From c2c37e796be3dfd71b77285faca325d15ad8d207 Mon Sep 17 00:00:00 2001 From: kl-i Date: Thu, 16 Sep 2021 16:20:35 +0100 Subject: [PATCH] Added Quest1Part0.md --- 1FundamentalGroup/Quest0Part3.md | 8 +-- 1FundamentalGroup/Quest1Part0.md | 51 ++++++++++++++++++ .../2.6.3/agda/1FundamentalGroup/Quest0.agdai | Bin 53617 -> 58708 bytes 3 files changed, 55 insertions(+), 4 deletions(-) create mode 100644 1FundamentalGroup/Quest1Part0.md diff --git a/1FundamentalGroup/Quest0Part3.md b/1FundamentalGroup/Quest0Part3.md index 79a152c..865ca75 100644 --- a/1FundamentalGroup/Quest0Part3.md +++ b/1FundamentalGroup/Quest0Part3.md @@ -24,7 +24,7 @@ see `1FundamentalGroup/Quest0SideQuests/SideQuest1`. 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` +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`, @@ -34,8 +34,8 @@ it will look like -The homotopy `h : refl ≡ loop` is 'lifted' -(starting at 'lifted `refl`') +The homotopy `h : Refl ≡ loop` is 'lifted' +(starting at 'lifted `Refl`') to some kind of surface @@ -83,7 +83,7 @@ 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`. +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`. diff --git a/1FundamentalGroup/Quest1Part0.md b/1FundamentalGroup/Quest1Part0.md new file mode 100644 index 0000000..54ba15f --- /dev/null +++ b/1FundamentalGroup/Quest1Part0.md @@ -0,0 +1,51 @@ +# Loop Space + +In this quest, +we continue to formalise the problem statement. + +> The fundamental group of `S¹` is `ℤ`. + +Intuitively, +the fundamental group of `S¹` at `base` is +consists of loops based as `base` up to homotopy of paths. +In homotopy type theory, +we have a native description of loops based at `base` : +it is the space `base ≡ base`. + +In general the _loop space_ of a space `A` at a point `a` is defined as follows : + +```agda +Ω : (A : Type) (a : A) → Type +Ω A a = a ≡ a +``` + +Warning : +the loop space can contain higher homotopical information that +the fundamental group does not capture. +For example, consider `S²`. +```agda +data S² : Type where + base : S² + loop : base ≡ base + northHemisphere : loop ≡ refl + southHemisphere : refl ≡ loop +``` + +

+

+What is `refl`? + +For any space `A` and point `a : A`, +`refl` is the constant path at `a`. +Technically speaking, we should write `refl a` to indicate the point we are at, +however `agda` is often smart enough to figure that out. +
+

+ +Intuitively, all loops in `S²` based at `base` is homotopic to +the constant path `refl`. +In other words, the fundamental group at `base` of `S²` is trivial. +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. diff --git a/_build/2.6.3/agda/1FundamentalGroup/Quest0.agdai b/_build/2.6.3/agda/1FundamentalGroup/Quest0.agdai index 15c98d4f5281b21bf386942c2bb82934cd3370fb..193257385f62fddbf52063ac30e5384b0f60e1da 100644 GIT binary patch literal 58708 zcmXVY3tUY3`~T;hGq-B0=|a=}M98%ZDy2G=G_7dLF1dEIl>4osRHqOTjab?x4WjAK zqTS>YHAv;MYh=E(6N*?F+mfv?O#PqF`0c!2d&Zn|p3n1np7-;9p3n0+r>|Q_uf6fI z^s@2CVP11;k9j$zI+}C7i9#kD&ReE}( z!u4(OYU_w;A+OF#2b%jHdtRLODcmXXhC@Jo+|4lyeij{DZ*6qh%Qa@bl|Ih>;2->h z;}M#~xUk4rg|_nqa>7rp@z{DdwxFE@s zW&;dgkf7r^o4@y0PZN>y8U9jbO0(g}&hM0nK}DR68zWs}NkK<6yLazs&J2S1$2_^K 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