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102
categories.html
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@ -94,16 +94,25 @@
</section>
<section class="slide">
<h2 id="general-overview">General Overview</h2>
<p>Recent Math Field ; 1942-45, Samuel Eilenberg <span class="and">&amp;</span> Saunders Mac Lane</p>
<div style="float:right; width: 20%">
<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>New math foundation<br /> formalism abstraction, package entire theory</li>
<li>Bridge between disciplines<br /> Physics, Quantum Physics, Topology, Logic, Computer Science</li>
<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="math-programming-relation">Math Programming relation</h2>
@ -117,9 +126,48 @@
<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, κατα-morphism,</p>
<p>Category, Morphism, Associativity, Preorder, Functor, Endofunctor, Categorial property, Commutative diagram, Isomorph, Initial, Dual, Monoid Natural transformation, Monad, κατα-morphism, ...</p>
</blockquote>
<p>Most will be translated for the programmer.</p>
<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>
</section>
<section class="slide">
<h2>Plan</h2>
@ -327,9 +375,15 @@ such that for each \(f:A→B\):</p>
<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>
<p>each preorder \((P,≤)\):</p>
<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>At most one morphism between two objects.</em></p>
<img src="categories/img/mp/preorder.png" alt="preorder category"/>
</section>
<section class="slide">
@ -345,10 +399,14 @@ such that for each \(f:A→B\):</p>
<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">&amp;</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
<ul><li> <em class="yellow">isomorphism</em>: \(f:A→B\) which can be "undone" <em class="yellow">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>.
</li><li> <em class="yellow">Dual</em>: reverse direction of arrows of \(\C\)
</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>
@ -357,14 +415,14 @@ such that for each \(f:A→B\):</p>
<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 \(\C\) to \(\D\):</p>
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: \(A\in\ob{\C}\) to <span class="yellow">\(\F(A)\)</span>\( \in\ob{\D}\) </li>
<li> Associate morphisms: \(f:A\to B\) to <span class="yellow">\(\F(f) : \F(A) \to \F(B)\)</span>
<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>\( \F (\id_X) = \id_{\F(X)} \),</li>
<li>\( \F (g \circ_\C f) = \F(g) \circ_\D \F(f) \)</li>
<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>
@ -446,18 +504,20 @@ CTree :: * -&gt; * -&gt; *
<section class="slide">
<h2 id="haskell-types">Haskell Types</h2>
<p>We can make function that can work for <em>all</em> type parameter. Such function can only work with the <em>topology</em> induced by the type. We know such function won't work <em>on</em> the elements.</p>
<p>Sometimes, the type determine a lot about the function:</p>
<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 duplicate/remove/reorder elements
-- can only rearrange: duplicate/remove/reorder elements
-- for example: the type of addOne isn't [a] -> [a]
addOne l = map (+1) l</code></pre>
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>

15
categories/10_Introduction/030_General_Overview.html
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@ -1,11 +1,20 @@
<h2 id="general-overview">General Overview</h2>
<p>Recent Math Field ; 1942-45, Samuel Eilenberg &amp; Saunders Mac Lane</p>
<div style="float:right; width: 20%">
<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 &amp; Saunders Mac Lane</p>
<p>Certainly one of the more abstract branches of math</p>
<ul>
<li>New math foundation<br /> formalism abstraction, package entire theory</li>
<li>Bridge between disciplines<br /> Physics, Quantum Physics, Topology, Logic, Computer Science</li>
<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>

19
categories/10_Introduction/030_General_Overview.md
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@ -1,15 +1,24 @@
General Overview
----------------
Recent Math Field ; 1942-45, Samuel Eilenberg &amp; Saunders Mac Lane
<div style="float:right; width: 20%">
<img src="categories/img/eilenberg.gif" alt="Samuel Eilenberg"/>
<img src="categories/img/maclaine.jpg" alt="Saunders Mac Lane"/>
</div>
_Recent Math Field_
1942-45, Samuel Eilenberg &amp; Saunders Mac Lane
Certainly one of the more abstract branches of math
- New math foundation
formalism abstraction, package entire theory
- Bridge between disciplines
Physics, Quantum Physics, Topology, Logic, Computer Science
- _New math foundation_
formalism abstraction, package entire theory<sup>★</sup>
- _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>

43
categories/10_Introduction/050_Vocabulary.html
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@ -2,6 +2,45 @@
<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, κατα-morphism,</p>
<p>Category, Morphism, Associativity, Preorder, Functor, Endofunctor, Categorial property, Commutative diagram, Isomorph, Initial, Dual, Monoid Natural transformation, Monad, κατα-morphism, ...</p>
</blockquote>
<p>Most will be translated for the programmer.</p>
<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>

15
categories/10_Introduction/050_Vocabulary.md
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@ -15,11 +15,20 @@ Math vocabulary used in this presentation:
> Isomorph,
> Initial,
> Dual,
> Monoid
> Monoid,
> Natural transformation,
> Monad,
> κατα-morphism,
> ...
<img class="right" src="categories/img/cute.gif" alt="mind blown"/>
<img class="right" src="categories/img/readingcat.jpg" alt="lolcat"/>
Most will be translated for the programmer.
<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/20_What/170_Degenerated_Categories_Preorders.html
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@ -1,6 +1,12 @@
<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>
<p>each preorder \((P,≤)\):</p>
<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>At most one morphism between two objects.</em></p>
<img src="categories/img/mp/preorder.png" alt="preorder category"/>

