view categorical/synthetic.html @ 11:1f112b4f9e8f tip

Fixed what was baroque.
author Dylan Holmes <ocsenave@gmail.com>
date Tue, 01 Nov 2011 02:30:49 -0500
parents b4de894a1e2e
children
line wrap: on
line source
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",
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>
124 <div id="content">
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>
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>
144 <h1 class="title">Synthetic Differential Geometry</h1>
145 <div class="author">Written by <author>Dylan Holmes</author></div>
152 <p>
153 (My notes on Anders Kock's <i>Synthetic Differential Geometry</i>)
154 </p>
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>
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">
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>
190 </div>
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">
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>
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>
210 \(\forall d\in D, g(d) = g(0)+ b\cdot d\)
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>
228 </div>
230 </div>
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">
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>
248 </div>
250 </div>
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">
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>
269 </blockquote>
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>