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Yann Esposito (Yogsototh) 2012-12-10 08:03:25 +01:00
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categories.html
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@ -67,7 +67,7 @@
<!-- Begin slides. Just make elements with a class of slide. -->
<section class="slide">
<div style="text-align:center; position:absolute; top: 2em;">
<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>
<author><em class="base01">by</em> Yann Esposito</author>
<div style="font-size:.5em">
@ -79,7 +79,7 @@
</googleplus>
</div>
<div class="base01" style="font-size: .5em; font-weight: 400; font-variant:italic">
<div class="button" style="margin: .5em auto;border: solid 2px; padding: 5px; width: 8em" onclick="javascript:gofullscreen();">ENTER FULLSCREEN</div>
<div class="button" style="margin: .5em auto;border: solid 2px; padding: 5px; width: 8em; border-radius: 1em; background:rgba(255,255,255,0.05);" onclick="javascript:gofullscreen();">ENTER FULLSCREEN</div>
HTML presentation: use arrows, space to navigate.
</div>
</div>
@ -104,15 +104,17 @@
<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>From a Programmer perspective:</p>
<blockquote>
<p>Category Theory is a new language/framework for Math</p>
</blockquote>
<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>
</section>
<section class="slide">
<h2 id="math-programming-relation">Math Programming relation</h2>
@ -128,6 +130,9 @@
<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:70%">
<tr><th>
@ -352,8 +357,7 @@ such that for each \(f:A→B\):</p>
<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>\(\hom{G}=\{ε,α,β,γ,αβ,βγ,...\}\)
</li><li>\(αβ∘γ=αβγ\)
</li></ul>
</section>
@ -374,7 +378,6 @@ such that for each \(f:A→B\):</p>
</li><li><code>(Integer,1,*)</code>,
</li><li><code>(Strings,"",++)</code>,
</li><li>for each <code>a</code>, <code>([a],[],++)</code>
</li><li>Couple of monoids \((M×N, (0_M,0_N), (⊙_M,⊙_N))\)
</li></ul>
</section>
<section class="slide">
@ -893,6 +896,65 @@ drawPoint p = do
</figure>
</section>
<section class="slide">
<h2 id="κατα-morphism-fold-generalization">κατα-morphism: fold generalization</h2>
<p>Typically, you need a recursive type (list, trees, ...)</p>
<pre><code>Str = Cons Char Str
1st: replace the recursive type by another type.
StrChar a = Cons Char a
Str&#39; a = Mu StrChar
Str&#39; = InF { outF :: StrChar (Mu ListElement) }
Str&#39; = InF { outF :: StrChar (Str&#39;) }</code></pre>
<p>Clearly <code>Str'</code> is isomorph <code>String</code>.</p>
<pre><code>type Algebra f a = f a -&gt; a
data Mu f = InF { outF :: f (Mu f) }
cata :: Functor f =&gt; Algebra f a -&gt; Mu f -&gt; a
cata f = f . fmap (cata f) . outF</code></pre>
</section>
<section class="slide">
<h2 id="κατα-morphism-example-with-strings">κατα-morphism: example with strings</h2>
<pre><code>type Algebra f a = f a -&gt; a
data Mu f = InF { outF :: f (Mu f) }
cata :: Functor f =&gt; Algebra f a -&gt; Mu f -&gt; a
cata f = f . fmap (cata f) . outF
-- Example with natural fold on String
data StrF x = Cons Char x | Nil
type Str = Mu StrF
instance Functor StrF where
fmap f (Cons a as) = Cons a (f as)
fmap _ Nil = Nil
length&#39; :: Str -&gt; Int
length&#39; = cata phi where
phi (Cons a b) = 1 + b
phi Nil = 0
toMu :: [Char] -&gt; Str
toMu (x:xs) = InF { outF = Cons x (toMu xs) }
toMu [] = InF { outF = Nil }</code></pre>
</section>
<section class="slide">
<h2 id="κατα-morphism-example-with-strings">κατα-morphism: example with strings</h2>
<pre><code>type Algebra f a = f a -&gt; a
data Mu f = InF { outF :: f (Mu f) }
cata :: Functor f =&gt; Algebra f a -&gt; Mu f -&gt; a
cata f = f . fmap (cata f) . outF
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 (Node x sons) = 1 + foldr max 0 sons</code></pre>
</section>
<!-- End slides. -->

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@ -1,4 +1,4 @@
<div style="text-align:center; position:absolute; top: 2em;">
<div style="text-align:center; position:absolute; top: 2em; font-size: .9em; width: 100%">
<h1 style="position: relative;">Category Theory &amp; Programming</h1>
<author><em class="base01">by</em> Yann Esposito</author>
<div style="font-size:.5em">
@ -10,7 +10,7 @@
</googleplus>
</div>
<div class="base01" style="font-size: .5em; font-weight: 400; font-variant:italic">
<div class="button" style="margin: .5em auto;border: solid 2px; padding: 5px; width: 8em" onclick="javascript:gofullscreen();">ENTER FULLSCREEN</div>
<div class="button" style="margin: .5em auto;border: solid 2px; padding: 5px; width: 8em; border-radius: 1em; background:rgba(255,255,255,0.05);" onclick="javascript:gofullscreen();">ENTER FULLSCREEN</div>
HTML presentation: use arrows, space to navigate.
</div>
</div>

4
categories/10_Introduction/030_General_Overview.html
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@ -9,10 +9,6 @@
<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>From a Programmer perspective:</p>
<blockquote>
<p>Category Theory is a new language/framework for Math</p>
</blockquote>
<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>

