temporary clean
This commit is contained in:
Yann Esposito
parent
04bc39ddcd
commit
8a5f22fb97
50 changed files with 137 additions and 126 deletions
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@ -16,7 +16,7 @@
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<link rel="stylesheet" href="extensions/navigation/deck.navigation.css">
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<link rel="stylesheet" href="extensions/status/deck.status.css">
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<link rel="stylesheet" href="extensions/hash/deck.hash.css">
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<link rel="stylesheet" href="extensions/scale/deck.scale.css">
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<!-- <link rel="stylesheet" href="extensions/scale/deck.scale.css"> -->
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||||
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||||
<!-- Transition theme. More available in /themes/transition/ or create your own. -->
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||||
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||||
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@ -134,7 +134,7 @@
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<section class="slide">
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<h2 id="programmer-translation">Programmer Translation</h2>
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<img class="right" src="categories/img/readingcat.jpg" alt="lolcat"/>
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<table style="width:70%">
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<table style="width:60%">
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<tr><th>
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Mathematician
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</th><th>
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@ -374,23 +374,19 @@ such that for each \(f:A→B\):</p>
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<p>Each Monoid \((M,e,⊙): \ob{M}=\{∙\},\hom{M}=M,\circ = ⊙\)</p>
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<p class="yellow">Only one object.</p>
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<p>Examples:</p>
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<ul><li><code>(Integer,0,+)</code>,
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</li><li><code>(Integer,1,*)</code>,
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</li><li><code>(Strings,"",++)</code>,
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</li><li>for each <code>a</code>, <code>([a],[],++)</code>
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<ul><li><code>(Integer,0,+)</code>, <code>(Integer,1,*)</code>,
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</li><li><code>(Strings,"",++)</code>, for each <code>a</code>, <code>([a],[],++)</code>
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</li></ul>
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</section>
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<section class="slide">
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<h2>Degenerated Categories: Preorders</h2>
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<p>each preorder \((P,≤)\):</p>
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<h2>Degenerated Categories: Preorders \((P,≤)\)</h2>
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<ul><li>\(\ob{P}={P}\),
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</li><li>\(\hom{x,y}=\{x≤y\} ⇔ x≤y\),
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</li><li>\((y≤z) \circ (x≤y) = (x≤z) \)
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</li></ul>
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<p><em>At most one morphism between two objects.</em></p>
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<p><em class="yellow">At most one morphism between two objects.</em></p>
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<img src="categories/img/mp/preorder.png" alt="preorder category"/>
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</section>
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@ -403,14 +399,11 @@ such that for each \(f:A→B\):</p>
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<p class="yellow">Only identities</p>
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</section>
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<section class="slide">
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<h2 class="base1">Categorical Property</h2>
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<h2 class="base1">Categorical Properties</h2>
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<p class="base1">Any property which can be expressed in term of category, objects, morphism and composition.</p>
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<ul><li> <em class="yellow">isomorphism</em>: \(f:A→B\) which can be "undone" <em>i.e.</em>
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<br/> \(∃g:B→A\), \(g∘f=id_A\) <span class="and">&</span> \(f∘g=id_B\)
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<br/> in this case, \(A\) <span class="and">&</span> \(B\) are <em class="yellow">isomorphic</em>.
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</li><li> <em class="yellow">Dual</em>: \(\D\) is \(\C\) with reversed morphisms.
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<ul><li> <em class="yellow">Dual</em>: \(\D\) is \(\C\) with reversed morphisms.
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</li><li> <em class="yellow">Initial</em>: \(Z\in\ob{\C}\) s.t. \(∀Y∈\ob{\C}, \#\hom{Z,Y}=1\)
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<br/> Unique ("up to isormophism")
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</li><li> <em class="yellow">Terminal</em>: \(T\in\ob{\C}\) s.t. \(T\) is initial in the dual of \(\C\)
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@ -419,6 +412,11 @@ such that for each \(f:A→B\):</p>
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</li></ul>
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</section>
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<section class="slide">
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<h2 id="isomorph">Isomorph</h2>
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<p><em class="yellow">isomorphism</em>: (f:A→B) which can be "undone" <em>i.e.</em><br />\(∃g:B→A\), \(g∘f=id_A\) <span class="and">&</span> \(f∘g=id_B\)<br />in this case, \(A\) <span class="and">&</span> \(B\) are <em class="yellow">isomorphic</em>.</p>
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<p>In Category Theory, <span class="yellow">=</span> is generally <span class="yellow">≌</span>. For example in commutative diagrams.</p>
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</section>
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<section class="slide">
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<h2>Functor</h2>
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<p> A functor is a mapping between two categories.
