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Michael Zhang 2024-09-15 19:38:05 -05:00
parent 67bb9a196c
commit 16c5e3ab81
6 changed files with 94 additions and 19 deletions

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@ -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 HTML code blocks replacing the Agda code blocks
const tree2 = fromMarkdown(doc);

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@ -6,13 +6,23 @@ tags: [algebraic-topology, hott]
draft: true
---
<details>
<summary>Imports</summary>
```
{-# 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
```
</details>
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_ : → 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.

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@ -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", "");
<meta slot="head" property="og:description" content={excerpt} />
<meta slot="head" property="og:type" content="article" />
<meta slot="head" property="og:locale" content="en_US" />
{heroImage && <meta slot="head" property="og:image" content={heroImage.src} />}
{heroAlt && <meta slot="head" property="og:image:alt" content={heroAlt} />}
{heroImage ?
<>
<meta slot="head" property="og:image" content={heroImage.src} />
{heroAlt && <meta slot="head" property="og:image:alt" content={heroAlt} />}
</>
:
<>
<meta slot="head" property="og:image" content={portrait.src} />
</>
}
<meta slot="head" property="keywords" content={post.data.tags.join(", ")} />
<meta slot="head" property="description" content={excerpt} />