#+TITLE: Synthetic Differential Geometry

 #+author: Dylan Holmes

 #+EMAIL:     rlm@mit.edu

 #+MATHJAX: align:"left" mathml:t path:"../MathJax/MathJax.js"

 #+STYLE: <link rel="stylesheet" type="text/css" href="../css/aurellem.css" />

 #+OPTIONS:   H:3 num:t toc:t \n:nil @:t ::t |:t ^:t -:t f:t *:t <:t

 #+SETUPFILE: ../templates/level-0.org

 #+INCLUDE: ../templates/level-0.org

 #+BABEL: :noweb yes :results silent

 

 (My notes on Anders Kock's /Synthetic Differential Geometry/)

 

 * Revisiting the real line

 

 *Lines*, the kind which Euclid talked about, each constitute a commutative

  ring: you choose any two points on the line to be 0 and 1, then add

  and multiply as if you were dealing with real numbers $\mathbb{R}$.

 

 Euclid moreover uses the axiom that for any two points, /either/ they are the

 same point /or/ there is a unique line between them. Algebraically,

 this amounts to saying that each line is not only a commutative ring

 but a *field*, as well. This marks our first departure from euclidean

 geometry, as our first axiom denies that each line is a field.

 

 

 ** The first anti-euclidean axiom

 A point in a ring is called *nilpotent* if its square is

 zero. Normally (that is, in $\mathbb{R}^n$), only $0$ is

 nilpotent. Here, as a consequence of the following axiom, there will

 exist other elements that are nilpotent. These elements will

 encapsulate our intuitive idea of \ldquo{}infinitesimally small\rdquo{} numbers.

 

 #+begin_quote

 *Axiom 1:* Let $R$ be the line, considered as a commutative ring, and

  let $D\subset R$ be the set of nilpotent elements on the line. Then for any

  morphism $g:D\rightarrow R$, there exists a unique $b\in R$ such that

 

 \(\forall d\in D, g(d) = g(0)+ b\cdot d\)

 

 Intuitively, this unique $b$ is the slope of the function $g$ near

 zero. Because every morphism $g$ has exactly one such $b$, we have the

 following results:

 

 1. The set $D$ of nilpotent elements contains more than

    just 0. Indeed, suppose the contrary: if $D=\{0\}$, then for any $g$, /every/ $b\in R$ has the

    property described above;\mdash{}$b$ isn't uniquely defined.

 2. Pick $b_1$ and $b_2$ in $R$. If every nilpotent $d$ satisfies $d\cdot

    b_1 = d\cdot b_2$, then $b_1$ and $b_2$ are equal.

 

 ** The first axiom $\ldots$ in terms of arrows

 

 Define $\xi:R\times R\rightarrow R^D$ by \(\xi:(a,b)\mapsto (d\mapsto

 a+b\cdot d)\). The first axiom is equivalent to the statement

 \ldquo{}\xi is invertible (i.e., a bijection)\rdquo{}

 

 We give $R\times R$ the structure of an $R$-algebra by defining

 multiplication: \( (a_1,b_1)\star(a_2,b_2) = (a_1\cdot a_2,\quad

 a_1\cdot b_2 + a_2\cdot b_1)\). This is called *dual-numbers

 multiplication*, and is similar to muliplication of complex numbers.

 

 

 ** Ex

 1. If $a$ and $b$ are nilpotent, then $ab$ is nilpotent.

 2. Even if $a$ and $b$ are nilpotent, the sum $a+b$ may not be.

 3. Even if $a+b$ is nilpotent, either summand $a$, $b$ may not be.

 4. 

  

 #+end_quote