301 lines
9.6 KiB
HTML
301 lines
9.6 KiB
HTML
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<title>Category Theory for Programming</title>
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<!-- Begin slides. Just make elements with a class of slide. -->
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<section class="slide">
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<h1>Category Theory <span class="and">&</span> Programming
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<div><author style="font-size: .4em"><em class="base01">by</em> Yann Esposito
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<div>
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<twitter style="font-size: .5em">
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<a href="http://twitter.com/yogsototh">@yogsototh</a>,
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</twitter>
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<googleplus style="font-size: .5em">
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<a href="https://plus.google.com/117858550730178181663">+yogsototh</a>
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</googleplus>
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</div>
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</author></div>
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</h1>
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</section>
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<section class="slide">
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<h2>Plan</h2>
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<ul style="font-size: 2em">
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<li class="yellow">Why?</li>
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<li>What?</li>
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<li>How?</li>
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</ul>
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</section>
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<section class="slide">
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<h2>Abstraction Example</h2>
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<p>Something you see very often but in different instances.</p>
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<div class="slide">
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<p>Numbers: 1,2,3,... <em class="small">3400 BC, real numbers 760 BC</em></p>
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<img src="img/tally-count.png" style="max-width: 40%; max-heiht: 30%; float:left; margin-right: 5%; vertical-align: center;" alt="Aboriginal Tally System"/>
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<img src="img/first-real-numbers.png" style="max-width: 40%; max-heiht: 30%; float:left; margin-right: 5%; vertical-align: center;" alt="Mesopotamian Numbers"/>
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<em style="font-size: 5em;vertical-align: center;">0</em>
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<div class="flush"></div>
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</section>
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<section class="slide">
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<h2>Abstraction Example: Numbers</h2>
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<p>Many different things can be understood with numbers</p>
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<ul>
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<li>How many apple, goats?</li>
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<li>Distance</li>
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<li>Time</li>
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<li>...</li>
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</ul>
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<p>Manipulated with operators:
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<span class="yellow">=</span>,
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<span class="yellow"><</span>,
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<span class="yellow">></span>,
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<span class="yellow">+</span>,
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<span class="yellow">×</span>, ...</p>
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</section>
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<section class="slide">
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<h2>Abstraction Example: Numbers</h2>
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<h3>Generalization: rational numbers</h3>
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<p>Rational numbers: \(\frac{p}{q}\) (concept is prehistoric): more precise.</p>
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<img src="img/egyptian-hieroglyphics.jpg" alt="Egyptian Fractions"/>
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</div>
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</section>
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<section class="slide">
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<h2>Abstraction Example: Numbers</h2>
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<h3>Generalization: negative numbers</h3>
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<p>Negative numbers: ..., -3, -2, -1, 0, 1, 2, ... (100-50 BC): Debts, temperature</p>
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<img src="img/negative-numbers.jpg" alt="Negative Numbers (Chinese)"/>
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</section>
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<section class="slide">
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<h2>Abstraction Example: Numbers</h2>
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<h3>Generalization: Irrational Numbers</h3>
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<p>Irrational numbers: \(\mathbb{A}\), \(\mathbb{R}\) (500 BC → Pythogoras killed Hippasus because of \(\sqrt{2}\)!)</p>
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<p>Complex numbers: \(\mathbb{C}\) (100 AD, then 16th century)</p>
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</section>
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<section class="slide">
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<h2>Abstraction Example: Numbers</h2>
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<h3>Generalization</h3>
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<ul>
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<li>More things enter into the notion</li>
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<li>More operator to manipulate the notion</li>
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</ul>
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</section>
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<section class="slide">
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<h2>Numbers ⇒ Set Theory (∞)</h2>
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<table>
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<tr>
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<th>Numbers</th>
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<th>Set Theory/Abstract Algebra</th>
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</tr>
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<tr>
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<td>\(\mathbb{N}\): \((+,0)\)</td>
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<td>Semigroups</td>
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</tr>
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<tr>
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<td>\(\mathbb{Z}\): \((+,0,\times,1)\)</td>
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<td>Rings</td>
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</tr>
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<tr>
