From f166392f35996fcb911c5b3204ae45f371346f8f Mon Sep 17 00:00:00 2001 From: Michael Zhang Date: Wed, 18 Sep 2024 14:43:34 -0500 Subject: [PATCH] more blog post wip --- .tokeignore | 3 +- src/content/posts/2024-09-18-hcomp/2d.jpg | Bin 0 -> 196533 bytes src/content/posts/2024-09-18-hcomp/fg.jpg | Bin 0 -> 216705 bytes src/content/posts/2024-09-18-hcomp/goal2.jpg | Bin 0 -> 240914 bytes .../posts/2024-09-18-hcomp/index.lagda.md | 108 ++++++++++++------ .../posts/2024-09-18-hcomp/numberline.jpg | Bin 0 -> 143787 bytes .../posts/2024-09-18-hcomp/suspbool.jpg | Bin 0 -> 319667 bytes src/styles/leftNav.scss | 6 +- src/styles/post.scss | 17 ++- 9 files changed, 93 insertions(+), 41 deletions(-) create mode 100644 src/content/posts/2024-09-18-hcomp/2d.jpg create mode 100644 src/content/posts/2024-09-18-hcomp/fg.jpg create mode 100644 src/content/posts/2024-09-18-hcomp/goal2.jpg create mode 100644 src/content/posts/2024-09-18-hcomp/numberline.jpg create mode 100644 src/content/posts/2024-09-18-hcomp/suspbool.jpg diff --git a/.tokeignore b/.tokeignore index 04849f7..698f6c8 100644 --- a/.tokeignore +++ b/.tokeignore @@ -1,3 +1,4 @@ public package-lock.json -src/styles/fork-awesome \ No newline at end of file +src/styles/fork-awesome +pnpm-lock.yaml diff --git a/src/content/posts/2024-09-18-hcomp/2d.jpg b/src/content/posts/2024-09-18-hcomp/2d.jpg new file mode 100644 index 0000000000000000000000000000000000000000..308def1505d145550af8ca076ef37c29bfd9cf4f GIT binary patch literal 196533 zcmeFYcT`hb*EbqPMVd(O<-iyrNo@=h(EbDCkY!z_T z*ucmDaPAxcaF_l8oZ$g!`cNM)0Km))AOipZE(0!};{-6!d*|pMz&Sy{#eeq!0ONB) z|D$htPU^qia~=Rl^Z{J>Z}<+rJL_$bUWk-&%%O=lj0NfBlSXpM*UPhekxk z#>FQjCMBn2W#{DPpuS%s{wscmX*X>Duo=e|{XAoSL4Qots}+ z#1n{L*VZ?_ZElf&?(H8O{-PWm|0Ne4&;JmMe*O>1{!enzK{$8*!Ucv4jDN{>?tJ)P zf^%HBcun~d=N$`1_h7E;H=keT)_qyt*vBNMVoBn85Hi8cdqWj3PX0@@e@piNOt2UK zuO#~)g8fgqmH>Km#r)qp&gg^u?~Ooz`s@o`Eep>U0jvz?=*q;v0nh=^Xi|Vz=bteH z{%HVG3}>JI!Q(&a0>jzFKY08NkI8@V_!}Nm|KRaAJf{D_<8OG({Da5e@Rc%&ND@d43NtE>oDc?6-^cT?n7@gUw^Lv zOz^a%n+xqmiNegC0a(LM4f|-al=9R3Q1R07jMg&%-?wmH*BdTEu0_&F(yxTN`u$S{ zAp5X~L@RRH@1~0h-%m*qN=3=k@8 zN4rLOPXsP<;s5AqKLf;}@W=rKmv5O`G{k6+C!m6~l1{y&fFkTNZhcx3$jH`4RF9YI zLEZE1NI#l=_~sFV_;GGf^EKRL?&CqOlKT;Jvr3w!5rodom1S#Xn^IGA!Pn`h#c^MF zZfO-nHkxyuN)BYcf=~SFFG&?j9EK{2lU7oC>a9-1QlNY?n1>b^qNY%V+|;$<^0<%9 zb;4WKa-LuV-aL0LVN&0Efnf6?`3!I;Z~=Sxf)aRQMdhSKkoWNF)r&(~7^tvi)f!k`A+4O%fKlaxO^NhSn@iDsKlf(Vi#}FJO?)tSsiSr6-c;gOb}FwJ zjRjbL2I$kII*^9(Kny%ll#}wDY&R&%mrao93?(k1ePAm1%nLp3)s%33`<7mIHy^Ds z6pE5*^ND)*<;da~!mr}aYo;5mGsTv_H&Hima8B7EFENIEPl}wjA~59@IoC5a*Uepz zd21~FM$JU2=5&o?rOaW>EXSJ~mw}3+L$^g8T|~Kp%YNZU3M#Dwo4l7Hj2Dho&t$tV z9dsbf+Uz{1ox#9XNGxfL$YpiqV~ibEDon7GzH>j=CU|PFW8f{DyK^J396>T4Uy_(R z{CS$weeOhGGVTURqQ2e*8jD?)Ays#0TQ*FAA$T*#Xfdm3KC3(ypTHKE-}M7)eRfq@ zg1vcm4?U+oyV#G1>dcb@IR{g3vPy(B?6O$ap)7Fs)5uzMo0qcj2wkt_C-)rUS8n?ly$&IfBpe!<1iU>Gu={|;8Nr4thnFSZd4wW zuzmRKK!9+6rSkL%NT?)1^Kawo6kPJ|SBOStHJ3UbY+L7owb?JmR|*i6JE$&>ATY0B zuCuvJtBmRf=nNpI!+!=?-5k{#?4J{ae}F95x?P1z43Q@lxu}-pZD^D$`Qq!lcB6@y zGJy#-6y(ZLQqMz9XG*b4ZkCP8{NhI;%*u;(S1^V@k+tGlSVLoOE-XfIeg-Y{qhVj# zCSB-9alOO4?aMz~;Akq>$&E()w4yRe9E9&$-p0(g8i)*DLBt`?E#EfB>T6AjS7*Zc zHt&y;!Sb

