[[!redirects cartesian dagger 2-posets]] ## Contents ## * table of contents {:toc} ## Idea ## A dagger 2-poset whose [[category of maps]] is a [[cartesian monoidal category]]. ## Definition ## A **cartesian dagger 2-poset** is a [[unital dagger 2-poset]] $C$ such that for every object $A:Ob(C)$ and $B:Ob(C)$, $Hom(A, B)$ is a [[meet-semilattice]] with top morphism $\top_{A,B}$ and meet operation $\wedge_{A,B}$, and there is an object $A \otimes B:Ob(C)$ such that there exist [[map in a dagger 2-poset|maps]] $p_A:A \otimes B \to A$ and $p_B:B \otimes B \to B$ such that $p_B^\dagger \circ p_A = \top_{A,B}$ and $u_B \circ p_B = u_A \circ p_A$, for every [[onto dagger morphism in a dagger 2-poset |onto morphism]] $u_A:A \to \mathbb{1}$ and $u_B:A \to \mathbb{1}$. ## Examples ## The dagger 2-poset of sets and relations is a cartesian dagger 2-poset. ## See also ## * [[dagger 2-poset]] * [[unital dagger 2-poset]] * [[tabular dagger 2-poset]]