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<h1>Category Theory <span class="and">&amp;</span> Programming
<div><author style="font-size: .4em"><em class="base01">by</em> Yann Esposito
<div>
<twitter style="font-size: .5em">
<a href="http://twitter.com/yogsototh">@yogsototh</a>,
</twitter>
<googleplus style="font-size: .5em">
<a href="https://plus.google.com/117858550730178181663">+yogsototh</a>
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</author></div>
</h1>
</section>
<section class="slide">
<h2>Plan</h2>
<ul style="font-size: 2em">
<li class="yellow">Why?</li>
<li>What?</li>
<li>How?</li>
</ul>
</section>
<section class="slide">
<h2>Abstraction Example</h2>
<p>Something you see very often but in different instances.</p>
<p>Once you recognize the notion, you see it almost everywhere.</p>
<ul>
<li class="slide">Numbers: <em>0</em>,1,2,3,... (3400 BC, real numbers 760 BC)</li>
<li class="slide">Rational numbers: \(\frac{p}{q}\) (concept is prehistoric)</li>
<li class="slide">Negative numbers: ..., -3, -2, -1, 0, 1, 2, ... (100-50 BC)</li>
<li class="slide">Irrational numbers: \(\mathbb{A}\), \(\mathbb{R}\) (500 BC → Pythogoras killed Hippasus because of \(\sqrt{2}\)!)</li>
<li class="slide">Complex numbers: \(\mathbb{C}\) (100 AD, then 16th century)</li>
</ul>
</section>
<section class="slide">
<h2>Numbers ⇒ Set Theory (\(\infty\))</h2>
<table>
<tr>
<th>Numbers</th>
<th>Set Theory</th>
</tr>
<tr>
<td>\(\mathbb{N}\): \((+,0)\)</td>
<td>Semigroups</td>
</tr>
<tr class="slide">
<td>\(\mathbb{Z}\): \((+,0,\times,1)\)</td>
<td>Rings</td>
</tr>
<tr class="slide">
<td>\(\mathbb{Q}\)</td>
<td>Fields</td>
</tr>
<tr class="slide">
<td>\(\mathbb{R}\)</td>
<td>Complete Fields (<em class="base01">topology</em>)</td>
</tr>
<tr class="slide">
<td>\(\mathbb{C}\)</td>
<td>Algebræ</td>
</tr>
<tr class="slide"><td></td><td>Modules,Vector Spaces, Monoids, ...</td></tr>
</table>
<p class="slide">More <strong>general</strong>: more things are sets.<br/>
<span class="and">&amp;</span> More <strong>precise</strong>: clear distinction between concepts)</p>
</section>
<section class="slide">
<h2>Set Theory ⇒? Category Theory</h2>
<table>
<tr>
<th>Numbers</th>
<th>Set Theory (\(\infty\))</th>
<th>Category Theory (relations)</th>
</tr>
<tr>
<td>\(\mathbb{N}\): \((+,0)\)</td>
<td>Semigroups</td>
<td>?</td>
</tr>
<tr>
<td>\(\mathbb{Z}\): \((+,0,\times,1)\)</td>
<td>Rings</td>
<td>?</td>
</tr>
<tr>
<td>\(\mathbb{Q}\)</td>
<td>Fields</td>
<td>?</td>
</tr>
<tr>
<td>\(\mathbb{R}\)</td>
<td>Complete Fields (<em class="base01">topology</em>)</td>
<td>?</td>
</tr>
<tr>
<td>\(\mathbb{C}\)</td>
<td>Algebræ</td>
<td>?</td>
</tr>
<tr><td></td><td>Modules,Vector Spaces, Monoids, ...</td><td>?</td></tr>
</table>
</section>
<section class="slide">
<h2><span class="yellow">/.*/</span> ⇒? Category Theory</h2>
<ul>
<li class="slide">Topology ⇒ Category Theory</li>
<li class="slide">Quantum Physics ⇒ Category Theory</li>
<li class="slide">Logic ⇒ Category Theory</li>
<li class="slide"><b>Programming</b> ⇒ Category Theory</li>
</ul>
<p class="slide">More <strong>general</strong>: more things are sets.<br/>
<span class="and">&amp;</span> More <strong>precise</strong>: clear distinction between concepts)</p>
</section>
<section class="slide">
<h2>too much abstraction?</h2>
<p>More general <span class="and">&amp;</span> more precise</p>
</section>
<section class="slide">
<h2>Animation, Math &amp; Code</h2>
<p>Some math \(\sum_{i=1}^\infty \frac{1}{2^x} = 1 \)</p>
<p class="slide">Another <b>math</b> <em>formula</em> $$\sum_{i=1}^\infty \frac{1}{2^x} = 1$$</p>
<pre class="slide"><code class="haskell">import Control.Monad
main :: IO()
main = do
putStrLn $ "Hello !"
</code></pre>
</section>
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