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@ -438,17 +436,17 @@ A <em>functor</em> <span class="yellow">\(\F\)</span> from <span class="blue">\(
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<section class="slide">
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<h2>Functor Example (ob → ob)</h2>
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<img src="categories/img/mp/functor.png" alt="Functor"/>
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<img width="70%" src="categories/img/mp/functor.png" alt="Functor"/>
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</section>
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<section class="slide">
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<h2>Functor Example (hom → hom)</h2>
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<img src="categories/img/mp/functor-morphism.png" alt="Functor"/>
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<img width="70%" src="categories/img/mp/functor-morphism.png" alt="Functor"/>
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</section>
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<section class="slide">
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<h2>Functor Example</h2>
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<img src="categories/img/mp/functor-morphism-color.png" alt="Functor"/>
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<img width="70%" src="categories/img/mp/functor-morphism-color.png" alt="Functor"/>
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</section>
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<section class="slide">
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<h2>Endofunctors</h2>
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@ -500,18 +498,14 @@ A <em>functor</em> <span class="yellow">\(\F\)</span> from <span class="blue">\(
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<section class="slide">
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<h2 id="haskell-kinds">Haskell Kinds</h2>
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<p>In Haskell some types can take type variable(s). Typically: <code>[a]</code>.</p>
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<pre><code>data Tree a = Node a [Tree a]
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data CTree a b = CNode a [b]</code></pre>
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<p>Types have <em>kind</em>; The kind is to type what type is to function. Kind are the types for types (so meta).</p>
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<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>
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<pre><code>Int, Char :: *
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[], Maybe, Tree :: * -> *
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CTree :: * -> * -> *
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[Int], Maybe Char, Tree [Int] :: *
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CTree [Int] :: * -> *</code></pre>
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[], Maybe :: * -> *
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(,) :: * -> * -> *
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[Int], Maybe Char, Maybe [Int] :: *</code></pre>
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</section>
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<section class="slide">
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<h2 id="haskell-types">Haskell Types</h2>
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<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>
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<p>Sometimes, the type determine a lot about the function<sup>★</sup>:</p>
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<pre class="haskell"><code>fst :: (a,b) -> a -- Only one choice
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snd :: (a,b) -> b -- Only one choice
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@ -530,12 +524,12 @@ addOne l = map <span class="red">(+1)</span> l
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<section class="slide">
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<h2>Haskell Functor vs \(\Hask\) Functor</h2>
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<p>Functor for Haskell language is a type <code>F :: * -> *</code> which belong to the type class <code>Functor</code>.<br/>
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It must implement <code>fmap :: (a -> b) -> (F a -> F b)</code>.
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<p>A Haskell Functor is a type <code>F :: * -> *</code> which belong to the type class <code>Functor</code> ; thus instantiate
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<code>fmap :: (a -> b) -> (F a -> F b)</code>.
