Merge branch 'main' of https://github.com/thehottgame/TheHoTTGame

1FundamentalGroup/Quest2.agda

@@ -0,0 +1,47 @@
+module 1FundamentalGroup.Quest2 where
+
+open import Cubical.Core.Everything
+open import Cubical.Data.Nat
+open import Cubical.Data.Int using (ℤ ; pos ; negsuc ; -_)
+open import Cubical.Foundations.Isomorphism
+open import Cubical.Foundations.Prelude renaming (subst to endPt)
+open import Cubical.HITs.S1 using (S¹ ; base ; loop)
+open import 1FundamentalGroup.Quest1
+
+sucℤ : ℤ → ℤ
+sucℤ (pos n) = pos (suc n)
+sucℤ (negsuc zero) = pos zero
+sucℤ (negsuc (suc n)) = negsuc n
+
+predℤ : ℤ → ℤ
+predℤ (pos zero) = negsuc zero
+predℤ (pos (suc n)) = pos n
+predℤ (negsuc n) = negsuc (suc n)
+
+sucℤIso : Iso ℤ ℤ
+sucℤIso = iso sucℤ predℤ s r where
+
+  s : section sucℤ predℤ
+  s (pos zero) = refl
+  s (pos (suc n)) = refl
+  s (negsuc zero) = refl
+  s (negsuc (suc n)) = refl
+
+  r : retract sucℤ predℤ
+  r (pos zero) = refl
+  r (pos (suc n)) = refl
+  r (negsuc zero) = refl
+  r (negsuc (suc n)) = refl
+
+sucℤPath : ℤ ≡ ℤ
+sucℤPath = isoToPath sucℤIso
+
+helix : S¹ → Type
+helix base = ℤ
+helix (loop i) = sucℤPath i
+
+spinCountBase : base ≡ base → ℤ
+spinCountBase p = endPt helix p 0
+
+spinCount : (x : S¹) → base ≡ x → helix x
+spinCount x p = endPt helix p 0

1FundamentalGroup/Quest2Part0.md

@@ -5,6 +5,8 @@ Creating the inverse map is difficult without access to the entire circle.
 Similarly to how we used `doubleCover` to distinguish `refl` and `base`,
 the idea is to replace `Bool` with `ℤ`, 
 allowing us to distinguish between all loops on `S¹`.
+In `Part0` and `Part1` we will construct one of the two comparison maps
+across the whole circle, called `spinCount`.
 
 The plan is :
 
@@ -14,11 +16,21 @@ The plan is :
 3. Turn `sucℤ` into a path `sucPath : ℤ ≡ ℤ` using `isoToPath`
 4. Define `helix : S¹ → Type` by mapping `base` to `ℤ` and
    a generic point `loop i` to `sucPath i`.
+<<<<<<< HEAD
 5. Use `helix` and `endPt` to define the map `base ≡ x → helix x`
    on all `x : S¹`, in particular giving us `Ω S¹ base → ℤ`
    when applied to `base`.
 
 In this part, we focus on `1` and `2`.
+=======
+5. Use `helix` and `endPt` to define the map 
+   `spinCountBase : base ≡ base → ℤ`
+   Intuitively it counts how many times a path loops around `S¹`. 
+   a generic point `loop i` to `sucPath i`.
+6. Generalize this across the circle.
+
+In this part, we focus on `1`, `2` and `3`.
+>>>>>>> df5d7c381b1adae1d2547df95f5d73bcf3447ac4
 
 ## `sucℤ`
 
@@ -77,4 +89,13 @@ In this part, we focus on `1` and `2`.
 - Imitating what we did with `flipIso` and 
   give a point `sucℤIso : ℤ ≅ ℤ`
   by using `predℤ` as the inverse and proving
+<<<<<<< HEAD
   `section sucℤ predℤ` and `retract sucℤ predℤ`.
+=======
+  `section sucℤ predℤ` and `retract sucℤ predℤ`. 
+
+## `sucℤ` is a path
+
+- Imitating what we did with `flipPath`, 
+  upgrade `sucℤIso` to `sucℤPath`.
+>>>>>>> df5d7c381b1adae1d2547df95f5d73bcf3447ac4

