Mercurial > dylan
diff categorical/synthetic.html @ 2:b4de894a1e2e
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| author | Robert McIntyre <rlm@mit.edu> |
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| date | Fri, 28 Oct 2011 00:03:05 -0700 |
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1.122 +/*]]>*///--> 1.123 +</script> 1.124 +</head> 1.125 +<body> 1.126 + 1.127 +<div id="content"> 1.128 + 1.129 + 1.130 + 1.131 +<div class="header"> 1.132 + <div class="float-right"> 1.133 + <!-- 1.134 + <form> 1.135 + <input type="text"/><input type="submit" value="search the blog »"/> 1.136 + </form> 1.137 + --> 1.138 + </div> 1.139 + 1.140 + <h1>aurellem <em>☉</em></h1> 1.141 + <ul class="nav"> 1.142 + <li><a href="/">read the blog »</a></li> 1.143 + <!-- li><a href="#">learn about us »</a></li--> 1.144 + </ul> 1.145 +</div> 1.146 + 1.147 +<h1 class="title">Synthetic Differential Geometry</h1> 1.148 +<div class="author">Written by <author>Dylan Holmes</author></div> 1.149 + 1.150 + 1.151 + 1.152 + 1.153 + 1.154 + 1.155 +<p> 1.156 +(My notes on Anders Kock's <i>Synthetic Differential Geometry</i>) 1.157 +</p> 1.158 + 1.159 +<div id="table-of-contents"> 1.160 +<h2>Table of Contents</h2> 1.161 +<div id="text-table-of-contents"> 1.162 +<ul> 1.163 +<li><a href="#sec-1">1 Revisiting the real line </a> 1.164 +<ul> 1.165 +<li><a href="#sec-1-1">1.1 The first anti-euclidean axiom </a></li> 1.166 +<li><a href="#sec-1-2">1.2 The first axiom \(\ldots\) in terms of arrows </a></li> 1.167 +<li><a href="#sec-1-3">1.3 Ex </a></li> 1.168 +</ul> 1.169 +</li> 1.170 +</ul> 1.171 +</div> 1.172 +</div> 1.173 + 1.174 +<div id="outline-container-1" class="outline-2"> 1.175 +<h2 id="sec-1"><span class="section-number-2">1</span> Revisiting the real line </h2> 1.176 +<div class="outline-text-2" id="text-1"> 1.177 + 1.178 + 1.179 +<p> 1.180 +<b>Lines</b>, the kind which Euclid talked about, each constitute a commutative 1.181 + ring: you choose any two points on the line to be 0 and 1, then add 1.182 + and multiply as if you were dealing with real numbers \(\mathbb{R}\). 1.183 +</p> 1.184 +<p> 1.185 +Euclid moreover uses the axiom that for any two points, <i>either</i> they are the 1.186 +same point <i>or</i> there is a unique line between them. Algebraically, 1.187 +this amounts to saying that each line is not only a commutative ring 1.188 +but a <b>field</b>, as well. This marks our first departure from euclidean 1.189 +geometry, as our first axiom denies that each line is a field. 1.190 +</p> 1.191 + 1.192 + 1.193 +</div> 1.194 + 1.195 +<div id="outline-container-1-1" class="outline-3"> 1.196 +<h3 id="sec-1-1"><span class="section-number-3">1.1</span> The first anti-euclidean axiom </h3> 1.197 +<div class="outline-text-3" id="text-1-1"> 1.198 + 1.199 +<p>A point in a ring is called <b>nilpotent</b> if its square is 1.200 +zero. Normally (that is, in \(\mathbb{R}^n\)), only \(0\) is 1.201 +nilpotent. Here, as a consequence of the following axiom, there will 1.202 +exist other elements that are nilpotent. These elements will 1.203 +encapsulate our intuitive idea of “infinitesimally small” numbers. 