99 lines
3.1 KiB
Markdown
99 lines
3.1 KiB
Markdown
# `refl ≡ loop` is empty - 'lifting' paths using the double cover
|
|
|
|
By the end of this page we will have shown that
|
|
`refl ≡ loop` is an empty space.
|
|
In `1FundamentalGroup/Quest0.agda` locate
|
|
|
|
```agda
|
|
Refl≢loop : Refl ≡ loop → ⊥
|
|
Refl≢loop h = ?
|
|
```
|
|
|
|
The cubical library has the result
|
|
`true≢false : true ≡ false → ⊥`
|
|
which says that the space of paths in `Bool`
|
|
from `true` to `false` is empty.
|
|
We will assume it here and leave the proof as a side quest,
|
|
see `1FundamentalGroup/Quest0SideQuests/SideQuest1`.
|
|
|
|
- Load the file with `C-c C-l` and navigate to the hole.
|
|
- Write `true≢false` in the hole and refine using `C-c C-r`,
|
|
`agda` knows `true≢false` maps to `⊥` so it automatically
|
|
will make a new hole.
|
|
- Check the goal in the new hole using `C-c C-,`
|
|
it should be asking for a path from `true` to `false`.
|
|
|
|
To give this path we need to visualise 'lifting' `Refl`, `loop`
|
|
and the homotopy `h : refl ≡ loop`
|
|
along the Boolean-bundle `doubleCover`.
|
|
When we 'lift' `Refl` - starting at the point `true : doubleCover base` -
|
|
it will still be a constant path at `true`,
|
|
drawn as a dot `true`.
|
|
When we 'lift' `loop` - starting at the point `true : doubleCover base` -
|
|
it will look like
|
|
|
|
<!-- [insert picture] -->
|
|
|
|
The homotopy `h : refl ≡ loop` is 'lifted'
|
|
(starting at 'lifted `refl`')
|
|
to some kind of surface
|
|
|
|
<!-- [insert picture] -->
|
|
|
|
According to the pictures the end point of the 'lifted'
|
|
`Refl` is `true` and the end point of the 'lifted' `loop` is `false`.
|
|
We are interested in the end points of each
|
|
'lifted paths' in the 'lifted homotopy',
|
|
since this forms a path in the endpoint fiber `doubleCover base`
|
|
from `true` to `false`
|
|
|
|
<!-- [insert picture] -->
|
|
|
|
We can evaluate the end points of both 'lifted paths' by using
|
|
something in the cubical library called `endPt`
|
|
(originally called `subst`).
|
|
|
|
```agda
|
|
endPt : (B : A → Type) (p : x ≡ y) (bx : B x) → B y
|
|
```
|
|
|
|
<p>
|
|
<details>
|
|
<summary>Try interpreting what it says</summary>
|
|
|
|
It says given a bundle `B` over space `A`,
|
|
a path `p` from `x : A` to `y : A`, and
|
|
a point `bx` above `x`,
|
|
we can get the end point of 'lifted `p` starting at `bx`'.
|
|
So let's make the function that takes
|
|
a path from `base` to `base` and spits out the end point
|
|
of the 'lifted path' starting at `true`.
|
|
|
|
</details>
|
|
</p>
|
|
|
|
```agda
|
|
endPtOfTrue : (p : base ≡ base) → doubleCover base
|
|
endPtOfTrue p = ?
|
|
```
|
|
|
|
Try filling in `endPtOfTrue` using `endPt`
|
|
and the skills you have developed so far.
|
|
You can verify our expectation that `endPtOfTrue Refl` is `true`
|
|
and `endPtOfTrue loop` is `false` using `C-c C-n`.
|
|
|
|
Lastly we need to make the function `endPtOfTrue`
|
|
take the path `h : refl ≡ loop` to a path from `true` to `false`.
|
|
In general if `f : A → B` is a function and `p` is a path
|
|
between points `x y : A` then we get a map `cong f p`
|
|
from `f x` to `f y`.
|
|
(Note that `p` here is actually a homotopy `h`.)
|
|
|
|
```agda
|
|
cong : (f : A → B) → (p : x ≡ y) → f x ≡ f y
|
|
```
|
|
|
|
Using `cong` and `endPtOfTrue` you should be able to complete Quest0.
|
|
If you have done everything correctly you can reload `agda` and see that
|
|
you have no remaining goals.
|