1FundamentalGroup/Quest2Part1.md

@@ -0,0 +1,54 @@
+# Comparison maps between `Ω S¹ base` and `ℤ` - `spinCount`
+
+## The `ℤ`-bundle `helix`
+
+We want to make a `ℤ`-bundle over `S¹` by 
+'copying ℤ across the loop via `sucℤPath`'.
+In `Quest2.agda` locate
+
+```agda
+helix : S¹ → Type
+helix = {!!}
+```
+
+Try to imitate the definition of `doubleCover` to define the bunlde `helix`.
+You should compare your definition to ours in `Quest2Solutions.agda`.
+Note that we have called this `helix`, since the picture of this `ℤ`-bundle 
+looks like
+
+<img src="images/helix.png"
+     alt="helix"
+     width="1000"
+     class="center"/>
+
+## Counting loops
+
+Now we can do what was originally difficult - constructing an inverse map 
+(over all points). 
+Now we want to be able to count how many times a path `base ≡ base` loops around
+`S¹`, which we can do now using `helix` and finding end points of 'lifted' paths.
+For example the path `loop` should loop around once, 
+counted by looking at the end point of 'lifted' `loop`, starting at `0`.
+Hence try to define
+
+```agda
+spinCountBase : base ≡ base → helix base
+spinCountBase = {!!}
+```
+
+Try computing a few values using `C-c C-n`,
+you can try it on `refl`, `loop`, `loop 3 times`, `loop (- 1) times` and so on.
+
+## Generalising
+
+The function `spinCountBase`
+can actually be improved without any extra work to a function on all of `S¹`
+
+```agda
+spinCount : (x : S¹) → base ≡ x → helix x
+spinCount = {!!}
+```
+
+We will show that this and a general version of `loop_times` are
+inverses of each other over `S¹`, in particular obtaining an isomorphism
+between `base ≡ base` and `ℤ`.

README.md

@@ -43,26 +43,29 @@ Try opening `Trinitarianism/Quest0.agda` in Emacs
 and do `Ctrl-c Ctrl-l`.
 Some text should be highlighted, and any `?` should turn into `{ }`.
 
-## How the game works
+## Where to start?
 
-Our Game is currently under development. 
-As of now you can try the _quests_ in the `Trinitarianism` folder.
-Each quest consists of three files, for example :
-- `Trinitarianism/Quest0.md` is the guide for the quest.
-  In there, you will find details of the tasks 
-  you must finish in order to complete the quest.
-  For now, it is recommended that
-  you view these online within github.
-- `Trinitarianism/Quest0.agda` is the agda file in which
-  you do the quest. 
-- `Trinitarianism/Quest0Solutions.agda` contains
-  solutions to the tasks in the quest.
+You can start with `0Trinitarianism` if you are interested in
+how logic, type theory and category theory come together 
+as different ways to view the same thing.
+If you are more interested in homotopy theory,
+try `1FundamentalGroup` where we show that the 
+fundamental group of `S¹` is `ℤ`.
 
-## Emacs and Agda usage
-We have a file with a list of [basic Emacs commands](
+## How to start?
+
+To start with `1FundamentalGroup` (for example),
+find the GitHub page [`1FundamentalGroup/Quest0Part0.md`](
+https://github.com/thehottgame/TheHoTTGame/blob/main/1FundamentalGroup/Quest0Part0.md
+)
+and follow the instructions there,
+then trying the following files in the same directory.
+Open up the corresponding `.agda` file in `emacs` to 
+follow along with the instructions in the quests.
+
+## Emacs issues
+
+If you can't figure out `emacs` or forget some command, then 
+try consulting our guide for [basic Emacs commands](
 https://github.com/thehottgame/TheHoTTGame/blob/main/EmacsCommands.md
-), 
-but you _should_ be able to learn how to use Agda as you go along.
-
-## Feedback
-If you have any feedback please contact the devs. 
+).