### Context

#### Higher category theory

higher category theory

## Definition

A semiadditive dagger 2-poset is a dagger 2-poset $C$ such that

• There exists an object $0:Ob(C)$ called the zero object such that for each object $A:Ob(C)$, there exists a morphism $Z_A:Hom(0, A)$ such that for each object $B:Ob(C)$ and morphism $R: Hom(A, B)$, $R \circ Z_A = Z_B$.

• For each object $A:Ob(C)$ and $B:Ob(C)$, there exists a object $A \oplus B:Ob(C)$ called the biproduct of $A$ and $B$, with morphisms $I_A: Hom(A,A \oplus B)$ and $I_B: Hom(B,A \oplus B)$, such that

• For each object $A:Ob(C)$, $B:Ob(C)$, and $C:Ob(C)$ and morphism $R_A:Hom(A,C)$ and $R_B:Hom(B,C)$, there exist a morphism $R: Hom(A \oplus B ,C)$ such that $R \circ I_A = R_A$ and $R \circ I_B = R_B$

• For each object $A:Ob(C)$, $B:Ob(C)$, $C:Ob(C)$, and $D:Ob(C)$, and morphism $R:Hom(A, B)$ and $S:Hom(C,D)$, there exists a morphism $R \oplus S:Hom(A \oplus B,C \oplus D)$ where $(R \oplus S)^\dagger = R^\dagger \oplus S^\dagger$.

## Examples

• The dagger 2-poset Rel of sets and relations is a semiadditive dagger 2-poset.