From d9ada0cd32f74dd06ec571587ace34c57c153aca Mon Sep 17 00:00:00 2001 From: Michael Zhang Date: Fri, 2 Aug 2024 10:35:50 -0500 Subject: [PATCH] lecture 7 --- Lecture7.typ | 199 +++++++++++++++++++++++++++++++++++++++++++++++++++ winding.jpeg | Bin 0 -> 107915 bytes 2 files changed, 199 insertions(+) create mode 100644 Lecture7.typ create mode 100644 winding.jpeg diff --git a/Lecture7.typ b/Lecture7.typ new file mode 100644 index 0000000..a084265 --- /dev/null +++ b/Lecture7.typ @@ -0,0 +1,199 @@ +#import "prooftree.typ": * +#import "@preview/showybox:2.0.1": showybox +#import "@preview/commute:0.2.0": node, arr, commutative-diagram +#import "@preview/cetz:0.2.2": * +#set page(width: 5.6in, height: 9in, margin: 0.4in) + +#let isofhlevel = $sans("isofhlevel")$ +#let PathOver = $sans("PathOver")$ +#let ua = $sans("ua")$ +#let idtoeqv = $sans("idtoeqv")$ +#let Helix = $sans("Helix")$ +#let UU = $cal(U)$ +#let transport = $sans("transport")$ +#let base = $sans("base")$ +#let ap = $sans("ap")$ +#let loop = $sans("loop")$ +#let Circle = $bb(S)^1$ +#let idtoiso = $sans("idtoiso")$ +#let Nat = $sans("Nat")$ +#let Vect = $sans("Vect")$ +#let Bool = $sans("Bool")$ +#let carrier = $sans("carrier")$ +#let iseqclass = $sans("iseqclass")$ +#let isInjective = $sans("isInjective")$ +#let Type = $sans("Type")$ +#let reflexive = $sans("reflexive")$ +#let Even = $sans("Even")$ +#let isEven = $sans("isEven")$ +#let Prop = $sans("Prop")$ +#let isProp = $sans("isProp")$ +#let Set = $sans("Set")$ +#let isContr = $sans("isContr")$ +#let isIso = $sans("isIso")$ +#let isEquiv = $sans("isEquiv")$ +#let isSet = $sans("isSet")$ +#let zero = $sans("zero")$ +#let suc = $sans("suc")$ +#let Monoid = $sans("Monoid")$ +#let MonoidStr = $sans("MonoidStr")$ +#let MonoidAxioms = $sans("MonoidAxioms")$ +#let refl = $sans("refl")$ +#let defeq = $equiv$ +#let propeq = $=$ + += Synthetic Homotopy Theory + +We are talking about _the_ circle today! + +What do all these words mean? + +- #[ + *Homotopy theory.* Study of topological spaces up to continuous deformation. You might have heard we don't distinguish coffee cup and donuts. But it's getting worse. + + We don't distinguish bagels and rubber bands, even if the rubber band doesn't have any thickness. Topologists distinguish them, but we don't. If you cut a hole in a piece of paper, that's still a bagel. + + They collapse many of the differences. The technical term is _continuous deformation_. As long as nothing is being torn apart, that's ok. (don't eat the bagel) +] + +- #[ + *Synthetic.* There's a coordinate system, and you can get the location of any objects in the system. You can just solve equations via algebra on the coordinate system instead. This is called the _analytic_ way of thinking. We are *not* doing that. + + Instead, we are going back to the Euclidean geometry. "There's a line between two points" doesn't specify what the line actually is. This is called _synthetic_, since it's just based on the axioms. +] + +=== Example + +The standard circle of size 1 is typically defined (_analytically_) as $ {(x, y) | x^2 + y^2 = 1} $ + +We are not doing this. Instead, we are defining the circle as the combination of + +- a point $base$ +- some abstract $loop$ from $base$ to itself + +We don't necessarily know what it looks like, or know how it works. This is a _synthetic_ approach. + +#align(center)[#line(length: 4cm)] + +One question is: this picture looks reasonable, but how do we do this in type theory? The type theory doesn't provide a reasonable way to talk about the $loop$. But we can talk about paths in the type, so we can represent $loop$ as a path. + +=== Definition using Coq notation + +``` +Inductive Circle : U := + | base : Circle + | loop : base ≡ base. +``` + +The $loop$ is a path constructor, which is not supported by Coq. How do we eliminate this circle (how do I define a function that uses a circle)? + +Mimicking the Coq notation again, we would have: + +``` +match (x : Circle) with +| base => ... +| loop => ... +end +``` + +What should we do with the second case? This is already kind of suspicious since it doesn't have type $Circle$. Also, what should be the output obligation? + +Suppose we define a function $f : Circle -> A$. We probably want the $base$ point to be sent somewhere such that $f(base) : A$. + +Intuitively, there should also be a loop in $A$, that is a path, that is "$f(loop)$". The path should be $f(base) propeq f(base)$. We put quotes around "$f(loop)$" because $f(loop)$ doesn't actually type-check in the theory. What we need right now is the map on paths $ap_f (loop)$. + +=== Dependent cases + +If $A$ depends on the circle, we can take a fibrational view. Imagine there are many fibers over the circle on the ground. In this case, $f(base) : A(base)$. Then "$f(loop)$" is a line travelling through the collection of fibers that goes back to $base$. However, $ap_f (loop)$ doesn't travel through the fibers, which we need. + +How can we specify something that is crossing different fibers? Suppose there are a collection of fibers $B : A -> U$ and a path $p : x propeq_A y$. There are endpoints $b_x : B(x)$ and $b_y : B(y)$ that travels through the collection of fibers. There are several ways to define this. + +*Definition (path over).