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<p><span style="visibility:hidden"><span class="and">&</span></span> <code>F</code>: \(\ob{\Hask}→\ob{\Hask}\)<br/> <span class="and">&</span> <code>fmap</code>: \(\hom{\Hask}→\hom{\Hask}\)
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<p>The couple <code>(F,fmap)</code> is a functor in the categorical sense for \(\Hask\) if for any <code>x :: F a</code>:</p>
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<p>The couple <code>(F,fmap)</code> is a \(\Hask\) endofunctor if for any <code>x :: F a</code>:</p>
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<ul><li><code>fmap id x = x</code>
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</li><li><code>fmap (f.g) x= (fmap f . fmap g) x</code>
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</li></ul>
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@ -556,22 +550,20 @@ fmap head (Just [1,2,3]) == Just 1</code></pre>
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<section class="slide">
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<h2>Haskell Functors Example: List</h2>
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<pre class="haskell"><code>
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instance Functor ([]) where
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<pre class="haskell"><code>instance Functor ([]) where
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fmap :: (a -> b) -> [a] -> [b]
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fmap = map
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fmap (+1) [1,2,3] == [2,3,4]
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fmap (+1) [] == []
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fmap head [[1,2,3],[4,5,6]] == [1,4]
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</code></pre>
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fmap head [[1,2,3],[4,5,6]] == [1,4]</code></pre>
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</section>
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<section class="slide">
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<h2 id="haskell-functors-for-the-programmer">Haskell Functors for the programmer</h2>
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<p><code>Functor</code> is a type class used for types that can be mapped over.</p>
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<ul>
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<li>Containers: <code>[]</code>, Trees, Map, HashMap...</li>
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<li>Smart containers:
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<li>"Feature Type":
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<ul>
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<li><code>Maybe a</code>: help to handle absence of <code>a</code>.<br />Ex: <code>safeDiv x 0 ⇒ Nothing</code></li>
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<li><code>Either String a</code>: help to handle errors<br />Ex: <code>reportDiv x 0 ⇒ Left "Division by 0!"</code></li>
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@ -633,7 +625,7 @@ Haskell types is fractal:</p>
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</section>
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<section class="slide">
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<h2 id="also-functor-inside-hask">Also Functor inside \(\Hask\)</h2>
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<p>\(\mathtt{[a]}∈\ob{\Hask}\)</code> but is also a complete category. The same can be said for <code>Int</code>.</p>
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<p>\(\mathtt{[a]}∈\ob{\Hask}\)</code> but is also a category. Idem for <code>Int</code>.</p>
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<p><code>length</code> is a Functor from the category <code>[a]</code> to the cateogry <code>Int</code>:</p>
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<ul class="left" style="max-width:40%">
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<li>\(\ob{\mathtt{[a]}}=\{∙\}\)</li>
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@ -653,26 +645,27 @@ Haskell types is fractal:</p>
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</section>
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<section class="slide">
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<h2 id="category-of-hask-endofunctors">Category of \(\Hask\) Endofunctors</h2>
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<img src="categories/img/mp/cat-hask-endofunctor.png" alt="Category of Hask endofunctors" />
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<img width="70%" src="categories/img/mp/cat-hask-endofunctor.png" alt="Category of Hask endofunctors" />
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</section>
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<section class="slide">
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<h2 id="category-of-functors">Category of Functors</h2>
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<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>
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<img src="categories/img/mp/natural-transformation.png" alt="Natural transformation commutative diagram" class="right"/>
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<ul>
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<li>\(\ob{\mathrm{Func}(\C,\D)}\): Functors \(F:\C→\D\)</li>
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<li>\(\hom{\mathrm{Func}(\C,\D)}\): <em>natural transformations</em>
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<ul>
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<li>η familly \(η_X\in\hom{\D}\) for \(X\in\ob{\C}\) s.t.</li>
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<li>ex: between Haskel functors; <code>F a -> G a</code><br />Rearragement functions only.</li>
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</ul></li>
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<li>\(\hom{\mathrm{Func}(\C,\D)}\): <em>natural transformations</em></li>
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<li>∘: Functor composition</li>
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</ul>
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<p>\(\mathrm{Func}(\C,\C)\) is the category of endofunctors of \(\C\).</p>
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</section>
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<section class="slide">
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<h2 id="natural-transformations">Natural Transformations</h2>
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<p>Let \(F\) and \(G\) be two functors from \(\C\) to \(\D\).</p>
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<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>
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<p>ex: between Haskell functors; <code>F a -> G a</code><br />Rearragement functions only.</p>
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</section>
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<section class="slide">
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<h2 id="natural-transformation-examples-14">Natural Transformation Examples (1/4)</h2>
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<pre><code class="haskell">data Tree a = Empty | Node a [Tree a]
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<pre><code class="haskell small">data Tree a = Empty | Node a [Tree a]
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toTree :: [a] -> Tree a
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toTree [] = Empty
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toTree (x:xs) = Node x [toTree xs]</pre>
