[[!redirects dagger 2-posets]] ## Contents ## * table of contents {:toc} ## Definition ## A dagger 2-poset is a [[dagger category]] $C$ such that * For each object $A:Ob(C)$ and $B:Ob(C)$ and morphism $R:Hom(A, B)$, $S:Hom(A, B)$, a propositional binary relation $R \leq_{A, B} S$ * For each object $A:Ob(C)$ and $B:Ob(C)$ and morphism $R:Hom(A, B)$, $R \leq_{A, B} R$. * For each object $A:Ob(C)$ and $B:Ob(C)$ and morphism $R:Hom(A, B)$, $S:Hom(A, B)$, $R \leq_{A, B} S$ and $S \leq_{A, B} R$ implies $R = S$. * For each object $A:Ob(C)$ and $B:Ob(C)$ and morphism $R:Hom(A, B)$, $S:Hom(A, B)$, $T:Hom(A, B)$, $R \leq_{A, B} S$ and $S \leq_{A, B} T$ implies $R \leq_{A, B} T$. * For each object $A:Ob(C)$ and $B:Ob(C)$ and morphism $R:Hom(A, B)$, $S:Hom(A, B)$, $R \leq_{A, B} S$ implies $R^\dagger \leq_{B, A} S^\dagger$. ## See also ## * [[dagger category]] * [[2-poset]] * [[dagger 2-preorder]]