Test rendering this page

src/content/posts/2024-06-28-boolean-equivalences.lagda.md

@@ -1,5 +1,5 @@
 ---
-title: "Boolean equivalences in HoTT"
+title: Boolean equivalences
 slug: 2024-06-28-boolean-equivalences
 date: 2024-06-28T21:37:04.299Z
 tags: ["agda", "type-theory", "hott"]

src/content/posts/2024-09-15-circle-is-a-suspension-over-booleans.lagda.md

@@ -0,0 +1,84 @@
+---
+title: The circle is a suspension over booleans
+slug: 2024-09-15-circle-is-a-suspension-over-booleans
+date: 2024-09-15T23:02:32.058Z
+tags: [algebraic-topology, hott]
+draft: true
+---
+
+```
+{-# OPTIONS --cubical #-}
+open import Cubical.Foundations.Prelude
+open import Data.Nat
+open import Data.Bool
+```
+
+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:
+
+$$
+x^2 + y^2 = 1
+$$
+
+Well, there are a lot more circles, like $x^2 + y^2 = 2$, and so on, but in topology land we don't really care.
+All of them are equivalent up to continuous deformation.
+
+What is interesting, though, is the fact that it can _not_ be deformed down to a single point at the origin.
+It has the giant hole in the middle.
+
+The circle, usually denoted $S^1$, is a special case of $n$-spheres.
+For some dimension $n \in \mathbb{N}$, the $n$-sphere can be defined analytically as:
+
+$$
+\lVert \bm{x} \rVert_2 = \bm{1}
+$$
+
+where $\lVert \bm{x} \rVert_2$ is the [Euclidean norm][1] of a point $\bm{x}$.
+
+[1]: https://en.wikipedia.org/wiki/Norm_(mathematics)
+
+However, in the synthetic world, circles look a lot different.
+We ditch the coordinate system and boil the circle down to its core components in terms of points and paths.
+The 1-sphere $S^1$ is defined with:
+
+$$
+\begin{align*}
+  \mathsf{base} &: S^1 \\
+  \mathsf{loop} &: \mathsf{base} \equiv \mathsf{base}
+\end{align*}
+$$
+
+What about the 2-sphere, aka what we normally think of as a sphere?
+We can technically define it as a 2-path over the base point:
+
+$$
+\begin{align*}
+  \mathsf{base} &: S^2 \\
+  \mathsf{surf} &: \mathsf{refl}_\mathsf{base} \equiv_{\mathsf{base} \equiv \mathsf{base}} \mathsf{refl}_\mathsf{base}
+\end{align*}
+$$
+
+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?
+
+If we take our original analytic definition of spheres and plug in $0$, we find out that this is just $| x | = 1$, which has two solutions: $-1$ and $1$.
+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
+```
+
+How about the iterative case? How can we take an $n$-sphere and get an $(n+1)$-sphere?
+
+```
+S (suc n) = Bool
+```