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Why? part first draft
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<title>Category Theory for Programming</title>
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functors
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<div style="display:none">
\(\newcommand{\F}{\mathbf{F}}\)
\(\newcommand{\C}{\mathcal{C}}\)
\(\newcommand{\D}{\mathcal{D}}\)
\(\newcommand{\id}{\mathrm{id}}\)
\(\newcommand{\ob}[1]{\mathrm{ob}(#1)}\)
\(\newcommand{\hom}[1]{\mathrm{hom}(#1)}\)
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\(\newcommand{\Set}{\mathbf{Set}}\)
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</div>
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<!-- Begin slides. Just make elements with a class of slide. -->
<section class="slide">
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<h1>Category Theory <span class="and"><span class="and">&amp;</span></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>
</googleplus>
</div>
</author></div>
</h1>
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</section>
<section class="slide">
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<h2>Plan</h2>
<ul style="font-size: 2em">
<li class="yellow">Why?</li>
<li>What?</li>
<li>How?</li>
</ul>
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</section>
<section class="slide">
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<h2>Abstraction</h2>
<p>A common concept you see often in multiple instances.</p>
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<div class="slide">
<p>Numbers: 1,2,3,... <em class="small">3400 BC, real numbers 760 BC</em></p>
<figure class="left">
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<img src="categories/img/tally-count.png" style="height:5em" alt="Aboriginal Tally System"/>
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<figcaption>Aboriginal Tally System</figcaption>
</figure>
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<div class="slide">
<div class="left">amelioration ⇒</div>
<figure class="left">
<img src="categories/img/first-real-numbers.png" style="height:5em" alt="Mesopotamian Numbers"/>
<figcaption>Mesopotamian base 60 system</figcaption>
</figure>
</div>
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<div class="flush"></div>
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<div class="slide">
Operators: <b>=, &lt;, &gt;, +, ×, ...</b>
</div>
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</section>
<section class="slide">
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<h2>Abstraction</h2>
<h3>Generalizations</h3>
<div class="right">
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<img src="categories/img/egyptian-hieroglyphics.jpg" alt="Egyptian Fractions"/>
<img src="categories/img/negative-numbers.jpg" alt="Negative Numbers (Chinese)"/>
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</div>
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<ul>
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<li>Rational (Weight/Distance/Time): \(\frac{p}{q}\)
</li>
<li>Negative (Debts): ..., -2, -1, ? , 1, 2, ...
</li>
<li>Irrational (Geometry): 500 BC, Pythogoras killed Hippasus for \(\sqrt{2}\)</li>
<li>Complex (Algebra): \(\mathbb{C}\) ; 100 AD, then 16th century</li>
<li>\(0\): nothing become a number!</li>
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</ul>
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<p><b>More "things" can be understood as numbers</b></p>
<p><b>More "operator" to manipulate numbers</b></p>
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</section>
<section class="slide">
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<h2>Numbers ⇒ Sets</h2>
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<table>
<tr>
<th>Numbers</th>
<th>Set Theory (∞)/Abstract Algebra/Topology</th>
</tr>
<tr>
<td>\(\mathbb{N}\): \((+,0)\)</td>
<td>Semigroups</td>
</tr>
<tr>
<td>\(\mathbb{Z}\): \((+,0,\times,1)\)</td>
<td>Rings</td>
</tr>
<tr>
<td>\(\mathbb{Q}\)</td>
<td>Fields</td>
</tr>
<tr>
<td>\(\mathbb{R}\)</td>
<td>Complete Fields (<em class="base01">topology</em>)</td>
</tr>
<tr>
<td>\(\mathbb{C}\)</td>
<td>Algebræ</td>
</tr>
<tr><td></td><td>Modules,Vector Spaces, Monoids, ...</td></tr>
</table>
<p><span class="and" style="visibility:hidden"><span class="and">&amp;</span></span> More <strong>general</strong>: more things are sets.<br/>
<span class="and"><span class="and">&amp;</span></span> More <strong>precise</strong>: clear distinction between concepts.</p>
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</section>
<section class="slide">
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<h2>Sets ⇒? <span class="yellow">Categories</span></h2>
<table>
<tr>
<th>Numbers</th>
<th>Sets</th>
<th>Categories</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>
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</section>
<section class="slide">
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<h2><span class="yellow">/.*/</span> ⇒? Category Theory</h2>
<p>Extend <span class="and"><span class="and">&amp;</span></span> Merge different scientific fields</p>
<ul>
<li>Topology</li>
<li>Quantum Physics</li>
<li>Logic</li>
<li><b>Programming</b></li>
</ul>
<p><span class="and" style="visibility:hidden"><span class="and">&amp;</span></span> More <strong>general</strong>: more things are Categories.<br/>
<span class="and"><span class="and">&amp;</span></span> More <strong>precise</strong>: better distinction between concepts.</p>
<p>Young field: <b>1942–45</b>, Samuel Eilenberg <span class="and"><span class="and">&amp;</span></span> Saunders Mac Lane
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</section>
<section class="slide">
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<h2>Type Theory ⇒ Categories</h2>
<ul>
<li>Type theory helped to remove paradoxes in Set Theory.</li>
<li>Prevent relations between different kind of objects.</li>
<li>Used in computer science</li>
</ul>
<ul>
<li>typed λ-calculus ⇒ cartesian closed categories</li>
<li>untyped λ-calculus ⇒ C-monoids (subclass of categories)</li>
<li>Martin-Löf type theories ⇒ locally cartesian closed categories</li>
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</ul>
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</section>
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<section class="slide">
better order
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<h2>Plan</h2>
<ul style="font-size: 2em">