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2024-09-18-hcomp.index where open import Cubical.Foundations.Prelude hiding (isProp→isSet) open import Cubical.Foundations.Equiv.Base @@ -33,6 +33,8 @@ In two dimensions, hcomp can be understood as the double-composition operation. "Single" composition (between two paths rather than three) is typically implemented as a double composition with the left leg as $\mathsf{refl}$. Without the double composition, this looks like: +![](./2d.jpg) + ``` path-comp : {A : Type} {x y z : A} → x ≡ y → y ≡ z → x ≡ z path-comp {x = x} p q i = @@ -51,11 +53,13 @@ This result exists in the cubical standard library, but let's go over it here. ``` isProp→isSet : {A : Type} → isProp A → isSet A -isProp→isSet {A} A-isProp = goal where +isProp→isSet {A} f = goal where goal : (x y : A) → (p q : x ≡ y) → p ≡ q goal x y p q j i = -- ... ``` +We're given a type $A$, some proof that it's a mere proposition, let's call it $f : \mathsf{isProp}(A)$. + Now let's construct an hcomp. In a set, we'd want paths $p$ and $q$ between the same points $x$ and $y$ to be equal. Suppose $p$ and $q$ operate over the same dimension, $i$. If we want to find a path between $p$ and $q$, we'll want another dimension. @@ -79,13 +83,18 @@ Before getting started, let's familiarize ourselves with the dimensions we're wo We can map both $p(i)$ and $q(i)$ down to a square that has $x$ on all corners and $\mathsf{refl}_x$ on all sides. Let's start with the left and right faces $(i = \{ \mathsf{i0} , \mathsf{i1} \})$. -These can be produced using the $\mathsf{isProp}(A)$ that we are given. -$\mathsf{isProp}(A)$ tells us that $x$ and $y$ are the same, so $x \equiv x$ and $x \equiv y$. -This means we can define the left face as $\mathsf{isProp}(A, x, x, k)$ and the right face as $\mathsf{isProp}(A, x, y, k)$. +These can be produced using the proof $f : \mathsf{isProp}(A)$ that we are given. +$f$ tells us that $x$ and $y$ are the same, so $f(x, x) = x \equiv x$ and $f(x, y) = x \equiv y$. +This means we can define the left face as $f(x, x, k)$ and the right face as $f(x, y, k)$. (Remember, $k$ is the direction going from bottom to top) ![](./sides.jpg) +We can write this down as a mapping: + +* $(i = \mathsf{i0}) \rightarrow f(x, x, k)$ +* $(i = \mathsf{i1}) \rightarrow f(x, y, k)$ + Since $k$ is only the bottom-to-top dimension, the front-to-back dimension isn't changing. So we can use $\mathsf{refl}$ for those bottom edges. @@ -102,10 +111,10 @@ Putting this all together, we can using $\mathsf{hcomp}$ to complete the top fac ``` let u = λ k → λ where - (i = i0) → A-isProp x x k - (i = i1) → A-isProp x y k - (j = i0) → A-isProp x (p i) k - (j = i1) → A-isProp x (q i) k + (i = i0) → f x x k + (i = i1) → f x y k + (j = i0) → f x (p i) k + (j = i1) → f x (q i) k in hcomp u x ``` @@ -118,7 +127,10 @@ Let's move on to a more complicated example. Suspensions are an example