nLab geometric dagger 2-poset

Contents

Idea

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

Definition

A geometric 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 $A:Ob(C)$, set $S$, and family of objects $B:S \to Ob(C)$, with monic maps $i_{B(s),A} \in Hom(B(s),A)$ for each element $s \in S$, there is an object

  $$\bigcup_{s \in S} B(s) \in Ob(C)$$

  with monic maps

  $$i_{\bigcup_{s \in S} B(s),A} \in Hom(\bigcup_{s \in S} B(s),A)$$

  and for each element $s \in S$, there is a monic map

  $$i_{B(s),\bigcup_{s \in S} B(s)} \in Hom(B(s),\bigcup_{s \in S} B(s))$$

  such that

  $$i_{\bigcup_{s \in S} B(s),A} \circ i_{B(s),\bigcup_{s \in S} B(s)} = i_{B(s),A}$$

  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

  $$i_{\bigcup_{s \in S} B(s),D} \in Hom(\bigcup_{s \in S} B(s),D)$$

  such that

  $$i_{D,A} \circ i_{\bigcup_{s \in S} B(s),D} = i_{\bigcup_{s \in S} B(s),A}$$

• 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 $A \in Ob(C)$, $E \in Ob(C)$, with monic maps $i_{E,A} \in Hom(E,A)$ and set $S$, and family of objects $B:S \to Ob(C)$, elements $t \in S$ and monic map $i_{B(s),A} \in Hom(B(s),A)$, there is a unitary isomorphism

  $$j_{B,C,E} \in E \cap (\bigcup_{s \in S} B(s)) \cong^\dagger \bigcup_{s \in S} E \cup B(s)$$

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 frame. Since every monic map is a map, the category of maps is a geometric category.

Examples

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

See also

• dagger 2-poset
• coherent dagger 2-poset
• geometric dagger 2-poset
• Heyting dagger 2-poset
• Boolean dagger 2-poset
• elementarily topical dagger 2-poset