774 lines
25 KiB
HTML
774 lines
25 KiB
HTML
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<title>Category Theory for Programming</title>
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<body class="deck-container">
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<div style="display:none">
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\(\newcommand{\F}{\mathbf{F}}\)
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\(\newcommand{\C}{\mathcal{C}}\)
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\(\newcommand{\D}{\mathcal{D}}\)
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\(\newcommand{\ob}[1]{\mathrm{ob}(#1)}\)
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</div>
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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 class="and">&</span></span> Programming
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<div><author style="font-size: .4em"><em class="base01">by</em> Yann Esposito
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<div style="font-size:.5em">
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<twitter>
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<a href="http://twitter.com/yogsototh">@yogsototh</a>,
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</twitter>
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<googleplus>
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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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<div class="base01" style="font-size: .25em; font-weight: 400; font-variant:italic">
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HTML presentation: use arrows, space to navigate.
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</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; font-weight:bold">
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<li><span class="yellow">Why?
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<ul class="base01" style="border-left: 2px solid; padding-left: 1em; font-size: .6em; float: right; font-weight: bold; margin: 0 0 0 1em">
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<li>Math <span class="and"><span class="and">&</span></span> Abstraction</li>
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<li>Programming <span class="and"><span class="and">&</span></span> Abstraction</li>
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<li>Categories <span class="and"><span class="and">&</span></span> Abstraction</li>
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</ul>
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</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</h2>
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<p>A common concept you see often in multiple instances.</p>
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<div>
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<p>Numbers: 1,2,3,... <em class="small">3400 BC, real numbers 760 BC</em></p>
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<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>
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</figure>
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<div>
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<div class="left" style="left-margin: 1em">amelioration ⇒</div>
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<figure class="left">
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<img src="categories/img/first-real-numbers.png" style="height:5em" alt="Mesopotamian Numbers"/>
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<figcaption>Mesopotamian base 60 system</figcaption>
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</figure>
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</div>
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<div class="flush"></div>
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<div>
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Operators: <b>=, <, >, +, ×, ...</b>
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</div>
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</section>
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<section class="slide">
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<h2>Generalization</h2>
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<div class="right">
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<img src="categories/img/egyptian-hieroglyphics.jpg" alt="Egyptian Fractions"/>
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<img src="categories/img/negative-numbers.jpg" alt="Negative Numbers (Chinese)"/>
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</div>
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<ul><li> Weight/Distance/Time ⇒ <em>Rational</em> \(\frac{p}{q}\)
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</li><li> Debts ⇒ <em>Negative</em> \(..., -2, -1, ? , 1, 2, ...\)
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</li><li> Geometry ⇒ <em>Irrational<sup style="vertical-align:middle">*</sup></em>
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</li><li> Algebra ⇒ <em>Complex</em>
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</li><li> \(0\) ⇒ "Nothing" become a number
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</li></ul>
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<p><span class="and" style="visibility:hidden"><span class="and">&</span></span> More <strong>things</strong> can be understood as numbers<br/>
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<span class="and"><span class="and">&</span></span> More <strong>operator</strong> to manipulate them.</p>
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</section>
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<section class="slide">
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<h2>Numbers ⇒ Sets</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/Topology</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 class="and">&</span></span> More <strong>general</strong>: more things are sets.<br/>
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<span class="and"><span class="and">&</span></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>Sets ⇒? <span class="yellow">Categories</span></h2>
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<table>
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<tr>
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<th>Numbers</th>
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<th>Sets</th>
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<th>Categories</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>Categories package entire mathematical theories.</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 class="and">&</span></span> More <strong>general</strong>: more things are Categories.<br/>
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<span class="and"><span class="and">&</span></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 class="and">&</span></span> Saunders Mac Lane
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</section>
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<section class="slide">
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<h2>Programming <span class="and"><span class="and">&</span></span> Abstraction</h2>
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<h3>Impure programming</h3>
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<ul>
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<li>Encouraged by imperative paradigms.
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</li><li>Actions <span class="and"><span class="and">&</span></span> mutable objects.
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</li><li>Time is <em>very</em> important: ex. linked list push
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</li><li>Synchronizing things is a challenge.