<li>Why?</li>
<li class="yellow">What?</li>
<li>How?</li>
</ul>
</section>
<section class="slide">
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<h2>Category Definition</h2>
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<p> A Category \(\mathcal{C}\) is defined by:</p>
<ul>
<li> <em>Objects (\(\ob{C}\))</em>,</li>
<li> <em>Morphisms (\(\hom{C}\))</em>,</li>
<li> a <em>Composition law (∘)</em></li>
<li> obeying some <em>Properties</em>.</li>
</ul>
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</section>
<section class="slide">
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<h2>Category Definition: Objects</h2>
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<img src="categories/img/mp/objects.png" alt="objects" />
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<p>\(\ob{\mathcal{C}}\) is a collection</p>
</section>
<section class="slide">
<h2>Category Definition: Morphisms</h2>
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<img src="categories/img/mp/morphisms.png" alt="morphisms"/>
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<p>\(\hom{A,B}\) is a collection</p>
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</section>
<section class="slide">
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<h2>Category Definition: Composition</h2>
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<p>Composition (∘): \(f:A\rightarrow B, g:B\rightarrow C\)
$$g\circ f:A\rightarrow C$$
</p>
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<img src="categories/img/mp/composition.png" alt="composition"/>
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</section>
<section class="slide">
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<h2>Category laws: neutral element</h2>
<p>for all \(X\), there is an \(\id_X\), s.t. for all \(f:A\to B\):</p>
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<img src="categories/img/mp/identity.png" alt="identity"/>
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</section>
<section class="slide">
<h2>Category laws: Associativity</h2>
<p> Composition is associative:</p>
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<img src="categories/img/mp/associativecomposition.png" alt="associative composition"/>
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</section>
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<section class="slide">
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<h2>Can this be a category? <span style="font-size: .5em">(\(\id_X\) implicit)</span></h2>
<figure class="left">
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<img src="categories/img/mp/cat-example1.png" alt="Category example 1"/>
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<figcaption class="slide">
<span class="green">OK</span>
</figcaption>
</figure>
<figure class="left">
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<img src="categories/img/mp/cat-example2.png" alt="Category example 2"/>
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<figcaption class="slide">
no candidate for \(g\circ f\)
<span class="red">NO</span>
</figcaption>
</figure>
<figure class="left">
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<img src="categories/img/mp/cat-example3.png" alt="Category example 3"/>
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<figcaption class="slide">
<span class="green">YES</span>
</figcaption>
</figure>
<figure class="left">
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<img src="categories/img/mp/cat-example4.png" alt="Category example 4"/>
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<figcaption class="slide">
no candidate for \(f:C\to B\)
<span class="red">NO</span>
</figcaption>
</figure>
<figure class="left">
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<img src="categories/img/mp/cat-example5.png" alt="Category example 5"/>
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<figcaption class="slide">
\((h∘g)∘f=\id_B∘f=f≠h=h∘\id_A=h∘(g∘f)\)
<span class="red">NO</span>
</figcaption>
</figure>
</section>
<section class="slide">
update
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<h2>Categories Everywhere?</h2>
<h3>\(\Set\)</h3>
<ul>
<li> \(\ob{\Set}\) are sets</li>
<li> \(\hom{\Set}\) are functions</li>
<li> ∘ is functions composition </li>
</ul>
<ul class="slide">
<li>\(\ob{\Set}\) is a proper class ; not a set</li>
<li>\(\hom{E,F}\) is a set</li>
<li>\(\Set\) is a <em>locally small category</em></li>
</ul>
</section>
<section class="slide">
<h2>Categories Everywhere?</h2>
<h3>Strings</h3>
<ul>
<li> \(\ob{Str}\) is a singleton </li>
<li> \(\hom{Str}\) each string </li>
<li> ∘ is concatenation <code>(++)</code> </li>
</ul>
<ul>
<li> <code>"" ++ u = u = u ++ ""</code> </li>
<li> <code>(u ++ v) ++ w = u ++ (v ++ w)</code> </li>
</ul>
</section>
<section class="slide">
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<h2>Definition: Functor</h2>
<p> A functor is a mapping between two categories.
Let \(\C\) and \(\D\) be two categories.
A <em>functor</em> \(\F\) from \(\C\) to \(\D\):</p>
<ul>
functors
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<li> Associate objects: \(A\in\ob{\C}\) to \(\F(A) \in\ob{\D}\) </li>
<li> Associate morphisms: \(f:A\to B\) to \(\F(f) : \F(A) \to \F(B)\)
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such that
<ul>
functors
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<li>\( \F (\id_X) = \id_{\F(X)} \),</li>
<li>\( \F (g \circ_\C f) = \F(g) \circ_\D \F(f) \)</li>
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</ul>
</li>
</ul>
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</section>
functors
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<section class="slide">
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<h2>Functor: Example</h2>
functors
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<img src="categories/img/mp/functor.png" alt="Functor"/>
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</section>
<section class="slide">
<h2>Functor: Example</h2>
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<img src="categories/img/mp/functor-morphism.png" alt="Functor"/>
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</section>
<section class="slide">
<h2>Functor: Example</h2>
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<img src="categories/img/mp/functor-morphism-color.png" alt="Functor"/>
functors
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</section>
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