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 CC such that:

  • Always Singleton: true there relation: is for an every object A 𝟙:Ob(C) A:Ob(C) \mathbb{1}:Ob(C) and such that for every morphism B f: Ob Hom( C 𝟙,𝟙) B:Ob(C) f:Hom(\mathbb{1},\mathbb{1}) , there is a morphism A,Bf :Hom1 𝟙(A,B) \top_{A,B}:Hom(A,B) f \leq 1_\mathbb{1} , such and that for every other object morphism a A: Hom Ob( A C,B) a:Hom(A, A:Ob(C) B) , there is ana A,Ba \leq \top_{A,B}onto dagger morphism ,u A:A𝟙u_A:A \to \mathbb{1}.

  • Singleton: Tabulations: there for is every an object 𝟙 A:Ob(C) \mathbb{1}:Ob(C) A:Ob(C) such and that 𝟙,𝟙B = :1 𝟙Ob(C) \top_{\mathbb{1},\mathbb{1}} B:Ob(C) = 1_\mathbb{1} , and for morphism every object A R: Ob Hom( C A,B) A:Ob(C) R:Hom(A,B) , there is an objectonto dagger morphism|R|:Ob(C)\vert R \vert:Ob(C) andu A:A𝟙u_A:A \to \mathbb{1}maps f:Hom(|R|,A)f:Hom(\vert R \vert, A), g:Hom(|R|,B)g:Hom(\vert R \vert, B), such that R=f gR = f^\dagger \circ g and for every object E:Ob(C)E:Ob(C) and maps h:Hom(E,|R|)h:Hom(E,\vert R \vert) and k:Hom(E,|R|)k:Hom(E,\vert R \vert), fh=fkf \circ h = f \circ k and gh=gkg \circ h = g \circ k imply h=kh = k.

  • Cartesian Power products: sets: for every objectA:Ob(C)A:Ob(C) , and there is an object B 𝒫:Ob( C A) B:Ob(C) \mathcal{P}(A) and a morphismR A:Hom(A, B 𝒫(A)) R:Hom(A,B) \in_A:Hom(A, \mathcal{P}(A)) , there such is that an for object each morphism A R×B: Ob Hom( C A,B) A R:Hom(A,B) \times B:Ob(C) , and there exists a maps map fχ R:Hom(A×B, A P(B)) f:Hom(A \chi_R:Hom(A,P(B)) \times B, A) , such that g R : =Hom(A B ×B,B)χ R g:Hom(A R \times = B, (\in_B^\dagger) B) \circ \chi_R, such that A,B=f g\top_{A,B} = f^\dagger \circ g.

  • Tabulations: Function extensionality: for every objectA:Ob(C)A:Ob(C) and B:Ob(C)B:Ob(C) and morphismR:Hom(A,B)R:Hom(A,B), there is an object |R|:Ob(C)\vert R \vert:Ob(C) and maps f:Hom(|R|,A,B) f:Hom(\vert f:Hom(A, R B) \vert, A), g:Hom(|R|,A,B) g:Hom(\vert g:Hom(A, R B) \vert, A) , such and that R x = :f Hom ( g 𝟙,A) R x:Hom(\mathbb{1}, = A) f^\dagger \circ g , and for two global elementsfx : = Hom g ( 𝟙 x,|R|) x:Hom(\mathbb{1},\vert f R \circ \vert) x = g \circ x and implies y f : = Hom g(𝟙,|R|) y:Hom(\mathbb{1},\vert f R = \vert) g, fx=fyf \circ x = f \circ y and gx=gyg \circ x = g \circ y imply x=yx = y.

  • Power Natural sets: numbers: for there every is an object A:Ob(C) A:Ob(C) \mathbb{N}:Ob(C) , there with is maps an object𝒫0 ( : A 𝟙 ) \mathcal{P}(A) 0:\mathbb{1} \to \mathbb{N} and a morphism As: Hom ( A,𝒫(A)) \in_A:Hom(A, s:\mathbb{N} \mathcal{P}(A)) \to \mathbb{N} , such that for each morphism objectR:Hom(A,B) R:Hom(A,B) A , there with exists maps amap0 A:𝟙A0_A:\mathbb{1} \to A and χ s R A:Hom(A , P A(B)) \chi_R:Hom(A,P(B)) s_A:A \to A , such there that is a map R f = :( B )Aχ R R f:\mathbb{N} = \to (\in_B^\dagger) A \circ \chi_R such that f0=0 Af \circ 0 = 0_A and fs=s Aff \circ s = s_A \circ f.

  • Function Choice: extensionality: for every objectA:Ob(C)A:Ob(C) and B:Ob(C)B:Ob(C) , and every maps entire f:Hom(A,B)f:Hom(A, B)dagger epimorphism , g R:Hom(A,B) g:Hom(A, R: B) Hom(A,B) and has a section.x:Hom(𝟙,A)x:Hom(\mathbb{1}, A), fx=gxf \circ x = g \circ x implies f=gf = g.

  • Natural numbers: there is an object :Ob(C)\mathbb{N}:Ob(C) with maps 0:𝟙0:\mathbb{1} \to \mathbb{N} and s:s:\mathbb{N} \to \mathbb{N}, such that for each object AA with maps 0 A:𝟙A0_A:\mathbb{1} \to A and s A:AAs_A:A \to A, there is a map f:Af:\mathbb{N} \to A such that f0=0 Af \circ 0 = 0_A and fs=s Aff \circ s = s_A \circ f.

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

See also

Revision on April 23, 2022 at 03:49:54 by Anonymous?. See the history of this page for a list of all contributions to it.