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</li>
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</ul>
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<p>Natural Abstractions: pointers, variables, loop, Objects, Classes...</p>
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<p>Representation: a data structure changing other time.</p>
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</section>
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<section class="slide">
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<h2>Untyped Pure Programming</h2>
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<ul><li> Time is irrelevant by default.
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</li><li> Mostly static constructions like pipes.
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</li><li> All pipes can be plugged ⇒ all error at runtime
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<ul><li> (+ ("foo" 27) 32)
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</li><li> Y = λf.(λx.f (x x)) (λx.f (x x))
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</li><li> Y g = g (Y g)
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</li></ul>
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</li></ul>
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<p>Natural abstraction: higher level functions <span class="and"><span class="and">&</span></span> equations</p>
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</section>
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<section class="slide">
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<h2>Typed Pure Programming</h2>
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<ul><li>Add shapes to pipes:
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<ul><li> <code class="red">4 + ["foo",27]</code> forbidden
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</li><li> <code class="red">["foo",27]</code> forbidden
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</li></ul>
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</li><li>Polymorphic (elastic) shapes:
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<ul><li> <code>data Maybe a = Just a | Nothing</code>
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</li><li> <code>[Just 32,Nothing,Just 12] :: [Maybe Integer]</code>
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</li></ul>
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</li></ul>
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<p>Natural abstraction: Polymorphic higher level functions.</p>
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</section>
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<section class="slide">
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<h2>Polymorphism: <code>mappend (<>)</code></h2>
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<pre class="haskell"><code>"abc" <> "def" = "abcdef" -- String
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("ab","xy") <> ("AB","XY") = ("abAB","xyXY") -- (String,String)
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3 <> 4 ⇒ ERROR which law? + or * -- Int
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-- Use a type to remove ambiguity
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type Sum = Sum {getSum :: a} -- Just a named box
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-- Monoid (N,+)
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(<>+) = getSum (Sum x <> Sum y)
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3 <>+ 4 = 7
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-- Monoid (N,*)
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(<>*) = getProduct (Product x <> Product y)
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3 <>* 4 = 12</code></pre>
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</section>
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<section class="slide">
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<h2>Polymorphism: <code>(>>=)</code></h2>
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<p>Example: <span class="red"><code>(>>=)</code></span> with <code>[a]</code> and <code>Maybe a</code></p>
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<pre class="haskell"><code>data Maybe a = Just a | Nothing</code></pre>
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<pre class="haskell"><code>-- Maybe Int >>= Int -> Maybe (Int -> Int) >>= (Int -> Int) -> Maybe Int
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(Just 2) <span class="red">>>=</span> \x -> (Just (\z->z*z)) <span class="red">>>=</span> \f -> Just (f x) = Just 4
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Nothing <span class="red">>>=</span> \x -> (Just (\z->z*z)) <span class="red">>>=</span> \f -> Just (f x) = Nothing
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-- [Int] >>= Int -> [Int -> Int] >>= (Int -> Int) -> [Int]
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[1,2] <span class="red">>>=</span> \x -> [(+10),(+20)] <span class="red">>>=</span> \f -> [f x] = [11,21,12,22]
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[] <span class="red">>>=</span> \x -> [(+10),(+20)] <span class="red">>>=</span> \f -> [f x] = []</code></pre>
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</section>
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<section class="slide">
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<h2>Programming Paradigms</h2>
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<table>
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<tr><td>Impure</td><td>Choose the data structure, find an algorithm.</td></tr>
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<tr><td>Untyped Pure</td><td>Choose the data structure, find an equation.</td></tr>
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<tr><td>Typed Pure</td><td>Choose the Types and their laws, find the right operator</td></tr>
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</table>
