From 682017ab752cd85c6b5d6c2eff13b6e424224e1a Mon Sep 17 00:00:00 2001 From: Jlh18 Date: Tue, 14 Sep 2021 18:12:03 +0100 Subject: [PATCH] added 1FundamentalGroup --- 1FundamentalGroup/Quest0.agda | 21 +++++++++++ 1FundamentalGroup/Quest0.md | 29 +++++++++++++++ 1FundamentalGroup/pi1S1.md | 33 ++++++++++++++++++ Plan.org | 19 +++++++--- .../2.6.2/agda/1FundamentalGroup/Quest0.agdai | Bin 0 -> 34177 bytes 5 files changed, 98 insertions(+), 4 deletions(-) create mode 100644 1FundamentalGroup/Quest0.agda create mode 100644 1FundamentalGroup/Quest0.md create mode 100644 1FundamentalGroup/pi1S1.md create mode 100644 _build/2.6.2/agda/1FundamentalGroup/Quest0.agdai diff --git a/1FundamentalGroup/Quest0.agda b/1FundamentalGroup/Quest0.agda new file mode 100644 index 0000000..6c9b8db --- /dev/null +++ b/1FundamentalGroup/Quest0.agda @@ -0,0 +1,21 @@ +module 1FundamentalGroup.Quest0 where + +open import Cubical.Core.Everything +open import Cubical.Data.Empty +open import Cubical.Foundations.Prelude +open import Cubical.Foundations.Path +open import Cubical.Foundations.GroupoidLaws + +data S¹ : Type where + base : S¹ + loop : base ≡ base + +¬ : Type → Type +¬ A = A → ⊥ + +doubleCover : type +doubleCover = ? + + +refl≢loop : ¬ ( refl ≡ loop ) +refl≢loop h = {!!} diff --git a/1FundamentalGroup/Quest0.md b/1FundamentalGroup/Quest0.md new file mode 100644 index 0000000..36f5a69 --- /dev/null +++ b/1FundamentalGroup/Quest0.md @@ -0,0 +1,29 @@ +Whoa very cool +======================= + +In this series of quests we will prove that the fundamental group +of `S¹` is `ℤ`. +In fact, our strategy will also show that the higher homotopy groups of +`S¹` are all trivial. +We begin by formalising the problem statement. + + + +## The Circle + +A contruction of 'the circle' is : + +- a point +- an edge from that point to itself + +Here is our definition of the circle in `agda`. + +```agda +data S¹ : Type where + base : S¹ + loop : base ≡ base +``` + +The `base \== base` is the _space of paths from `base` to `base`_. +An "edge" is the same as a path. + diff --git a/1FundamentalGroup/pi1S1.md b/1FundamentalGroup/pi1S1.md new file mode 100644 index 0000000..dc51339 --- /dev/null +++ b/1FundamentalGroup/pi1S1.md @@ -0,0 +1,33 @@ +Fundamental Group of S¹ +================ + +Prerequisites : +- circle +- loop space +- have useless maps S¹ → ℤ and ℤ → S¹ + + + + +- we make helix from ℤ ≡ ℤ + +- + +Motivating Steps : +0. After sufficient pondering, you guess that + loops are determined by how many times they go around +1. Count number of times loops go around using ℤ + by setting the single loop to +1. +2. You can make a comparison maps between loop space and ℤ + but can't yet show an equivalence +3. You realize you need to use the definition of S¹, + so you go from `base ≡ base ≃ Z` to `base ≡ x ≃ Bundle x`. + i.e. You try to make the equivalence _over_ S¹ + + +Random thought : +to prove `a = b`, we realise that `a = f(x0)` +and `b = g(x0)` for `x0 : X`, +and instead show `(x : X) → f x = g x`. +This turns out to be easier since we now get +access to the recursor of `X`. diff --git a/Plan.org b/Plan.org index c819cdf..b37660a 100644 --- a/Plan.org +++ b/Plan.org @@ -100,7 +100,18 @@ + Membership not the same as : + typing is unique (doesn't make sense to intersect two types) + -+ Though set theory had fewer axioms type theory's assumptions are more intuitive (hence intionistic type theory) - There is no fiddling about with membership to construct things e.g. cartesian product -+ 'we cannot use LEM' ~ not assuming law of excluded middle _globally_ means type theory theorems are stronger! -+ ++ Though set theory had fewer axioms type theory's assumptions are + more intuitive (hence intionistic type theory). + There is no fiddling about with membership to construct things + e.g. cartesian product ++ 'we cannot use LEM' ~ not assuming law of excluded middle _globally_ means + type theory theorems are stronger! + +** Fundamental group of S1 ++ def of S¹ ++ a bunch of stuff about ℤ ++ isoToPath to make ℤ ≡ ℤ ++ subst ++ funExt ++ set-truncation ++ paths as equality diff --git a/_build/2.6.2/agda/1FundamentalGroup/Quest0.agdai b/_build/2.6.2/agda/1FundamentalGroup/Quest0.agdai new file mode 100644 index 0000000000000000000000000000000000000000..c9eca070afd5456c61356373dc8c0c7bdfe12d9e GIT binary patch literal 34177 zcmXV130MYQHX*dU#%>rO5=ivt$-pfEE){_=fSYfnle>;V=*V6JwFAn+DJ92R?J(K76Uu&1t?G5q|Nnt5;KMHA`*I zU2U6Sx8slhT1*Mr+9p2!`9DK`ng599ZvDaar&`zS)cv;Cnt5joW{(!Hy?0==)arvJ z`5}Re&7dbR7{3ME&*3R?A7zUN#*$z==rd6~o}8m3`F5TdewRQWI8lrpW+&^;WAo(3 zN}O-TVE+>1m1QKwoW_l0c}6_B6-}D|ixGJy(af!*3}y+pC00V!MEE#?V3qm8w|q$= zkC;KPwWk<=_*Oexs9J_@vsq=fpv5N!8{)9aFrjJ@O-`k`thfleOjwN*?N%)<<-Ef< 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