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

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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 $C$ such that

• For each object $A:Ob(C)$, $B:Ob(C)$, $E:Ob(C)$ with functional monic map dagger monomorphisms$i_{B,A}:Hom(B,A)$ ,$i_{\mathrm{<span><del\; class="diffmod">\; B</del><ins\; class="diffmod">\; E</ins></span>},A}:\mathrm{Hom}(\mathrm{<span><del\; class="diffmod">\; B</del><ins\; class="diffmod">\; E</ins></span>},A)$ i_{B,A}:Hom(B,A) i_{E,A}:Hom(E,A) , there is an object$i_{E,A}B\cap E:\mathrm{<span><del\; class="diffmod">\; Hom</del><ins\; class="diffmod">\; Ob</ins></span>}(\mathrm{<span><del\; class="diffmod">\; E</del><ins\; class="diffmod">\; C</ins></span>},A)$ i_{E,A}:Hom(E,A) B \cap E:Ob(C) , there with is monic an maps object$i_{B\cap E,A}:\mathrm{Hom}(B\cap E<span><del\; class="diffmod">\; :</del><ins\; class="diffmod">\; ,</ins></span>\mathrm{<span><del\; class="diffmod">\; Ob</del><ins\; class="diffmod">\; A</ins></span>}(C)$ B i_{B \cap E:Ob(C) E,A}:Hom(B \cap E,A) , with functional dagger monomorphisms$i_{B\cap E,\mathrm{<span><del\; class="diffmod">\; A</del><ins\; class="diffmod">\; B</ins></span>}}:\mathrm{Hom}(B\cap E,\mathrm{<span><del\; class="diffmod">\; A</del><ins\; class="diffmod">\; B</ins></span>})$ i_{B \cap E,A}:Hom(B E,B}:Hom(B \cap E,A) E,B), $i_{B\cap E,\mathrm{<span><del\; class="diffmod">\; B</del><ins\; class="diffmod">\; E</ins></span>}}:\mathrm{Hom}(B\cap E,\mathrm{<span><del\; class="diffmod">\; B</del><ins\; class="diffmod">\; E</ins></span>})$ i_{B \cap E,B}:Hom(B E,E}:Hom(B \cap E,B) E,E) , such that$i_{B\cap E,\mathrm{<span><del\; class="diffmod">\; E</del><ins\; class="diffmod">\; A</ins></span>}}<span><del\; class="diffmod">\; :</del><ins\; class="diffmod">\; \circ </ins></span>\mathrm{Hom}{i}_{B\cap E,B}<span><del\; class="diffmod">\; (</del><ins\; class="diffmod">\; =</ins></span>B{i}_{B\cap E,A}\cap E,E)$ i_{B,A} \circ i_{B \cap E,E}:Hom(B E,B} = i_{B \cap E,E) E,A} , such and that$i_{\mathrm{<span><del\; class="diffmod">\; B</del><ins\; class="diffmod">\; E</ins></span>},A}\circ i_{B\cap E,\mathrm{<span><del\; class="diffmod">\; B</del><ins\; class="diffmod">\; E</ins></span>}}=i_{B\cap E,A}$ i_{B,A} i_{E,A} \circ i_{B \cap E,B} E,E} = i_{B \cap E,A} , and for every object$i_{E,A}D<span><del\; class="diffmod">\; \circ </del><ins\; class="diffmod">\; :</ins></span>i_{B\cap E,E}\mathrm{Ob}<span><del\; class="diffmod">\; =</del><ins\; class="diffmod">\; (</ins></span>i_{B\cap E,A}C)$ i_{E,A} D:Ob(C) \circ i_{B \cap E,E} = i_{B \cap E,A} , and with for monic every maps object$D{i}_{D,A}:\mathrm{<span><del\; class="diffmod">\; Ob</del><ins\; class="diffmod">\; Hom</ins></span>}(\mathrm{<span><del\; class="diffmod">\; C</del><ins\; class="diffmod">\; D</ins></span>},A)$ D:Ob(C) i_{D,A}:Hom(D,A) with functional dagger monomorphisms$i_{D,\mathrm{<span><del\; class="diffmod">\; A</del><ins\; class="diffmod">\; B</ins></span>}}:\mathrm{Hom}(D,\mathrm{<span><del\; class="diffmod">\; A</del><ins\; class="diffmod">\; B</ins></span>})$ i_{D,A}:Hom(D,A) i_{D,B}:Hom(D,B) ,$i_{D,\mathrm{<span><del\; class="diffmod">\; B</del><ins\; class="diffmod">\; E</ins></span>}}:\mathrm{Hom}(D,\mathrm{<span><del\; class="diffmod">\; B</del><ins\; class="diffmod">\; E</ins></span>})$ i_{D,B}:Hom(D,B) i_{D,E}:Hom(D,E) , such that$i_{\mathrm{<span><del\; class="diffmod">\; D</del><ins\; class="diffmod">\; B</ins></span>},\mathrm{<span><del\; class="diffmod">\; E</del><ins\; class="diffmod">\; A</ins></span>}}<span><del\; class="diffmod">\; :</del><ins\; class="diffmod">\; \circ </ins></span>\mathrm{Hom}{i}_{D,B}<span><del\; class="diffmod">\; (</del><ins\; class="diffmod">\; =</ins></span>D{i}_{D,A},E)$ i_{D,E}:Hom(D,E) i_{B,A} \circ i_{D,B} = i_{D,A} such and that$i_{\mathrm{<span><del\; class="diffmod">\; B</del><ins\; class="diffmod">\; E</ins></span>},A}\circ i_{D,\mathrm{<span><del\; class="diffmod">\; B</del><ins\; class="diffmod">\; E</ins></span>}}=i_{D,A}$ i_{B,A} i_{E,A} \circ i_{D,B} i_{D,E} = i_{D,A} , and there is a monic map$i_{\mathrm{<span><del\; class="diffmod">\; E</del><ins\; class="diffmod">\; D</ins></span>},\mathrm{<span><del\; class="diffmod">\; A</del><ins\; class="diffmod">\; B</ins></span>}\cap E}<span><del\; class="diffmod">\; \circ </del><ins\; class="diffmod">\; :</ins></span>i_{D,E}\mathrm{Hom}<span><del\; class="diffmod">\; =</del><ins\; class="diffmod">\; (</ins></span>i_{D,A}D,B\cap E)$ 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 monomorphism$i_{B\cap E,A}\circ i_{D,B\cap E}<span><del\; class="diffmod">\; :</del><ins\; class="diffmod">\; =</ins></span>\mathrm{Hom}{i}_{D,A}(D,B\cap E)$ i_{B \cap E,A} \circ i_{D,B \cap E}:Hom(D,B E} \cap = E) i_{D,A} such that $i_{B \cap E,A} \circ i_{D,B \cap E} = i_{D,A}$.

Properties

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

• The unitary isomorphism classes of functional monic dagger maps monomorphisms into every object$A$ is a meet-semilattice.

Examples

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

See also

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