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<title>Category Theory for Programming</title>
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\(\newcommand{\F}{\mathbf{F}}\)
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\(\newcommand{\C}{\mathcal{C}}\)
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<!-- Begin slides. Just make elements with a class of slide. -->
<section class="slide">
<div style="text-align:center; position:absolute; top: 2em; font-size: .9em; width: 100%">
<h1 style="position: relative;">Category Theory <span class="and">&amp;</span> Programming</h1>
<div><em class="base01">for</em> <a href="http://www.meetup.com/riviera-scala-clojure">Rivieria Scala Clojure</a> (Note this presentation uses Haskell)</div>
<author><em class="base01">by</em> <a href="http://yannesposito.com">Yann Esposito</a></author>
<div style="font-size:.8em">
<twitter>
<a href="http://twitter.com/yogsototh">@yogsototh</a>,
</twitter>
<googleplus>
<a href="https://plus.google.com/117858550730178181663">+yogsototh</a>
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</div>
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HTML presentation: use arrows, space, swipe to navigate.
</div>
</div>
</section>
<section class="slide">
<h2>Plan</h2>
<ul style="font-size: 2em; font-weight:bold">
<li><span class="yellow">General overview</li>
<li>Definitions</li>
<li>Applications</li>
</ul>
</section>
<section class="slide">
<h2 id="not-really-about-cat-glory">Not really about: Cat <span class="and">&amp;</span> glory</h2>
<figure>
<img src="categories/img/categlory.jpg" alt="Cat n glory" /> <figcaption>credit to Tokuhiro Kawai (川井徳寛)</figcaption>
</figure>
</section>
<section class="slide">
<h2 id="general-overview">General Overview</h2>
<div style="float:right; width: 18%">
<img src="categories/img/eilenberg.gif" alt="Samuel Eilenberg"/> <img src="categories/img/maclaine.jpg" alt="Saunders Mac Lane"/>
</div>
<p><em>Recent Math Field</em><br />1942-45, Samuel Eilenberg <span class="and">&amp;</span> Saunders Mac Lane</p>
<p>Certainly one of the more abstract branches of math</p>
<ul>
<li><em>New math foundation</em><br /> formalism abstraction, package entire theory<sup></sup></li>
<li><em>Bridge between disciplines</em><br /> Physics, Quantum Physics, Topology, Logic, Computer Science<sup></sup></li>
</ul>
<p class="smaller base01" style="border-top: solid 1px">
★: <a href="http://www.math.harvard.edu/~mazur/preprints/when_is_one.pd">When is one thing equal to some other thing?, Barry Mazur, 2007</a><br/> ☆: <a href="http://math.ucr.edu/home/baez/rosetta.pdf">Physics, Topology, Logic and Computation: A Rosetta Stone, John C. Baez, Mike Stay, 2009</a>
</p>
</section>
<section class="slide">
<h2 id="from-a-programmer-perspective">From a Programmer perspective</h2>
<blockquote>
<p>Category Theory is a new language/framework for Math</p>
</blockquote>
<ul>
<li>Another way of thinking</li>
<li>Extremely efficient for generalization</li>
</ul>
</section>
<section class="slide">
<h2 id="math-programming-relation">Math Programming relation</h2>
<img class="right" src="categories/img/buddha.gif" alt="Buddha Fractal"/>
<p>Programming <em><span class="yellow">is</span></em> doing Math</p>
<p>Strong relations between type theory and category theory.</p>
<p>Not convinced?<br />Certainly a <em>vocabulary</em> problem.</p>
<p>One of the goal of Category Theory is to create a <em>homogeneous vocabulary</em> between different disciplines.</p>
</section>
<section class="slide">
<h2 id="vocabulary">Vocabulary</h2>
<img class="right" src="categories/img/mindblown.gif" alt="mind blown"/>
<p>Math vocabulary used in this presentation:</p>
<blockquote>
<p>Category, Morphism, Associativity, Preorder, Functor, Endofunctor, Categorial property, Commutative diagram, Isomorph, Initial, Dual, Monoid, Natural transformation, Monad, Klesli arrows, κατα-morphism, ...</p>
</blockquote>
</section>
<section class="slide">
<h2 id="programmer-translation">Programmer Translation</h2>
<img class="right" src="categories/img/readingcat.jpg" alt="lolcat"/>
<table style="width:60%">
<tr><th>
Mathematician
</th><th>
Programmer
</th></tr>
<tr><td>
