# nLab W-topical dagger 2-poset

### Context

#### Higher category theory

higher category theory

## Idea

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 $\mathbb{N} \in Ob(C)$ and maps $0 \in Map_C(\mathcal{P}(0),\mathcal{N})$ and $s \in Map_C(\mathbb{N},\mathbb{N})$, such that for every object $A$ with maps $0_A \in Map_C(\mathcal{P}(0),A)$ and $s_A \in Map_C(A,A)$, there is a map $f \in Map_C(\mathbb{N},A)$ such that $f \circ 0 = 0_A$ and $f \circ s = s_A \circ f$.

## Examples

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