Homotopy Type Theory conjunctive dagger 2-poset > history (Rev #7, changes)

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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.


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 monic map dagger monomorphismsi B,A:Hom(B,A)i_{B,A}:Hom(B,A) ,i B E,A:Hom( B E,A) i_{B,A}:Hom(B,A) i_{E,A}:Hom(E,A) , there is an objecti E,ABE: Hom Ob( E C,A) i_{E,A}:Hom(E,A) B \cap E:Ob(C) , there with is monic an maps objecti BE,A:Hom(BE : , Ob A(C) B i_{B \cap E:Ob(C) E,A}:Hom(B \cap E,A) , with functional dagger monomorphismsi BE, A B:Hom(BE, A B) i_{B \cap E,A}:Hom(B E,B}:Hom(B \cap E,A) E,B), i BE, B E:Hom(BE, B E) i_{B \cap E,B}:Hom(B E,E}:Hom(B \cap E,B) E,E) , such thati BE, E A :Homi BE,B ( =Bi BE,AE,E) i_{B,A} \circ i_{B \cap E,E}:Hom(B E,B} = i_{B \cap E,E) E,A} , such and thati B E,Ai BE, B E=i BE,A i_{B,A} i_{E,A} \circ i_{B \cap E,B} E,E} = i_{B \cap E,A} , and for every objecti E,AD :i BE,EOb = (i BE,AC) i_{E,A} D:Ob(C) \circ i_{B \cap E,E} = i_{B \cap E,A} , and with for monic every maps objectDi D,A: Ob Hom( C D,A) D:Ob(C) i_{D,A}:Hom(D,A) with functional dagger monomorphismsi D, A B:Hom(D, A B) i_{D,A}:Hom(D,A) i_{D,B}:Hom(D,B) ,i D, B E:Hom(D, B E) i_{D,B}:Hom(D,B) i_{D,E}:Hom(D,E) , such thati D B, E A :Homi D,B ( =Di D,A,E) i_{D,E}:Hom(D,E) i_{B,A} \circ i_{D,B} = i_{D,A} such and thati B E,Ai D, B E=i D,A i_{B,A} i_{E,A} \circ i_{D,B} i_{D,E} = i_{D,A} , and there is a monic mapi E D, A BE :i D,EHom = (i D,AD,BE) i_{E,A} i_{D,B \circ \cap i_{D,E} E}:Hom(D,B = \cap i_{D,A} E) , there such is that a functional dagger monomorphismi BE,Ai D,BE : =Homi D,A(D,BE) i_{B \cap E,A} \circ i_{D,B \cap E}:Hom(D,B E} \cap = E) i_{D,A} 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 A:Ob(C)A:Ob(C), the identity function 1 A:Hom(A,A)1_A:Hom(A,A) is a functional monic dagger map, monomorphism, and for each objectB:Ob(C)B:Ob(C) with a functional monic dagger map monomorphismi 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 monic dagger maps monomorphisms into every objectAA is a meet-semilattice.


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

See also

Revision on April 25, 2022 at 14:40:18 by Anonymous?. See the history of this page for a list of all contributions to it.