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<!doctype html>
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<title>Introduction to Haskell - Lecture 12</title>
<meta name="description" content="A course on the world's fastest growing functional programming language">
<meta name="author" content="Nishant Shukla">
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<section>
<section>
<h1>Introduction To Haskell</h1>
<p>Lecture 12</p>
<p>
<br>
</p>
<p>Category Theory</p>
<br>
<p><i><a href="https://twitter.com/superninjarobot/status/321663231842189312" target="_blank">Proposed title</a>:</i> Do You Even <a href="http://en.wikipedia.org/wiki/Lift_(mathematics)" target="_blank">Lift</a>?</p>
</section>
<section>
<h2>Category Theory</h2>
<a href="http://yogsototh.github.io/Category-Theory-Presentation/#slide-2" target="_blank"> Not really about Cat and Glory
<img src="L12_files/categlory.jpg">
</a>
</section>
<section>
<small><q>...allows one to see the <b>forest</b> rather than the <b>individual trees</b>, and offers the possibility for study of the structure of the <b>entire forest</b>, in preparation for the next stage of abstraction - <b>comparing forests</b>.</q></small>
<p><small><a href="http://www.amazon.com/Category-theory-introduction-advanced-mathematics/dp/B0006D0EXY" target="_blank">-- Herrlich, Strecker, <b>Category Theory</b></a></small></p>
<img src="L12_files/forest.jpg">
</section>
<section>
<h3>Terminology</h3>
<a href="http://yogsototh.github.io/Category-Theory-Presentation/#slide-7" target="_blank"><img src="L12_files/terms.png"></a>
</section>
<section>
<h3>History</h3>
<p>Roots in <b>algebraic topology</b> in the early 1940s by Eilenberg and Mac Lane</p>
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</section>
<section>
<h3>Why Category Theory?</h3>
<p>It provides a <b>powerful language</b>.</p>
<br>
<p>It can <b>translate</b> difficult problems to easy ones.</p>
<br>
<p>It features strong <b>abstractions</b> and attempts to unify separate ideas.</p>
<p></p>
</section>
<section>
<h3>Brouwer Fixed-Point Theorem</h3>
<small>Category Theory in Action</small>
<p><b>A continuous function from a unit circle to itself must have a fixed point.</b></p>
<p><a href="http://investigations1213.blogspot.com/2012/11/brouwers-fixed-point-theorem-proof-in_13.html" target="_blank"><img src="L12_files/brouwer.gif"></a></p>
<p><small>(Image © <a href="http://investigations1213.blogspot.com/2012/11/brouwers-fixed-point-theorem-proof-in_13.html" target="_blank">Jack E.</a>)</small></p>
</section>
<section>
<h3>Implications of Brouwer's Theorem</h3>
<small>
<ul>
<li><p>If you crumple up one sheet of paper and place it on top of a flat sheet, then at least one point will be directly over it's corresponding point.</p></li>
<li><p>After sloshing around a cup of coffee, at least one point will always remain in the same spot.</p></li>
<li><p>Brouwer's Theorem assures existence of solutions to some differential equations.</p></li>
<li><p>Brouwer's Theorem assures existence of equilibria in Game Theory.</p></li>
</ul>
</small>
<center>
<table>
<tr>
<td width="25%">
<img style="width:150px; height:150px" src="L12_files/paper.png">
</td>
<td width="25%">
<img style="width:150px; height:150px" src="L12_files/coffee.png">
</td>
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<img style="width:150px; height:150px" src="L12_files/diffyeq.gif">
</td>
<td width="25%">
<img style="width:150px; height:150px" src="L12_files/gametheory.png">
</td>
</tr>
</table>
</center>
</section>
<section>
<h3>Proof (Step 1)</h3>
<p>Let <b>D</b> be the unit-circle disk.</p>
<p>Let <b>S</b> be the surface of the unit-circle.</p>
<br>
<p><u>Lemma</u>: There is no continuous function</p><p> <i>h: D → S</i> that leaves each point on S fixed.</p>
<br>
<center><img src="L12_files/cat_top.png"></center>