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</section>
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<section class="slide">
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<h2>Type Theory ⇒ Categories</h2>
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<ul>
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<li>Type theory helped to remove paradoxes in Set Theory.</li>
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<li>Prevent relations between different kind of objects.</li>
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<li>Used in computer science</li>
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</ul>
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<ul>
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<li>typed λ-calculus ⇒ cartesian closed categories</li>
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<li>untyped λ-calculus ⇒ C-monoids (subclass of categories)</li>
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<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">
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<h2>Plan</h2>
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<ul style="font-size: 2em; font-weight: bold">
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<li>Why?</li>
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<li> <span class="yellow">What?</span>
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<ul class="base01" style="border-left: 2px solid; padding-left: 1em; font-size: .6em; float: right; font-weight: bold; margin: 0 0 0 1em">
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<li>Category</li>
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<li>Intuition</li>
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<li>Examples</li>
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<li>Functor</li>
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<li>Examples</li>
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</ul>
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</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>Category</h2>
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<p>A way of representing <strong><em>things</em></strong> and <strong><em>ways to go between things</em></strong>.</p>
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<p> A Category \(\mathcal{C}\) is defined by:</p>
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<ul>
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<li> <em>Objects (\(\ob{C}\))</em>,</li>
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<li> <em>Morphisms (\(\hom{C}\))</em>,</li>
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<li> a <em>Composition law (∘)</em></li>
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<li> obeying some <em>Properties</em>.</li>
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</ul>
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</section>
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<section class="slide">
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<h2>Category: 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>
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</section>
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<section class="slide">
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<h2>Category: 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>
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<section class="slide">
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<h2>Category: Composition</h2>
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<p>Composition (∘): \(f:A→B, g:B→C\)
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$$g∘f:A\rightarrow C$$
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</p>
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<img src="categories/img/mp/composition.png" alt="composition"/>
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</section>
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<section class="slide">
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<h2>Category laws: neutral element</h2>
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<p>for all \(X\), there is an \(\id_X\), s.t. for all \(f:A→B\):</p>
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<img src="categories/img/mp/identity.png" alt="identity"/>
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</section>
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<section class="slide">
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<h2>Category laws: Associativity</h2>
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<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>Commutative diagrams</h2>
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<p>Two path with the same source and destination are equal.</p>
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<figure class="left" style="max-width: 40%;margin-left: 10%;">
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<img
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src="categories/img/mp/commutative-diagram-assoc.png"
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alt="Commutative Diagram (Associativity)"/>
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<figcaption>
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\((h∘g)∘f = h∘(g∘f) \)
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</figcaption>
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</figure>
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<figure class="right" style="max-width:31%;margin-right: 10%;">
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||
<img
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||
src="categories/img/mp/commutative-diagram-id.png"
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||
alt="Commutative Diagram (Identity law)"/>
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<figcaption>
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\(id_B∘f = f = f∘id_A \)
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</figcaption>
