added natural transformation explanation

categories.html

@@ -659,6 +659,69 @@ Haskell types is fractal:</p>
 <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>
 </ul></li>
 </ul>
+</section>
+<section class="slide">
+<h2 id="natural-transformation-examples">Natural Transformation Examples</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>
+</code>
+<p><code>toTree</code> is a natural transformation. It is also a morphism from <code>[]</code> to <code>Tree</code> in the Category of \(\) endofunctors.</p>
+<img style="float:left;width:50%" src="categories/img/mp/nattrans-list-tree.png" alt="natural transformation commutative diagram"/>
+<figure style="float:right;width:40%">
+<img src="categories/img/mp/list-tree-endofunctor-morphism.png" alt="natural transformation commutative diagram"/>
+</figure>
+
+
+</section>
+<section class="slide">
+<h2 id="natural-transformation-examples">Natural Transformation Examples</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>
+</code>
+<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>
+</figure>
+
+
+</section>
+<section class="slide">
+<h2 id="another-nat.-trans.-example">Another Nat. Trans. Example</h2>
+<pre><code class="haskell">
+toMaybe :: [a] -> Maybe a   ;   mToList :: Maybe a -> [a]
+toMaybe [] = Nothing        ;   mToList Nothing = []
+toMaybe (x:xs) = Just x     ;   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 \(\) endofunctors.</p>
+<img style="float:left;width:50%" src="categories/img/mp/nattrans-list-maybe.png" alt="natural transformation commutative diagram"/>
+<figure style="float:right;width:40%">
+<img src="categories/img/mp/list-maybe-endofunctor-morphism.png" alt="natural transformation commutative diagram"/> <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="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>

categories/30_How/170_Natural_Transformation_Examples.html

@@ -0,0 +1,14 @@
+<h2 id="natural-transformation-examples">Natural Transformation Examples</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>
+</code>
+<p><code>toTree</code> is a natural transformation. It is also a morphism from <code>[]</code> to <code>Tree</code> in the Category of \(\) endofunctors.</p>
+<img style="float:left;width:50%" src="categories/img/mp/nattrans-list-tree.png" alt="natural transformation commutative diagram"/>
+<figure style="float:right;width:40%">
+<img src="categories/img/mp/list-tree-endofunctor-morphism.png" alt="natural transformation commutative diagram"/>
+</figure>
+
+

categories/30_How/170_Natural_Transformation_Examples.md

@@ -0,0 +1,17 @@
+Natural Transformation Examples
+-------------------------------
+
+<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>
+
+
+`toTree` is a natural transformation.
+It is also a morphism from `[]` to `Tree` in the Category of \\(\Hask\\) endofunctors.
+
+<img style="float:left;width:50%" src="categories/img/mp/nattrans-list-tree.png" alt="natural transformation commutative diagram"/>
+<figure style="float:right;width:40%">
+<img src="categories/img/mp/list-tree-endofunctor-morphism.png" alt="natural transformation commutative diagram"/>
+</figure>

categories/30_How/180_Natural_Transformation_Examples.html

@@ -0,0 +1,14 @@
+<h2 id="natural-transformation-examples">Natural Transformation Examples</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>
+</code>
+<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>
+</figure>
+
+

categories/30_How/180_Natural_Transformation_Examples.md

@@ -0,0 +1,20 @@
+Natural Transformation Examples
+-------------------------------
+
+<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>
+
+
+`toList` is a natural transformation.
+It is also a morphism from `Tree` to `[]` in the Category of \\(\Hask\\) endofunctors.
+
+<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>
+</figure>

categories/30_How/190_Another_Nat_Trans_Example.html

@@ -0,0 +1,14 @@
+<h2 id="another-nat.-trans.-example">Another Nat. Trans. Example</h2>
+<pre><code class="haskell">
+toMaybe :: [a] -> Maybe a   ;   mToList :: Maybe a -> [a]
+toMaybe [] = Nothing        ;   mToList Nothing = []
+toMaybe (x:xs) = Just x     ;   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 \(\) endofunctors.</p>
+<img style="float:left;width:50%" src="categories/img/mp/nattrans-list-maybe.png" alt="natural transformation commutative diagram"/>
+<figure style="float:right;width:40%">
+<img src="categories/img/mp/list-maybe-endofunctor-morphism.png" alt="natural transformation commutative diagram"/> <figcaption>There is <span class="red">no isomorphism</span>.<br/> Hint: <code>Bool</code> lists longer than 1. </figcaption>
+</figure>
+
+

categories/30_How/190_Another_Nat_Trans_Example.md

@@ -0,0 +1,20 @@
+Another Nat. Trans. Example
+---------------------------
+
+<pre><code class="haskell">
+toMaybe :: [a] -> Maybe a   ;   mToList :: Maybe a -> [a]
+toMaybe [] = Nothing        ;   mToList Nothing = []
+toMaybe (x:xs) = Just x     ;   mToList Just x  = [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-list-maybe.png" alt="natural transformation commutative diagram"/>
+<figure style="float:right;width:40%">
+<img src="categories/img/mp/list-maybe-endofunctor-morphism.png" alt="natural transformation commutative diagram"/>
+<figcaption>There is <span class="red">no isomorphism</span>.<br/>
+Hint: <code>Bool</code> lists longer than 1.
+</figcaption>
+</figure>