</section>
<section>
<h3>Proof of Lemma (Step 2)</h3>
<p>Using a Functor, we can transform one Category to another by preserving identities and compositions.</p>
<br>
<center>
<table>
<tr>
<td width="48%">
<center>
Topologies
</center>
</td>
<td width="4%">
<center>
→
</center>
</td>
<td width="48%">
<center>
Groups
</center>
</td>
</tr>
<tr>
<td>
<center>
<img src="L12_files/cat_top.png">
</center>
</td>
<td>
</td>
<td>
<center>
<img src="L12_files/cat_gro.png">
</center>
</td>
</tr>
</table>
</center>
</section>
<section>
<h3>Proof of Lemma (Step 3)</h3>
<img src="L12_files/cat_gro.png">
<p>By instead examining the Category of Groups, we can conclude that no such group homomorphism <i>g: 0 → Z</i> can exist.</p>
<br>
<p class="fragment roll-in">We have proved the Lemma.</p>
</section>
<section data-state="soothe">
<h3>Proof (The Build Up)</h3>
<p><b>GIVEN: </b><u>Lemma</u>: There is no continuous function</p><p> <i>F: D → S</i> that leaves each point on S fixed.</p>
<br>
<p><b>SHOW</b>: A continuous function <i>f: D → D</i> must have a fixed point</p>
<br>
<small>Take a moment, and try this by yourself.</small>
</section>
<section>
<h3>Proof (Fin)</h3>
<p><small>...and now, THE DROP!</small></p>
<p>By way of contradiction, assume ∀x∈D: x ≠ f(x)</p>
<img src="L12_files/proof.png">
<p>Then we can always form the function <i>F: D → S</i> that leaves each point on S fixed.</p>
<p>Contradiction. GG.</p>
</section>
<section>
<h3>Definition: Category</h3>
<small class="fragment roll-in"><p>A <b>Category</b> <i>C</i> is a collection of <b>Ob(<i>C</i>)</b> and <b>Ar(<i>C</i></b>).</p></small>
<p><img src="L12_files/category.png"></p>
<small class="fragment roll-in"><b>Ob(<i>C</i>)</b> are the objects of <b><i>C</i></b>. <b>Ar(<i>C</i>)</b> are the "arrows" or morphisms of <b><i>C</i></b>.</small>
<small class="fragment roll-in"><p>Each <i><b>f</b>:A→B</i> ∈ <b>Ar(<i>C</i>)</b> has it's <b>A</b> and <b>B</b> chosen from <b>Ob(<i>C</i>)</b>.</p></small>
<small class="fragment roll-in">If <b><i>f</i></b>:A→B and <b><i>g</i></b>:B→C, then there always exists <b><i>h</i></b> = <b><i>g∘f</i></b>: A→C</small>
<small class="fragment roll-in">For every <b>A</b> ∈ <b>Ob(<i>C</i>)</b>, there is an identity function <b>id<sub>A</sub></b>: A→A.</id></small>
</section>
<section>
<h2>Axioms</h2>
<p><small>Left and right identity: <b><i>f</i>∘id<sub>A</sub></b> = <b>id<sub>A</sub>∘<i>f</i></b></small></p>
<div class="fragment roll-in">
<p><small>Associativity: <b><i>h</i>∘(<i>g∘f</i>)</b> = <b>(<i>h∘g</i>)∘<i>f</i></b></small></p>
<p><img style="width:500px;" src="L12_files/categoryexample.png"></p>
</div>
</section>
<section data-state="soothe">
<h1><a href="http://yogsototh.github.io/Category-Theory-Presentation/#slide-16" target="_blank">Question Time!</a></h1>
</section>
<section>
<h3>Examples of Categories</h3>
<ul>
<li><b><a href="http://en.wikipedia.org/wiki/Set_theory" target="_blank">Set</a></b> (with set functions)</li>
<li><b><a href="http://shuklan.com/haskell/lec09.html#/0/9" target="_blank">Monoids</a></b> are one-object categories</li>
<li><b><a href="http://simple.wikipedia.org/wiki/Group_theory" target="_blank">Grp</a></b> (groups with group morphisms)</li>
<li><b><a href="http://en.wikipedia.org/wiki/Ring_theory" target="_blank">Rng</a></b> (rings with ring morphisms)</li>
<li><b><a href="http://www.haskell.org/haskellwiki/Hask">Hask</a></b> (Haskell types and functions)</li>
</ul>
</section>
<section>
<h2>Hask</h2>
<p><b>Ob(<i>Hask</i>)</b> = the Haskell types. (Bool, [Char], ...)</p>
<p><b>Ar(<i>Hask</i>)</b> = the Haskell functions. (head, not, ...)</p>
<p>The identity function is <code>id :: a -> a</code></p>
<br>
<p>The axioms are satisfied</p>
<p><small>Left and right identity: <b><i>f</i>∘id</b> = <b>id∘<i>f</i></b></small></p>
<p><small>Associativity: <b><i>h</i>∘(<i>g∘f</i>)</b> = <b>(<i>h∘g</i>)∘<i>f</i></b></small></p>
