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| author | Robert McIntyre <rlm@mit.edu> |
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| date | Fri, 28 Oct 2011 00:03:05 -0700 |
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| 1:8d8278e09888 | 2:b4de894a1e2e |
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| 1 #+TITLE: Synthetic Differential Geometry | |
| 2 #+author: Dylan Holmes | |
| 3 #+EMAIL: rlm@mit.edu | |
| 4 #+MATHJAX: align:"left" mathml:t path:"../MathJax/MathJax.js" | |
| 5 #+STYLE: <link rel="stylesheet" type="text/css" href="../css/aurellem.css" /> | |
| 6 #+OPTIONS: H:3 num:t toc:t \n:nil @:t ::t |:t ^:t -:t f:t *:t <:t | |
| 7 #+SETUPFILE: ../templates/level-0.org | |
| 8 #+INCLUDE: ../templates/level-0.org | |
| 9 #+BABEL: :noweb yes :results silent | |
| 10 | |
| 11 (My notes on Anders Kock's /Synthetic Differential Geometry/) | |
| 12 | |
| 13 * Revisiting the real line | |
| 14 | |
| 15 *Lines*, the kind which Euclid talked about, each constitute a commutative | |
| 16 ring: you choose any two points on the line to be 0 and 1, then add | |
| 17 and multiply as if you were dealing with real numbers $\mathbb{R}$. | |
| 18 | |
| 19 Euclid moreover uses the axiom that for any two points, /either/ they are the | |
| 20 same point /or/ there is a unique line between them. Algebraically, | |
| 21 this amounts to saying that each line is not only a commutative ring | |
| 22 but a *field*, as well. This marks our first departure from euclidean | |
| 23 geometry, as our first axiom denies that each line is a field. | |
| 24 | |
| 25 | |
| 26 ** The first anti-euclidean axiom | |
| 27 A point in a ring is called *nilpotent* if its square is | |
| 28 zero. Normally (that is, in $\mathbb{R}^n$), only $0$ is | |
| 29 nilpotent. Here, as a consequence of the following axiom, there will | |
| 30 exist other elements that are nilpotent. These elements will | |
| 31 encapsulate our intuitive idea of \ldquo{}infinitesimally small\rdquo{} numbers. | |
| 32 | |
| 33 #+begin_quote | |
| 34 *Axiom 1:* Let $R$ be the line, considered as a commutative ring, and | |
| 35 let $D\subset R$ be the set of nilpotent elements on the line. Then for any | |
| 36 morphism $g:D\rightarrow R$, there exists a unique $b\in R$ such that | |
| 37 | |
| 38 \(\forall d\in D, g(d) = g(0)+ b\cdot d\) | |
| 39 | |
| 40 Intuitively, this unique $b$ is the slope of the function $g$ near | |
| 41 zero. Because every morphism $g$ has exactly one such $b$, we have the | |
| 42 following results: | |
| 43 | |
| 44 1. The set $D$ of nilpotent elements contains more than | |
| 45 just 0. Indeed, suppose the contrary: if $D=\{0\}$, then for any $g$, /every/ $b\in R$ has the | |
| 46 property described above;\mdash{}$b$ isn't uniquely defined. | |
| 47 2. Pick $b_1$ and $b_2$ in $R$. If every nilpotent $d$ satisfies $d\cdot | |
| 48 b_1 = d\cdot b_2$, then $b_1$ and $b_2$ are equal. | |
| 49 | |
| 50 ** The first axiom $\ldots$ in terms of arrows | |
| 51 | |
| 52 Define $\xi:R\times R\rightarrow R^D$ by \(\xi:(a,b)\mapsto (d\mapsto | |
| 53 a+b\cdot d)\). The first axiom is equivalent to the statement | |
| 54 \ldquo{}\xi is invertible (i.e., a bijection)\rdquo{} | |
| 55 | |
| 56 We give $R\times R$ the structure of an $R$-algebra by defining | |
| 57 multiplication: \( (a_1,b_1)\star(a_2,b_2) = (a_1\cdot a_2,\quad | |
| 58 a_1\cdot b_2 + a_2\cdot b_1)\). This is called *dual-numbers | |
| 59 multiplication*, and is similar to muliplication of complex numbers. | |
| 60 | |
| 61 | |
| 62 ** Ex | |
| 63 1. If $a$ and $b$ are nilpotent, then $ab$ is nilpotent. | |
| 64 2. Even if $a$ and $b$ are nilpotent, the sum $a+b$ may not be. | |
| 65 3. Even if $a+b$ is nilpotent, either summand $a$, $b$ may not be. | |
| 66 4. | |
| 67 | |
| 68 #+end_quote |