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category theory

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Definition

A cartesian monoidal dagger category is category which is both a dagger category and a cartesian monoidal category in a compatible way, namely a monoidal dagger category $(C, \times, 1)$ with

• a morphism $p_A \in Hom_C(A \times B,A)$ for $A \in Ob(C)$ and $B \in Ob(C)$.
• a morphism $p_B \in Hom_C(A \times B,B)$ for $A \in Ob(C)$ and $B \in Ob(C)$.
• a morphism $p_{A \times B} \in Hom_C(D,A \times B)$ for an object $D \in Ob(C)$ and morphisms $d_A \in Hom_C(D,A)$ and $d_B \in Hom_C(D,B)$
• a morphism $!_A \in Hom_C(A,1)$ for every object $A \in C$

such that

• for every object $D \in Ob(C)$ and morphisms $d_A \in Hom_C(D,B)$ and $d_B \in Hom_C(D,B)$, $p_A \circ d_{A \times B} = d_A$
• for every object $D \in Ob(C)$ and morphisms $d_A \in Hom_C(D,B)$ and $d_B \in Hom_C(D,B)$, $p_B \circ d_{A \otimes B} = d_B$
• for every object $A \in Ob(C)$ and $B \in Ob(C)$ and morphism $f \in Hom_C(A,B)$, $!_B \circ f = !_A$.

In a cartesian monoidal dagger category, the tensor product is called a cartesian product and the tensor unit is called a terminal object.