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@ -687,7 +680,7 @@ toTree (x:xs) = Node x [toTree xs]</pre>
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</section>
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<section class="slide">
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<h2 id="natural-transformation-examples-24">Natural Transformation Examples (2/4)</h2>
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<pre><code class="haskell">data Tree a = Empty | Node a [Tree a]
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<pre><code class="haskell small">data Tree a = Empty | Node a [Tree a]
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toList :: Tree a -> [a]
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toList Empty = []
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toList (Node x l) = [x] ++ concat (map toList l)</pre>
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@ -702,7 +695,7 @@ toList (Node x l) = [x] ++ concat (map toList l)</pre>
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</section>
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<section class="slide">
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<h2 id="natural-transformation-examples-34">Natural Transformation Examples (3/4)</h2>
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<pre><code class="haskell">toMaybe :: [a] -> Maybe a
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<pre><code class="haskell small">toMaybe :: [a] -> Maybe a
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toMaybe [] = Nothing
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toMaybe (x:xs) = Just x</pre>
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</code>
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@ -716,7 +709,7 @@ toMaybe (x:xs) = Just x</pre>
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</section>
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<section class="slide">
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<h2 id="natural-transformation-examples-44">Natural Transformation Examples (4/4)</h2>
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<pre><code class="haskell">mToList :: Maybe a -> [a]
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<pre><code class="haskell small">mToList :: Maybe a -> [a]
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mToList Nothing = []
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mToList Just x = [x]</pre>
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</code>
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@ -997,7 +990,7 @@ depth = cata phi where
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<script src="extensions/goto/deck.goto.js"></script>
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<script src="extensions/status/deck.status.js"></script>
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<script src="extensions/navigation/deck.navigation.js"></script>
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<script src="extensions/scale/deck.scale.js"></script>
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<!-- <script src="extensions/scale/deck.scale.js"></script> -->
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<!-- Initialize the deck. You can put this in an external file if desired. -->
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<script>
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|
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@ -1,6 +1,6 @@
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<h2 id="programmer-translation">Programmer Translation</h2>
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<img class="right" src="categories/img/readingcat.jpg" alt="lolcat"/>
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<table style="width:70%">
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<table style="width:60%">
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<tr><th>
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Mathematician
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</th><th>
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@ -3,7 +3,7 @@ Programmer Translation
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<img class="right" src="categories/img/readingcat.jpg" alt="lolcat"/>
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<table style="width:70%">
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<table style="width:60%">
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<tr><th>Mathematician</th><th>Programmer</th></tr>
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<tr><td>Morphism</td><td>Arrow</td></tr>
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<tr><td>Monoid</td><td>String-like</td></tr>
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@ -4,8 +4,6 @@
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<p>Each Monoid \((M,e,⊙): \ob{M}=\{∙\},\hom{M}=M,\circ = ⊙\)</p>
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<p class="yellow">Only one object.</p>
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<p>Examples:</p>
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<ul><li><code>(Integer,0,+)</code>,
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</li><li><code>(Integer,1,*)</code>,
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</li><li><code>(Strings,"",++)</code>,
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</li><li>for each <code>a</code>, <code>([a],[],++)</code>
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<ul><li><code>(Integer,0,+)</code>, <code>(Integer,1,*)</code>,
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</li><li><code>(Strings,"",++)</code>, for each <code>a</code>, <code>([a],[],++)</code>
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</li></ul>
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|
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@ -1,12 +1,10 @@
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<h2>Degenerated Categories: Preorders</h2>
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<p>each preorder \((P,≤)\):</p>
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<h2>Degenerated Categories: Preorders \((P,≤)\)</h2>
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<ul><li>\(\ob{P}={P}\),
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</li><li>\(\hom{x,y}=\{x≤y\} ⇔ x≤y\),
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</li><li>\((y≤z) \circ (x≤y) = (x≤z) \)
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</li></ul>
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<p><em>At most one morphism between two objects.</em></p>
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<p><em class="yellow">At most one morphism between two objects.</em></p>
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<img src="categories/img/mp/preorder.png" alt="preorder category"/>
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|
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@ -1,11 +1,8 @@
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<h2 class="base1">Categorical Property</h2>
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<h2 class="base1">Categorical Properties</h2>
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<p class="base1">Any property which can be expressed in term of category, objects, morphism and composition.</p>
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<ul><li> <em class="yellow">isomorphism</em>: \(f:A→B\) which can be "undone" <em>i.e.</em>
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<br/> \(∃g:B→A\), \(g∘f=id_A\) & \(f∘g=id_B\)
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<br/> in this case, \(A\) & \(B\) are <em class="yellow">isomorphic</em>.
