From 16c5e3ab8161a42a172cfdcafb15785b17dca41e Mon Sep 17 00:00:00 2001 From: Michael Zhang Date: Sun, 15 Sep 2024 19:38:05 -0500 Subject: [PATCH] wip --- plugin/remark-agda.ts | 15 +++- .../1.png | Bin 0 -> 122656 bytes .../boolsusp.png | Bin 0 -> 100252 bytes .../index.lagda.md} | 83 +++++++++++++++--- .../susp.png | Bin 0 -> 134048 bytes src/pages/posts/[...slug].astro | 15 +++- 6 files changed, 94 insertions(+), 19 deletions(-) create mode 100644 src/content/posts/2024-09-15-circle-is-a-suspension-over-booleans/1.png create mode 100644 src/content/posts/2024-09-15-circle-is-a-suspension-over-booleans/boolsusp.png rename src/content/posts/{2024-09-15-circle-is-a-suspension-over-booleans.lagda.md => 2024-09-15-circle-is-a-suspension-over-booleans/index.lagda.md} (50%) create mode 100644 src/content/posts/2024-09-15-circle-is-a-suspension-over-booleans/susp.png diff --git a/plugin/remark-agda.ts b/plugin/remark-agda.ts index 762fc05..8714e57 100644 --- a/plugin/remark-agda.ts +++ b/plugin/remark-agda.ts @@ -52,21 +52,28 @@ const remarkAgda: RemarkPlugin = ({ base, publicDir }: Options) => { {}, ); - if (childOutput.error || !existsSync(agdaOutFile)) { + // Locate output file + const directory = readdirSync(agdaOutDir); + const outputFilename = directory.find((name) => name.endsWith(".md")); + + if (childOutput.status !== 0 || outputFilename === undefined) { throw new Error( - `Agda error: + `Agda output: Stdout: ${childOutput.stdout} Stderr: - ${childOutput.stderr}`, + ${childOutput.stderr} + `, { cause: childOutput.error, }, ); } + const outputFile = join(agdaOutDir, outputFilename); + // // TODO: Handle child output // console.error("--AGDA OUTPUT--"); // console.error(childOutput); @@ -106,7 +113,7 @@ const remarkAgda: RemarkPlugin = ({ base, publicDir }: Options) => { const htmlname = parse(path).base.replace(/\.lagda.md/, ".html"); - const doc = readFileSync(agdaOutFile); + const doc = readFileSync(outputFile); // This is the post-processed markdown with 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+Imports + ``` -{-# OPTIONS --cubical #-} -open import Cubical.Foundations.Prelude -open import Data.Nat -open import Data.Bool +{-# OPTIONS --cubical --allow-unsolved-metas #-} +module 2024-09-15-circle-is-a-suspension-over-booleans.index where +open import Cubical.Core.Primitives +open import Cubical.Foundations.Function +open import Cubical.Foundations.Isomorphism +open import Cubical.Foundations.Equiv.Base +open import Cubical.HITs.S1.Base +open import Data.Bool.Base +open import Data.Nat.Base ``` +
+ One of the simpler yet still very interesting space in algebraic topology is the **circle**. Analytically, a circle can described as the set of all points that satisfy: @@ -61,10 +71,6 @@ $$ It would be nice to have an iterative definition of spheres; one that doesn't rely on us using our intuition to form new ones. Ideally, it would be a function $S^n : \mathbb{N} \rightarrow \mathcal{U}$, where we could plug in an $n$ of our choosing. -``` -S_ : ℕ → Type -``` - For an iterative definition, we'd like some kind of base case. What's the base case of spheres? What is a $0$-sphere? @@ -73,12 +79,65 @@ If we take our original analytic definition of spheres and plug in $0$, we find The space of solutions is just a space with two elements! In other words, the type of booleans is actually the $0$-sphere, $S^0$. -``` -S zero = Bool -``` +![](./1.png) How about the iterative case? How can we take an $n$-sphere and get an $(n+1)$-sphere? +This is where we need something called a _suspension_. + +Suspensions are a form of higher inductive type (HIT). It can be defined like this: ``` -S (suc n) = Bool +data Susp (A : Type) : Type where + north : Susp A + south : Susp A + merid : A → north ≡ south ``` + +Intuitively, you take the entirety of the type $A$, and imagine it as some kind of plane. +Then, add two points, $\mathsf{north}$ and $\mathsf{south}$, to the top and bottom respectively. +Now, each original point in $A$ defines a _path_ between $\mathsf{north}$ and $\mathsf{south}$. + +![](./susp.png) + +It may not be too obvious, but the circle $S^1$, can be represented as a suspension of the booleans $S^0$. +This is presented as Theorem 6.5.1 in the [HoTT book]. I'll go over it briefly here. + +[hott book]: https://homotopytypetheory.org/book/ + +The main idea is to consider the paths generated by both the $\mathsf{true}$ and the $\mathsf{false}$ points to add up to a single $\mathsf{loop}$. +Here's a visual guide: + +![](./boolsusp.png) + +We can write functions going back and forth: + +``` +Σ2→S¹ : Susp Bool → S¹ +Σ2→S¹ north = base -- map the north point to the base point +Σ2→S¹ south = base -- also map the south point to the base point +Σ2→S¹ (merid false i) = loop i -- for the path going through false, let's map this to the loop +Σ2→S¹ (merid true i) = base -- for the path going through true, let's map this to refl + +S¹→Σ2 : S¹ → Susp Bool +S¹→Σ2 base = north +S¹→Σ2 (loop i) = {! !} +``` + +Now, to finish showing the equivalence, we need to prove that these functions concatenate into the identity in both directions: + +``` +rightInv : section Σ2→S¹ S¹→Σ2 +rightInv = {! !} + +leftInv : retract Σ2→S¹ S¹→Σ2 +leftInv = {! !} +``` + +And this gives us our equivalence! + +``` +Σ2≃S¹ : Susp Bool ≃ S¹ +Σ2≃S¹ = isoToEquiv (iso Σ2→S¹ S¹→Σ2 rightInv leftInv) +``` + +In fact, we can also show that the $2$-sphere is the suspension of the $1$-sphere. diff --git a/src/content/posts/2024-09-15-circle-is-a-suspension-over-booleans/susp.png b/src/content/posts/2024-09-15-circle-is-a-suspension-over-booleans/susp.png new file mode 100644 index 0000000000000000000000000000000000000000..d2752c5c4a2bf2a748ad5e534cf33584005ecb0b GIT binary patch literal 134048 zcmeEu`9G9>-@ZgdXdy+Gl5ANjl3mH3ov{okI%f()mCMoW22*>pkPo}d#F!A zaY}`P;^ZRjDd0P>I~(VK|4z8;t3IG8@4d1Nyg2QmX6jBsL4V=+-wBG;^h*>JiWKS( z?;HA@Se-b1#9(lJ)+JCj!+o0bU=XMAS?(0a=^BR1kvYyz`a)dnN*1b z|LnMtruefuOjwcf#DgHkyO@_Z6=@wP)5*SjV{_iEw(_>?-^b_ua2dnCvM%cxGVTlC z-4`65E!zNlKgeD_~|OvT2dX!ZV9 zSSp9|f4Pd1C;jAqxQpZ4>%Vk@k|#=PXqEpz-79d_*-JB5{=+>U-+b^F;ELzpbDpXG zm&XZeqh0)u_5gk$t`gLy_JmPFCj7r#MWu3&=08~rxRdN#lvFBg?nNVd|IwyYXSu~b zO8sZgP*6sl;}-jotke^I?Y~?Ve6IID`;F?XT_iE8{%JJ-j*tEc zk$*zuAGrFjtWf+1uKpXC|7W84w@L1w5c&5>?w=6(Cq(`OFQV8U1?aZD1Q_MN@-5TJfSe1`1q{Huun%NzZjlsIX7asKm^*SXS%IvF_I3J^YD z8|7@J_FA805NePaeh2Hl$W4kxdGA#BP6%$CM~87*n?dhgU&}}%>$>tAUFF~5`l+S*%rd8pKo6o)DuC2eFDz`AVK*mnPK z!zfPC%I>ARr)5d(K3?b3LF`fN6*+WbE^uPVt8ZIJCW0Nmo>90tD70JY6yhoyfMU{@ zTjv@wfbnHhSovR>#c|11d9OXzq~(rfSyOkw=MY%=Z*MhuwcZc~O2^c{w8}wCnJl#r7iH-@jl2-F&x-m+I^# zAw{dFWi(U&o8<~tp(~P9J(|AjuW;D3a(yh_cG!C?1dX*nWUEP%+6&Wb4R>+udPG*> zy3**bHOS)EOt#&>!h?wJu?>*FbZnlOWAnJ$mi`~+u{@oQ6eDu)S;mx34r#KE;W?JW zMsnj>bw$F5_kXPvhpmdHPLCJz$uz7T^dh&_7MZE45;Mdrj_#UP-=T!h_^Fw8Ui(g% zvq&FPV@&R~GN7kqp{9*Aqzvo{4My~8 zE0~j4py^-^-px$!3gl6+al+IMO)r({kaw$3&QbEDLXf$4#&R!Goh`Zn(4VQSnmqr8 zLxGn{!j*G|gvaYjI)+ngD202L!+tTv5FLZzWOx=w(T|gOi-IvhEN5Su!$H8}&+a=7 zew5_E+<=?a!RMki_l}A$2VVnnLs|ML^j{hAuRkLGNKr||hHE^u&(qD&{|nP5Glpq! 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HcmV?d00001 diff --git a/src/pages/posts/[...slug].astro b/src/pages/posts/[...slug].astro index 7d8d9e0..30a3673 100644 --- a/src/pages/posts/[...slug].astro +++ b/src/pages/posts/[...slug].astro @@ -7,6 +7,7 @@ import Timestamp from "../../components/Timestamp.astro"; import { getImage } from "astro:assets"; import TocWrapper from "../../components/TocWrapper.astro"; import TagList from "../../components/TagList.astro"; +import portrait from "../../assets/self.png"; export async function getStaticPaths() { const posts = await getCollection("posts"); @@ -22,7 +23,7 @@ const post = Astro.props; const { Content, remarkPluginFrontmatter, headings } = await post.render(); const { title, toc, heroImage: heroImagePath, heroAlt, draft } = post.data; -let heroImage; +let heroImage = undefined; if (heroImagePath) { heroImage = await getImage({ src: heroImagePath }); } @@ -40,8 +41,16 @@ const excerpt = remarkPluginFrontmatter.excerpt?.replaceAll("\n", ""); - {heroImage && } - {heroAlt && } + {heroImage ? + <> + + {heroAlt && } + + : + <> + + + }