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</figure>
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</section>
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<section class="slide">
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||
<h2>Can this be a category?</h2>
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||
<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">
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||
<span class="green">YES</span>
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</figcaption>
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</figure>
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<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">
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no candidate for \(g∘f\)
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<br/><span class="red">NO</span>
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</figcaption>
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</figure>
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<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">
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||
<span class="green">YES</span>
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</figcaption>
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||
</figure>
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</section>
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||
<section class="slide">
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<h2>Can this be a category?</h2>
|
||
<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">
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||
no candidate for \(f:C→B\)
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<br/><span class="red">NO</span>
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</figcaption>
|
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</figure>
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||
<figure class="right" style="min-width: 59%">
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<img src="categories/img/mp/cat-example5.png" alt="Category example 5"/>
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||
<figcaption class="slide">
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||
\((h∘g)∘f=\id_B∘f=f\)<br/>
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\(h∘(g∘f)=h∘\id_A=h\)<br/>
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||
but \(h≠f\)<br/>
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<span class="red">NO</span>
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</figcaption>
|
||
</figure>
|
||
</section>
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<section class="slide">
|
||
<h2>Category \(\Set\)</h2>
|
||
|
||
<ul>
|
||
<li> \(\ob{\Set}\) are sets</li>
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<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>
|
||
|
||
<ul>
|
||
<li>\(\Mon\): (monoids, monoid morphisms,∘)</li>
|
||
<li>\(\Vec\): (Vectorial spaces, linear functions,∘)</li>
|
||
<li>\(\Grp\): (groups, group morphisms,∘)</li>
|
||
<li>\(\Rng\): (rings, ring morphisms,∘)</li>
|
||
<li>\( \ML\): (types, terms, \(λg. λf. λx. g f x\) )</li>
|
||
<li>\( \Hask\): (Haskell types, functions, <code>(.)</code> )</li>
|
||
<li>...</li>
|
||
</ul>
|
||
</section>
|
||
<section class="slide">
|
||
<h2>Smaller Examples</h2>
|
||
|
||
<h3>Strings</h3>
|
||
<img class="right" style="max-width:17%" src="categories/img/mp/strings.png" alt="Monoids are one object categories"/>
|
||
<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">
|
||
<h2>Finite Example?</h2>
|
||
|
||
<h3>Graph</h3>
|
||
<figure class="right" style="max-width:40%" >
|
||
<img src="categories/img/mp/graph-category.png" alt="Each graph is a category"/>
|
||
</figure>
|
||
<ul>
|
||
<li> \(\ob{G}\) are vertices</li>
|
||
<li> \(\hom{G}\) each path</li>
|
||
<li> ∘ is path concatenation</li>
|
||
</ul>
|
||
<ul><li>\(\ob{G}=\{X,Y,Z\}\),
|
||
</li><li>\(\hom{G}=\{ε,α,β,γ,αβ,βγ,...\}\)<br/>
|
||
\(\phantom{\hom{G}}=(β?γ)?(αβγ)^*(αβ?)?\),
|
||
</li><li>\(αβ∘γ=αβγ\)
|
||
</li></ul>
|
||
</section>
|
||
<section class="slide">
|
||
<h2>Degenerated Categories</h2>
|
||
|
||
<img class="right" style="max-width:17%" src="categories/img/mp/monoid.png" alt="Monoids are one object categories"/>
|
||
<h3>Monoids</h3>
|
||
<p>each Monoid \((M,e,◎): \ob{M}=\{∙\},\hom{M}=M,\circ = ◎\)</p>
|
||
<p>one object</p>
|
||
<p>Examples: <code>(Integer,0,+)</code>, <code>(Integer,1,*)</code>, <code>(Strings,"",++)</code>, <code>(Lists,[],++)</code>, ...
|
||
</section>
|
||
<section class="slide">
|
||
<h2>Number construction</h2>
|
||
|
||
<h3>Each Numbers as a whole category</h3>
|
||
<img src="categories/img/mp/numbers.png" alt="Each number as a category"/>
|
||
</section>
|
||
<section class="slide">
|
||
<h2>Degenerated Categories: Preorders</h2>
|
||
|
||
<h3>Preorders</h3>
|
||
<p>each preorder \((P,≤): \ob{P}={P},\hom{x,y}=\{{x≤y}\} ⇔ x≤y,f_{y,z} \circ f_{x,y} = f_{x,z} \)</p>
|
||
<em>At most one morphism between two objects.</em>
|
||
<img src="categories/img/mp/preorder.png" alt="preorder category"/>
|
||
</section>
|
||
<section class="slide">
|
||
<h2>Degenerated Categories</h2>
|
||
|
||
<img class="right" src="categories/img/mp/set.png" alt="Any set can be a category"/>
|
||
<h3>Any Set</h3>
|
||
<p>Any set \(E: \ob{E}=E, \hom{x,y}=\{x\} ⇔ x=y \)</p>
|
||
<p><em>Only identities ; not so interesting</em></p>
|
||
</section>
|
||
<section class="slide">
|
||
<h2>Categorical Property</h2>
|
||
|
||
<p>Any property which can be expressed in term of category, objects, morphism and composition</p>
|
||
|
||
<ul><li> <em>isomorphism</em>: \(f:A→B\) s.t. ∃g:B→A, \(g∘f=id_A\) <span class="and">&</span> \(f∘g=id_B\)
|
||
</li><li> <em>Initial</em>: \(Z\in\ob{C}\) s.t. \(∀Y∈\ob{C}, \#\hom{Z,Y}=1\)
|
||
</li><li> <em>Dual</em>: reverse direction of arrows of \(\C\)
|
||
</li><li> <em>Functor</em>: structure preserving mapping between categories
|
||
</li><li> ...