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</li><li> <em class="yellow">Dual</em>: \(\D\) is \(\C\) with reversed morphisms.
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<ul><li> <em class="yellow">Dual</em>: \(\D\) is \(\C\) with reversed morphisms.
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</li><li> <em class="yellow">Initial</em>: \(Z\in\ob{\C}\) s.t. \(∀Y∈\ob{\C}, \#\hom{Z,Y}=1\)
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<br/> Unique ("up to isormophism")
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</li><li> <em class="yellow">Terminal</em>: \(T\in\ob{\C}\) s.t. \(T\) is initial in the dual of \(\C\)
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3
categories/20_What/192.html
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3
categories/20_What/192.html
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@ -0,0 +1,3 @@
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<h2 id="isomorph">Isomorph</h2>
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<p><em class="yellow">isomorphism</em>: (f:A→B) which can be "undone" <em>i.e.</em><br />\(∃g:B→A\), \(g∘f=id_A\) & \(f∘g=id_B\)<br />in this case, \(A\) & \(B\) are <em class="yellow">isomorphic</em>.</p>
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<p>In Category Theory, <span class="yellow">=</span> is generally <span class="yellow">≌</span>. For example in commutative diagrams.</p>
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9
categories/20_What/192.md
Normal file
9
categories/20_What/192.md
Normal file
|
|
@ -0,0 +1,9 @@
|
|||
Isomorph
|
||||
--------
|
||||
|
||||
<em class="yellow">isomorphism</em>:
|
||||
\(f:A→B\) which can be "undone" _i.e._
|
||||
\\(∃g:B→A\\), \\(g∘f=id\_A\\) & \\(f∘g=id\_B\\)
|
||||
in this case, \\(A\\) & \\(B\\) are <em class="yellow">isomorphic</em>.
|
||||
|
||||
In Category Theory, <span class="yellow">=</span> is generally <span class="yellow">≌</span>. For example in commutative diagrams.
|
||||
|
|
@ -1,3 +1,3 @@
|
|||
<h2>Functor Example (ob → ob)</h2>
|
||||
|
||||
<img src="categories/img/mp/functor.png" alt="Functor"/>
|
||||
<img width="70%" src="categories/img/mp/functor.png" alt="Functor"/>
|
||||
|
|
|
|||
|
|
@ -1,3 +1,3 @@
|
|||
<h2>Functor Example (hom → hom)</h2>
|
||||
|
||||
<img src="categories/img/mp/functor-morphism.png" alt="Functor"/>
|
||||
<img width="70%" src="categories/img/mp/functor-morphism.png" alt="Functor"/>
|
||||
|
|
|
|||
|
|
@ -1,3 +1,3 @@
|
|||
<h2>Functor Example</h2>
|
||||
|
||||
<img src="categories/img/mp/functor-morphism-color.png" alt="Functor"/>
|
||||
<img width="70%" src="categories/img/mp/functor-morphism-color.png" alt="Functor"/>
|
||||
|
|
|
|||
|
|
@ -1,10 +1,7 @@
|
|||
<h2 id="haskell-kinds">Haskell Kinds</h2>
|
||||
<p>In Haskell some types can take type variable(s). Typically: <code>[a]</code>.</p>
|
||||
<pre><code>data Tree a = Node a [Tree a]
|
||||
data CTree a b = CNode a [b]</code></pre>
|
||||
<p>Types have <em>kind</em>; The kind is to type what type is to function. Kind are the types for types (so meta).</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, Tree :: * -> *
|
||||
CTree :: * -> * -> *
|
||||
[Int], Maybe Char, Tree [Int] :: *
|
||||
CTree [Int] :: * -> *</code></pre>
|
||||
[], Maybe :: * -> *
|
||||
(,) :: * -> * -> *
|
||||
[Int], Maybe Char, Maybe [Int] :: *</code></pre>
|
||||
|
|
|
|||
|
|
@ -4,19 +4,13 @@ Haskell Kinds
|
|||
In Haskell some types can take type variable(s).
|
||||
Typically: `[a]`.
|
||||
|
||||
~~~
|
||||
data Tree a = Node a [Tree a]
|
||||
data CTree a b = CNode a [b]
|
||||
~~~
|
||||
|
||||
Types have _kind_;
|
||||
Types have _kinds_;
|
||||
The kind is to type what type is to function.