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<td>\(\mathbb{Q}\)</td>
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<td>Fields</td>
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</tr>
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<tr>
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<td>\(\mathbb{R}\)</td>
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<td>Complete Fields (<em class="base01">topology</em>)</td>
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</tr>
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<tr>
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<td>\(\mathbb{C}\)</td>
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<td>Algebræ</td>
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</tr>
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<tr><td></td><td>Modules,Vector Spaces, Monoids, ...</td></tr>
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</table>
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<p><span class="and" style="visibility:hidden">&</span> More <strong>general</strong>: more things are sets.<br/>
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<span class="and">&</span> More <strong>precise</strong>: clear distinction between concepts.</p>
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</section>
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<section class="slide">
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<h2>Set Theory ⇒? <span class="yellow">Category Theory</span></h2>
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<table>
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<tr>
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<th>Numbers</th>
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<th>Set Theory (∞)</th>
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<th>Category Theory (relations)</th>
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</tr>
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<tr>
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<td>\(\mathbb{N}\): \((+,0)\)</td>
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<td>Semigroups</td>
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<td>?</td>
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</tr>
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<tr>
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<td>\(\mathbb{Z}\): \((+,0,\times,1)\)</td>
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<td>Rings</td>
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<td>?</td>
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</tr>
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<tr>
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<td>\(\mathbb{Q}\)</td>
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<td>Fields</td>
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<td>?</td>
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</tr>
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<tr>
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<td>\(\mathbb{R}\)</td>
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<td>Complete Fields (<em class="base01">topology</em>)</td>
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<td>?</td>
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</tr>
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<tr>
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<td>\(\mathbb{C}\)</td>
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<td>Algebræ</td>
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<td>?</td>
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</tr>
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<tr><td></td><td>Modules,Vector Spaces, Monoids, ...</td><td>?</td></tr>
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</table>
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</section>
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<section class="slide">
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<h2><span class="yellow">/.*/</span> ⇒? Category Theory</h2>
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<p>Extend <span class="and">&</span> Merge different scientific fields</p>
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<ul>
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<li>Topology</li>
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<li>Quantum Physics</li>
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<li>Logic</li>
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<li><b>Programming</b></li>
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</ul>
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<p><span class="and" style="visibility:hidden">&</span> More <strong>general</strong>: more things are Categories.<br/>
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<span class="and">&</span> More <strong>precise</strong>: better distinction between concepts.</p>
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<p>Young field: <b>1942–45</b>, Samuel Eilenberg <span class="and">&</span> Saunders Mac Lane
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</section>
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<section class="slide">
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<h2>Plan</h2>
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<ul style="font-size: 2em">
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<li>Why?</li>
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<li class="yellow">What?</li>
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<li>How?</li>
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</ul>
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</section>
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<section class="slide">
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<h2>Definition: Category [Objects]</h2>
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<img src="img/mp/objects.png" alt="objects"/>
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</section>
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<section class="slide">
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<h2>Definition: Category [Morphisms]</h2>
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<img src="img/mp/morphisms.png" alt="morphisms"/>
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</section>
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<section class="slide">
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<h2>Definition: Category [Composition]</h2>
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<p>Composition (∘): \(f:A\rightarrow B, g:B\rightarrow C\)
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$$g\circ f:A\rightarrow C$$
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</p>
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<img src="img/mp/composition.png" alt="composition"/>
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</section>
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<section class="slide">
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<h2>Definition: Category [Laws]</h2>
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<ul>
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<li>for all \(X\), there is an \(id_X\), s.t. for all \(f:A\to B\)
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<img src="img/mp/identity.png" alt="identity"/>
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</li>
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<li> Composition is associative:
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<img src="img/mp/associativecomposition.png" alt="associative composition"/>
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</li>
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</ul>
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</section>
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