Morphism
</td><td>
Arrow
</td></tr>
<tr><td>
Monoid
</td><td>
String-like
</td></tr>
<tr><td>
Preorder
</td><td>
Acyclic graph
</td></tr>
<tr><td>
Isomorph
</td><td>
The same
</td></tr>
<tr><td>
Natural transformation
</td><td>
rearrangement function
</td></tr>
<tr><td>
Funny Category
</td><td>
LOLCat
</td></tr>
</table>
</section>
<section class="slide">
<h2>Plan</h2>
<ul style="font-size: 2em; font-weight: bold">
<li>General overview</li>
<li> <span class="yellow">Definitions</span>
<ul class="base01" style="border-left: 2px solid; padding-left: 1em; font-size: .6em; float: right; font-weight: bold; margin: 0 0 0 1em">
<li>Category</li>
<li>Intuition</li>
<li>Examples</li>
<li>Functor</li>
<li>Examples</li>
</ul>
</li>
<li>Applications</li>
</ul>
</section>
<section class="slide">
<h2>Category</h2>
<p>A way of representing <strong><em>things</em></strong> and <strong><em>ways to go between things</em></strong>.</p>
<p> A Category \(\mathcal{C}\) is defined by:</p>
<ul>
<li> <em>Objects <span class="yellow">\(\ob{C}\)</span></em>,</li>
<li> <em>Morphisms <span class="yellow">\(\hom{C}\)</span></em>,</li>
<li> a <em>Composition law <span class="yellow">(∘)</span></em></li>
<li> obeying some <em>Properties</em>.</li>
</ul>
</section>
<section class="slide">
<h2>Category: Objects</h2>
<img src="categories/img/mp/objects.png" alt="objects" />
<p>\(\ob{\mathcal{C}}\) is a collection</p>
</section>
<section class="slide">
<h2>Category: Morphisms</h2>
<img src="categories/img/mp/morphisms.png" alt="morphisms"/>
<p>\(A\) and \(B\) objects of \(\C\)<br/>
\(\hom{A,B}\) is a collection of morphisms<br/>
\(f:A→B\) denote the fact \(f\) belongs to \(\hom{A,B}\)</p>
<p>\(\hom{\C}\) the collection of all morphisms of \(\C\)</p>
</section>
<section class="slide">
<h2>Category: Composition</h2>
<p>Composition (∘): associate to each couple \(f:A→B, g:B→C\)
$$g∘f:A\rightarrow C$$
</p>
<img src="categories/img/mp/composition.png" alt="composition"/>
</section>
<section class="slide">
<h2>Category laws: neutral element</h2>
<p>for each object \(X\), there is an \(\id_X:X→X\),<br/>
such that for each \(f:A→B\):</p>
<img src="categories/img/mp/identity.png" alt="identity"/>
</section>
<section class="slide">
<h2>Category laws: Associativity</h2>
<p> Composition is associative:</p>
<img src="categories/img/mp/associativecomposition.png" alt="associative composition"/>
</section>
<section class="slide">
<h2>Commutative diagrams</h2>
<p>Two path with the same source and destination are equal.</p>
<figure class="left" style="max-width: 40%;margin-left: 10%;">
<img
src="categories/img/mp/commutative-diagram-assoc.png"
alt="Commutative Diagram (Associativity)"/>
<figcaption>
\((h∘g)∘f = h∘(g∘f) \)
</figcaption>
</figure>
<figure class="right" style="max-width:31%;margin-right: 10%;">
<img
src="categories/img/mp/commutative-diagram-id.png"
alt="Commutative Diagram (Identity law)"/>
<figcaption>
\(id_B∘f = f = f∘id_A \)
</figcaption>
</figure>
</section>
<section class="slide">
<h2>Question Time!</h2>
<figure style="width:70%; margin:0 auto">
<img src="categories/img/batquestion.jpg" width="100%"/>
<figcaption>
<em>- French-only joke -</em>
</figcaption>
</figure>
</section>
<section class="slide">
<h2>Can this be a category?</h2>
<p>\(\ob{\C},\hom{\C}\) fixed, is there a valid ∘?</p>
<figure class="left">
<img src="categories/img/mp/cat-example1.png" alt="Category example 1"/>
<figcaption class="slide">
<span class="green">YES</span>
</figcaption>
</figure>
<figure class="left">
<img src="categories/img/mp/cat-example2.png" alt="Category example 2"/>
<figcaption class="slide">
no candidate for \(g∘f\)
<br/><span class="red">NO</span>
</figcaption>
</figure>
<figure class="left">
<img src="categories/img/mp/cat-example3.png" alt="Category example 3"/>
<figcaption class="slide">
<span class="green">YES</span>
</figcaption>
</figure>
</section>
<section class="slide">
<h2>Can this be a category?</h2>
<figure class="left">
<img src="categories/img/mp/cat-example4.png" alt="Category example 4"/>
<figcaption class="slide">
no candidate for \(f:C→B\)
<br/><span class="red">NO</span>
</figcaption>
</figure>
<figure class="right" style="min-width: 59%">
