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

## Idea

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

## Definition

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

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

• 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 $f:Hom(\vert R \vert, A)$, $g:Hom(\vert R \vert, B)$, 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 $h = k$.

• 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.