* + +$ PathOver(p, b_x, b_y) :defeq (transport^B (p, b_x) propeq b_y) $ + +This is a path _in_ $B(y)$ between the transported $b_x$, which does the travelling between fibers. + +``` +match (x : Circle) with +| base => (fb : B base) +| loop => (... : PathOver loop fb fb) +``` + +=== Other examples of HITs + +List of things to ask ChatGPT: + ++ Spheres in arbitrary dimensions (circle is dimension 1) ++ Suspension ++ Arbitrary dimension of truncation ++ Homotopy pushouts + +=== Algebraic topology + +Why the word "algebraic"? Algebraic means you want to find some algebraic _invariants_ under continuous deformation. "If I'm allowing myself to equate bagels and rubber bands, what are the essential properties that are still the same?" For example, the bagel has a hole, and the rubber band also has a hole in it. We can turn this into a number we can calculate. + +Why do we care about this property? How can we prove that a bagel is different from a solid ball? If you can prove that the number of holes is an invariant on the deformation process, then it's a good way to classify different objects. + +One important thing is called the *fundamental group*. + +=== Fundamental group + +Fundamental group only works when u have an element in the type. Suppose a space $X$ with a point $x_0 : X$. We can define a _group_ on the paths. + +$ { "paths" x_0 propeq x_0 } \/ "homotopy" $ + +You can prove it satisfies all the group laws (refl, concatenation, inverse). We will reproduce this in type theory. + +#let trunc(x) = $bar.double #x bar.double$ + +*Definition (fundamental group).* $ pi_1 (X, x_0) &:defeq trunc(x_0 propeq x_0)_0 \ +&:defeq trunc(Omega(X, x_0))_0 $ + +We need the $x_0 propeq x_0$ to be a set, since it may have some higher structure. + +*Definition (loop space).* Loop space does not forget the interesting higher dimensional structures. + +$ Omega(X, x_0) :defeq (x_0 propeq x_0) $ + +Now we can state the theorem. + +*Theorem.* $pi_1 (Circle, base) tilde.eq ZZ$. + +*Theorem.* $Omega(Circle, base) tilde.eq ZZ$. + +Quick proof sketch: expanding the definition, we are trying to prove: $base propeq base tilde.eq ZZ$. + +What do we need for equivalence? Function going right, function going left, and prove that there are inverses going both directions. Consider the easy case: $g : ZZ -> base propeq base$ + +- $g(-1) = loop^(-1)$ +- $g(0) = refl$ +- $g(1) = loop$ +- $g(2) = loop dot loop$ + +Intuitively, this represents "how many times are you going around the circle"? With the opposite direction being the negative numbers. There is a name for this, called *winding numbers*. + +The other direction $f : (p : base propeq base) -> ZZ$ is more complicated. We can't really do induction on the loop like the integers. The trick is to construct a *covering space*. In the case of the circle, this looks like a helix going up. +The fiber of $base$ is $Z$. Then, you can follow the unknown path $p$ around the circle and simultaneously wind up and down the circular stairs. + +#image("winding.jpeg", height: 3in) + ++ We want to build a helix. ++ Transport zero along the path $p$ ++ The final number is the output, the *winding number* + +Define: $Helix : Circle -> UU$ (non-dependent case, so slightly easier) + +- $Helix(base) :defeq ZZ$ + +We can define the result of the path $ZZ propeq ZZ$. But due to univalence, we can use an equivalence here instead. + +The successor function is a good equivalence to give us this winding functionality. + +- $Helix(loop) :defeq ua(suc) : ZZ propeq ZZ$ + +Now, we can define $f$: + +$ f :defeq lambda p. transport^Helix (p, 0) $ + +Then, prove that $f$ and $g$ are inverses of each other, and then you would prove that $base propeq base tilde.eq ZZ$. \ No newline at end of file diff --git a/winding.jpeg b/winding.jpeg new file mode 100644 index 0000000000000000000000000000000000000000..a7a4c18674e2e26a3966f11bdb22f78fbc1333ed GIT binary patch literal 107915 zcmcG#cR*9kvoO2~34~A#7!Z&WDH3{-E?_|E0s)aOy(%44P)a~i=~a*-M2dhQU8RGC z8jvE2s0h+QMVfT*Jv`6v{+@g9_ucP%-#^}kbIxXW&+hE(%-{~*xUU2sPr{rI_@^Hr;BHqW|&)?Z3$j4cQ*Vx(L z`>LO#^S^ukR}cU6e~5s8_xxAwVTsZOXHR>t0Cz_fUZS77r@dd$KXCukNZ;KP8Uw0! zb(Xm7`u0BmTVVdb%=f&rmur9o-0e@;(6X#Quji|6j$l zP7nt|4VIyc-wzrb5+ztDk9GY!k3Y;i{GC@m%wG@i34q!V5A%K|#%H0t6O_lh{-?ad 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