yogsototh/ycategories-presentation
1
0
Fork 0
Code Issues Pull requests Projects Releases Packages Wiki Activity Actions

some fixes

Browse source
This commit is contained in:
Yann Esposito 2012-12-07 12:17:02 +01:00
parent dceac9e1e4
commit 6e08bf59be
14 changed files with 32 additions and 83 deletions
Show stats Download patch file Download diff file Expand all files Collapse all files

29
categories.html
View file

@ -630,7 +630,7 @@ Haskell types is fractal:</p>
</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>
<p><code>length</code> can be seen as a Functor from the category <code>[a]</code> to the cateogry <code>Int</code>:</p>
<ul class="left">
<li>\(\ob{\mathtt{[a]}}=\{∙\}\)</li>
<li>\(\hom{\mathtt{[a]}}=\mathtt{[a]}\)</li>
@ -656,14 +656,13 @@ Haskell types is fractal:</p>
<li>\(\hom{\E_\C}\): natural transformations
<ul>
<li>η familly \(η_X\in\hom{\C}\) for \(X\in\ob{\C}\) s.t.</li>
<li>for Haskell functors: <code>F a -&gt; G a</code> are the natural transformations.<br />List to Trees, Tree to List, Tree to Maybe...<br />Rearragement functions only.</li>
<li>for Haskell functors: <code>F a -&gt; G a</code>.<br />Rearragement functions only.</li>
</ul></li>
</ul>
</section>
<section class="slide">
<h2 id="natural-transformation-examples">Natural Transformation Examples</h2>
<h2 id="natural-transformation-examples-13">Natural Transformation Examples (1/3)</h2>
<pre><code class="haskell">data Tree a = Empty | Node a [Tree a]
deriving (Show)
toTree :: [a] -> Tree a
toTree [] = Empty
toTree (x:xs) = Node x [toTree xs]</pre>
@ -677,9 +676,8 @@ toTree (x:xs) = Node x [toTree xs]</pre>
</section>
<section class="slide">
<h2 id="natural-transformation-examples">Natural Transformation Examples</h2>
<h2 id="natural-transformation-examples-23">Natural Transformation Examples (2/3)</h2>
<pre><code class="haskell">data Tree a = Empty | Node a [Tree a]
deriving (Show)
toList :: Tree a -> [a]
toList Empty = []
toList (Node x l) = [x] ++ concat (map toList l)</pre>
@ -687,13 +685,13 @@ toList (Node x l) = [x] ++ concat (map toList l)</pre>
<p><code>toList</code> is a natural transformation. It is also a morphism from <code>Tree</code> to <code>[]</code> in the Category of \(\) endofunctors.</p>
<img style="float:left;width:50%" src="categories/img/mp/nattrans-tree-list.png" alt="natural transformation commutative diagram"/>
<figure style="float:right;width:40%">
<img src="categories/img/mp/tree-list-endofunctor-morphism.png" alt="natural transformation commutative diagram"/> <figcaption><code>toList.toTree=id</code> <span class="and">&amp;</span> <code>toTree.toList=id</code>.<br/> Therefore <code>[]</code> <span class="and">&amp;</span> <code>Tree</code> are <span class="yellow">isomorph</span>. </figcaption>
<img src="categories/img/mp/tree-list-endofunctor-morphism.png" alt="natural transformation commutative diagram"/> <figcaption><code>toList . toTree = id</code> <span class="and">&amp;</span><br/> <code>toTree . toList = id</code> <span style="visibility:hidden"><span class="and">&amp;</span></span><br/> therefore <code>[]</code> <span class="and">&amp;</span> <code>Tree</code> are <span class="yellow">isomorph</span>. </figcaption>
</figure>
</section>
<section class="slide">
<h2 id="another-nat.-trans.-example">Another Nat. Trans. Example</h2>
<h2 id="natural-transformation-examples-33">Natural Transformation Examples (3/3)</h2>
<pre><code class="haskell">
toMaybe :: [a] -> Maybe a ; mToList :: Maybe a -> [a]
toMaybe [] = Nothing ; mToList Nothing = []
@ -707,21 +705,6 @@ toMaybe (x:xs) = Just x ; mToList Just x = [x]
</figure>
</section>
<section class="slide">
<h2 id="another-nat.-trans.-example">Another Nat. Trans. Example</h2>
<pre><code class="haskell">
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 \(\) endofunctors.</p>
<img style="float:left;width:50%" src="categories/img/mp/nattrans-maybe-list.png" alt="natural transformation commutative diagram"/>
<figure style="float:right;width:40%">
<img src="categories/img/mp/maybe-list-endofunctor-morphism.png" alt="natural transformation commutative diagram"/> <figcaption>There is no isomorphism.<br/> Hint: List with more than one element. </figcaption>
</figure>
</section>
<section class="slide">
<h2 id="monads">Monads</h2>