|
||||
Kind are the types for types (so meta).
|
||||
|
||||
~~~
|
||||
Int, Char :: *
|
||||
[], Maybe, Tree :: * -> *
|
||||
CTree :: * -> * -> *
|
||||
[Int], Maybe Char, Tree [Int] :: *
|
||||
CTree [Int] :: * -> *
|
||||
[], Maybe :: * -> *
|
||||
(,) :: * -> * -> *
|
||||
[Int], Maybe Char, Maybe [Int] :: *
|
||||
~~~
|
||||
|
|
|
|||
|
|
@ -1,5 +1,4 @@
|
|||
<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<sup>★</sup>:</p>
|
||||
<pre class="haskell"><code>fst :: (a,b) -> a -- Only one choice
|
||||
snd :: (a,b) -> b -- Only one choice
|
||||
|
|
|
|||
|
|
@ -1,10 +1,6 @@
|
|||
Haskell Types
|
||||
-------------
|
||||
|
||||
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<sup>★</sup>:
|
||||
|
||||
<pre class="haskell"><code>fst :: (a,b) -> a -- Only one choice
|
||||
|
|
|
|||
|
|
@ -1,11 +1,11 @@
|
|||
<h2>Haskell Functor vs \(\Hask\) Functor</h2>
|
||||
|
||||
<p>Functor for Haskell language is a type <code>F :: * -> *</code> which belong to the type class <code>Functor</code>.<br/>
|
||||
It must implement <code>fmap :: (a -> b) -> (F a -> F b)</code>.
|
||||
<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> <code>F</code>: \(\ob{\Hask}→\ob{\Hask}\)<br/> & <code>fmap</code>: \(\hom{\Hask}→\hom{\Hask}\)
|
||||
|
||||
<p>The couple <code>(F,fmap)</code> is a functor in the categorical sense for \(\Hask\) if for any <code>x :: F a</code>:</p>
|
||||
<p>The couple <code>(F,fmap)</code> is a \(\Hask\) endofunctor 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>
|
||||
|
|
|
|||
|
|
@ -1,11 +1,9 @@
|
|||
<h2>Haskell Functors Example: List</h2>
|
||||
|
||||
<pre class="haskell"><code>
|
||||
instance Functor ([]) where
|
||||
<pre class="haskell"><code>instance Functor ([]) where
|
||||
fmap :: (a -> b) -> [a] -> [b]
|
||||
fmap = map
|
||||
|
||||
fmap (+1) [1,2,3] == [2,3,4]
|
||||
fmap (+1) [] == []
|
||||
fmap head [[1,2,3],[4,5,6]] == [1,4]
|
||||
</code></pre>
|
||||
fmap head [[1,2,3],[4,5,6]] == [1,4]</code></pre>
|
||||
|
|
|
|||
|
|
@ -2,7 +2,7 @@
|
|||
<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>Smart containers:
|
||||
<li>"Feature Type":
|
||||
<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 "Division by 0!"</code></li>
|
||||
|
|
|
|||
|
|
@ -4,7 +4,7 @@ Haskell Functors for the programmer
|
|||
`Functor` is a type class used for types that can be mapped over.
|
||||
|
||||
- Containers: `[]`, Trees, Map, HashMap...
|
||||
- Smart containers:
|
||||
- "Feature Type":
|
||||
- `Maybe a`: help to handle absence of `a`.
