Homotopy Type Theory ETCR > history (Rev #4, changes)

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Idea

A model of the Elementary Theory of the Category of RelationsRel (ETRel) (ETCR) is thedagger 2-poset whose category of maps is a model of ETCS?.

Definition

A model of ETRel ETCR is adagger 2-poset $C$ such that:

• Always Singleton: true there relation: is for an every object$\mathrm{<span><del\; class="diffmod">\; A</del><ins\; class="diffmod">\; 𝟙</ins></span>}:\mathrm{Ob}(C)$ A:Ob(C) \mathbb{1}:Ob(C) and such that for every morphism$\mathrm{<span><del\; class="diffmod">\; B</del><ins\; class="diffmod">\; f</ins></span>}:\mathrm{<span><del\; class="diffmod">\; Ob</del><ins\; class="diffmod">\; Hom</ins></span>}(\mathrm{<span><del\; class="diffmod">\; C</del><ins\; class="diffmod">\; 𝟙</ins></span>},𝟙)$ B:Ob(C) f:Hom(\mathbb{1},\mathbb{1}) , there is a morphism${\top }_{A,B}f<span><del\; class="diffmod">\; :</del><ins\; class="diffmod">\; \le </ins></span>\mathrm{Hom}{1}_{𝟙}(A,B)$ \top_{A,B}:Hom(A,B) f \leq 1_\mathbb{1} , such and that for every other object morphism$\mathrm{<span><del\; class="diffmod">\; a</del><ins\; class="diffmod">\; A</ins></span>}:\mathrm{<span><del\; class="diffmod">\; Hom</del><ins\; class="diffmod">\; Ob</ins></span>}(\mathrm{<span><del\; class="diffmod">\; A</del><ins\; class="diffmod">\; C</ins></span>},B)$ a:Hom(A, A:Ob(C) B) , there is an$a \leq \top_{A,B}$onto dagger morphism ,$u_A:A \to \mathbb{1}$.

• Singleton: Tabulations: there for is every an object$\mathrm{<span><del\; class="diffmod">\; 𝟙</del><ins\; class="diffmod">\; A</ins></span>}:\mathrm{Ob}(C)$ \mathbb{1}:Ob(C) A:Ob(C) such and that${\top }_{𝟙,𝟙}B<span><del\; class="diffmod">\; =</del><ins\; class="diffmod">\; :</ins></span>1_{𝟙}\mathrm{Ob}(C)$ \top_{\mathbb{1},\mathbb{1}} B:Ob(C) = 1_\mathbb{1} , and for morphism every object$\mathrm{<span><del\; class="diffmod">\; A</del><ins\; class="diffmod">\; R</ins></span>}:\mathrm{<span><del\; class="diffmod">\; Ob</del><ins\; class="diffmod">\; Hom</ins></span>}(\mathrm{<span><del\; class="diffmod">\; C</del><ins\; class="diffmod">\; A</ins></span>},B)$ A:Ob(C) R:Hom(A,B) , there is an objectonto dagger morphism$\vert R \vert:Ob(C)$ and$u_A:A \to \mathbb{1}$maps $f:Hom(\vert R \vert, A)$, $g:Hom(\vert R \vert, B)$, such that $R = f^\dagger \circ g$ and for every object $E:Ob(C)$ and maps $h:Hom(E,\vert R \vert)$ and $k:Hom(E,\vert R \vert)$, $f \circ h = f \circ k$ and $g \circ h = g \circ k$ imply $h = k$.

• Cartesian Power products: sets: for every object$A:Ob(C)$ , and there is an object$\mathrm{<span><del\; class="diffmod">\; B</del><ins\; class="diffmod">\; 𝒫</ins></span>}:\mathrm{Ob}(\mathrm{<span><del\; class="diffmod">\; C</del><ins\; class="diffmod">\; A</ins></span>})$ B:Ob(C) \mathcal{P}(A) and a morphism$R{\in }_A:\mathrm{Hom}(A,\mathrm{<span><del\; class="diffmod">\; B</del><ins\; class="diffmod">\; 𝒫</ins></span>}(A))$ R:Hom(A,B) \in_A:Hom(A, \mathcal{P}(A)) , there such is that an for object each morphism$\mathrm{<span><del\; class="diffmod">\; A</del><ins\; class="diffmod">\; R</ins></span>}\times B:\mathrm{<span><del\; class="diffmod">\; Ob</del><ins\; class="diffmod">\; Hom</ins></span>}(\mathrm{<span><del\; class="diffmod">\; C</del><ins\; class="diffmod">\; A</ins></span>},B)$ A R:Hom(A,B) \times B:Ob(C) , and there exists a maps map $f{\chi }_R:\mathrm{Hom}(A\times B,\mathrm{<span><del\; class="diffmod">\; A</del><ins\; class="diffmod">\; P</ins></span>}(B))$ f:Hom(A \chi_R:Hom(A,P(B)) \times B, A) , such that$\mathrm{<span><del\; class="diffmod">\; g</del><ins\; class="diffmod">\; R</ins></span>}<span><del\; class="diffmod">\; :</del><ins\; class="diffmod">\; =</ins></span>\mathrm{Hom}(A{\in }_B^{†}\times B,B)\circ {\chi }_R$ g:Hom(A R \times = B, (\in_B^\dagger) B) \circ \chi_R, such that $\top_{A,B} = f^\dagger \circ g$.

