nLab ETCR

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.

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

Singleton: there is an object $\mathbb{1} \in Ob(C)$ such that for every morphism $f \in Hom(\mathbb{1},\mathbb{1})$ , $f \leq 1_\mathbb{1}$ , and for every object $A \in Ob(C)$ there is an onto morphism $u_A \in Hom(A,\mathbb{1})$ .
Tabulations: for every object $A \in Ob(C)$ and $B \in Ob(C)$ and morphism $R \in Hom(A,B)$ , there is an object $\vert R \vert \in Ob(C)$ and jointly monic maps $f \in Hom(\vert R \vert, A)$ , $g \in Hom(\vert R \vert, B)$ , such that $R = f^\dagger \circ g$ .
Power sets: for every object $A \in Ob(C)$ , there is an object $\mathcal{P}(A)$ and a morphism $\in_A:Hom(A, \mathcal{P}(A))$ such that for each morphism $R \in Hom(A,B)$ , there exists a map $\chi_R \in Hom(A,P(B))$ such that $R = (\in_B^\dagger) \circ \chi_R$ .
Function extensionality: for every object $A \in Ob(C)$ and $B \in Ob(C)$ and maps $f \in Hom(A, B)$ , $g \in Hom(A, B)$ and $x \in Hom(\mathbb{1}, A)$ , $f \circ x = g \circ x$ implies $f = g$ .
Natural numbers: there is an object $\mathbb{N} \in Ob(C)$ with maps $0 \in Hom(\mathbb{1},\mathbb{N})$ and $s \in Hom(\mathbb{N},\mathbb{N})$ , such that for each object $A$ with maps $0_A \in Hom(\mathbb{1},A)$ and $s_A \in Hom(A,A)$ , there is a map $f \in Hom(\mathbb{N},A)$ such that $f \circ 0 = 0_A$ and $f \circ s = s_A \circ f$ .
Choice: for every object $A \in Ob(C)$ and $B \in Ob(C)$ , every epic map $R \in Hom(A,B)$ has a section.