of a higher inductive type. It can be shown that spheres can be iteratively defined in terms of suspensions. -Since the $0$-sphere is just two points (solutions to $\| \bm{x} \|_2 = 1$ in 1 dimension), we can show that a suspension over this is equivalent to the classic $1$-sphere, or the circle. +![](./numberline.jpg) + +Since the $0$-sphere is just two points (solutions to $\| \bm{x} \|_2 = 1$ in 1 dimension), we represent this as $S^0 :\equiv 2$. +We can show that a suspension over this, $\Sigma 2$, is equivalent to the classic $1$-sphere, the circle $S^1$. Let's state the lemma: @@ -136,11 +148,23 @@ In this model, we're going to define $f$ and $g$ by having both the north and so The choice of side is arbitrary, so I'll choose $\mathsf{true}$. This way, $\mathsf{true}$ is suspended into the $\mathsf{refl}$ path, and $\mathsf{false}$ is suspended into the $\mathsf{loop}$. - - - +Here's a picture of our function $f$: - - - + - - -
    +![](./suspbool.jpg) + +The left setup is presented in the traditional suspension layout, with meridians going through $\mathsf{true}$ and $\mathsf{false}$ while the right side "squishes" the north and south poles using $\mathsf{refl}$, while having the other path represent the $\mathsf{loop}$. + +On the other side, $g$, we want to map back from $S^1$ into $\Sigma 2$. +We can map the $\mathsf{loop}$ back into $\mathsf{merid} \; \mathsf{false}$, but the types mismatch, since $\mathsf{loop} : \mathsf{base} \equiv \mathsf{base}$ but $\mathsf{merid} \; \mathsf{false} : \mathsf{north} \equiv \mathsf{south}$. +But since $\mathsf{merid} \; \mathsf{true}$ is $\mathsf{refl}$, we can just concatenate with its inverse to get the full path: + +$$ +\mathsf{merid} \; \mathsf{false} \cdot (\mathsf{merid} \; \mathsf{true})^{-1} : \mathsf{north} \equiv \mathsf{north} +$$ + +In Agda, we can write it like this: + +
    ``` f : Susp Bool → S¹ @@ -150,34 +174,24 @@ This way, $\mathsf{true}$ is suspended into the $\mathsf{refl}$ path, and $\math f (merid false i) = loop i ``` -
    - ``` g : S¹ → Susp Bool g base = north g (loop i) = (merid false ∙ sym (merid true)) i - - ``` -
    - -Now, the fun part is to show the isomorphisms. +Now, the fun part is to show the extra requirements that is needed to show that these two functions indeed form an isomorphism. Starting with the first, let's show $f(g(s)) \equiv s$. -The base case is easily handled by $\mathsf{refl}$. +The base case is easily handled by $\mathsf{refl}$, since $f(g(\mathsf{base})) = f(\mathsf{north}) = \mathsf{base}$ definitionally by the reduction rules we gave above. ``` fg : (s : S¹) → f (g s) ≡ s fg base = refl ``` -The loop case is trickier. Let's solve it algebraically first: +The loop case is trickier. If we were using book HoTT, here's how we would solve this (this result is given in Lemma 6.5.1 of the HoTT book): $$ \begin{align*} @@ -191,21 +205,43 @@ $$ \end{align*} $$ -Between the second and third steps, I used functoriality of the $\mathsf{ap}$ operation. +Between the second and third