nLab Heyting dagger 2-poset

Context

Higher category theory

higher category theory

Idea

A Heyting dagger 2-poset is a dagger 2-poset whose category of maps is a Heyting category.

Definition

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

• There is an object $0 \in Ob(C)$ such that for each object $A \in Ob(C)$, there is a monic map $i_{0,A} \in Hom(0,A)$ such that for each object $B \in Ob(C)$ with a monic map $i_{B,A} \in Hom(B,A)$, there is a monic map $i_{0,B} \in Hom(0,B)$ such that $i_{B,A} \circ i_{0,B} = i_{0,A}$.

• For each object $A \in Ob(C)$, $B \in Ob(C)$, $E \in Ob(C)$ with monic maps $i_{B,A} \in Hom(B,A)$, $i_{E,A} \in Hom(E,A)$, there is an object $B \cup E \in Ob(C)$ with monic maps $i_{B \cup E,A} \in Hom(B \cup E,A)$, $i_{B,B \cup E} \in Hom(B,B \cup E)$, $i_{E,B \cup E} \in Hom(E,B \cup E)$, such that $i_{B \cap E,A} \circ i_{B,B \cap E} = i_{B,A}$ and $i_{B \cap E,A} \circ i_{E,B \cap E} = i_{E,A}$,and for every object $D \in Ob(C)$ with monic maps $i_{D,A} \in Hom(D,A)$ $i_{B,D} \in Hom(B,D)$, $i_{E,D} \in Hom(E,D)$ such that $i_{D,A} \circ i_{B,D} = i_{B,A}$ and $i_{D,A} \circ i_{E,D} = i_{E,A}$, there is a monic map $i_{B \cup E,D} \in Hom(B \cup E,D)$ such that $i_{D,A} \circ i_{B \cup E,D} = i_{B \cup E,A}$

• For each object $A \in Ob(C)$, $B \in Ob(C)$, $E \in Ob(C)$ with monic maps $i_{B,A} \in Hom(B,A)$, $i_{E,A} \in Hom(E,A)$, there is an object $B \cap E \in Ob(C)$ with monic maps $i_{B \cap E,A} \in Hom(B \cap E,A)$, $i_{B \cap E,B} \in Hom(B \cap E,B)$, $i_{B \cap E,E} \in Hom(B \cap E,E)$, such that $i_{B,A} \circ i_{B \cap E,B} = i_{B \cap E,A}$ and $i_{E,A} \circ i_{B \cap E,E} = i_{B \cap E,A}$, and for every object $D \in Ob(C)$ with monic maps $i_{D,A} \in Hom(D,A)$ $i_{D,B} \in Hom(D,B)$, $i_{D,E} \in Hom(D,E)$ such that $i_{B,A} \circ i_{D,B} = i_{D,A}$ and $i_{E,A} \circ i_{D,E} = i_{D,A}$, there is a monic map $i_{D,B \cap E} \in Hom(D,B \cap E)$ such that $i_{B \cap E,A} \circ i_{D,B \cap E} = i_{D,A}$.

• For each object $A \in Ob(C)$, $B \in Ob(C)$, $D \in Ob(C)$ with monic maps $i_{B,A} \in Hom(B,A)$, $i_{D,A} \in Hom(D,A)$, there is an object $B \Rightarrow D \in Ob(C)$ with monic map $i_{D,A} \in Hom(D,A)$,

$i_{(B \Rightarrow D) \cap D, B}:Hom((B \Rightarrow D) \cap D, B)$

and

$i_{D, B \Rightarrow (D \cap B)}:Hom(D, B \Rightarrow (D \cap B))$

Properties

• For each object $A \in Ob(C)$, the identity function $1_A \in Hom(A,A)$ is a monic map, and for each object $B:Ob(C)$ with a monic map $i_{B,A} \in Hom(B,A)$, $1_A \circ i_{B,A} = i_{B,A}$.

• The isomorphism classes of monic maps into every object $A$ is a Heyting algebra. Since every monic map is a map, the category of maps is a Heyting category. As a result, Heyting dagger 2-posets are the same as division allegories.

Examples

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