12
categories/20_What/190_Categorical_Property.html
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@ -2,9 +2,13 @@
<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, \(g∘f=id_A\) &amp; \(f∘g=id_B\)
</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
<ul><li> <em class="yellow">isomorphism</em>: \(f:A→B\) which can be "undone" <em class="yellow">i.e.</em>
<br/> \(∃g:B→A\), \(g∘f=id_A\) &amp; \(f∘g=id_B\)
<br/> in this case, \(A\) &amp; \(B\) are <em class="yellow">isomorphic</em>.
</li><li> <em class="yellow">Dual</em>: reverse direction of arrows of \(\C\)
</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>

10
categories/20_What/200_Functor.html
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@ -2,14 +2,14 @@
<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 \(\C\) to \(\D\):</p>
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: \(A\in\ob{\C}\) to <span class="yellow">\(\F(A)\)</span>\( \in\ob{\D}\) </li>
<li> Associate morphisms: \(f:A\to B\) to <span class="yellow">\(\F(f) : \F(A) \to \F(B)\)</span>
<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>\( \F (\id_X) = \id_{\F(X)} \),</li>
<li>\( \F (g \circ_\C f) = \F(g) \circ_\D \F(f) \)</li>
<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>

10
categories/30_How/040_Haskell_Types.html
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@ -1,14 +1,16 @@
<h2 id="haskell-types">Haskell Types</h2>
<p>We can make function that can work for <em>all</em> type parameter. Such function can only work with the <em>topology</em> induced by the type. We know such function won't work <em>on</em> the elements.</p>
<p>Sometimes, the type determine a lot about the function:</p>
<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 duplicate/remove/reorder elements
-- can only rearrange: duplicate/remove/reorder elements
-- for example: the type of addOne isn't [a] -> [a]
addOne l = map (+1) l</code></pre>
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>

9
categories/30_How/040_Haskell_Types.md
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@ -5,7 +5,7 @@ We can make function that can work for _all_ type parameter.
Such function can only work with the _topology_ induced by the type.
We know such function won't work _on_ the elements.
Sometimes, the type determine a lot about the function:
Sometimes, the type determine a lot about the function<sup>★</sup>:
<pre class="haskell"><code>fst :: (a,b) -> a -- Only one choice
snd :: (a,b) -> b -- Only one choice
@ -13,6 +13,9 @@ f :: a -> [a] -- Many choices
-- Possibilities: f x=[], or [x], or [x,x] or [x,...,x]
? :: [a] -> [a] -- Many choices
-- can only duplicate/remove/reorder elements
-- can only rearrange: duplicate/remove/reorder elements
-- for example: the type of addOne isn't [a] -> [a]
addOne l = map (+1) l</code></pre>
addOne l = map <span class="red">(+1)</span> l
-- The (+1) force a to be a Num.</code></pre>
<p><span class="small base01">★:<a href="http://ttic.uchicago.edu/~dreyer/course/papers/wadler.pdf">Theorems for free!, Philip Wadler, 1989</a></span>

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@ -68,10 +68,18 @@ body.deck-container {
color: #6c71c4; }
.deck-container .blue {
color: #268bd2; }
.deck-container .backblue {
border: solid 2px #268bd2;
border-radius: 2px;
background-color: rgba(38, 139, 210, 0.2); }
.deck-container .cyan {
color: #2aa198; }
.deck-container .green {
color: #859900; }
.deck-container .backgreen {
border: solid 2px #859900;
border-radius: 2px;
background-color: rgba(133, 153, 0, 0.2); }
.deck-container div {
font-family: "ComputerModernSansSerif", Helvetica, sans-serif; }
.deck-container .and {
@ -260,6 +268,8 @@ body.deck-container {
margin-top: 0; }
.deck-container .small {
font-size: 0.8em; }
.deck-container .smaller {
font-size: 0.6em; }
.deck-container .sc {
text-transform: uppercase;
font-size: 0.8em; }
@ -469,10 +479,9 @@ body.deck-container {
margin-bottom: 0; }
.deck-container img, .deck-container .likeimg {
max-width: 80%;
border: 2px solid #586e75;
background-color: #073642;
padding: 0.5em;
box-shadow: 0 0 10px #002b36 inset;
border: 2px solid #073642;
padding: 0.3em;
box-shadow: 0 0 0.5em black inset;
border-radius: 0.5em;
text-align: center; }
.deck-container a:hover img {

17
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@ -213,10 +213,18 @@ body.deck-container
color: $violet
.blue
color: $blue
.backblue
border: solid 2px $blue
border-radius: 2px
background-color: rgba(38,139,210,.2)
.cyan
color: $cyan
.green
color: $green
.backgreen
border: solid 2px $green
border-radius: 2px
background-color: rgba(133,153,0,.2)
font-family: "ComputerModernSansSerif",Helvetica,sans-serif
div
@ -399,6 +407,8 @@ body.deck-container
// -- TYPOGRAPHY --
.small
font-size: 0.8em
.smaller
font-size: 0.6em
.sc
+sc
@ -627,10 +637,9 @@ body.deck-container
img,.likeimg
max-width: 80%
border: 2px solid $borderColor
background-color: $imageBackgroundColor
padding: 0.5em
box-shadow: 0 0 10px $base03 inset
border: 2px solid $base02
padding: 0.3em
box-shadow: 0 0 .5em #000 inset
border-radius: 0.5em
text-align: center
a:hover img