A W-topical dagger 2-poset is a dagger 2-poset whose category of maps is a W-topos?.

Definition

A W-topical dagger 2-poset$C$ is an elementarily topical dagger 2-poset with an object $\mathbb{N}:Ob(C)$ and maps $0:Hom(\mathcal{P}(0),\mathcal{N})$ and $s:Hom(\mathbb{N},\mathbb{N})$, such that for every object $A$ with maps $0_A:Hom(\mathcal{P}(0),A)$ and $s_A:Hom(A,A)$, there is a map $f:Hom(\mathbb{N},A)$ such that $f \circ 0 = 0_A$ and $f \circ s = s_A \circ f$.

Examples

The dagger 2-poset of sets and relations is a W-topical dagger 2-poset.