# Homotopy Type Theory Boolean dagger 2-poset > history (Rev #5)

## Idea

A Boolean dagger 2-poset is a dagger 2-poset whose category of maps is an Boolean category?.

## Definition

A Boolean dagger 2-poset $C$ is a Heyting dagger 2-poset where for each objects $A:Ob(C)$, $B:Ob(C)$ and monic map $i_{B,A}:Hom(B,A)$, there is a unitary isomorphism $j_{B,A}:A \cong^\dagger B \cup (B \Rightarrow 0)$.

## Properties

• The unitary isomorphism classes of monic maps into every object $A$ is a Boolean algebra?. Since every monic map is a map, the category of maps is a Boolean category?.

## Examples

The dagger 2-poset of decidable sets and relations is an Boolean dagger 2-poset.