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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>

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author	Robert McIntyre <rlm@mit.edu>
date	Fri, 28 Oct 2011 04:56:48 -0700
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