aurellem ☉
Synthetic Differential Geometry
(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:
- 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.
- 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
- If \(a\) and \(b\) are nilpotent, then \(ab\) is nilpotent.
- Even if \(a\) and \(b\) are nilpotent, the sum \(a+b\) may not be.
- Even if \(a+b\) is nilpotent, either summand \(a\), \(b\) may not be.