nLab Heyting dagger 2-poset

Context

Higher category theory

higher category theory

Basic concepts

Basic theorems

Applications

Models

Morphisms

Functors

Universal constructions

Extra properties and structure

1-categorical presentations

Contents

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 CC such that

  • There is an object 0Ob(C)0 \in Ob(C) such that for each object AOb(C)A \in Ob(C), there is a monic map i 0,AHom(0,A)i_{0,A} \in Hom(0,A) such that for each object BOb(C)B \in Ob(C) with a monic map i B,AHom(B,A)i_{B,A} \in Hom(B,A), there is a monic map i 0,BHom(0,B)i_{0,B} \in Hom(0,B) such that i B,Ai 0,B=i 0,Ai_{B,A} \circ i_{0,B} = i_{0,A}.

  • For each object AOb(C)A \in Ob(C), BOb(C)B \in Ob(C), EOb(C)E \in Ob(C) with monic maps i B,AHom(B,A)i_{B,A} \in Hom(B,A), i E,AHom(E,A)i_{E,A} \in Hom(E,A), there is an object BEOb(C)B \cup E \in Ob(C) with monic maps i BE,AHom(BE,A)i_{B \cup E,A} \in Hom(B \cup E,A), i B,BEHom(B,BE)i_{B,B \cup E} \in Hom(B,B \cup E), i E,BEHom(E,BE)i_{E,B \cup E} \in Hom(E,B \cup E), such that i BE,Ai B,BE=i B,Ai_{B \cap E,A} \circ i_{B,B \cap E} = i_{B,A} and i BE,Ai E,BE=i E,Ai_{B \cap E,A} \circ i_{E,B \cap E} = i_{E,A},and for every object DOb(C)D \in Ob(C) with monic maps i D,AHom(D,A)i_{D,A} \in Hom(D,A) i B,DHom(B,D)i_{B,D} \in Hom(B,D), i E,DHom(E,D)i_{E,D} \in Hom(E,D) such that i D,Ai B,D=i B,Ai_{D,A} \circ i_{B,D} = i_{B,A} and i D,Ai E,D=i E,Ai_{D,A} \circ i_{E,D} = i_{E,A}, there is a monic map i BE,DHom(BE,D)i_{B \cup E,D} \in Hom(B \cup E,D) such that i D,Ai BE,D=i BE,Ai_{D,A} \circ i_{B \cup E,D} = i_{B \cup E,A}

  • For each object AOb(C)A \in Ob(C), BOb(C)B \in Ob(C), EOb(C)E \in Ob(C) with monic maps i B,AHom(B,A)i_{B,A} \in Hom(B,A), i E,AHom(E,A)i_{E,A} \in Hom(E,A), there is an object BEOb(C)B \cap E \in Ob(C) with monic maps i BE,AHom(BE,A)i_{B \cap E,A} \in Hom(B \cap E,A), i BE,BHom(BE,B)i_{B \cap E,B} \in Hom(B \cap E,B), i BE,EHom(BE,E)i_{B \cap E,E} \in Hom(B \cap E,E), such that i B,Ai BE,B=i BE,Ai_{B,A} \circ i_{B \cap E,B} = i_{B \cap E,A} and i E,Ai BE,E=i BE,Ai_{E,A} \circ i_{B \cap E,E} = i_{B \cap E,A}, and for every object DOb(C)D \in Ob(C) with monic maps i D,AHom(D,A)i_{D,A} \in Hom(D,A) i D,BHom(D,B)i_{D,B} \in Hom(D,B), i D,EHom(D,E)i_{D,E} \in Hom(D,E) such that i B,Ai D,B=i D,Ai_{B,A} \circ i_{D,B} = i_{D,A} and i E,Ai D,E=i D,Ai_{E,A} \circ i_{D,E} = i_{D,A}, there is a monic map i D,BEHom(D,BE)i_{D,B \cap E} \in Hom(D,B \cap E) such that i BE,Ai D,BE=i D,Ai_{B \cap E,A} \circ i_{D,B \cap E} = i_{D,A}.

  • For each object AOb(C)A \in Ob(C), BOb(C)B \in Ob(C), DOb(C)D \in Ob(C) with monic maps i B,AHom(B,A)i_{B,A} \in Hom(B,A), i D,AHom(D,A)i_{D,A} \in Hom(D,A), there is an object BDOb(C)B \Rightarrow D \in Ob(C) with monic map i D,AHom(D,A)i_{D,A} \in Hom(D,A),

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

    and

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

Properties

  • For each object AOb(C)A \in Ob(C), the identity function 1 AHom(A,A)1_A \in Hom(A,A) is a monic map, and for each object B:Ob(C)B:Ob(C) with a monic map i B,AHom(B,A)i_{B,A} \in Hom(B,A), 1 Ai B,A=i B,A1_A \circ i_{B,A} = i_{B,A}.

  • The isomorphism classes of monic maps into every object AA 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.

See also

Created on May 3, 2022 at 18:03:13. See the history of this page for a list of all contributions to it.