4
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@ -16,9 +16,5 @@ Certainly one of the more abstract branches of math
- _Bridge between disciplines_
Physics, Quantum Physics, Topology, Logic, Computer Science<sup>☆</sup>
From a Programmer perspective:
> Category Theory is a new language/framework for Math
<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>

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@ -0,0 +1,4 @@
<h2 id="from-a-programmer-perspective">From a Programmer perspective</h2>
<blockquote>
<p>Category Theory is a new language/framework for Math</p>
</blockquote>

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@ -0,0 +1,4 @@
From a Programmer perspective
----------------
> Category Theory is a new language/framework for Math

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@ -4,43 +4,3 @@
<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>
<img class="right" src="categories/img/readingcat.jpg" alt="lolcat"/>
<table style="width:70%">
<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>

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categories/10_Introduction/050_Vocabulary.md
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@ -21,15 +21,3 @@ Math vocabulary used in this presentation:
> Klesli arrows,
> κατα-morphism,
> ...
<img class="right" src="categories/img/readingcat.jpg" alt="lolcat"/>
<table style="width:70%">
<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>

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@ -0,0 +1,41 @@
<h2 id="programmer-translation">Programmer Translation</h2>
<img class="right" src="categories/img/readingcat.jpg" alt="lolcat"/>
<table style="width:70%">
<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>

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@ -0,0 +1,14 @@
Programmer Translation
----------------------
<img class="right" src="categories/img/readingcat.jpg" alt="lolcat"/>
<table style="width:70%">
<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>

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@ -10,7 +10,6 @@
<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>\(\hom{G}=\{ε,α,β,γ,αβ,βγ,...\}\)
</li><li>\(αβ∘γ=αβγ\)
</li></ul>

1
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@ -8,5 +8,4 @@
</li><li><code>(Integer,1,*)</code>,
</li><li><code>(Strings,"",++)</code>,
</li><li>for each <code>a</code>, <code>([a],[],++)</code>
</li><li>Couple of monoids \((M×N, (0_M,0_N), (⊙_M,⊙_N))\)
</li></ul>

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@ -0,0 +1,16 @@
<h2 id="κατα-morphism-fold-generalization">κατα-morphism: fold generalization</h2>
<p>Typically, you need a recursive type (list, trees, ...)</p>
<pre><code>Str = Cons Char Str
1st: replace the recursive type by another type.
StrChar a = Cons Char a
Str&#39; a = Mu StrChar
Str&#39; = InF { outF :: StrChar (Mu ListElement) }
Str&#39; = InF { outF :: StrChar (Str&#39;) }</code></pre>
<p>Clearly <code>Str'</code> is isomorph <code>String</code>.</p>
<pre><code>type Algebra f a = f a -&gt; a
data Mu f = InF { outF :: f (Mu f) }
cata :: Functor f =&gt; Algebra f a -&gt; Mu f -&gt; a
cata f = f . fmap (cata f) . outF</code></pre>

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@ -3,6 +3,7 @@
Typically, you need a recursive type (list, trees, ...)
~~~
Str = Cons Char Str
1st: replace the recursive type by another type.
@ -11,8 +12,9 @@ Str' a = Mu StrChar
Str' = InF { outF :: StrChar (Mu ListElement) }
Str' = InF { outF :: StrChar (Str') }
~~~
Clearly isomorph to List.
Clearly `Str'` is isomorph `String`.
~~~
type Algebra f a = f a -> a
@ -20,20 +22,4 @@ data Mu f = InF { outF :: f (Mu f) }
cata :: Functor f => Algebra f a -> Mu f -> a
cata f = f . fmap (cata f) . outF
-- Example with natural fold on String
data StrF x = Cons Char x | Nil
type Str = Mu StrF
instance Functor StrF where
fmap f (Cons a as) = Cons a (f as)
fmap _ Nil = Nil
length' :: Str -> Int
length' = cata phi where
phi (Cons a b) = 1 + b
phi Nil = 0
toMu :: [Char] -> Str
toMu (x:xs) = InF { outF = Cons x (toMu xs) }
toMu [] = InF { outF = Nil }
~~~

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@ -0,0 +1,22 @@
<h2 id="κατα-morphism-example-with-strings">κατα-morphism: example with strings</h2>
<pre><code>type Algebra f a = f a -&gt; a
data Mu f = InF { outF :: f (Mu f) }
cata :: Functor f =&gt; Algebra f a -&gt; Mu f -&gt; a
cata f = f . fmap (cata f) . outF
-- Example with natural fold on String
data StrF x = Cons Char x | Nil
type Str = Mu StrF
instance Functor StrF where
fmap f (Cons a as) = Cons a (f as)
fmap _ Nil = Nil
length&#39; :: Str -&gt; Int
length&#39; = cata phi where
phi (Cons a b) = 1 + b
phi Nil = 0
toMu :: [Char] -&gt; Str
toMu (x:xs) = InF { outF = Cons x (toMu xs) }
toMu [] = InF { outF = Nil }</code></pre>

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@ -0,0 +1,15 @@
<h2 id="κατα-morphism-example-with-strings">κατα-morphism: example with strings</h2>
<pre><code>type Algebra f a = f a -&gt; a
data Mu f = InF { outF :: f (Mu f) }
cata :: Functor f =&gt; Algebra f a -&gt; Mu f -&gt; a
cata f = f . fmap (cata f) . outF
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 (Node x sons) = 1 + foldr max 0 sons</code></pre>