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typos, gender reassignment
author Dylan Holmes <ocsenave@gmail.com>
date Thu, 27 Oct 2011 23:02:59 -0500
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1 #+TITLE: Synthetic Differential Geometry
2 #+author: Dylan Holmes
3 #+EMAIL: rlm@mit.edu
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11 (My notes on Anders Kock's /Synthetic Differential Geometry/)
13 * Revisiting the real line
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 $\mathbb{R}$.
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.
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.
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
38 \(\forall d\in D, g(d) = g(0)+ b\cdot d\)
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:
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.
50 ** The first axiom $\ldots$ in terms of arrows
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{}
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.
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.
68 #+end_quote