1.204 +</p> 1.205 +<blockquote> 1.206 + 1.207 +<p><b>Axiom 1:</b> Let \(R\) be the line, considered as a commutative ring, and 1.208 + let \(D\subset R\) be the set of nilpotent elements on the line. Then for any 1.209 + morphism \(g:D\rightarrow R\), there exists a unique \(b\in R\) such that 1.210 +</p> 1.211 + 1.212 + 1.213 +\(\forall d\in D, g(d) = g(0)+ b\cdot d\) 1.214 + 1.215 +<p> 1.216 +Intuitively, this unique \(b\) is the slope of the function \(g\) near 1.217 +zero. Because every morphism \(g\) has exactly one such \(b\), we have the 1.218 +following results: 1.219 +</p> 1.220 +<ol> 1.221 +<li>The set \(D\) of nilpotent elements contains more than 1.222 + just 0. Indeed, suppose the contrary: if \(D=\{0\}\), then for any \(g\), <i>every</i> \(b\in R\) has the 1.223 + property described above;—\(b\) isn't uniquely defined. 1.224 +</li> 1.225 +<li>Pick \(b_1\) and \(b_2\) in \(R\). If every nilpotent \(d\) satisfies \(d\cdot 1.226 + b_1 = d\cdot b_2\), then \(b_1\) and \(b_2\) are equal. 1.227 +</li> 1.228 +</ol> 1.229 + 1.230 + 1.231 +</div> 1.232 + 1.233 +</div> 1.234 + 1.235 +<div id="outline-container-1-2" class="outline-3"> 1.236 +<h3 id="sec-1-2"><span class="section-number-3">1.2</span> The first axiom \(\ldots\) in terms of arrows </h3> 1.237 +<div class="outline-text-3" id="text-1-2"> 1.238 + 1.239 + 1.240 +<p> 1.241 +Define \(\xi:R\times R\rightarrow R^D\) by \(\xi:(a,b)\mapsto (d\mapsto 1.242 +a+b\cdot d)\). The first axiom is equivalent to the statement 1.243 +“ξ is invertible (i.e., a bijection)” 1.244 +</p> 1.245 +<p> 1.246 +We give \(R\times R\) the structure of an \(R\)-algebra by defining 1.247 +multiplication: \( (a_1,b_1)\star(a_2,b_2) = (a_1\cdot a_2,\quad 1.248 +a_1\cdot b_2 + a_2\cdot b_1)\). This is called <b>dual-numbers multiplication</b>, and is similar to muliplication of complex numbers. 1.249 +</p> 1.250 + 1.251 +</div> 1.252 + 1.253 +</div> 1.254 + 1.255 +<div id="outline-container-1-3" class="outline-3"> 1.256 +<h3 id="sec-1-3"><span class="section-number-3">1.3</span> Ex </h3> 1.257 +<div class="outline-text-3" id="text-1-3"> 1.258 + 1.259 +<ol> 1.260 +<li>If \(a\) and \(b\) are nilpotent, then \(ab\) is nilpotent. 1.261 +</li> 1.262 +<li>Even if \(a\) and \(b\) are nilpotent, the sum \(a+b\) may not be. 1.263 +</li> 1.264 +<li>Even if \(a+b\) is nilpotent, either summand \(a\), \(b\) may not be. 1.265 +</li> 1.266 +<li> 1.267 +</li> 1.268 +</ol> 1.269 + 1.270 + 1.271 + 1.272 +</blockquote> 1.273 + 1.274 +</div> 1.275 +</div> 1.276 +</div> 1.277 +<div id="postamble"> 1.278 +<p class="date">Date: 2011-08-15 22:42:41 EDT</p> 1.279 +<p class="author">Author: Dylan Holmes</p> 1.280 +<p class="creator">Org version 7.6 with Emacs version 23</p> 1.281 +<a href="http://validator.w3.org/check?uri=referer">Validate XHTML 1.0</a> 1.282 +</div> 1.283 +</div> 1.284 +</body> 1.285 +</html>