2
categories/30_How/150_Also_Functor_inside_Hask.html
View file

@ -1,5 +1,5 @@
<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>
<p><code>length</code> can be seen as a Functor from the category <code>[a]</code> to the cateogry <code>Int</code>:</p>
<ul class="left">
<li>\(\ob{\mathtt{[a]}}=\{∙\}\)</li>
<li>\(\hom{\mathtt{[a]}}=\mathtt{[a]}\)</li>

2
categories/30_How/160_Category_of_Endofunctors.html
View file

@ -6,6 +6,6 @@
<li>\(\hom{\E_\C}\): natural transformations
<ul>
<li>η familly \(η_X\in\hom{\C}\) for \(X\in\ob{\C}\) s.t.</li>
<li>for Haskell functors: <code>F a -&gt; G a</code> are the natural transformations.<br />List to Trees, Tree to List, Tree to Maybe...<br />Rearragement functions only.</li>
<li>for Haskell functors: <code>F a -&gt; G a</code>.<br />Rearragement functions only.</li>
</ul></li>
</ul>

3
categories/30_How/160_Category_of_Endofunctors.md
View file

@ -11,6 +11,5 @@ of endofunctors of \\(\\C\\).
- \\(\\ob{\\E\_\\C}\\): endofunctors of \\(\\C\\) ; \\(F:\\C→\\C\\)
- \\(\\hom{\\E\_\\C}\\): natural transformations
- η familly \\(η\_X\\in\\hom{\\C}\\) for \\(X\\in\\ob{\\C}\\) s.t.
- for Haskell functors: `F a -> G a` are the natural transformations.
List to Trees, Tree to List, Tree to Maybe...
- for Haskell functors: `F a -> G a`.
Rearragement functions only.

3
categories/30_How/170_Natural_Transformation_Examples.html
View file

@ -1,6 +1,5 @@
<h2 id="natural-transformation-examples">Natural Transformation Examples</h2>
<h2 id="natural-transformation-examples-13">Natural Transformation Examples (1/3)</h2>
<pre><code class="haskell">data Tree a = Empty | Node a [Tree a]
deriving (Show)
toTree :: [a] -> Tree a
toTree [] = Empty
toTree (x:xs) = Node x [toTree xs]</pre>

5
categories/30_How/170_Natural_Transformation_Examples.md
View file

@ -1,8 +1,7 @@
Natural Transformation Examples
-------------------------------
Natural Transformation Examples (1/3)
------------------------------------
<pre><code class="haskell">data Tree a = Empty | Node a [Tree a]
deriving (Show)
toTree :: [a] -> Tree a
toTree [] = Empty
toTree (x:xs) = Node x [toTree xs]</pre></code>

5
categories/30_How/180_Natural_Transformation_Examples.html
View file

@ -1,6 +1,5 @@
<h2 id="natural-transformation-examples">Natural Transformation Examples</h2>
<h2 id="natural-transformation-examples-23">Natural Transformation Examples (2/3)</h2>
<pre><code class="haskell">data Tree a = Empty | Node a [Tree a]
deriving (Show)
toList :: Tree a -> [a]
toList Empty = []
toList (Node x l) = [x] ++ concat (map toList l)</pre>
@ -8,7 +7,7 @@ toList (Node x l) = [x] ++ concat (map toList l)</pre>
<p><code>toList</code> is a natural transformation. It is also a morphism from <code>Tree</code> to <code>[]</code> in the Category of \(\) endofunctors.</p>
<img style="float:left;width:50%" src="categories/img/mp/nattrans-tree-list.png" alt="natural transformation commutative diagram"/>
<figure style="float:right;width:40%">
<img src="categories/img/mp/tree-list-endofunctor-morphism.png" alt="natural transformation commutative diagram"/> <figcaption><code>toList.toTree=id</code> &amp; <code>toTree.toList=id</code>.<br/> Therefore <code>[]</code> &amp; <code>Tree</code> are <span class="yellow">isomorph</span>. </figcaption>
<img src="categories/img/mp/tree-list-endofunctor-morphism.png" alt="natural transformation commutative diagram"/> <figcaption><code>toList . toTree = id</code> &amp;<br/> <code>toTree . toList = id</code> <span style="visibility:hidden">&amp;</span><br/> therefore <code>[]</code> &amp; <code>Tree</code> are <span class="yellow">isomorph</span>. </figcaption>
</figure>