• Tabulations: Function extensionality: for every object$A:Ob(C)$ and $B:Ob(C)$ and morphism$R:Hom(A,B)$, there is an object $\vert R \vert:Ob(C)$ and maps $f:\mathrm{Hom}(|R|,A,B)$ f:Hom(\vert f:Hom(A, R B) \vert, A), $g:\mathrm{Hom}(|R|,A,B)$ g:Hom(\vert g:Hom(A, R B) \vert, A) , such and that$\mathrm{<span><del\; class="diffmod">\; R</del><ins\; class="diffmod">\; x</ins></span>}<span><del\; class="diffmod">\; =</del><ins\; class="diffmod">\; :</ins></span>f^{†}\mathrm{Hom}<span><del\; class="diffmod">\; \circ </del><ins\; class="diffmod">\; (</ins></span>\mathrm{<span><del\; class="diffmod">\; g</del><ins\; class="diffmod">\; 𝟙</ins></span>},A)$ R x:Hom(\mathbb{1}, = A) f^\dagger \circ g , and for two global elements$f\circ x<span><del\; class="diffmod">\; :</del><ins\; class="diffmod">\; =</ins></span>\mathrm{<span><del\; class="diffmod">\; Hom</del><ins\; class="diffmod">\; g</ins></span>}<span><del\; class="diffmod">\; (</del><ins\; class="diffmod">\; \circ </ins></span>\mathrm{<span><del\; class="diffmod">\; 𝟙</del><ins\; class="diffmod">\; x</ins></span>},|R|)$ x:Hom(\mathbb{1},\vert f R \circ \vert) x = g \circ x and implies$\mathrm{<span><del\; class="diffmod">\; y</del><ins\; class="diffmod">\; f</ins></span>}<span><del\; class="diffmod">\; :</del><ins\; class="diffmod">\; =</ins></span>\mathrm{<span><del\; class="diffmod">\; Hom</del><ins\; class="diffmod">\; g</ins></span>}(𝟙,|R|)$ y:Hom(\mathbb{1},\vert f R = \vert) g, $f \circ x = f \circ y$ and $g \circ x = g \circ y$ imply $x = y$.

• Power Natural sets: numbers: for there every is an object$\mathrm{<span><del\; class="diffmod">\; A</del><ins\; class="diffmod">\; \mathbb{N}</ins></span>}:\mathrm{Ob}(C)$ A:Ob(C) \mathbb{N}:Ob(C) , there with is maps an object$𝒫0<span><del\; class="diffmod">\; (</del><ins\; class="diffmod">\; :</ins></span>\mathrm{<span><del\; class="diffmod">\; A</del><ins\; class="diffmod">\; 𝟙</ins></span>}<span><del\; class="diffmod">\; )</del><ins\; class="diffmod">\; \to </ins></span>\mathbb{N}$ \mathcal{P}(A) 0:\mathbb{1} \to \mathbb{N} and a morphism${\in }_{A}s:\mathrm{<span><del\; class="diffmod">\; Hom</del><ins\; class="diffmod">\; \mathbb{N}</ins></span>}<span><del\; class="diffmod">\; (</del><ins\; class="diffmod">\; \to </ins></span>\mathrm{<span><del\; class="diffmod">\; A</del><ins\; class="diffmod">\; \mathbb{N}</ins></span>},𝒫(A))$ \in_A:Hom(A, s:\mathbb{N} \mathcal{P}(A)) \to \mathbb{N} , such that for each morphism object$R:\mathrm{Hom}(A,B)$ R:Hom(A,B) A , there with exists maps amap$0_A:\mathbb{1} \to A$ and${\mathrm{<span><del\; class="diffmod">\; \chi </del><ins\; class="diffmod">\; s</ins></span>}}_{\mathrm{<span><del\; class="diffmod">\; R</del><ins\; class="diffmod">\; A</ins></span>}}:\mathrm{Hom}(A<span><del\; class="diffmod">\; ,</del><ins\; class="diffmod">\; \to </ins></span>\mathrm{<span><del\; class="diffmod">\; P</del><ins\; class="diffmod">\; A</ins></span>}(B))$ \chi_R:Hom(A,P(B)) s_A:A \to A , such there that is a map$\mathrm{<span><del\; class="diffmod">\; R</del><ins\; class="diffmod">\; f</ins></span>}<span><del\; class="diffmod">\; =</del><ins\; class="diffmod">\; :</ins></span>(\mathbb{N}{\in }_B^{†}\to )A\circ {\chi }_R$ R f:\mathbb{N} = \to (\in_B^\dagger) A \circ \chi_R such that $f \circ 0 = 0_A$ and $f \circ s = s_A \circ f$.

• Function Choice: extensionality: for every object$A:Ob(C)$ and $B:Ob(C)$ , and every maps entire $f:Hom(A, B)$dagger epimorphism ,$\mathrm{<span><del\; class="diffmod">\; g</del><ins\; class="diffmod">\; R</ins></span>}:\mathrm{Hom}(A,B)$ g:Hom(A, R: B) Hom(A,B) and has a section.$x:Hom(\mathbb{1}, A)$, $f \circ x = g \circ x$ implies $f = g$.

• Natural numbers: there is an object $\mathbb{N}:Ob(C)$ with maps $0:\mathbb{1} \to \mathbb{N}$ and $s:\mathbb{N} \to \mathbb{N}$, such that for each object $A$ with maps $0_A:\mathbb{1} \to A$ and $s_A:A \to A$, there is a map $f:\mathbb{N} \to A$ such that $f \circ 0 = 0_A$ and $f \circ s = s_A \circ f$.

• Choice: for every object $A:Ob(C)$ and $B:Ob(C)$, every entire dagger epimorphism $R: Hom(A,B)$ has a section.

See also

• Rel
• ETCS?
• Categorical SEAR
• SEAR