<img src="categories/img/mp/cat-example5.png" alt="Category example 5"/>
<figcaption class="slide">
\((h∘g)∘f=\id_B∘f=f\)<br/>
\(h∘(g∘f)=h∘\id_A=h\)<br/>
but \(h≠f\)<br/>
<span class="red">NO</span>
</figcaption>
</figure>
</section>
<section class="slide">
<h2>Categories Examples</h2>
<figure style="width:70%; margin:0 auto">
<img src="categories/img/basket_of_cats.jpg" alt="Basket of cats"/>
<figcaption>
<em>- Basket of Cats -</em>
</figcaption>
</figure>
</section>
<section class="slide">
<h2>Category \(\Set\)</h2>
<ul>
<li> \(\ob{\Set}\) are <em>all</em> the sets</li>
<li> \(\hom{E,F}\) are <em>all</em> functions from \(E\) to \(F\)</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 then a <em>locally <b>small</b> category</em></li>
</ul>
</section>
<section class="slide">
<h2>Categories Everywhere?</h2>
<img class="right" src="categories/img/cats-everywhere.jpg" alt="Cats everywhere"/>
<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>Any deductive system <i>T</i>: (theorems, proofs, proof concatenation)</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>\(\hom{G}=\{ε,α,β,γ,αβ,βγ,...\}\)
</li><li>\(αβ∘γ=αβγ\)
</li></ul>
</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: Monoids</h2>
<img class="right" style="max-width:17%" src="categories/img/mp/monoid.png" alt="Monoids are one object categories"/>
<p>Each Monoid \((M,e,⊙): \ob{M}=\{∙\},\hom{M}=M,\circ = ⊙\)</p>
<p class="yellow">Only one object.</p>
<p>Examples:</p>
<ul><li><code>(Integer,0,+)</code>, <code>(Integer,1,*)</code>,
</li><li><code>(Strings,"",++)</code>, for each <code>a</code>, <code>([a],[],++)</code>
</li></ul>
</section>
<section class="slide">
<h2>Degenerated Categories: Preorders \((P,≤)\)</h2>
<ul><li>\(\ob{P}={P}\),
</li><li>\(\hom{x,y}=\{x≤y\} ⇔ x≤y\),
</li><li>\((y≤z) \circ (x≤y) = (x≤z) \)
</li></ul>
<p><em class="yellow">At most one morphism between two objects.</em></p>
<img src="categories/img/mp/preorder.png" alt="preorder category"/>
</section>
<section class="slide">
<h2>Degenerated Categories: Discrete 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 class="yellow">Only identities</p>
</section>
<section class="slide">
<h2 id="choice">Choice</h2>
<p>The same object can be seen in many different way as a category.</p>
<p>You can choose what are object, morphisms and composition.</p>
<p>ex: <strong>Str</strong> and discrete(Σ<sup>*</sup>)</p>
</section>
<section class="slide">
<h2 class="base1">Categorical Properties</h2>
<p class="base1">Any property which can be expressed in term of category, objects, morphism and composition.</p>
<ul><li> <em class="yellow">Dual</em>: \(\D\) is \(\C\) with reversed morphisms.
</li><li> <em class="yellow">Initial</em>: \(Z\in\ob{\C}\) s.t. \(∀Y∈\ob{\C}, \#\hom{Z,Y}=1\)
<br/> Unique ("up to isormophism")
</li><li> <em class="yellow">Terminal</em>: \(T\in\ob{\C}\) s.t. \(T\) is initial in the dual of \(\C\)
</li><li> <em class="yellow">Functor</em>: structure preserving mapping between categories
</li><li> ...
</li></ul>
</section>
<section class="slide">
<h2 id="isomorph">Isomorph</h2>
<p><img class="right" alt="isomorph cats" src="categories/img/isomorph-cats.jpg" /> <em class="yellow">isomorphism</em>: \(f:A→B\) which can be &quot;undone&quot; <em>i.e.</em><br />\(∃g:B→A\), \(g∘f=id_A\) <span class="and">&amp;</span> \(f∘g=id_B\)<br />in this case, \(A\) <span class="and">&amp;</span> \(B\) are <em class="yellow">isomorphic</em>.</p>
<p><span class="yellow">A≌B</span> means A and B are essentially the same.<br />In Category Theory, <span class="yellow">=</span> is in fact mostly <span class="yellow"></span>.<br />For example in commutative diagrams.</p>
</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> <span class="yellow">\(\F\)</span> from <span class="blue">\(\C\)</span> to <span class="green">\(\D\)</span>:</p>
<ul>
<li> Associate objects: <span class="backblue">\(A\in\ob{\C}\)</span> to <span class="backgreen">\(\F(A)\in\ob{\D}\)</span> </li>
<li> Associate morphisms: <span class="backblue">\(f:A\to B\)</span> to <span class="backgreen">\(\F(f) : \F(A) \to \F(B)\)</span>
such that
<ul>