9
categories/30_How/180_Natural_Transformation_Examples.md
View file

@ -1,8 +1,7 @@
Natural Transformation Examples
-------------------------------
Natural Transformation Examples (2/3)
------------------------------------
<pre><code class="haskell">data Tree a = Empty | Node a [Tree a]
deriving (Show)
toList :: Tree a -> [a]
toList Empty = []
toList (Node x l) = [x] ++ concat (map toList l)</pre></code>
@ -14,7 +13,7 @@ It is also a morphism from `Tree` to `[]` in the Category of \\(\Hask\\) endofun
<img style="float:left;width:50%" src="categories/img/mp/nattrans-tree-list.png" alt="natural transformation commutative diagram"/>
<figure style="float:right;width:40%">
<img src="categories/img/mp/tree-list-endofunctor-morphism.png" alt="natural transformation commutative diagram"/>
<figcaption><code>toList.toTree=id</code> &amp; <code>toTree.toList=id</code>.<br/>
Therefore <code>[]</code> &amp; <code>Tree</code> are <span class="yellow">isomorph</span>.
<figcaption><code>toList . toTree = id</code> &amp;<br/> <code>toTree . toList = id</code> <span style="visibility:hidden">&amp;</span><br/>
therefore <code>[]</code> &amp; <code>Tree</code> are <span class="yellow">isomorph</span>.
</figcaption>
</figure>

2
categories/30_How/190_Another_Nat_Trans_Example.html
View file

@ -1,4 +1,4 @@
<h2 id="another-nat.-trans.-example">Another Nat. Trans. Example</h2>
<h2 id="natural-transformation-examples-33">Natural Transformation Examples (3/3)</h2>
<pre><code class="haskell">
toMaybe :: [a] -> Maybe a ; mToList :: Maybe a -> [a]
toMaybe [] = Nothing ; mToList Nothing = []

4
categories/30_How/190_Another_Nat_Trans_Example.md
View file

@ -1,5 +1,5 @@
Another Nat. Trans. Example
---------------------------
Natural Transformation Examples (3/3)
-------------------------------------
<pre><code class="haskell">
toMaybe :: [a] -> Maybe a ; mToList :: Maybe a -> [a]

13
categories/30_How/200_Another_Nat_Trans_Example.html
View file

@ -1,13 +0,0 @@
<h2 id="another-nat.-trans.-example">Another Nat. Trans. Example</h2>
<pre><code class="haskell">
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 \(\) endofunctors.</p>
<img style="float:left;width:50%" src="categories/img/mp/nattrans-maybe-list.png" alt="natural transformation commutative diagram"/>
<figure style="float:right;width:40%">
<img src="categories/img/mp/maybe-list-endofunctor-morphism.png" alt="natural transformation commutative diagram"/> <figcaption>There is no isomorphism.<br/> Hint: List with more than one element. </figcaption>
</figure>

19
categories/30_How/200_Another_Nat_Trans_Example.md
View file

@ -1,19 +0,0 @@
Another Nat. Trans. Example
---------------------------
<pre><code class="haskell">
toMaybe :: [a] -> Maybe a
toMaybe [] = Nothing
toMaybe (x:xs) = Just x</pre></code>
`toMaybe` is a natural transformation.
It is also a morphism from `[]` to `Maybe` in the Category of \\(\Hask\\) endofunctors.
<img style="float:left;width:50%" src="categories/img/mp/nattrans-maybe-list.png" alt="natural transformation commutative diagram"/>
<figure style="float:right;width:40%">
<img src="categories/img/mp/maybe-list-endofunctor-morphism.png" alt="natural transformation commutative diagram"/>
<figcaption>There is no isomorphism.<br/>
Hint: List with more than one element.
</figcaption>
</figure>

18
themes/style/y/main.css
View file

@ -838,14 +838,16 @@ body.deck-container {
background-color: #073642; }
.deck-container table tr:nth-child(even) {
background-color: #002b36; }
.deck-container p pre code, .deck-container ul li pre code, .deck-container ol li pre code {
background: none;
border: none;
padding: 0; }
.deck-container p code, .deck-container ul li code, .deck-container ol li code {
background: #053440;
border: solid 1px #586e75;
padding: 2px; }
.deck-container p, .deck-container ul li, .deck-container ol li {
line-height: 1.5em; }
.deck-container p pre code, .deck-container ul li pre code, .deck-container ol li pre code {
background: none;
border: none;
padding: 0; }
.deck-container p code, .deck-container ul li code, .deck-container ol li code {
background: #053440;
border: solid 1px #586e75;
padding: 2px; }
.deck-container ul.sameline {
list-style: none; }
.deck-container ul.sameline li {

1
themes/style/y/main.sass
View file

@ -1075,6 +1075,7 @@ body.deck-container
background-color: $mainBackgroundColor
p, ul li, ol li
line-height: 1.5em
pre code
background: none
border: none