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New article: Inductive lattices
author Dylan Holmes <ocsenave@gmail.com>
date Tue, 01 Nov 2011 01:55:26 -0500
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rlm@2 143
rlm@2 144 <h1 class="title">Synthetic Differential Geometry</h1>
rlm@2 145 <div class="author">Written by <author>Dylan Holmes</author></div>
rlm@2 146
rlm@2 147
rlm@2 148
rlm@2 149
rlm@2 150
rlm@2 151
rlm@2 152 <p>
rlm@2 153 (My notes on Anders Kock's <i>Synthetic Differential Geometry</i>)
rlm@2 154 </p>
rlm@2 155
rlm@2 156 <div id="table-of-contents">
rlm@2 157 <h2>Table of Contents</h2>
rlm@2 158 <div id="text-table-of-contents">
rlm@2 159 <ul>
rlm@2 160 <li><a href="#sec-1">1 Revisiting the real line </a>
rlm@2 161 <ul>
rlm@2 162 <li><a href="#sec-1-1">1.1 The first anti-euclidean axiom </a></li>
rlm@2 163 <li><a href="#sec-1-2">1.2 The first axiom \(\ldots\) in terms of arrows </a></li>
rlm@2 164 <li><a href="#sec-1-3">1.3 Ex </a></li>
rlm@2 165 </ul>
rlm@2 166 </li>
rlm@2 167 </ul>
rlm@2 168 </div>
rlm@2 169 </div>
rlm@2 170
rlm@2 171 <div id="outline-container-1" class="outline-2">
rlm@2 172 <h2 id="sec-1"><span class="section-number-2">1</span> Revisiting the real line </h2>
rlm@2 173 <div class="outline-text-2" id="text-1">
rlm@2 174
rlm@2 175
rlm@2 176 <p>
rlm@2 177 <b>Lines</b>, the kind which Euclid talked about, each constitute a commutative
rlm@2 178 ring: you choose any two points on the line to be 0 and 1, then add
rlm@2 179 and multiply as if you were dealing with real numbers \(\mathbb{R}\).
rlm@2 180 </p>
rlm@2 181 <p>
rlm@2 182 Euclid moreover uses the axiom that for any two points, <i>either</i> they are the
rlm@2 183 same point <i>or</i> there is a unique line between them. Algebraically,
rlm@2 184 this amounts to saying that each line is not only a commutative ring
rlm@2 185 but a <b>field</b>, as well. This marks our first departure from euclidean
rlm@2 186 geometry, as our first axiom denies that each line is a field.
rlm@2 187 </p>
rlm@2 188
rlm@2 189
rlm@2 190 </div>
rlm@2 191
rlm@2 192 <div id="outline-container-1-1" class="outline-3">
rlm@2 193 <h3 id="sec-1-1"><span class="section-number-3">1.1</span> The first anti-euclidean axiom </h3>
rlm@2 194 <div class="outline-text-3" id="text-1-1">
rlm@2 195
rlm@2 196 <p>A point in a ring is called <b>nilpotent</b> if its square is
rlm@2 197 zero. Normally (that is, in \(\mathbb{R}^n\)), only \(0\) is
rlm@2 198 nilpotent. Here, as a consequence of the following axiom, there will
rlm@2 199 exist other elements that are nilpotent. These elements will
rlm@2 200 encapsulate our intuitive idea of &ldquo;infinitesimally small&rdquo; numbers.
rlm@2 201 </p>
rlm@2 202 <blockquote>
rlm@2 203
rlm@2 204 <p><b>Axiom 1:</b> Let \(R\) be the line, considered as a commutative ring, and
rlm@2 205 let \(D\subset R\) be the set of nilpotent elements on the line. Then for any
rlm@2 206 morphism \(g:D\rightarrow R\), there exists a unique \(b\in R\) such that
rlm@2 207 </p>
rlm@2 208
rlm@2 209
rlm@2 210 \(\forall d\in D, g(d) = g(0)+ b\cdot d\)
rlm@2 211
rlm@2 212 <p>
rlm@2 213 Intuitively, this unique \(b\) is the slope of the function \(g\) near
rlm@2 214 zero. Because every morphism \(g\) has exactly one such \(b\), we have the
rlm@2 215 following results:
rlm@2 216 </p>
rlm@2 217 <ol>
rlm@2 218 <li>The set \(D\) of nilpotent elements contains more than
rlm@2 219 just 0. Indeed, suppose the contrary: if \(D=\{0\}\), then for any \(g\), <i>every</i> \(b\in R\) has the
rlm@2 220 property described above;&mdash;\(b\) isn't uniquely defined.
rlm@2 221 </li>
rlm@2 222 <li>Pick \(b_1\) and \(b_2\) in \(R\). If every nilpotent \(d\) satisfies \(d\cdot
rlm@2 223 b_1 = d\cdot b_2\), then \(b_1\) and \(b_2\) are equal.
rlm@2 224 </li>
rlm@2 225 </ol>
rlm@2 226
rlm@2 227
rlm@2 228 </div>
rlm@2 229
rlm@2 230 </div>
rlm@2 231
rlm@2 232 <div id="outline-container-1-2" class="outline-3">
rlm@2 233 <h3 id="sec-1-2"><span class="section-number-3">1.2</span> The first axiom \(\ldots\) in terms of arrows </h3>
rlm@2 234 <div class="outline-text-3" id="text-1-2">
rlm@2 235
rlm@2 236
rlm@2 237 <p>
rlm@2 238 Define \(\xi:R\times R\rightarrow R^D\) by \(\xi:(a,b)\mapsto (d\mapsto
rlm@2 239 a+b\cdot d)\). The first axiom is equivalent to the statement
rlm@2 240 &ldquo;&xi; is invertible (i.e., a bijection)&rdquo;
rlm@2 241 </p>
rlm@2 242 <p>
rlm@2 243 We give \(R\times R\) the structure of an \(R\)-algebra by defining
rlm@2 244 multiplication: \( (a_1,b_1)\star(a_2,b_2) = (a_1\cdot a_2,\quad
rlm@2 245 a_1\cdot b_2 + a_2\cdot b_1)\). This is called <b>dual-numbers multiplication</b>, and is similar to muliplication of complex numbers.
rlm@2 246 </p>
rlm@2 247
rlm@2 248 </div>
rlm@2 249
rlm@2 250 </div>
rlm@2 251
rlm@2 252 <div id="outline-container-1-3" class="outline-3">
rlm@2 253 <h3 id="sec-1-3"><span class="section-number-3">1.3</span> Ex </h3>
rlm@2 254 <div class="outline-text-3" id="text-1-3">
rlm@2 255
rlm@2 256 <ol>
rlm@2 257 <li>If \(a\) and \(b\) are nilpotent, then \(ab\) is nilpotent.
rlm@2 258 </li>
rlm@2 259 <li>Even if \(a\) and \(b\) are nilpotent, the sum \(a+b\) may not be.
rlm@2 260 </li>
rlm@2 261 <li>Even if \(a+b\) is nilpotent, either summand \(a\), \(b\) may not be.
rlm@2 262 </li>
rlm@2 263 <li>
rlm@2 264 </li>
rlm@2 265 </ol>
rlm@2 266
rlm@2 267
rlm@2 268
rlm@2 269 </blockquote>
rlm@2 270
rlm@2 271 </div>
rlm@2 272 </div>
rlm@2 273 </div>
rlm@2 274 <div id="postamble">
rlm@2 275 <p class="date">Date: 2011-08-15 22:42:41 EDT</p>
rlm@2 276 <p class="author">Author: Dylan Holmes</p>
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