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

Contents

Idea

An conjunctive dagger 2-poset is a dagger 2-poset whose internal logic of the category of maps consists only of conjunction \wedge and true \top, or equivalently, whose category of maps has all pullback?s of monic maps.

Definition

An conjunctive dagger 2-poset is a dagger 2-poset CC such that

  • For each object A:Ob(C)A:Ob(C), B:Ob(C)B:Ob(C), E:Ob(C)E:Ob(C) with functional dagger monomorphisms i B,A:Hom(B,A)i_{B,A}:Hom(B,A), i E,A:Hom(E,A)i_{E,A}:Hom(E,A), there is an object BE:Ob(C)B \cap E:Ob(C) with functional dagger monomorphisms i BE,A:Hom(BE,A)i_{B \cap E,A}:Hom(B \cap E,A), i BE,B:Hom(BE,B)i_{B \cap E,B}:Hom(B \cap E,B), i BE,E:Hom(BE,E)i_{B \cap E,E}: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 D:Ob(C)D:Ob(C) with functional dagger monomorphisms i D,A:Hom(D,A)i_{D,A}:Hom(D,A) i D,B:Hom(D,B)i_{D,B}:Hom(D,B), i D,E:Hom(D,E)i_{D,E}: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 functional dagger monomorphism i D,BE:Hom(D,BE)i_{D,B \cap E}: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}.

Properties

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

  • The unitary isomorphism classes of functional dagger monomorphisms into every object AA is a meet-semilattice.

Examples

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

See also

Revision on April 21, 2022 at 12:20:02 by Anonymous?. See the history of this page for a list of all contributions to it.