From 16f39b5bf68b45226702c7a61c4ada40d3afb7a4 Mon Sep 17 00:00:00 2001 From: Jlh18 Date: Sun, 19 Sep 2021 13:47:41 +0100 Subject: [PATCH] 1FundamentalGroup/Quest1 edits --- 1FundamentalGroup/Quest0Part0.md | 2 +- 1FundamentalGroup/Quest0Part1.md | 13 +- 1FundamentalGroup/Quest0Part2.md | 6 +- 1FundamentalGroup/Quest0Part3.md | 19 ++- 1FundamentalGroup/Quest1.agda | 16 ++ 1FundamentalGroup/Quest1Part0.md | 139 ++++++++++++++++++ 1FundamentalGroup/images/doubleCover.png | Bin 0 -> 66483 bytes 1FundamentalGroup/images/lifted_homotopy.png | Bin 0 -> 65725 bytes 1FundamentalGroup/images/lifted_loops.png | Bin 0 -> 69817 bytes .../2.6.2/agda/1FundamentalGroup/Quest1.agdai | Bin 0 -> 55203 bytes 10 files changed, 183 insertions(+), 12 deletions(-) create mode 100644 1FundamentalGroup/Quest1.agda create mode 100644 1FundamentalGroup/images/doubleCover.png create mode 100644 1FundamentalGroup/images/lifted_homotopy.png create mode 100644 1FundamentalGroup/images/lifted_loops.png create mode 100644 _build/2.6.2/agda/1FundamentalGroup/Quest1.agdai diff --git a/1FundamentalGroup/Quest0Part0.md b/1FundamentalGroup/Quest0Part0.md index 5933a58..7556e89 100644 --- a/1FundamentalGroup/Quest0Part0.md +++ b/1FundamentalGroup/Quest0Part0.md @@ -66,7 +66,7 @@ We will fill the hole `{ }0`. ... ``` - This says you have some unfilled __holes_. + This says you have some unfilled _holes_. - navigate to the hole `{ }0` using `C-c C-f` (forward) or `C-c C-b` (backward) - enter `C-c C-r`. The `r` stands for _refine_. Whenever you do this whilst having your cursor in a hole, diff --git a/1FundamentalGroup/Quest0Part1.md b/1FundamentalGroup/Quest0Part1.md index c3bf512..f8153dd 100644 --- a/1FundamentalGroup/Quest0Part1.md +++ b/1FundamentalGroup/Quest0Part1.md @@ -41,7 +41,12 @@ you can hover over `Bool` in `agda` and use `M-SPC c d`.) We will create a map `doubleCover : S¹ → Type` that sends `base` to `Bool` and the path `loop` to a non-trivial path `flipPath : Bool ≡ Bool` in the space of spaces. -(Insert gif of double cover.) + +doubleCover + Viewing the picture vertically, for each point `x : S¹`, we call `doubleCover x` the _fiber of `doubleCover` over `x`_. @@ -52,7 +57,11 @@ in the fiber of `doubleCover` over `base` by 'lifting the homotopy' `h : Refl ≡ loop` and considering the end points of the 'lifted paths'. `Refl` will 'lift' to a 'constant path' and `loop` will 'lift' to -(Insert picture of 'lift' of `loop`) + +lifted_loops Let's assume for the moment that we have `flipPath` already and define `doubleCover`. diff --git a/1FundamentalGroup/Quest0Part2.md b/1FundamentalGroup/Quest0Part2.md index 561d189..ecf93b2 100644 --- a/1FundamentalGroup/Quest0Part2.md +++ b/1FundamentalGroup/Quest0Part2.md @@ -2,7 +2,11 @@ In this part, we will define the path `flipPath : Bool ≡ Bool`. Recall the picture of `doubleCover`. -(Insert gif.) + +doubleCover This means we need `flipPath` to correspond to the unique non-identity permutation of `Bool` diff --git a/1FundamentalGroup/Quest0Part3.md b/1FundamentalGroup/Quest0Part3.md index 865ca75..78aece5 100644 --- a/1FundamentalGroup/Quest0Part3.md +++ b/1FundamentalGroup/Quest0Part3.md @@ -32,26 +32,29 @@ drawn as a dot `true`. When we 'lift' `loop` - starting at the point `true : doubleCover base` - it will look like - +lifted_loops The homotopy `h : Refl ≡ loop` is 'lifted' (starting at 'lifted `Refl`') to some kind of surface - +lifted_homotopy 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` - - +from `true` to `false`. We can evaluate the end points of both 'lifted paths' by using -something in the cubical library called `endPt` -(originally called `subst`). +something in the cubical library (called `subst`) which we call `endPt`. ```agda endPt : (B : A → Type) (p : x ≡ y) (bx : B x) → B y @@ -93,6 +96,6 @@ from `f x` to `f y`. cong : (f : A → B) → (p : x ≡ y) → f x ≡ f y ``` -Using `cong` and `endPtOfTrue` you should be able to complete Quest0. +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. diff --git a/1FundamentalGroup/Quest1.agda b/1FundamentalGroup/Quest1.agda new file mode 100644 index 0000000..1a7bf8e --- /dev/null +++ b/1FundamentalGroup/Quest1.agda @@ -0,0 +1,16 @@ +module 1FundamentalGroup.Quest1 where + +open import Cubical.Core.Everything +open import Cubical.HITs.S1 +open import Cubical.Data.Nat +open import Cubical.Data.Int +open import Cubical.Foundations.Prelude + +Ω : (A : Type) (a : A) → Type +Ω A a = a ≡ a + +loop_times : ℤ → Ω S¹ base +loop pos zero times = refl +loop pos (suc n) times = loop pos n times ∙ loop +loop negsuc zero times = sym loop +loop negsuc (suc n) times = loop negsuc n times ∙ sym loop diff --git a/1FundamentalGroup/Quest1Part0.md b/1FundamentalGroup/Quest1Part0.md index 54ba15f..ec5d6ed 100644 --- a/1FundamentalGroup/Quest1Part0.md +++ b/1FundamentalGroup/Quest1Part0.md @@ -19,6 +19,81 @@ In general the _loop space_ of a space `A` at a point `a` is defined as follows Ω A a = a ≡ a ``` +Clearly for each integer `n : ℤ` we have a path +that is '`loop` around `n` times'. +Locate `loop_times` in the `Quest1.agda` +(note how `agda` treats underscores) + +```agda +loop_times : ℤ → Ω S¹ base +loop n times = {!!} +``` + +Try casing on `n`, you should see + +```agda +loop_times : ℤ → Ω S¹ base +loop pos n times = {!!} +loop negsuc n times = {!!} +``` + +It says to map out of `ℤ` it suffices to +map the non-negative integers (`pos`) +and the negative integers (`negsuc`). + +```agda +data ℤ : Type where + pos : (n : ℕ) → ℤ + negsuc : (n : ℕ) → ℤ +``` + +This definition of `ℤ` uses the naturals, so try +casing on `n` again, you should see + +```agda +loop_times : ℤ → Ω S¹ base +loop pos zero times = {!!} +loop pos (suc n) times = {!!} +loop negsuc n times = {!!} +``` + +It says to map out of `ℕ` it suffices to map `zero` and +map each succesive integer `suc n` inductively. +When we loop `zero` (`pos zero`) times what should we get? +Try filling it in. +For looping `pos (suc n)` times we loop `n` times and +loop once more. +For this we need composition of paths. + +```agda +_∙_ : x ≡ y → y ≡ z → x ≡ z +``` + +Try typing `_∙_` or `? ∙ ?` in the hole (input `/.`) +and refining. +Checking the new holes you should see that now you need +to give two loops. +Try giving it '`loop n times`' composed with `loop`. +Then try to also define the map on the negative integers. +You will need to invert paths using `sym`. + +