<li>\( \F (\)<span class="backblue blue">\(\id_X\)</span>\()= \)<span class="backgreen"><span class="green">\(\id\)</span>\(\vphantom{\id}_{\F(}\)<span class="blue">\(\vphantom{\id}_X\)</span>\(\vphantom{\id}_{)} \)</span>,</li>
<li>\( \F (\)<span class="backblue blue">\(g∘f\)</span>\()= \)<span class="backgreen">\( \F(\)<span class="blue">\(g\)</span>\() \)<span class="green">\(\circ\)</span>\( \F(\)<span class="blue">\(f\)</span>\() \)</span></li>
</ul>
</li>
</ul>
</section>
<section class="slide">
<h2>Functor Example (ob → ob)</h2>
<img width="65%" src="categories/img/mp/functor.png" alt="Functor"/>
</section>
<section class="slide">
<h2>Functor Example (hom → hom)</h2>
<img width="65%" src="categories/img/mp/functor-morphism.png" alt="Functor"/>
</section>
<section class="slide">
<h2>Functor Example</h2>
<img width="65%" 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 width="75%" src="categories/img/mp/endofunctor.png" alt="Endofunctor"/>
</section>
<section class="slide">
<h2>Category of Categories</h2>
<img style="min-width:43%; width: 43%" class="right" src="categories/img/fractalcat.jpg" />
<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>General overview</li>
<li>Definitions</li>
<li><span class="yellow">Applications
<ul class="base01" style="border-left: 2px solid; padding-left: 1em; font-size: .6em; float: right; font-weight: bold; margin: -4em 0 0 1em">
<li>\(\Hask\) category
</li><li> Functors
</li><li> Natural transformations
</li><li> Monads
</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 id="haskell-kinds">Haskell Kinds</h2>
<p>In Haskell some types can take type variable(s). Typically: <code>[a]</code>.</p>
<p>Types have <em>kinds</em>; The kind is to type what type is to function. Kind are the types for types (so meta).</p>
<pre><code>Int, Char :: *
[], Maybe :: * -&gt; *
(,), (-&gt;) :: * -&gt; * -&gt; *
[Int], Maybe Char, Maybe [Int] :: *</code></pre>
</section>
<section class="slide">
<h2 id="haskell-types">Haskell Types</h2>
<p>Sometimes, the type determine a lot about the function<sup></sup>:</p>
<pre class="haskell"><code>fst :: (a,b) -> a -- Only one choice
snd :: (a,b) -> b -- Only one choice
f :: a -> [a] -- Many choices
-- Possibilities: f x=[], or [x], or [x,x] or [x,...,x]
? :: [a] -> [a] -- Many choices
-- can only rearrange: duplicate/remove/reorder elements
-- for example: the type of addOne isn't [a] -> [a]
addOne l = map <span class="red">(+1)</span> l
-- The (+1) force 'a' to be a Num.</code></pre>
<p>
<p><span class="small base01">★:<a href="http://ttic.uchicago.edu/~dreyer/course/papers/wadler.pdf">Theorems for free!, Philip Wadler, 1989</a></span></p>
</section>
<section class="slide">
<h2>Haskell Functor vs \(\Hask\) Functor</h2>
<p>A Haskell Functor is a type <code>F :: * -> *</code> which belong to the type class <code>Functor</code> ; thus instantiate
<code>fmap :: (a -> b) -> (F a -> F b)</code>.
<p><span style="visibility:hidden"><span class="and">&amp;</span></span> <code>F</code>: \(\ob{\Hask}→\ob{\Hask}\)<br/> <span class="and">&amp;</span> <code>fmap</code>: \(\hom{\Hask}→\hom{\Hask}\)
<p>The couple <code>(F,fmap)</code> is a \(\Hask\)'s functor if 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) = Just (f a)
fmap f Nothing = Nothing</code></pre>
<pre class="haskell"><code>fmap (+1) (Just 1) == Just 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</pre></code>
<pre class="haskell"><code>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 id="haskell-functors-for-the-programmer">Haskell Functors for the programmer</h2>
<p><code>Functor</code> is a type class used for types that can be mapped over.</p>
<ul>
<li>Containers: <code>[]</code>, Trees, Map, HashMap...</li>
<li>&quot;Feature Type&quot;:
<ul>
<li><code>Maybe a</code>: help to handle absence of <code>a</code>.<br />Ex: <code>safeDiv x 0 ⇒ Nothing</code></li>
<li><code>Either String a</code>: help to handle errors<br />Ex: <code>reportDiv x 0 ⇒ Left &quot;Division by 0!&quot;</code></li>
</ul></li>
</ul>
</section>
<section class="slide">
<h2>Haskell Functor intuition</h2>
<p>Put normal function inside a container. Ex: list, trees...<p>
<img width="70%" 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> in \(\Hask\).
