# Homotopy Type Theory dagger 2-poset > history (Rev #4, changes)

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## 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 $R^\dagger \leq_{B, A} S^\dagger$.