# Homotopy Type Theory Categorical SEAR > history (Rev #4, changes)

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## Idea

< SEAR

A general (non-concrete) categorical version of Mike Shulman’s dependently typed first-order theory SEAR.

## Definition

Assuming excluded middle, a model of categorical SEAR is a dagger 2-poset $C$ such that:

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

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

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

• 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 $R = (\in_B^\dagger) \circ \chi_R$.

• Natural numbers: there is an object $\mathbb{N}:Ob(C)$ with maps $0:Hom(\mathbb{1},\mathbb{N})$ and $s:Hom(\mathbb{N},\mathbb{N})$, such that for every object $A$ with maps $0_A:Hom(\mathbb{1},A)$ and $s_A:Hom(A,A)$, there is a map $f:\mathbb{N} \to A$ such that $f \circ 0 = 0_A$ and $f \circ s = s_A \circ f$.

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