From 041367daa0670703e18683c3330a6b99f0b77000 Mon Sep 17 00:00:00 2001 From: kl-i Date: Wed, 15 Sep 2021 19:15:12 +0100 Subject: [PATCH] Added Quest0Part2.md, Quest0Part3.md --- 1FundamentalGroup/Quest0.agda | 67 ++------ 1FundamentalGroup/Quest0Part0.md | 1 + 1FundamentalGroup/Quest0Part2.md | 155 ++++++++++++++++++ 1FundamentalGroup/Quest0Part3.md | 87 ++++++++++ .../2.6.3/agda/1FundamentalGroup/Quest0.agdai | Bin 54066 -> 53617 bytes 5 files changed, 258 insertions(+), 52 deletions(-) create mode 100644 1FundamentalGroup/Quest0Part2.md create mode 100644 1FundamentalGroup/Quest0Part3.md diff --git a/1FundamentalGroup/Quest0.agda b/1FundamentalGroup/Quest0.agda index fd12ba5..a8c4364 100644 --- a/1FundamentalGroup/Quest0.agda +++ b/1FundamentalGroup/Quest0.agda @@ -3,14 +3,10 @@ module 1FundamentalGroup.Quest0 where open import Cubical.Data.Empty open import Cubical.Data.Unit renaming ( Unit to ⊤ ) open import Cubical.Data.Bool -open import Cubical.Foundations.Prelude +open import Cubical.Foundations.Prelude renaming ( subst to endPt ) open import Cubical.Foundations.Isomorphism renaming ( Iso to _≅_ ) open import Cubical.Foundations.Path -private - variable - u : Level - data S¹ : Type where base : S¹ loop : base ≡ base @@ -18,33 +14,21 @@ data S¹ : Type where Refl : base ≡ base Refl = λ i → base +{- transport + +To follow a point in `a : A` along a path `p : A ≡ B` +we use + + transport : {A B : Type u} → A ≡ B → A → B + +Why do we propify? Discuss. + +-} + Flip : Bool → Bool Flip false = true Flip true = false -{- Iso - -We show that Flip is an isomorphism from Bool → Bool -with inverse Flip. - -A proof of `A ≅ B` (input \cong or write Iso A B) is given by - - iso f i s r - -where - - f : A → B and i : B → A - -are the map and its inverse, -here both `f` and `i` are Flip - -`s` is a proof that `f` is a section with -right inverse `i` and -`r` is a proof that `f` is a retraction -with left inverse `i` - --} - flipIso : Bool ≅ Bool flipIso = iso Flip Flip s r where s : section Flip Flip @@ -55,30 +39,9 @@ flipIso = iso Flip Flip s r where r false = refl r true = refl -{- Path ≡ - -A corollary of univalence is -`isoToPath` which takes an isomorphism -`f : A ≅ B` and gives a path -`fPath : A ≡ B`. -The resulting path has the important property -that when you follow (transport/subst) -a point in `A` along the path -you will get the point `f(a)` in `B` - --} - flipPath : Bool ≡ Bool flipPath = isoToPath flipIso -{- - -Try out `transport` on `true : Bool` and -`flipPath` by doing `C-c C-n` -and typing in `transport flipPath true` - --} - {- bundle over S¹ We want to construct a bundle over S¹ @@ -117,8 +80,8 @@ Note that `doubleCover base` is just `Bool` (externally). -} -SubstTrue : (p : base ≡ base) → doubleCover base -SubstTrue p = subst doubleCover p true +endPtOfTrue : (p : base ≡ base) → doubleCover base +endPtOfTrue p = endPt doubleCover p true {- @@ -145,4 +108,4 @@ by -} refl≢loop : refl ≡ loop → ⊥ -refl≢loop p = true≢false (cong SubstTrue p) +refl≢loop p = true≢false (cong endPtOfTrue p) diff --git a/1FundamentalGroup/Quest0Part0.md b/1FundamentalGroup/Quest0Part0.md index 65138a9..c49fcc7 100644 --- a/1FundamentalGroup/Quest0Part0.md +++ b/1FundamentalGroup/Quest0Part0.md @@ -86,3 +86,4 @@ We will fill the hole `{ }0`. - if you want to play around with this you can start again by replacing what you wrote with `?