|
||
</li></ul>
|
||
</section>
|
||
<section class="slide">
|
||
<h2>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>
|
||
<li> Associate objects: \(A\in\ob{\C}\) to \(\F(A) \in\ob{\D}\) </li>
|
||
<li> Associate morphisms: \(f:A\to B\) to \(\F(f) : \F(A) \to \F(B)\)
|
||
such that
|
||
<ul>
|
||
<li>\( \F (\id_X) = \id_{\F(X)} \),</li>
|
||
<li>\( \F (g \circ_\C f) = \F(g) \circ_\D \F(f) \)</li>
|
||
</ul>
|
||
</li>
|
||
</ul>
|
||
</section>
|
||
<section class="slide">
|
||
<h2>Functor Example (ob → ob)</h2>
|
||
|
||
<img src="categories/img/mp/functor.png" alt="Functor"/>
|
||
</section>
|
||
<section class="slide">
|
||
<h2>Functor Example (hom → hom)</h2>
|
||
|
||
<img src="categories/img/mp/functor-morphism.png" alt="Functor"/>
|
||
</section>
|
||
<section class="slide">
|
||
<h2>Functor Example</h2>
|
||
|
||
<img src="categories/img/mp/functor-morphism-color.png" alt="Functor"/>
|
||
</section>
|
||
<section class="slide">
|
||
<h2>Endofunctors</h2>
|
||
|
||
<p>An <em>endofunctor</em> for \(\C\) is a functor \(F:\C→\C\).</p>
|
||
<img src="categories/img/mp/endofunctor.png" alt="Endofunctor"/>
|
||
</section>
|
||
<section class="slide">
|
||
<h2>Category of Categories</h2>
|
||
|
||
<p>Categories and functors form a category: \(\Cat\)</p>
|
||
<ul><li>\(\ob{\Cat}\) are categories
|
||
</li><li>\(\hom{\Cat}\) are functors
|
||
</li><li>∘ is functor composition
|
||
</li></ul>
|
||
</section>
|
||
<section class="slide">
|
||
<h2>Plan</h2>
|
||
<ul style="font-size: 2em; font-weight:bold">
|
||
<li>Why?</li>
|
||
<li>What?</li>
|
||
<li><span class="yellow">How?
|
||
<ul class="base01" style="border-left: 2px solid; padding-left: 1em; font-size: .6em; float: right; font-weight: bold; margin: -2em 0 0 1em">
|
||
<li>\(\Hask\) category
|
||
</li><li> Functors
|
||
</li><li> Monads
|
||
</li><li> Arrows
|
||
</li><li> κατα-morphisms
|
||
</li></ul>
|
||
</li>
|
||
</ul>
|
||
</section>
|
||
<section class="slide">
|
||
<h2>Hask</h2>
|
||
|
||
<p>Category \(\Hask\):</p>
|
||
|
||
<img class="right" style="max-width:30%" src="categories/img/mp/hask.png" alt="Haskell Category Representation"/>
|
||
|
||
<ul><li>
|
||
\(\ob{\Hask} = \) Haskell types
|
||
</li><li>
|
||
\(\hom{\Hask} = \) Haskell functions
|
||
</li><li>
|
||
∘ = <code>(.)</code> Haskell function composition
|
||
</li></ul>
|
||
|
||
<p>Forget glitches because of <code>undefined</code>.</p>
|
||
</section>
|
||
<section class="slide">
|
||
<h2>Haskell Functor vs \(\Hask\) Functor</h2>
|
||
|
||
<p>Functor for Haskell language is a type <code>F</code> which belong to the type class <code>Functor</code></p>
|
||
<p>It must therefore implement an <code>fmap</code> function.
|
||
|
||
<p>The couple <code>(F,fmap)</code> is then a real functor in the categorical sense for \(\Hask\) if</p>
|
||
|
||
<p>for any <code>x :: F a</code>:</p>
|
||
<ul><li><code>fmap id x = x</code>
|
||
</li><li><code>fmap (f.g) x= (fmap f . fmap g) x</code>
|
||
</li></ul>
|
||
</section>
|
||
<section class="slide">
|
||
<h2>Haskell Functors Example: Maybe</h2>
|
||
|
||
<pre class="haskell"><code>data Maybe a = Just a | Nothing
|
||
instance Functor Maybe where
|
||
fmap :: (a -> b) -> (Maybe a -> Maybe b)
|
||
fmap f (Just a) = f a
|
||
fmap f Nothing = Nothing
|
||
|
||
fmap (+1) (Just 1) == 2
|
||
fmap (+1) Nothing == Nothing
|
||
fmap head (Just [1,2,3]) == Just 1</code></pre>
|
||
</section>
|
||
<section class="slide">
|
||
<h2>Haskell Functors Example: List</h2>
|
||
|
||
<pre class="haskell"><code>
|
||
instance Functor ([]) where
|
||
fmap :: (a -> b) -> [a] -> [b]
|
||
fmap = map
|
||
|
||
fmap (+1) [1,2,3] == [2,3,4]
|
||
fmap (+1) [] == []
|
||
fmap head [[1,2,3],[4,5,6]] == [1,4]
|
||
</code></pre>
|
||
</section>
|
||
<section class="slide">
|
||
<h2>String like type</h2>
|
||
|
||
<pre class="haskell"><code>
|
||
data F a = Cons Char a | Nil
|
||
-- examples :
|
||
-- Cons 'c' 32 :: F Int
|
||
-- Cons 'c' (\x -> x*x) :: F (Int -> Int)
|
||
-- Cons 'c' (Cons 'a' (\x -> x*x)) :: F (F (Int -> Int))
|
||
-- Cons 'c' (Cons 'c' Nil) :: F (F (F))
|
||
-- note String is the fixed point of F: F(F(F(...)))
