rlm@2: #+TITLE: Synthetic Differential Geometry
rlm@2: #+author: Dylan Holmes
rlm@2: #+EMAIL:     rlm@mit.edu
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rlm@2: 
rlm@2: (My notes on Anders Kock's /Synthetic Differential Geometry/)
rlm@2: 
rlm@2: * Revisiting the real line
rlm@2: 
rlm@2: *Lines*, the kind which Euclid talked about, each constitute a commutative
rlm@2:  ring: you choose any two points on the line to be 0 and 1, then add
rlm@2:  and multiply as if you were dealing with real numbers $\mathbb{R}$.
rlm@2: 
rlm@2: Euclid moreover uses the axiom that for any two points, /either/ they are the
rlm@2: same point /or/ there is a unique line between them. Algebraically,
rlm@2: this amounts to saying that each line is not only a commutative ring
rlm@2: but a *field*, as well. This marks our first departure from euclidean
rlm@2: geometry, as our first axiom denies that each line is a field.
rlm@2: 
rlm@2: 
rlm@2: ** The first anti-euclidean axiom
rlm@2: A point in a ring is called *nilpotent* if its square is
rlm@2: zero. Normally (that is, in $\mathbb{R}^n$), only $0$ is
rlm@2: nilpotent. Here, as a consequence of the following axiom, there will
rlm@2: exist other elements that are nilpotent. These elements will
rlm@2: encapsulate our intuitive idea of \ldquo{}infinitesimally small\rdquo{} numbers.
rlm@2: 
rlm@2: #+begin_quote
rlm@2: *Axiom 1:* Let $R$ be the line, considered as a commutative ring, and
rlm@2:  let $D\subset R$ be the set of nilpotent elements on the line. Then for any
rlm@2:  morphism $g:D\rightarrow R$, there exists a unique $b\in R$ such that
rlm@2: 
rlm@2: \(\forall d\in D, g(d) = g(0)+ b\cdot d\)
rlm@2: 
rlm@2: Intuitively, this unique $b$ is the slope of the function $g$ near
rlm@2: zero. Because every morphism $g$ has exactly one such $b$, we have the
rlm@2: following results:
rlm@2: 
rlm@2: 1. The set $D$ of nilpotent elements contains more than
rlm@2:    just 0. Indeed, suppose the contrary: if $D=\{0\}$, then for any $g$, /every/ $b\in R$ has the
rlm@2:    property described above;\mdash{}$b$ isn't uniquely defined.
rlm@2: 2. Pick $b_1$ and $b_2$ in $R$. If every nilpotent $d$ satisfies $d\cdot
rlm@2:    b_1 = d\cdot b_2$, then $b_1$ and $b_2$ are equal.
rlm@2: 
rlm@2: ** The first axiom $\ldots$ in terms of arrows
rlm@2: 
rlm@2: Define $\xi:R\times R\rightarrow R^D$ by \(\xi:(a,b)\mapsto (d\mapsto
rlm@2: a+b\cdot d)\). The first axiom is equivalent to the statement
rlm@2: \ldquo{}\xi is invertible (i.e., a bijection)\rdquo{}
rlm@2: 
rlm@2: We give $R\times R$ the structure of an $R$-algebra by defining
rlm@2: multiplication: \( (a_1,b_1)\star(a_2,b_2) = (a_1\cdot a_2,\quad
rlm@2: a_1\cdot b_2 + a_2\cdot b_1)\). This is called *dual-numbers
rlm@2: multiplication*, and is similar to muliplication of complex numbers.
rlm@2: 
rlm@2: 
rlm@2: ** Ex
rlm@2: 1. If $a$ and $b$ are nilpotent, then $ab$ is nilpotent.
rlm@2: 2. Even if $a$ and $b$ are nilpotent, the sum $a+b$ may not be.
rlm@2: 3. Even if $a+b$ is nilpotent, either summand $a$, $b$ may not be.
rlm@2: 4. 
rlm@2:  
rlm@2: #+end_quote