categories/30_How/200_Another_Nat_Trans_Example.html

@@ -0,0 +1,13 @@
+<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>
+
+

categories/30_How/200_Another_Nat_Trans_Example.md

@@ -0,0 +1,19 @@
+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>

categories/img/mp/list-maybe-endofunctor-morphism.mp

@@ -0,0 +1,8 @@
+pair tree,list;
+resize(.8cm);
+list=origin;
+tree=list shifted (gu,0);
+drawState(list,"\mathtt{[]}");
+drawState(tree,"\mathtt{Maybe}");
+drawEdgeAngle(list,tree,"\mathtt{toMaybe}",30);
+drawEdgeAngle(tree,list,"\mathtt{mToList}",30);

categories/img/mp/list-tree-endofunctor-morphism.mp

@@ -0,0 +1,7 @@
+pair tree,list;
+resize(.8cm);
+list=origin;
+tree=list shifted (gu,0);
+drawState(list,"\mathtt{[]}");
+drawState(tree,"\mathtt{Tree}");
+drawEdgeAngle(list,tree,"\mathtt{toTree}",30);

categories/img/mp/nattrans-list-maybe.mp

@@ -0,0 +1,28 @@
+z0=(0,0);
+z1=(1.5gu,0);
+z2=(0,-gu);
+z3=(1.5gu,-gu);
+path ac,ab;
+
+label(btex $\mathtt{[a]}$ etex,z0);
+label(btex $\mathtt{[b]}$ etex,z1);
+label(btex $\mathtt{Maybe\ a}$ etex,z2 shifted (-u,0));
+label(btex $\mathtt{Maybe\ b}$ etex,z3 shifted (u,0));
+
+drawEdge(z0,z1,"\mathtt{fmap_{[]}\ f}");
+drawEdge(z2,z3,"\mathtt{fmap_{Maybe}\ f}");
+ab:=edge(z0,z2);
+drawarrow ab;
+label.lft(btex $\mathtt{toMaybe}$ etex,midpoint(ab));
+
+ac:=edge(z1,z3);
+drawarrow ac;
+label.rt(btex $\mathtt{toMaybe}$ etex,midpoint(ac));
+
+path abb,acb;
+abb:=edge(z2 shifted (.2u,0),z0 shifted (.2u,0));
+acb:=edge(z3 shifted (-.2u,0),z1 shifted (-.2u,0));
+drawarrow abb;
+drawarrow acb;
+label.rt(btex $\mathtt{mToList}$ etex,midpoint(abb));
+label.lft(btex $\mathtt{mToList}$ etex,midpoint(acb));

categories/img/mp/nattrans-list-tree.mp

@@ -0,0 +1,21 @@
+z0=(0,0);
+z1=(1.5gu,0);
+z2=(0,-gu);
+z3=(1.5gu,-gu);
+path ac,ab;
+
+label(btex $\mathtt{[a]}$ etex,z0);
+label(btex $\mathtt{[b]}$ etex,z1);
+label(btex $\mathtt{Tree\ a}$ etex,z2 shifted (-u,0));
+label(btex $\mathtt{Tree\ b}$ etex,z3 shifted (u,0));
+
+drawEdge(z0,z1,"\mathtt{fmap_{[]}\ f}");
+drawEdge(z2,z3,"\mathtt{fmap_{Tree}\ f}");
+ab:=edge(z0,z2);
+drawarrow ab;
+label.lft(btex $\mathtt{toTree}$ etex,midpoint(ab));
+
+ac:=edge(z1,z3);
+drawarrow ac;
+label.rt(btex $\mathtt{toTree}$ etex,midpoint(ac));
+

categories/img/mp/nattrans-tree-list.mp

@@ -0,0 +1,29 @@
+z0=(0,0);
+z1=(1.5gu,0);
+z2=(0,-gu);
+z3=(1.5gu,-gu);
+path ac,ab;
+
+label(btex $\mathtt{[a]}$ etex,z0);
+label(btex $\mathtt{[b]}$ etex,z1);
+label(btex $\mathtt{Tree\ a}$ etex,z2 shifted (-u,0));
+label(btex $\mathtt{Tree\ b}$ etex,z3 shifted (u,0));
+
+drawEdge(z0,z1,"\mathtt{fmap_{[]}\ f}");
+drawEdge(z2,z3,"\mathtt{fmap_{Tree}\ f}");
+ab:=edge(z0,z2);
+drawarrow ab;
+label.lft(btex $\mathtt{toTree}$ etex,midpoint(ab));
+
+ac:=edge(z1,z3);
+drawarrow ac;
+label.rt(btex $\mathtt{toTree}$ etex,midpoint(ac));
+
+drawoptions(withcolor yellow);
+path abb,acb;
+abb:=edge(z2 shifted (.2u,0),z0 shifted (.2u,0));
+acb:=edge(z3 shifted (-.2u,0),z1 shifted (-.2u,0));
+drawarrow abb;
+drawarrow acb;
+label.rt(btex $\mathtt{toList}$ etex,midpoint(abb));
+label.lft(btex $\mathtt{toList}$ etex,midpoint(acb));

categories/img/mp/tree-list-endofunctor-morphism.mp

@@ -0,0 +1,9 @@
+pair tree,list;
+resize(.8cm);
+list=origin;
+tree=list shifted (gu,0);
+drawState(list,"\mathtt{[]}");
+drawState(tree,"\mathtt{Tree}");
+drawEdgeAngle(list,tree,"\mathtt{toTree}",30);
+drawoptions(withcolor yellow);
+drawEdgeAngle(tree,list,"\mathtt{toList}",30);