</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 width="70%" src="categories/img/mp/hask-endofunctor.png" alt="Haskell functor representation"/>
</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 width="70%" src="categories/img/mp/hask-endofunctor-objects.png" alt="Haskell functor representation"/>
</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 width="70%" src="categories/img/mp/hask-endofunctor-morphisms.png" alt="Haskell functor representation"/>
</section>
<section class="slide">
<h2 id="non-haskell-hasks-functors">&quot;Non Haskell&quot; 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::* -&gt; *</code></li>
<li><code>fmap :: (a -&gt; b) -&gt; (F a) -&gt; (F b)</code></li>
</ul>
<p>Another example:</p>
<ul>
<li>F(<code>T</code>)=<code>Int</code></li>
<li>F(<code>f</code>)=<code>\_-&gt;0</code></li>
</ul>
</section>
<section class="slide">
<h2 id="also-functor-inside-hask">Also Functor inside \(\Hask\)</h2>
<p>\(\mathtt{[a]}∈\ob{\Hask}\)</code> but is also a category. Idem for <code>Int</code>.</p>
<p><code>length</code> is a Functor from the category <code>[a]</code> to the cateogry <code>Int</code>:</p>
<ul class="left" style="max-width:40%">
<li>\(\ob{\mathtt{[a]}}=\{∙\}\)</li>
<li>\(\hom{\mathtt{[a]}}=\mathtt{[a]}\)</li>
<li>\(∘=\mathtt{(++)}\)</li>
</ul>
<p class="left" style="margin:2em 3em"></p>
<ul class="left" style="max-width:40%">
<li>\(\ob{\mathtt{Int}}=\{∙\}\)</li>
<li>\(\hom{\mathtt{Int}}=\mathtt{Int}\)</li>
<li>\(∘=\mathtt{(+)}\)</li>
</ul>
<div class="flush"></div>
<ul><li>id: <code>length [] = 0</code>
</li><li>comp: <code>length (l ++ l') = (length l) + (length l')</code>
</li></ul>
</section>
<section class="slide">
<h2 id="category-of-hask-endofunctors">Category of \(\Hask\) Endofunctors</h2>
<img width="60%" src="categories/img/mp/cat-hask-endofunctor.png" alt="Category of Hask endofunctors" />
</section>
<section class="slide">
<h2 id="category-of-functors">Category of Functors</h2>
<p>If \(\C\) is <em>small</em> (\(\hom{\C}\) is a set). All functors from \(\C\) to some category \(\D\) form the category \(\mathrm{Func}(\C,\D)\).</p>
<ul>
<li>\(\ob{\mathrm{Func}(\C,\D)}\): Functors \(F:\C→\D\)</li>
<li>\(\hom{\mathrm{Func}(\C,\D)}\): <em>natural transformations</em></li>
<li>∘: Functor composition</li>
</ul>
<p>\(\mathrm{Func}(\C,\C)\) is the category of endofunctors of \(\C\).</p>
</section>
<section class="slide">
<h2 id="natural-transformations">Natural Transformations</h2>
<p>Let \(F\) and \(G\) be two functors from \(\C\) to \(\D\).</p>
<p><img src="categories/img/mp/natural-transformation.png" alt="Natural transformation commutative diagram" class="right"/> <em>A natural transformation:</em> familly η ; \(η_X\in\hom{\D}\) for \(X\in\ob{\C}\) s.t.</p>
<p>ex: between Haskell functors; <code>F a -&gt; G a</code><br />Rearragement functions only.</p>
</section>
<section class="slide">
<h2 id="natural-transformation-examples-14">Natural Transformation Examples (1/4)</h2>
<pre><code class="haskell small">data List a = Nil | Cons a (List a)
toList :: [a] -> List a
toList [] = Nil
toList (x:xs) = Cons x (toList xs)</pre>
</code>
<p><code>toList</code> is a natural transformation. It is also a morphism from <code>[]</code> to <code>List</code> in the Category of \(\Hask\) endofunctors.</p>
<img style="float:left;width:30%" src="categories/img/mp/nattrans-list-tree.png" alt="natural transformation commutative diagram"/>
<figure style="float:right;width:60%">
<img style="width:40%" src="categories/img/mp/list-tree-endofunctor-morphism.png" alt="natural transformation commutative diagram"/>
</figure>
</section>
<section class="slide">
<h2 id="natural-transformation-examples-24">Natural Transformation Examples (2/4)</h2>
<pre><code class="haskell small">data List a = Nil | Cons a (List a)
toHList :: List a -> [a]
toHList Nil = []
toHList (Cons x xs) = x:toHList xs</pre>
</code>
<p><code>toHList</code> is a natural transformation. It is also a morphism from <code>List</code> to <code>[]</code> in the Category of \(\Hask\) endofunctors.</p>
<img style="float:left;width:30%" src="categories/img/mp/nattrans-tree-list.png" alt="natural transformation commutative diagram"/>