|
||
instance Functor F where
|
||
fmap :: (a -> b) -> [a] -> [b]
|
||
fmap f (Cons c x) = Cons c (f x)
|
||
fmap f Nil = Nil
|
||
|
||
fmap (+1) (Cons 'c' 3) == Cons 'c' 4
|
||
fmap (+1) Nil == Nil
|
||
fmap head (Cons 'c' [1,2,3])== Cons 'c' 1
|
||
</code></pre>
|
||
</section>
|
||
<section class="slide">
|
||
<h2>Haskell Functor intuition</h2>
|
||
|
||
<p>Put normal function inside Box-like types</p>
|
||
|
||
<img src="categories/img/mp/boxfunctor.png" alt="Haskell Functor as a box play"/>
|
||
</section>
|
||
<section class="slide">
|
||
<h2>Haskell Functor properties</h2>
|
||
|
||
<p>Haskell Functors are:</p>
|
||
|
||
<ul><li><em>endofunctors</em> ; \(F:\C→\C\) here \(\C = \Hask\),
|
||
</li><li>a couple <b>(Object,Morphism)</b> of Hask.
|
||
</li></ul>
|
||
</section>
|
||
<section class="slide">
|
||
<h2>Functor as boxes</h2>
|
||
|
||
<p>Haskell functor can be seen as boxes containing all Haskell types and functions.
|
||
Haskell types is fractal:</p>
|
||
|
||
<img src="categories/img/mp/hask-endofunctor.png" alt="Haskell functor representation"/>
|
||
</section>
|
||
<section class="slide">
|
||
<h2 id="non-haskell-hasks-functors">"Non Haskell" Hask's Functors</h2>
|
||
<p>A simple basic example is the \(id_\Hask\) functor. It simply cannot be expressed as a couple (<code>F</code>,<code>fmap</code>) where</p>
|
||
<ul>
|
||
<li><code>F::* -> *</code></li>
|
||
<li><code>fmap :: (a -> b) -> (F a) -> (F b)</code></li>
|
||
</ul>
|
||
<p>Also other degenerated examples:</p>
|
||
<ul>
|
||
<li>F(<code>T</code>)=<code>Int</code></li>
|
||
<li>F(<code>f</code>)=<code>\x->0</code></li>
|
||
</ul>
|
||
</section>
|
||
<section class="slide">
|
||
<h2 id="also-functor-inside-hask">Also Functor inside \(\Hask\)</h2>
|
||
<p><code>length</code> can be seen as a Functor from the category <code>[a]</code> to <code>Int</code>. More precisely:</p>
|
||
<ul>
|
||
<li>\(\ob{\mathtt{[a]}}=\{∙\}\)</li>
|
||
<li>\(\hom{\mathtt{[a]}}=\mathtt{[a]}\)</li>
|
||
<li>\(∘=\mathtt{(++)}\)</li>
|
||
</ul>
|
||
<p><span class="and">&</span></p>
|
||
<ul>
|
||
<li>\(\ob{\mathtt{Int}}=\{∙\}\)</li>
|
||
<li>\(\hom{\mathtt{Int}}=\mathtt{Int}\)</li>
|
||
<li>\(∘=\mathtt{(+)}\)</li>
|
||
</ul>
|
||
</section>
|
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