steps, I used functoriality of the $\mathsf{ap}$ operation (equation _(i)_ of Lemma 2.2.2 in the HoTT book). + +How can we construct a cube to solve this? Like in the first example, let's start out by writing down the face we want to end up with: + +![](./goal2.jpg) + +We're looking for the path in the bottom-to-top $j$ dimension. +We would like to have $\mathsf{hcomp}$ give us this face. +This time, let's see the whole cube first and then pick apart how it was constructed. + +![](./fg.jpg) + +First of all, note that unrolling $f(g(\mathsf{loop} \; i))$ would give us $f((\mathsf{merid} \; \mathsf{false} \cdot (\mathsf{merid} \; \mathsf{true})^{-1}) \; i)$, which is the same as $f(\mathsf{merid} \; \mathsf{false} \; i) \cdot f(\mathsf{merid} \; \mathsf{true} \; (\sim i))$. +Since this is a composition, it deserves its own face of the cube, with our goal $f(g(\mathsf{loop} \; i))$ being the lid. + +We can express this as an $\mathsf{hfill}$ operation, which is similar to $\mathsf{hcomp}$ in that it takes the same face arguments, but produces the contents of the face rather than just the upper lid. + +For the rest of the faces, we can just fill in opposite sides until the cube is complete. +The Agda translation looks like this: ``` - fg (loop i) k = + fg (loop i) j = let - u = λ j → λ where + u = λ k → λ where (i = i0) → base - (i = i1) → f (merid true (~ j)) - (k = i0) → compPath-filler - (λ i → f (merid false i)) - (λ j → f (merid true (~ j))) j i - (k = i1) → loop i + (i = i1) → f (merid true (~ k)) + (j = i0) → + let u = λ k' → λ where + (i = i0) → base + (i = i1) → f (merid true (~ k')) + in hfill u (inS (f (merid false i))) k + (j = i1) → loop i in hcomp u (f (merid false i)) ``` +Nothing should be too surprising here, other than the use of a nested $\mathsf{hfill}$ which is needed to describe the face that contains the composition. + ``` gf : (b : Susp Bool) → g (f b) ≡ b gf north = refl diff --git a/src/content/posts/2024-09-18-hcomp/numberline.jpg b/src/content/posts/2024-09-18-hcomp/numberline.jpg new file mode 100644 index 0000000000000000000000000000000000000000..b73b52a10d3d9eca964fbddd6c5322c04137ccb5 GIT 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} .portrait { - aspect-ratio: 1; + aspect-ratio: 1 / 1; } a.portrait { @@ -47,8 +47,8 @@ a.portrait img { border-radius: 100%; - width: 100%; - height: 100%; + width: auto; + height: auto; } .bio { diff --git a/src/styles/post.scss b/src/styles/post.scss index 51d19e9..d94323a 100644 --- a/src/styles/post.scss +++ b/src/styles/post.scss @@ -99,7 +99,7 @@ p>img { margin: auto; max-width: 75%; - max-height: 240px; + max-height: 280px; width: auto; height: auto; @@ -280,4 +280,19 @@ hr.endline { --admonition-color: var(--warning-color); --admonition-bg-color: var(--warning-bg-color); } +} + +.halfSplit { + display: flex; + flex-direction: row; + gap: 12px; + overflow-y: auto; + + @media screen and (max-width: 960px) { + flex-direction: column; + + pre.Agda { + margin: 0; + } + } } \ No newline at end of file