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     1.4 +#+TITLE: Synthetic Differential Geometry
     1.5 +#+author: Dylan Holmes
     1.6 +#+EMAIL:     rlm@mit.edu
     1.7 +#+MATHJAX: align:"left" mathml:t path:"../MathJax/MathJax.js"
     1.8 +#+STYLE: <link rel="stylesheet" type="text/css" href="../css/aurellem.css" />
     1.9 +#+OPTIONS:   H:3 num:t toc:t \n:nil @:t ::t |:t ^:t -:t f:t *:t <:t
    1.10 +#+SETUPFILE: ../templates/level-0.org
    1.11 +#+INCLUDE: ../templates/level-0.org
    1.12 +#+BABEL: :noweb yes :results silent
    1.13 +
    1.14 +(My notes on Anders Kock's /Synthetic Differential Geometry/)
    1.15 +
    1.16 +* Revisiting the real line
    1.17 +
    1.18 +*Lines*, the kind which Euclid talked about, each constitute a commutative
    1.19 + ring: you choose any two points on the line to be 0 and 1, then add
    1.20 + and multiply as if you were dealing with real numbers $\mathbb{R}$.
    1.21 +
    1.22 +Euclid moreover uses the axiom that for any two points, /either/ they are the
    1.23 +same point /or/ there is a unique line between them. Algebraically,
    1.24 +this amounts to saying that each line is not only a commutative ring
    1.25 +but a *field*, as well. This marks our first departure from euclidean
    1.26 +geometry, as our first axiom denies that each line is a field.
    1.27 +
    1.28 +
    1.29 +** The first anti-euclidean axiom
    1.30 +A point in a ring is called *nilpotent* if its square is
    1.31 +zero. Normally (that is, in $\mathbb{R}^n$), only $0$ is
    1.32 +nilpotent. Here, as a consequence of the following axiom, there will
    1.33 +exist other elements that are nilpotent. These elements will
    1.34 +encapsulate our intuitive idea of \ldquo{}infinitesimally small\rdquo{} numbers.
    1.35 +
    1.36 +#+begin_quote
    1.37 +*Axiom 1:* Let $R$ be the line, considered as a commutative ring, and
    1.38 + let $D\subset R$ be the set of nilpotent elements on the line. Then for any
    1.39 + morphism $g:D\rightarrow R$, there exists a unique $b\in R$ such that
    1.40 +
    1.41 +\(\forall d\in D, g(d) = g(0)+ b\cdot d\)
    1.42 +
    1.43 +Intuitively, this unique $b$ is the slope of the function $g$ near
    1.44 +zero. Because every morphism $g$ has exactly one such $b$, we have the
    1.45 +following results:
    1.46 +
    1.47 +1. The set $D$ of nilpotent elements contains more than
    1.48 +   just 0. Indeed, suppose the contrary: if $D=\{0\}$, then for any $g$, /every/ $b\in R$ has the
    1.49 +   property described above;\mdash{}$b$ isn't uniquely defined.
    1.50 +2. Pick $b_1$ and $b_2$ in $R$. If every nilpotent $d$ satisfies $d\cdot
    1.51 +   b_1 = d\cdot b_2$, then $b_1$ and $b_2$ are equal.
    1.52 +
    1.53 +** The first axiom $\ldots$ in terms of arrows
    1.54 +
    1.55 +Define $\xi:R\times R\rightarrow R^D$ by \(\xi:(a,b)\mapsto (d\mapsto
    1.56 +a+b\cdot d)\). The first axiom is equivalent to the statement
    1.57 +\ldquo{}\xi is invertible (i.e., a bijection)\rdquo{}
    1.58 +
    1.59 +We give $R\times R$ the structure of an $R$-algebra by defining
    1.60 +multiplication: \( (a_1,b_1)\star(a_2,b_2) = (a_1\cdot a_2,\quad
    1.61 +a_1\cdot b_2 + a_2\cdot b_1)\). This is called *dual-numbers
    1.62 +multiplication*, and is similar to muliplication of complex numbers.
    1.63 +
    1.64 +
    1.65 +** Ex
    1.66 +1. If $a$ and $b$ are nilpotent, then $ab$ is nilpotent.
    1.67 +2. Even if $a$ and $b$ are nilpotent, the sum $a+b$ may not be.
    1.68 +3. Even if $a+b$ is nilpotent, either summand $a$, $b$ may not be.
    1.69 +4. 
    1.70 + 
    1.71 +#+end_quote