<figure style="float:right;width:60%">
<img style="width:40%" src="categories/img/mp/tree-list-endofunctor-morphism.png" alt="natural transformation commutative diagram"/> <figcaption><code>toList . toHList = id</code> <span class="and">&amp;</span> <code>toHList . toList = id</code> <span style="visibility:hidden"><span class="and">&amp;</span></span><br/> therefore <code>[]</code> <span class="and">&amp;</span> <code>List</code> are <span class="orange">isomorph</span>. </figcaption>
</figure>
</section>
<section class="slide">
<h2 id="natural-transformation-examples-34">Natural Transformation Examples (3/4)</h2>
<pre><code class="haskell small">toMaybe :: [a] -> Maybe a
toMaybe [] = Nothing
toMaybe (x:xs) = Just x</pre>
</code>
<p><code>toMaybe</code> is a natural transformation. It is also a morphism from <code>[]</code> to <code>Maybe</code> in the Category of \(\Hask\) endofunctors.</p>
<img style="float:left;width:30%" src="categories/img/mp/nattrans-list-maybe.png" alt="natural transformation commutative diagram"/>
<figure style="float:right;width:60%">
<img style="width:40%" src="categories/img/mp/list-maybe-endofunctor-morphism.png" alt="natural transformation commutative diagram"/>
</figure>
</section>
<section class="slide">
<h2 id="natural-transformation-examples-44">Natural Transformation Examples (4/4)</h2>
<pre><code class="haskell small">mToList :: Maybe a -> [a]
mToList Nothing = []
mToList Just x = [x]</pre>
</code>
<p><code>toMaybe</code> is a natural transformation. It is also a morphism from <code>[]</code> to <code>Maybe</code> in the Category of \(\Hask\) endofunctors.</p>
<img style="float:left;width:30%" src="categories/img/mp/nattrans-maybe-list.png" alt="natural transformation commutative diagram"/>
<figure style="float:right;width:60%">
<img style="width:40%" src="categories/img/mp/maybe-list-endofunctor-morphsm.png" alt="relation between [] and Maybe"/> <figcaption>There is <span class="red">no isomorphism</span>.<br/> Hint: <code>Bool</code> lists longer than 1. </figcaption>
</figure>
</section>
<section class="slide">
<h2 id="composition-problem">Composition problem</h2>
<p>The Problem; example with lists:</p>
<pre class="haskell"><code>f x = [x] ⇒ f 1 = [1] ⇒ (f.f) 1 = [[1]] ✗
g x = [x+1] ⇒ g 1 = [2] ⇒ (g.g) 1 = ERROR [2]+1 ✗
h x = [x+1,x*3] ⇒ h 1 = [2,3] ⇒ (h.h) 1 = ERROR [2,3]+1 ✗ </code></pre>
<p>The same problem with most <code>f :: a -&gt; F a</code> functions and functor <code>F</code>.</p>
</section>
<section class="slide">
<h2 id="composition-fixable">Composition Fixable?</h2>
<p>How to fix that? We want to construct an operator which is able to compose:</p>
<p><code>f :: a -&gt; F b</code> <span class="and">&amp;</span> <code>g :: b -&gt; F c</code>.</p>
<p>More specifically we want to create an operator ◎ of type</p>
<p><code>◎ :: (b -&gt; F c) -&gt; (a -&gt; F b) -&gt; (a -&gt; F c)</code></p>
<p>Note: if <code>F</code> = I, ◎ = <code>(.)</code>.</p>
</section>
<section class="slide">
<h2 id="fix-composition-12">Fix Composition (1/2)</h2>
<p>Goal, find: <code>◎ :: (b -&gt; F c) -&gt; (a -&gt; F b) -&gt; (a -&gt; F c)</code><br /><code>f :: a -&gt; F b</code>, <code>g :: b -&gt; F c</code>:</p>
<ul>
<li><code>(g ◎ f) x</code> ???</li>
<li>First apply <code>f</code> to <code>x</code><code>f x :: F b</code></li>
<li>Then how to apply <code>g</code> properly to an element of type <code>F b</code>?</li>
</ul>
</section>
<section class="slide">
<h2 id="fix-composition-22">Fix Composition (2/2)</h2>
<p>Goal, find: <code>◎ :: (b -&gt; F c) -&gt; (a -&gt; F b) -&gt; (a -&gt; F c)</code><br /><code>f :: a -&gt; F b</code>, <code>g :: b -&gt; F c</code>, <span class="yellow"><code>f x :: F b</code></span>:</p>
<ul>
<li>Use <code>fmap :: (t -&gt; u) -&gt; (F t -&gt; F u)</code>!</li>
<li><code>(fmap g) :: F b -&gt; F (F c)</code> ; (<code>t=b</code>, <code>u=F c</code>)</li>
<li><code>(fmap g) (f x) :: F (F c)</code> it almost WORKS!</li>
<li>We lack an important component, <code>join :: F (F c) -&gt; F c</code></li>
