Homotopy Type Theory ETCR > history (Rev #3)

Idea

A model of the Elementary Theory of Rel (ETRel) is the dagger 2-poset whose category of maps is a model of ETCS?.

Definition

A model of ETRel is a dagger 2-poset $C$ such that:

• Always true relation: for every object $A:Ob(C)$ and $B:Ob(C)$, there is a morphism $\top_{A,B}:Hom(A,B)$ such that for every other morphism $a:Hom(A, B)$, $a \leq \top_{A,B}$,

• Singleton: there is an object $\mathbb{1}:Ob(C)$ such that $\top_{\mathbb{1},\mathbb{1}} = 1_\mathbb{1}$, and for every object $A:Ob(C)$ there is an onto dagger morphism $u_A:A \to \mathbb{1}$.

• Cartesian products: for every object $A:Ob(C)$ and $B:Ob(C)$ and morphism $R:Hom(A,B)$, there is an object $A \times B:Ob(C)$ and maps $f:Hom(A \times B, A)$, $g:Hom(A \times B, B)$, such that $\top_{A,B} = f^\dagger \circ g$.

• Tabulations: 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 $f:Hom(\vert R \vert, A)$, $g:Hom(\vert R \vert, A)$, such that $R = f^\dagger \circ g$ and for two global elements $x:Hom(\mathbb{1},\vert R \vert)$ and $y:Hom(\mathbb{1},\vert R \vert)$, $f \circ x = f \circ y$ and $g \circ x = g \circ y$ imply $x = y$.

• Power sets: for every object $A:Ob(C)$, there is an object $\mathcal{P}(A)$ and a morphism $\in_A:Hom(A, \mathcal{P}(A))$ such that for each morphism $R:Hom(A,B)$, there exists a map $\chi_R:Hom(A,P(B))$ such that $R = (\in_B^\dagger) \circ \chi_R$.

• Function extensionality: for every object $A:Ob(C)$ and $B:Ob(C)$ and maps $f:Hom(A, B)$, $g:Hom(A, B)$ and $x:Hom(\mathbb{1}, A)$, $f \circ x = g \circ x$ 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