# Homotopy Type Theory geometric dagger 2-poset > history (Rev #1)

## Idea

A geometric dagger 2-poset is a dagger 2-poset whose category of maps is a geometric category.

## Definition

A geometric dagger 2-poset is a dagger 2-poset $C$ such that

• There is an object $0:Ob(C)$ such that for each object $A:Ob(C)$, there is a monic map $i_{0,A}:Hom(0,A)$ such that for each object $B:Ob(C)$ with a monic map $i_{B,A}:Hom(B,A)$, there is a monic map $i_{0,B}:Hom(0,B)$ such that $i_{B,A} \circ i_{0,B} = i_{0,A}$.

• For each object $A:Ob(C)$, type $T$, and family of objects $B:T \to Ob(C)$, with monic maps

$t:T \vdash i_{B(t),A}:Hom(B(t),A)$

there is an object

$\bigcup_{t:T} B(t):Ob(C)$

with monic maps

$i_{\bigcup_{t:T} B(t),A}:Hom(\bigcup_{t:T} B(t),A)$
$t:T \vdash i_{B(t),\bigcup_{t:T} B(t)}:Hom(B(t),\bigcup_{t:T} B(t))$

such that

$t:T \vdash i_{\bigcup_{t:T} B(t),A} \circ i_{B(t),\bigcup_{t:T} B(t)} = i_{B(t),A}$

and for every object $D:Ob(C)$ with monic maps $i_{D,A}:Hom(D,A)$ $i_{B,D}:Hom(B,D)$, $i_{E,D}:Hom(E,D)$ such that $i_{D,A} \circ i_{B,D} = i_{B,A}$ and $i_{D,A} \circ i_{E,D} = i_{E,A}$, there is a monic map

$i_{\bigcup_{t:T} B(t),D}:Hom(\bigcup_{t:T} B(t),D)$

such that

$i_{D,A} \circ i_{\bigcup_{t:T} B(t),D} = i_{\bigcup_{t:T} B(t),A}$
• For each object $A:Ob(C)$, $B:Ob(C)$, $E:Ob(C)$ with monic maps $i_{B,A}:Hom(B,A)$, $i_{E,A}:Hom(E,A)$, there is an object $B \cap E:Ob(C)$ with monic maps $i_{B \cap E,A}:Hom(B \cap E,A)$, $i_{B \cap E,B}:Hom(B \cap E,B)$, $i_{B \cap E,E}:Hom(B \cap E,E)$, such that $i_{B,A} \circ i_{B \cap E,B} = i_{B \cap E,A}$ and $i_{E,A} \circ i_{B \cap E,E} = i_{B \cap E,A}$, and for every object $D:Ob(C)$ with monic maps $i_{D,A}:Hom(D,A)$ $i_{D,B}:Hom(D,B)$, $i_{D,E}:Hom(D,E)$ such that $i_{B,A} \circ i_{D,B} = i_{D,A}$ and $i_{E,A} \circ i_{D,E} = i_{D,A}$, there is a monic map $i_{D,B \cap E}:Hom(D,B \cap E)$ such that $i_{B \cap E,A} \circ i_{D,B \cap E} = i_{D,A}$.

• For each object $A:Ob(C)$, $E:Ob(C)$, with monic maps $i_{E,A}:Hom(E,A)$ and type $T$, and family of objects $B:T \to Ob(C)$, with monic maps

$t:T \vdash i_{B(t),A}:Hom(B(t),A)$

## Properties

• For each object $A:Ob(C)$, the identity function $1_A:Hom(A,A)$ is a monic map, and for each object $B:Ob(C)$ with a monic map $i_{B,A}:Hom(B,A)$, $1_A \circ i_{B,A} = i_{B,A}$.

• The unitary isomorphisms classes of monic maps into every object $A$ is a frame. Since every monic map is a map, the category of maps is a geometric category?.

## Examples

The dagger 2-poset of sets and relations is a coherent dagger 2-poset.