Homotopy Type Theory SEAR > history (Rev #1)

Idea

Mike Shulman’s dependently typed first-order theory SEAR.

Definition

A model of SEAR is a dagger 2-poset $C$ with a set $El(A)$ of elements in $A$ for every object $A:Ob(C)$ such that

• Nontriviality: There exists an object $A:Ob(C)$ with a term $p:\left[El(A)\right]$, where $\left[T\right]$ is the propositional truncation of the type $T$.

• Relational comprehension: for every object $A:Ob(C)$ and $B:Ob(C)$ there is a function $P:Hom(A,B) \to (El(A) \times El(B) \to Prop_\mathcal{U})$

• Function evaluation: For every map $f:Hom(A,B)$, there is a function $(-)((-)): Hom(A,B) \times El(A) \to El(B)$ such that $P(f)(a, f(a))$ is contractible.

• 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 elements $x:El(\vert R \vert)$ and $y:El(\vert R \vert)$, $f(x) = f(y)$ and $g(x) = g(y)$ 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 $R = (\in_B^\dagger) \circ \chi_R$.

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

• Collection: for every object $A:Ob(C)$ and function $P:El(A) \times Ob(C) \to Prop_\mathcal{U}$, there exists an object $B:Ob(C)$ with a map $p:Hom(A,B)$ and a $B$-indexed family of objects $M:Hom(\mathbb{1},B) \to \mathcal{U}$ such that (1) for every element $b:El(B)$, $P(p(b),M(b))$ is contractible and (2) for every element $a:El(A)$, if there exists an object $X:Ob(C)$ such that $P(a,X)$ is contractible, then $a$ is in the image of $p$.

See also

• Rel
• Categorical SEAR