</section>
<section>
<h2>Functor</h2>
<small><p>Functor <b>F</b>: C→D is a transformation from Category <b><i>C</i></b> to Category <b><i>D</i></b></p></small>
<a target="_blank" href="http://en.wikibooks.org/wiki/Haskell/Category_theory#Functors"><img src="L12_files/functor.png"></a>
<small>It maps objects in C to objects in D, and functions in C to functions in D</small>
<br>
<p>Functor Axioms:</p>
<small>
<ol>
<p><li>F(id<sub>A</sub>) = id<sub>F(A)</sub></li></p>
<p><li>F(f∘g) = F(f)∘F(g)</li></p>
</ol>
</small>
</section>
<section>
<h3>Functors in Haskell</h3>
<p>In Haskell, a Functor is a typeclass for things that can be mapped over.</p>
<pre><code class="haskell">
Prelude> fmap odd (Just 3) -- Maybe is a Functor
Just True
</code></pre>
<pre><code class="haskell">
Prelude> fmap odd [1..5] -- a list is a Functor
[True,False,True,False,True]
</code></pre>
</section>
<section>
<h3><code>Maybe</code> Functor</h3>
<small><p>1. The <i>type constructor</i> transforms anything of type <code>a</code> to <code>Maybe a</code></p></small>
<small><p>Like transforming an object in <b><i>C</i></b> to an object in <b><i>D</i></b>.</p></small>
<p><code>Maybe</code> derives the <code>Functor</code> typeclass as follows:</p>
<pre><code>
instance Functor Maybe where
fmap f (Just x) = Just (f x)
fmap _ Nothing = Nothing
</code></pre>
<br>
<small><p>2. fmap transforms a function <b><i>f</i></b>: a→b</p> to Maybe a → Maybe b.</small>
</section>
<section>
<h3><code>Maybe</code> Functor (Cont.)</h3>
<p>We just showed that Maybe transforms objects and functions over from the <b>Hask</b> category to the <b>Maybe</b> subcategory.</p>
<br>
<p>The Functor Axioms are satisfied:</p>
<ol>
<p><li>fmap id = id</li></p>
<p><li>fmap (f . g) = fmap f . fmap g</li></p>
</ol>
</section>
<section>
<h2>Monads</h2>
<p>A Monad is a functor from a Category to itself: <b>M: C→C</b></p>
<small>
<p>And, for every <b>X</b> ∈ Ob(<b><i>C</i></b>)</p>
<ul>
<li><p>unit: <b>X</b>→M(<b>X</b>)</p></li>
<li><p>join: M(M(<b>X</b>))→M(<b>X</b>)</p></li>
</ul>
</small>
<pre><code class="haskell">
class Functor m => Monad m where
return :: a -> m a
(>>=) :: m a -> (a -> m b) -> m b
</code></pre>
<pre><code>
Prelude> import Control.Monad
Prelude Control.Monad> :t join
join :: Monad m => m (m a) -> m a
</code></pre>
</section>
<section>
<h2><a href="http://en.wikibooks.org/wiki/Haskell/Category_theory#Monads" target="_blank">The Monad Laws</a></h2>
<ol>
<li>
<p><i>join</i> ∘ M(<i>join</i>) = <i>join</i> ∘ <i>join</i></p>
<small><q>Collapsing the inner two layers first, then that with the outer layer is exactly the same as collapsing the outer layers first, then that with the innermost layer.</q></small>
</li>
<br>
<li>
<p><i>join</i> ∘ M(<i>unit</i>) = <i>join</i> ∘ <i>unit</i> = <i>id</i></p>
<small><q>Applying return to a monadic value, then joining the result should have the same effect whether you perform the return from inside the top layer or from outside it.</q></small>
</li>
<br>
<li><i>unit</i> ∘ <i>f</i> = M(<i>f</i>) ∘ <i>unit</i></li>
<br>
<li><i>join</i> ∘ M(M(<i>f</i>)) = M(<i>f</i>) ∘ <i>join</i></li>
</ol>
</section>
<section>
<h3>The Power of Abstraction</h3>
<p>Category theory powers Haskell's generalizability.</p>
<ul>
<li><a href="http://shuklan.com/haskell/lec09.html#/0/5" target="_blank">Functors generalize maps</a></li>
<li><a href="http://yogsototh.github.io/Category-Theory-Presentation/#slide-77" target="_blank">Catamorphisms generalize folds</a></li>
</ul>
</section>
<section>
<h2>Thanks for your time!</h2>
<img src="L12_files/cat.gif"</img>
<p>Homework: <a href="https://docs.google.com/a/virginia.edu/forms/d/16H3j_LdorJ4wAgRZRchGD0PGXTaYOIwXAfJwL4grRtM/viewform" target="_blank">Fill out the Course Evaluation</a></p>
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