Homotopy Type Theory SEAR > history (Rev #4)

Idea

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

A model of SEAR $C$ consists of

A type $Ob(C)$ , whose terms are called “sets”.
For each “set” $A:Ob(C)$ , a set $El(A)$ , whose terms are called “elements”.
For each “set” $A:Ob(C)$ and $B:Ob(C)$ , a set $Hom(A,B)$ , whose terms are called “relations”.
For each “set” $A:Ob(C)$ and $B:Ob(C)$ , “relation” $R:Hom(A,B)$ , and “element” $a:El(A)$ and $b:El(B)$ , a type $R(a,b)$ with a term $p:isProp(R(a,b))$
For each “set” $A:Ob(C)$ , $B:Ob(C)$ , $D:Ob(C)$ , a function

$(-)\circ(-):Hom(B,D) \times Hom(A,B) \to Hom(A,D)$

and a term

$\gamma:\prod_{A:Ob(C)} \prod_{B:Ob(C)} \prod_{D:Ob(C)} \prod_{R:Hom(A,B)} \prod_{S:Hom(B,C)} \prod_{a:El(A)} \prod_{d:El(D)} (S \circ R)(a, d) = \left[\sum_{b:B} S(b,d) \times R(a,b)\right]$

where $\left[T\right]$ is the propositional truncation of the type $T$ .
For each “set” $A:Ob(C)$ , $B:Ob(C)$ , a relation

$id_{A,B}:Hom(A,B)$

and terms

$p:\prod_{A:Ob(C)} \prod_{B:Ob(C)} \prod_{R:Hom(A,B)} \prod_{a:El(A)} \prod_{a^{'}:El(A)} \prod_{b:El(B)} \prod_{b^{'}:El(B)} (R(a,b) \times id_A(a,a^{'}) \times id_B(b,b^{'})) \to R(a^{'},b^{'})$

Let the type of all functional entire “relations” between “sets” $A:Ob(C)$ and $B:Ob(C)$ in $C$ be defined as

Map(A, B) \coloneqq \sum_{f:Hom(A,B)} \prod_{a:El(A)} isContr\left(\sum_{b:El(B)} P(f)(a,b)\right)

For each “set” $A:Ob(C)$ and $B:Ob(C)$ , a function $(-)((-)): Map(A,B) \times El(A) \to El(B)$ and a term $\delta:isContr(P(f)(a, f(a)))$ .
A “set” $A:Ob(C)$ with a term $\epsilon:\left[El(A)\right]$ .
For each “set” $A:Ob(C)$ and $B:Ob(C)$ and “relation” $R:Hom(A,B)$ , a “set” $\vert R \vert:Ob(C)$ and functional entire “relation” $f:Hom(\vert R \vert, A)$ , $g:Hom(\vert R \vert, B)$ , and a term $q:R = f^\dagger \circ g$ and a term

$\eta:\prod_{x:El(\vert R \vert)} \prod_{y:El(\vert R \vert)} (f(x) = f(y)) \times (g(x) = g(y)) \to (x = y)$
For each “set” $A:Ob(C)$ , a “set” $\mathcal{P}(A)$ and a “relation” $\in_A:Hom(A, \mathcal{P}(A))$ and a term

$\iota:\prod_{B:Ob(C)} \prod_{R:Hom(A,B)} \left[\sum_{\chi_R:Hom(A,P(B))} isMap(\chi_R) \times (R = (\in_B^\dagger) \circ \chi_R)\right]$
A “set” $\mathbb{N}:Ob(C)$ with functional entire “relations” $0:El(\mathbb{N})$ and $s:Hom(\mathbb{N},\mathbb{N})$ and a term

\kappa:\prod_{A:Ob(C)} \prod_{0_A:El(A)} \prod_{s_A:Hom(A,A)} \left[\sum_{f:Hom(\mathbb{N},A)} (f(0) = 0_A) \times \prod_{n:El(\mathbb{N})} (f(s(n)) = s_A(f(n)))\right]

For each “set” $A:Ob(C)$ and function $Q:El(A) \times Ob(C) \to Prop_\mathcal{U}$ , a “set” $B:Ob(C)$ with a functional entire “relation” $p:Hom(B,A)$ , a function $M:El(B) \to Ob(C)$ and terms
$\lambda:\prod_{b:El(B)} isContr(Q(p(b),M(b)))$
$\mu:\prod_{a:El(A)} \left[\sum_{X:Ob(C)} isContr(Q(a,X))\right] \to \left[\sum_{b:B} p(b) = a\right]$