Mercurial > dylan

     1 <?xml version="1.0" encoding="utf-8"?>

     2 <!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"

     3                "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">

     4 <html xmlns="http://www.w3.org/1999/xhtml"

     5 lang="en" xml:lang="en">

     6 <head>

     7 <title>Synthetic Differential Geometry</title>

     8 <meta http-equiv="Content-Type" content="text/html;charset=utf-8"/>

     9 <meta name="generator" content="Org-mode"/>

    10 <meta name="generated" content="2011-08-15 22:42:41 EDT"/>

    11 <meta name="author" content="Dylan Holmes"/>

    12 <meta name="description" content=""/>

    13 <meta name="keywords" content=""/>

    14 <style type="text/css">

    15  <!--/*--><![CDATA[/*><!--*/

    16   html { font-family: Times, serif; font-size: 12pt; }

    17   .title  { text-align: center; }

    18   .todo   { color: red; }

    19   .done   { color: green; }

    20   .tag    { background-color: #add8e6; font-weight:normal }

    21   .target { }

    22   .timestamp { color: #bebebe; }

    23   .timestamp-kwd { color: #5f9ea0; }

    24   .right  {margin-left:auto; margin-right:0px;  text-align:right;}

    25   .left   {margin-left:0px;  margin-right:auto; text-align:left;}

    26   .center {margin-left:auto; margin-right:auto; text-align:center;}

    27   p.verse { margin-left: 3% }

    28   pre {

    29 	border: 1pt solid #AEBDCC;

    30 	background-color: #F3F5F7;

    31 	padding: 5pt;

    32 	font-family: courier, monospace;

    33         font-size: 90%;

    34         overflow:auto;

    35   }

    36   table { border-collapse: collapse; }

    37   td, th { vertical-align: top;  }

    38   th.right  { text-align:center;  }

    39   th.left   { text-align:center;   }

    40   th.center { text-align:center; }

    41   td.right  { text-align:right;  }

    42   td.left   { text-align:left;   }

    43   td.center { text-align:center; }

    44   dt { font-weight: bold; }

    45   div.figure { padding: 0.5em; }

    46   div.figure p { text-align: center; }

    47   textarea { overflow-x: auto; }

    48   .linenr { font-size:smaller }

    49   .code-highlighted {background-color:#ffff00;}

    50   .org-info-js_info-navigation { border-style:none; }

    51   #org-info-js_console-label { font-size:10px; font-weight:bold;

    52                                white-space:nowrap; }

    53   .org-info-js_search-highlight {background-color:#ffff00; color:#000000;

    54                                  font-weight:bold; }

    55   /*]]>*/-->

    56 </style>

    57 <link rel="stylesheet" type="text/css" href="../css/aurellem.css" />

    58 <script type="text/javascript">

    59 <!--/*--><![CDATA[/*><!--*/

    60  function CodeHighlightOn(elem, id)

    61  {

    62    var target = document.getElementById(id);

    63    if(null != target) {

    64      elem.cacheClassElem = elem.className;

    65      elem.cacheClassTarget = target.className;

    66      target.className = "code-highlighted";

    67      elem.className   = "code-highlighted";

    68    }

    69  }

    70  function CodeHighlightOff(elem, id)

    71  {

    72    var target = document.getElementById(id);

    73    if(elem.cacheClassElem)

    74      elem.className = elem.cacheClassElem;

    75    if(elem.cacheClassTarget)

    76      target.className = elem.cacheClassTarget;

    77  }

    78 /*]]>*///-->

    79 </script>

    80 <script type="text/javascript" src="../MathJax/MathJax.js">

    81 <!--/*--><![CDATA[/*><!--*/

    82     MathJax.Hub.Config({

    83         // Only one of the two following lines, depending on user settings

    84         // First allows browser-native MathML display, second forces HTML/CSS

    85             config: ["MMLorHTML.js"], jax: ["input/TeX"],

    86         //  jax: ["input/TeX", "output/HTML-CSS"],

    87         extensions: ["tex2jax.js","TeX/AMSmath.js","TeX/AMSsymbols.js",

    88                      "TeX/noUndefined.js"],

    89         tex2jax: {

    90             inlineMath: [ ["\\(","\\)"] ],

    91             displayMath: [ ['$$','$$'], ["\\[","\\]"], ["\\begin{displaymath}","\\end{displaymath}"] ],

    92             skipTags: ["script","noscript","style","textarea","pre","code"],

    93             ignoreClass: "tex2jax_ignore",

    94             processEscapes: false,

    95             processEnvironments: true,

    96             preview: "TeX"

    97         },

    98         showProcessingMessages: true,

    99         displayAlign: "left",

   100         displayIndent: "2em",

   101 

   102         "HTML-CSS": {

   103              scale: 100,

   104              availableFonts: ["STIX","TeX"],

   105              preferredFont: "TeX",

   106              webFont: "TeX",

   107              imageFont: "TeX",

   108              showMathMenu: true,

   109         },

   110         MMLorHTML: {

   111              prefer: {

   112                  MSIE:    "MML",

   113                  Firefox: "MML",

   114                  Opera:   "HTML",

   115                  other:   "HTML"

   116              }

   117         }

   118     });

   119 /*]]>*///-->

   120 </script>

   121 </head>

   122 <body>

   123 

   124 <div id="content">

   125 

   126 

   127 

   128 <div class="header">

   129   <div class="float-right">	

   130     <!-- 

   131     <form>

   132       <input type="text"/><input type="submit" value="search the blog &raquo;"/> 

   133     </form>

   134     -->

   135   </div>

   136 

   137   <h1>aurellem <em>&#x2609;</em></h1>

   138   <ul class="nav">

   139     <li><a href="/">read the blog &raquo;</a></li>

   140     <!-- li><a href="#">learn about us &raquo;</a></li-->

   141   </ul>

   142 </div>

   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>

   279 </div>

   280 </div>

   281 </body>

   282 </html>
author	Robert McIntyre <rlm@mit.edu>
date	Fri, 28 Oct 2011 00:03:05 -0700
parents
children