<li><code>(g ◎ f) x = join ((fmap g) (f x))</code><br />◎ is the Kleisli composition; in Haskell: <code>&lt;=&lt;</code> (in <code>Control.Monad</code>).</li>
</ul>
</section>
<section class="slide">
<h2 id="necessary-laws">Necessary laws</h2>
<p>For ◎ to work like composition, we need join to hold the following properties:</p>
<ul>
<li><code>join (join (F (F (F a))))=join (F (join (F (F a))))</code></li>
<li>abusing notations denoting <code>join</code> by ⊙; this is equivalent to<br /><span class="yellow"><code>(F ⊙ F) ⊙ F = F ⊙ (F ⊙ F)</code></span></li>
<li>There exists <code>η :: a -&gt; F a</code> s.t.<br /><span class="yellow"><code>η⊙F=F=F⊙η</code></span></li>
</ul>
</section>
<section class="slide">
<h2 id="klesli-composition">Klesli composition</h2>
<p>Now the composition works as expected. In Haskell ◎ is <code>&lt;=&lt;</code> in <code>Control.Monad</code>.</p>
<p><code>g &lt;=&lt; f = \x -&gt; join ((fmap g) (f x))</code></p>
<pre class="haskell"><code>f x = [x] ⇒ f 1 = [1] ⇒ (f <=< f) 1 = [1]
g x = [x+1] g 1 = [2] (g <=< g) 1 = [3]
h x = [x+1,x*3] h 1 = [2,3] (h <=< h) 1 = [3,6,4,9] </code></pre>
</section>
<section class="slide">
<h2 id="we-reinvented-monads">We reinvented Monads!</h2>
<p>A monad is a triplet <code>(M,⊙,η)</code> where</p>
<ul>
<li>\(M\) an <span class="yellow">Endofunctor</span> (to type <code>a</code> associate <code>M a</code>)</li>
<li>\(⊙:M×M→M\) a <span class="yellow">nat. trans.</span> (i.e. <code>⊙::M (M a) → M a</code> ; <code>join</code>)</li>
<li>\(η:I→M\) a <span class="yellow">nat. trans.</span> (\(I\) identity functor ; <code>η::a → M a</code>)</li>
</ul>
<p>Satisfying</p>
<ul>
<li>\(M ⊙ (M ⊙ M) = (M ⊙ M) ⊙ M\)</li>
<li>\(η ⊙ M = M = M ⊙ η\)</li>
</ul>
</section>
<section class="slide">
<h2 id="compare-with-monoid">Compare with Monoid</h2>
<p>A Monoid is a triplet \((E,∙,e)\) s.t.</p>
<ul>
<li>\(E\) a set</li>
<li>\(∙:E×E→E\)</li>
<li>\(e:1→E\)</li>
</ul>
<p>Satisfying</p>
<ul>
<li>\(x∙(y∙z) = (x∙y)∙z, ∀x,y,z∈E\)</li>
<li>\(e∙x = x = x∙e, ∀x∈E\)</li>
</ul>
</section>
<section class="slide">
<h2 id="monads-are-just-monoids">Monads are just Monoids</h2>
<blockquote>
<p>A Monad is just a monoid in the category of endofunctors, what's the problem?</p>
</blockquote>
<p>The real sentence was:</p>
<blockquote>
<p>All told, a monad in X is just a monoid in the category of endofunctors of X, with product × replaced by composition of endofunctors and unit set by the identity endofunctor.</p>
</blockquote>
</section>
<section class="slide">
<h2 id="example-list">Example: List</h2>
<ul>
<li><code>[] :: * -&gt; *</code> an <span class="yellow">Endofunctor</span></li>
<li>\(⊙:M×M→M\) a nat. trans. (<code>join :: M (M a) -&gt; M a</code>)</li>
<li>\(η:I→M\) a nat. trans.</li>
</ul>
<pre class="haskell"><code>-- In Haskell ⊙ is "join" in "Control.Monad"
join :: [[a]] -> [a]
join = concat
-- In Haskell the "return" function (unfortunate name)
η :: a -> [a]
η x = [x]</code></pre>
</section>
<section class="slide">
<h2 id="example-list-law-verification">Example: List (law verification)</h2>
<p>Example: <code>List</code> is a functor (<code>join</code> is ⊙)</p>
<ul>
<li>\(M ⊙ (M ⊙ M) = (M ⊙ M) ⊙ M\)</li>
<li>\(η ⊙ M = M = M ⊙ η\)</li>
</ul>
<pre class="nohighlight small"><code>join [ join [[x,y,...,z]] ] = join [[x,y,...,z]]
= join (join [[[x,y,...,z]]])
join (η [x]) = [x] = join [η x]</code></pre>
<p>Therefore <code>([],join,η)</code> is a monad.</p>
</section>
<section class="slide">
<h2 id="monads-utility">Monads useful?</h2>
<p>A <em>LOT</em> of monad tutorial on the net. Just one example; the State Monad</p>
<p><code>DrawScene</code> to <code><span class="yellow">State Screen</span> DrawScene</code> ; still <b>pure</b>.</p>
<pre class="haskell left smaller" style="width:40%"><code>main = drawImage (width,height)
drawImage :: Screen -&gt; DrawScene
drawImage <span class="orange">screen</span> = do
drawPoint p <span class="orange">screen</span>
drawCircle c <span class="orange">screen</span>
drawRectangle r <span class="orange">screen</span>
drawPoint point <span class="orange">screen</span> = ...
drawCircle circle <span class="orange">screen</span> = ...