` and doing `C-c C-l` + diff --git a/1FundamentalGroup/Quest0Part2.md b/1FundamentalGroup/Quest0Part2.md new file mode 100644 index 0000000..417dd8c --- /dev/null +++ b/1FundamentalGroup/Quest0Part2.md @@ -0,0 +1,155 @@ +# `refl ≡ loop` is empty - Defining `flipPath` via Univalence + +In this part, we will define the path `flipPath : Bool ≡ Bool`. +Recall the picture of `doubleCover`. +(Insert gif.) + +This means we need `flipPath` to correspond to +the unique non-identity permutation of `Bool` +that flips `true` and `false`. + +We proceed in steps : + +1. Define the function `Flip : Bool → Bool`. +2. Promote this to an isomorphism `flipIso : Bool ≅ Bool`. +3. The intuition is that the univalence axiom asserts that + paths in the space of spaces correspond to + homotopy-equivalences of spaces. + As a corollary, + we can make paths in `Type` from isomorphisms of types. + We use this to turn `flipIso` into + a path `flipPath : Bool ≡ Bool`. + +## The function + +- In `1FundamentalGroup/Quest0.agda`, navigate to : + + ```agda + Flip : Bool → Bool + Flip x = {!!} + ``` +- Write `x` inside the hole, + and do `C-c C-c` with your cursor still inside. + The `c` stands for _cases_. + You should now see : + ```agda + Flip : Bool → Bool + Flip false = {!!} + Flip true = {!!} + ``` + What this is saying is that + the space `Bool` is made of two points `false, true` and nothing else, + so to map out of it, + it suffices to give something to map `false` and `true` to respectively. +- Since we want `Flip` to flip `true` and `false`, + fill the first hole with `true` and the second with `false`. +- To check things have worked, + try `C-c C-d`. (`d` stands for _deduce_.) + Then `agda` will ask you to input an expression. + Enter `Flip`. + In the `*Agda Information*` window, + you should see + + ```agda + Bool → Bool + ``` + + This means `agda` recognises `Flip` as a well-formulated term + and is a point in the space of maps from `Bool` to `Bool`. +- We can also ask `agda` to compute outputs of `Flip`. + Try `C-c C-n`. (`n` stands for _normalise_.) + `agda` should again be asking for an expression. + Enter `Flip true`. + In the `*Agda Information*` window, you should see `false`, as desired. + +## The isomorphism + +- Navigate to + ```agda + flipIso : Bool ≅ Bool + flipIso = {!!} + ``` +- Write `iso` in the hole and refine with `C-c C-r`. + You should now see + ```agda + flipIso : Bool ≅ Bool + flipIso = iso {!!} {!!} {!!} {!!} + ``` +- Check that what `agda` is expecting in the first two holes + are functions `Bool → Bool`. + These are our maps back and forth which will constitute the isomorphism + so write `Flip` and `Flip` in the first two holes. +- Check the goal of the next two holes. + They should be + ```agda + section Flip Flip + ``` + and + ```agda + retract Flip Flip + ``` + This means we need to prove + `Flip` is a right inverse and a left inverse of `Flip`. +- Write the following so that your code looks like + ```agda + flipIso : Bool ≅ Bool + flipIso = iso Flip Flip s r where + + s : section Flip Flip + s b = {!!} + + r : retract Flip Flip + r b = {!!} + ``` + The `where` allows you to make definitions local to the current definition, + in the sense that you will not be able to access `s` and `r` outside this proof. + Note that what follows `where` must be indented. +- Check the goal of the hole `s b = {!!}`. + In the `*Agda Information*` window, you should see + ```agda + Goal: Flip (Flip b) ≡ b + ————————————————————————————————— + b : Bool + ``` + Try to prove this. +

+

+ Hint + + You need to do cases on what `b` can be. + Then for the case of `true` and `false`, + try `C-c C-r` to see if `agda` can help. +
+

