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