimes B}: hom(D,A \otimes B)$ for an object $D:C$ and morphisms $d_A: hom(D,A)$ and $d_B: hom(D,B)$ * an identity $u_A: p_A \circ d_{A \otimes B} = d_A$ for an object $D:C$ and morphisms $d_A: hom(D,A)$ and $d_B: hom(D,B)$ * an identity $u_B: p_B \circ d_{A \otimes B} = d_B$ for an object $D:C$ and morphisms $d_A: hom(D,A)$ and $d_B: hom(D,B)$ * a morphism $a_!: hom(A,\Iota)$ for every object $A:C$ In a cartesian monoidal dagger category, the tensor product is called a **cartesian product** and the tensor unit is called a **terminal object**. ## Examples ## * [[semiadditive dagger category]] ## See also ## * [[Category theory]] * [[dagger category]]