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aurellem ☉

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Synthetic Differential Geometry

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Written by Dylan Holmes
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rlm@2: (My notes on Anders Kock's Synthetic Differential Geometry) rlm@2:

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Table of Contents

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1 Revisiting the real line

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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:

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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:

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1.1 The first anti-euclidean axiom

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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 “infinitesimally small” numbers. rlm@2:

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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: rlm@2: rlm@2: \(\forall d\in D, g(d) = g(0)+ b\cdot d\) rlm@2: 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:

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  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;—\(b\) isn't uniquely defined. rlm@2:
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  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:
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1.2 The first axiom \(\ldots\) in terms of arrows

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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: “ξ is invertible (i.e., a bijection)” rlm@2:

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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 multiplication, and is similar to muliplication of complex numbers. rlm@2:

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1.3 Ex

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  1. If \(a\) and \(b\) are nilpotent, then \(ab\) is nilpotent. rlm@2:
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  2. Even if \(a\) and \(b\) are nilpotent, the sum \(a+b\) may not be. rlm@2:
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  3. Even if \(a+b\) is nilpotent, either summand \(a\), \(b\) may not be. rlm@2:
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Date: 2011-08-15 22:42:41 EDT

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Author: Dylan Holmes

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