105 lines
3.8 KiB
Markdown
105 lines
3.8 KiB
Markdown
# `Refl ≡ loop` is empty
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To get a better feel of `S¹`, we show that the space of paths (homotopies) between
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`Refl` and `loop`, written `Refl ≡ loop`, is empty.
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First, we define the empty space and what it means for a space to be empty.
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Here is what this looks like in `agda` :
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```agda
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data ⊥ : Type where
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```
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This says "the empty space `⊥` is a space with no points in it".
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Here are three candidate definitions for a space `A` to be empty :
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- there is a point `f : A → ⊥` in the space of functions from `A` to the empty space
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- there is a path `p : A ≡ ⊥` in the space of spaces `Type` from `A` to the empty space
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- there is an isomorphism `i : A ≅ ⊥` of spaces
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These turn out to be 'the same' (see `1FundamentalGroup/Quest0SideQuests/SideQuest0`),
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however for our present purposes we will use the first definition.
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Our goal is therefore to produce a point in the function space
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```agda
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( Refl ≡ loop ) → ⊥
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```
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The authors of this series have thought long and hard
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about how one would come up with the following argument.
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Unfortunately, sometimes mathematics is in need of a new trick
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and this was one of them.
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> The trick is to make a path `p : true ≡ false` from the assumed path (homotopy) `h : Refl ≡ loop` by
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> constructing a non-trivial `Bool`-bundle over the circle,
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> hence obtaining a map `( Refl ≡ loop ) → ⊥`.
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To elaborate :
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`Bool` here refers to the discrete space with two points `true, false`.
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(To find out the definition of `Bool` in `agda`
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you can hover over `Bool` in `agda` and use `M-SPC c d`.)
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We will create a map `doubleCover : S¹ → Type` that sends
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`base` to `Bool` and the path `loop` to a non-trivial path `flipPath : Bool ≡ Bool`
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in the space of spaces.
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<img src="images/doubleCover.png"
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alt="doubleCover"
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width="500"
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class="center"/>
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Viewing the picture vertically,
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for each point `x : S¹`,
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we call `doubleCover x` the _fiber of `doubleCover` over `x`_.
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All the fibers look like `Bool`, hence our choice of the name _`Bool`-bundle_.
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We will get a path from `true` to `false`
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in the fiber of `doubleCover` over `base`
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by 'lifting the homotopy' `h : Refl ≡ loop` and considering the end points of
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the 'lifted paths'.
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`Refl` will 'lift' to a 'constant path' and `loop` will 'lift' to
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<img src="images/lifted_loops.png"
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alt="lifted_loops"
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width="500"
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class="center"/>
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Let's assume for the moment that we have `flipPath` already and
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define `doubleCover`.
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- Navigate to the definition of `doubleCover` and make sure
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you have loaded the file with `C-c C-l`.
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```agda
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doubleCover : S¹ → Type
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doubleCover x = {!!}
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```
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- Navigate your cursor to the hole,
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write `x` and do `C-c C-c`.
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The `c` stands for _cases_.
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You should now see two new holes :
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```agda
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doubleCover : S¹ → Type
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doubleCover base = {!!}
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doubleCover (loop i) = {!!}
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```
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This means :
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`S¹` is made from a point `base` and an edge `loop`,
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so a map out of `S¹` to a space is the same as choosing
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a point and an edge to map `base` and `loop` to respectively.
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Since `loop` is a path from `base` to itself,
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its image must also be a path from the image of `base` to itself.
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- Use `C-c C-f` and/or `C-c C-b` to navigate to the first hole.
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We want to map `base` to `Bool` so
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fill the hole with `Bool` using `C-c C-SPC`.
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- Navigate to the second hole.
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Here `loop i` is a generic point in the path `loop`,
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where `i : I` is a generic point of the 'unit interval'.
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We want to map `loop` to `flipPath`,
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so `loop i` should map to a generic point in the path `flipPath`.
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Try filling the hole.
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- Once you think you are done, reload the `agda` file with `C-c C-l`
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and if it doesn't complain this means there are no problems with your definition.
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Defining `flipPath` is quite involved and we will do so in the next quest!
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