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Synthetic Differential Geometry

Written by Dylan Holmes

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

Table of Contents

1 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.

1.1 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 “infinitesimally small” numbers.

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;—\(b\) 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.

1.2 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 “ξ is invertible (i.e., a bijection)”

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.

1.3 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.

Date: 2011-08-15 22:42:41 EDT

Author: Dylan Holmes

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