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author Robert McIntyre <rlm@mit.edu>
date Fri, 28 Oct 2011 00:03:05 -0700
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143
144 <h1 class="title">Synthetic Differential Geometry</h1>
145 <div class="author">Written by <author>Dylan Holmes</author></div>
146
147
148
149
150
151
152 <p>
153 (My notes on Anders Kock's <i>Synthetic Differential Geometry</i>)
154 </p>
155
156 <div id="table-of-contents">
157 <h2>Table of Contents</h2>
158 <div id="text-table-of-contents">
159 <ul>
160 <li><a href="#sec-1">1 Revisiting the real line </a>
161 <ul>
162 <li><a href="#sec-1-1">1.1 The first anti-euclidean axiom </a></li>
163 <li><a href="#sec-1-2">1.2 The first axiom \(\ldots\) in terms of arrows </a></li>
164 <li><a href="#sec-1-3">1.3 Ex </a></li>
165 </ul>
166 </li>
167 </ul>
168 </div>
169 </div>
170
171 <div id="outline-container-1" class="outline-2">
172 <h2 id="sec-1"><span class="section-number-2">1</span> Revisiting the real line </h2>
173 <div class="outline-text-2" id="text-1">
174
175
176 <p>
177 <b>Lines</b>, the kind which Euclid talked about, each constitute a commutative
178 ring: you choose any two points on the line to be 0 and 1, then add
179 and multiply as if you were dealing with real numbers \(\mathbb{R}\).
180 </p>
181 <p>
182 Euclid moreover uses the axiom that for any two points, <i>either</i> they are the
183 same point <i>or</i> there is a unique line between them. Algebraically,
184 this amounts to saying that each line is not only a commutative ring
185 but a <b>field</b>, as well. This marks our first departure from euclidean
186 geometry, as our first axiom denies that each line is a field.
187 </p>
188
189
190 </div>
191
192 <div id="outline-container-1-1" class="outline-3">
193 <h3 id="sec-1-1"><span class="section-number-3">1.1</span> The first anti-euclidean axiom </h3>
194 <div class="outline-text-3" id="text-1-1">
195
196 <p>A point in a ring is called <b>nilpotent</b> if its square is
197 zero. Normally (that is, in \(\mathbb{R}^n\)), only \(0\) is
198 nilpotent. Here, as a consequence of the following axiom, there will
199 exist other elements that are nilpotent. These elements will
200 encapsulate our intuitive idea of &ldquo;infinitesimally small&rdquo; numbers.
201 </p>
202 <blockquote>
203
204 <p><b>Axiom 1:</b> Let \(R\) be the line, considered as a commutative ring, and
205 let \(D\subset R\) be the set of nilpotent elements on the line. Then for any
206 morphism \(g:D\rightarrow R\), there exists a unique \(b\in R\) such that
207 </p>
208
209
210 \(\forall d\in D, g(d) = g(0)+ b\cdot d\)
211
212 <p>
213 Intuitively, this unique \(b\) is the slope of the function \(g\) near
214 zero. Because every morphism \(g\) has exactly one such \(b\), we have the
215 following results:
216 </p>
217 <ol>
218 <li>The set \(D\) of nilpotent elements contains more than
219 just 0. Indeed, suppose the contrary: if \(D=\{0\}\), then for any \(g\), <i>every</i> \(b\in R\) has the
220 property described above;&mdash;\(b\) isn't uniquely defined.
221 </li>
222 <li>Pick \(b_1\) and \(b_2\) in \(R\). If every nilpotent \(d\) satisfies \(d\cdot
223 b_1 = d\cdot b_2\), then \(b_1\) and \(b_2\) are equal.
224 </li>
225 </ol>
226
227
228 </div>
229
230 </div>
231
232 <div id="outline-container-1-2" class="outline-3">
233 <h3 id="sec-1-2"><span class="section-number-3">1.2</span> The first axiom \(\ldots\) in terms of arrows </h3>
234 <div class="outline-text-3" id="text-1-2">
235
236
237 <p>
238 Define \(\xi:R\times R\rightarrow R^D\) by \(\xi:(a,b)\mapsto (d\mapsto
239 a+b\cdot d)\). The first axiom is equivalent to the statement
240 &ldquo;&xi; is invertible (i.e., a bijection)&rdquo;
241 </p>
242 <p>
243 We give \(R\times R\) the structure of an \(R\)-algebra by defining
244 multiplication: \( (a_1,b_1)\star(a_2,b_2) = (a_1\cdot a_2,\quad
245 a_1\cdot b_2 + a_2\cdot b_1)\). This is called <b>dual-numbers multiplication</b>, and is similar to muliplication of complex numbers.
246 </p>
247
248 </div>
249
250 </div>
251
252 <div id="outline-container-1-3" class="outline-3">
253 <h3 id="sec-1-3"><span class="section-number-3">1.3</span> Ex </h3>
254 <div class="outline-text-3" id="text-1-3">
255
256 <ol>
257 <li>If \(a\) and \(b\) are nilpotent, then \(ab\) is nilpotent.
258 </li>
259 <li>Even if \(a\) and \(b\) are nilpotent, the sum \(a+b\) may not be.
260 </li>
261 <li>Even if \(a+b\) is nilpotent, either summand \(a\), \(b\) may not be.
262 </li>
263 <li>
264 </li>
265 </ol>
266
267
268
269 </blockquote>
270
271 </div>
272 </div>
273 </div>
274 <div id="postamble">
275 <p class="date">Date: 2011-08-15 22:42:41 EDT</p>
276 <p class="author">Author: Dylan Holmes</p>
277 <p class="creator">Org version 7.6 with Emacs version 23</p>
278 <a href="http://validator.w3.org/check?uri=referer">Validate XHTML 1.0</a>
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