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

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

    with monic maps

    i sSB(s),AHom( sSB(s),A)i_{\bigcup_{s \in S} B(s),A} \in Hom(\bigcup_{s \in S} B(s),A)

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

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

    such that

    i sSB(s),Ai B(s), sSB(s)=i B(s),Ai_{\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 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 sSB(s),DHom( sSB(s),D)i_{\bigcup_{s \in S} B(s),D} \in Hom(\bigcup_{s \in S} B(s),D)

    such that

    i D,Ai sSB(s),D=i sSB(s),Ai_{D,A} \circ i_{\bigcup_{s \in S} B(s),D} = i_{\bigcup_{s \in S} B(s),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), EOb(C)E \in Ob(C), with monic maps i E,AHom(E,A)i_{E,A} \in Hom(E,A) and set SS, and family of objects B:SOb(C)B:S \to Ob(C), elements tSt \in S and monic map i B(s),AHom(B(s),A)i_{B(s),A} \in Hom(B(s),A), there is a unitary isomorphism

    j B,C,EE( sSB(s)) sSEB(s)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 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 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

Created on May 3, 2022 at 18:19:32. See the history of this page for a list of all contributions to it.