+- Do the same for `r b = {!!}`. +- Use `C-c C-d` to check that `agda` is okay with `flipIso`. + +## The path + +- Navigate to + ```agda + flipPath : Bool ≡ Bool + flipPath = {!!} + ``` +- In the hole, write in `isoToPath` and refine with `C-c C-r`. + You should now have + ```agda + flipPath : Bool ≡ Bool + flipPath = isoToPath {!!} + ``` + If you check the new hole, you should see that + `agda` is expecting an isomorphism `Bool ≅ Bool`. + + `isoToPath` is the function from the cubical library + that converts isomorphisms between spaces + into paths between the corresponding points in the space of spaces `Type`. +- Fill in the hole with `flipIso` + and use `C-c C-d` to check `agda` is happy with `flipPath`. +- Try `C-c C-n` with `transport flipPath false`. + You should get `true` in the `*Agda Information*` window. + + What `transport` did is it took the path `flipPath` in the + space of spaces `Type` and followed the point `false` + as `Bool` is transformed along `flipPath`. + The end result is of course `true`, + since `flipPath` is the path obtained from `flip`! diff --git a/1FundamentalGroup/Quest0Part3.md b/1FundamentalGroup/Quest0Part3.md new file mode 100644 index 0000000..740dd3c --- /dev/null +++ b/1FundamentalGroup/Quest0Part3.md @@ -0,0 +1,87 @@ +# `refl ≡ loop` is empty - transporting paths using the double cover + +By the end of this page we will have shown that +`refl ≡ loop` is an empty space, +we start at the end, moving backwards to what we need, +as we would often do in practice. + +In `Quest0.agda` you should see + +```agda +Refl≢loop : Refl ≡ loop → ⊥ +Refl≢loop h = ? +``` + +In the library we have +`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 it as a side quest, +see `1FundamentalGroup/Quest0SideQuests/SideQuest0`. + +- 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` and `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`, +which we can just draw as a dot `true`. +When we 'lift' `loop` - starting at the point `true : doubleCover base` - +it will look like + + + +We can find the end points of both 'lifted paths' by using `subst`. +We should be able to see that the end point of the 'lifted' +`Refl` is just `true` and the end point of the 'lifted' `loop` is `false`. +Now a homotopy `h : refl ≡ loop` is 'lifted' to some kind of surface + + + +The end points of each 'lifted paths' in the 'lifted homotopy' +form a path in the endpoint fiber `doubleCover base` +from the endpoint of 'lifted `Refl`' to the endpoint of 'lifted `base`', +i.e. a path from `true` to `false` in `Bool`, which is what we need. + +We use `endPt` to pick out the end points of 'lifted paths', +given to us in the library (originally called `subst`): + +```agda +endPt : (B : A → Type) (p : x ≡ y) (bx : B x) → B y +``` + +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'. + +```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 check that `endPtOfTrue Refl` is `true` +and that `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. diff --git a/_build/2.6.3/agda/1FundamentalGroup/Quest0.agdai b/_build/2.6.3/agda/1FundamentalGroup/Quest0.agdai index 99fa6817216af6e1a6b58af3b86823c404ffb682..15c98d4f5281b21bf386942c2bb82934cd3370fb 100644 GIT binary patch literal 53617 zcmXVXcR&;A^Y^p6X@mfw2%)bM1y7WK1*I+`5>JF6cZ!wfDHh~xSP)oHR1`6yr`%Zx z$O*jaDc?gZsGuO=*`nsuiv<-6MzIpmVBnnxf98)w_St#nnfc6T=9y475BX1C^`*D2c(<~=^1-M(K97E1c;Lrv zKmMXUc}PCU>WB2)FUpQPKJzAT^je;0*>pRw{Nhi4_>AB9&s6D^Jt2bz_h`EVBH~05 zb&g1lauMmlP&N)U%WG%HHzRR9QWLoa18LTT;oJBPLBtY-FE}x11>vk{zYik<;D#Xt zu|$;2UYgCIeaJy|0&bW>m??_*g7|?dLg+}4_nkocb{J*?!)$2w7&24b=ZNbQ;qI3d z87qW8A7)VvF^WkbXto0`#|i}Td=LmXTdL8%5{kTUFJ4kG91(BHm5O2`qO9@jR1*nbI*eyE(o&gA~+w$fC?AZi~^-ki{Wo3 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