+

+Looking up definitions + +If you don't know the definition of something +you can look up the definition by sticking your cursor +on it and pressing `M-SPC c d` in _insert mode_ +or `SPC c d` in _evil mode_. + +You can use it to find out the definition of `ℤ` and `ℕ`. + +
+

+ + + + Warning : the loop space can contain higher homotopical information that the fundamental group does not capture. @@ -49,3 +124,67 @@ However, the 'composition' of the path `southHemisphere` with `northHemisphere` in `base ≡ base` gives the surface of `S²`, which intuitively is not homotopic to the constant point `base`. So `base ≡ base` has non-trivial path structure. + +S2 + +Let's be more precise about homotopical data : +We can check that a space is 'homotopically trivial' (h-trivial) +from dimension `n` +by checking if spheres of dimension `n` can be filled. +To be h-trivial from `0` is for any two points +to have a line in between; to fill `S⁰`. +This data is captured in + +```agda +isProp : Type → Type +isProp A = (x y : A) → x ≡ y +``` + +

+

+All maps are continuous in HoTT + +There is a subtlety in the definition `isProp`. +This is _stronger_ than saying that the space `A` is path connected. +Since `A` is equipped with a continuous map taking pairs `x y : A` +to a path between them. + +We will show that `isProp S¹` is _empty_ despite `S¹` being path connected. + +
+

+ +Similarly, to be h-trivial from `1` is for any two points `x y : A` +and any two paths `p q : x ≡ y` to have a homotopy from `p` to `q`; +to fill `S¹`. This is captured in + +```agda +isSet : Type → Type +isSet A = (x y : A) → isProp (x ≡ y) +``` + +We may therefore want to remove homotopical data +beyond a certain dimension. +We define the fundamental group to be +the homotopical data of `S¹` up to (and +including) dimension `1`, i.e. +the homotopical data of the loop space of `S¹` +up to dimension `0`. + +We make the definitions for a space to +have homotopical data only up to dimension `0` +and `1` respectively + + +```agda +isProp : Type → Type +isProp A = (x y : A) → x ≡ y + +isSet : Type → Type +isSet A = (x y : A) → isProp (x ≡ y) + +isProp→isSet : (A : Type) → isProp A → isSet A +``` diff --git a/1FundamentalGroup/images/doubleCover.png b/1FundamentalGroup/images/doubleCover.png new file mode 100644 index 0000000000000000000000000000000000000000..fda55a4bb6025e14d2482d91d8413ac6fe69a259 GIT binary patch literal 66483 zcmeGDg;!h8_XP}7$BPswrIZf^ihFTMOK}Pmw*W0t916iL6{L7^hvHV8;0Z169vo8K 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