drawRectangle rectangle <span class="orange">screen</span> = ...</code></pre>
<pre class="haskell right smaller" style="width:45%"><code>main = do
<span class="orange">put (Screen 1024 768)</span>
drawImage
drawImage :: State Screen DrawScene
drawImage = do
drawPoint p
drawCircle c
drawRectangle r
drawPoint :: Point -&gt; State Screen DrawScene
drawPoint p = do
<span class="orange">Screen width height &lt;- get</span>
...</code></pre>
</section>
<section class="slide">
<h2 id="fold"><code>fold</code></h2>
<img src="categories/img/tower_folded.gif" alt="fold" style="width:50%;max-width:50%"/>
</section>
<section class="slide">
<h2 id="κατα-morphism">κατα-morphism</h2>
<img src="categories/img/earth_catamorphed.gif" alt="catamorphism" style="width:90%;max-width:90%"/>
</section>
<section class="slide">
<h2 id="κατα-morphism-fold-generalization">κατα-morphism: fold generalization</h2>
<p><code>acc</code> type of the &quot;accumulator&quot;:<br /><code>fold :: (acc -&gt; a -&gt; acc) -&gt; acc -&gt; [a] -&gt; acc</code></p>
<p>Idea: put the accumulated value inside the type.</p>
<pre class="haskell"><code>-- Equivalent to fold (+1) 0 "cata"
(Cons 'c' (Cons 'a' (Cons 't' (Cons 'a' Nil))))
(Cons 'c' (Cons 'a' (Cons 't' (Cons 'a' <span style="border: solid 1px;">0</span>))))
(Cons 'c' (Cons 'a' (Cons 't' <span style="border: solid 1px;">1</span>)))
(Cons 'c' (Cons 'a' <span style="border: solid 1px;">2</span>))
(Cons 'c' <span style="border: solid 1px;">3</span>)
<span style="border: solid 1px;">4</span></code></pre>
<p>But where are all the informations? <code>(+1)</code> and <code>0</code>?</p>
</section>
<section class="slide">
<h2 id="κατα-morphism-missing-information">κατα-morphism: Missing Information</h2>
<p>Where is the missing information?</p>
<ul>
<li>Functor operator <code>fmap</code></li>
<li>Algebra representing the <code>(+1)</code> and also knowing about the <code>0</code>.</li>
</ul>
<p>First example, make <code>length</code> on <code>[Char]</code></p>
</section>
<section class="slide">
<h2 id="κατα-morphism-type-work">κατα-morphism: Type work</h2>
<pre class="haskell"><code>
data StrF a = Cons Char a | Nil
data Str' = StrF Str'
-- generalize the construction of Str to other datatype
-- Mu: type fixed point
-- Mu :: (* -> *) -> *
data Mu f = InF { outF :: f (Mu f) }
data Str = Mu StrF
-- Example
foo=InF { outF = Cons 'f'
(InF { outF = Cons 'o'
(InF { outF = Cons 'o'
(InF { outF = Nil })})})}</code></pre>
</section>
<section class="slide">
<h2 id="κατα-morphism-missing-information-retrieved">κατα-morphism: missing information retrieved</h2>
<pre class="haskell"><code>type Algebra f a = f a -> a
instance Functor (StrF a) =
fmap f (Cons c x) = Cons c (f x)
fmap _ Nil = Nil</code></pre>
<pre class="haskell"><code>cata :: Functor f => Algebra f a -> Mu f -> a
cata f = f . fmap (cata f) . outF</code></pre>
</section>
<section class="slide">
<h2 id="κατα-morphism-finally-length">κατα-morphism: Finally length</h2>
<p>All needed information for making length.</p>
<pre><code>instance Functor (StrF a) =
fmap f (Cons c x) = Cons c (f x)
fmap _ Nil = Nil
length&#39; :: Str -&gt; Int
length&#39; = cata phi where
phi :: Algebra StrF Int -- StrF Int -&gt; Int
phi (Cons a b) = 1 + b
phi Nil = 0
main = do
l &lt;- length&#39; $ stringToStr &quot;Toto&quot;
...</code></pre>
</section>
<section class="slide">
<h2 id="κατα-morphism-extension-to-trees">κατα-morphism: extension to Trees</h2>
<p>Once you get the trick, it is easy to extent to most Functor.</p>
<pre><code>type Tree = Mu TreeF
data TreeF x = Node Int [x]
instance Functor TreeF where
fmap f (Node e xs) = Node e (fmap f xs)
depth = cata phi where
phi :: Algebra TreeF Int -- TreeF Int -&gt; Int
phi (Node x sons) = 1 + foldr max 0 sons</code></pre>
</section>
<section class="slide">
<h2 id="conclusion">Conclusion</h2>
<p>Category Theory oriented Programming:</p>
<ul>
<li>Focus on the type and operators</li>
<li>Extreme generalisation</li>
<li>Better modularity</li>
<li>Better control through properties of types</li>
</ul>
<p><span class="smaller">No cat were harmed in the making of this presentation.</span></p>
</section>
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function analytics() {
// add an event to all link for google analytics
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// tell analytics to save event
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var href=$(this).attr('href')
var label="";
if ( typeof( identifier ) != 'undefined' ) {
label=label+'[id]:'+identifier
category='JSLink'
}
if ( typeof( href ) != 'undefined' ) {
label=label+' [href]:'+href
if ( href[0] == '#' ) {
category='Anchor';
} else {
category='Link';
}
}
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// console.log('[tracked]: ' + category + ' ; clicked ; ' + label );
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catch (err) {
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