|
||||
Ex: `safeDiv x 0 ⇒ Nothing`
|
||||
- `Either String a`: help to handle errors
|
||||
|
|
|
|||
|
|
@ -1,5 +1,5 @@
|
|||
<h2 id="also-functor-inside-hask">Also Functor inside \(\Hask\)</h2>
|
||||
<p>\(\mathtt{[a]}∈\ob{\Hask}\)</code> but is also a complete category. The same can be said for <code>Int</code>.</p>
|
||||
<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>
|
||||
|
|
|
|||
|
|
@ -1,2 +1,2 @@
|
|||
<h2 id="category-of-hask-endofunctors">Category of \(\Hask\) Endofunctors</h2>
|
||||
<img src="categories/img/mp/cat-hask-endofunctor.png" alt="Category of Hask endofunctors" />
|
||||
<img width="70%" src="categories/img/mp/cat-hask-endofunctor.png" alt="Category of Hask endofunctors" />
|
||||
|
|
|
|||
|
|
@ -1,4 +1,4 @@
|
|||
Category of \\(\\Hask\\) Endofunctors
|
||||
------------------------------------
|
||||
|
||||
<img src="categories/img/mp/cat-hask-endofunctor.png" alt="Category of Hask endofunctors" />
|
||||
<img width="70%" src="categories/img/mp/cat-hask-endofunctor.png" alt="Category of Hask endofunctors" />
|
||||
|
|
|
|||
|
|
@ -1,13 +1,8 @@
|
|||
<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>
|
||||
<img src="categories/img/mp/natural-transformation.png" alt="Natural transformation commutative diagram" class="right"/>
|
||||
<ul>
|
||||
<li>\(\ob{\mathrm{Func}(\C,\D)}\): Functors \(F:\C→\D\)</li>
|
||||
<li>\(\hom{\mathrm{Func}(\C,\D)}\): <em>natural transformations</em>
|
||||
<ul>
|
||||
<li>η familly \(η_X\in\hom{\D}\) for \(X\in\ob{\C}\) s.t.</li>
|
||||
<li>ex: between Haskel functors; <code>F a -> G a</code><br />Rearragement functions only.</li>
|
||||
</ul></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>
|
||||
|
|
|
|||
|
|
@ -5,14 +5,8 @@ If \\(\\C\\) is _small_ (\\(\\hom{\\C}\\) is a set).
|
|||
All functors from \\(\\C\\) to some category \\(\\D\\)
|
||||
form the category \\(\\mathrm{Func}(\\C,\\D)\\).
|
||||
|
||||
<img src="categories/img/mp/natural-transformation.png" alt="Natural transformation commutative diagram" class="right"/>
|
||||
|
||||
- \\(\\ob{\\mathrm{Func}(\\C,\\D)}\\): Functors \\(F:\\C→\\D\\)
|
||||
- \\(\\hom{\\mathrm{Func}(\\C,\\D)}\\): _natural transformations_
|
||||
- η familly \\(η\_X\\in\\hom{\\D}\\) for \\(X\\in\\ob{\\C}\\) s.t.
|
||||
- ex: between Haskel functors; `F a -> G a`
|
||||
Rearragement functions only.
|
||||
- ∘: Functor composition
|
||||
|
||||
|
||||
\\(\\mathrm{Func}(\\C,\\C)\\) is the category of endofunctors of \\(\\C\\).
|
||||
|
|
|
|||
4
categories/30_How/100_Functors/140.html
Normal file
4
categories/30_How/100_Functors/140.html
Normal file
|
|
@ -0,0 +1,4 @@
|
|||
<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 -> G a</code><br />Rearragement functions only.</p>
|
||||
10
categories/30_How/100_Functors/140.md
Normal file
10
categories/30_How/100_Functors/140.md
Normal file
|
|
@ -0,0 +1,10 @@
|
|||
Natural Transformations
|
||||
-----------------------
|
||||
|
||||
Let \\(F\\) and \\(G\\) be two functors from \\(\\C\\) to \\(\\D\\).
|
||||
|
||||
<img src="categories/img/mp/natural-transformation.png" alt="Natural transformation commutative diagram" class="right"/>
|
||||
_A natural transformation:_ familly η ; \\(η\_X\\in\\hom{\\D}\\) for \\(X\\in\\ob{\\C}\\) s.t.
|
||||
|
||||
ex: between Haskell functors; `F a -> G a`
|
||||
Rearragement functions only.
|
||||
|
|
@ -1,5 +1,5 @@
|
|||
<h2 id="natural-transformation-examples-14">Natural Transformation Examples (1/4)</h2>
|
||||
<pre><code class="haskell">data Tree a = Empty | Node a [Tree a]
|
||||
<pre><code class="haskell small">data Tree a = Empty | Node a [Tree a]
|
||||
toTree :: [a] -> Tree a
|
||||
toTree [] = Empty
|
||||
toTree (x:xs) = Node x [toTree xs]</pre>
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
Natural Transformation Examples (1/4)
|
||||
------------------------------------
|
||||
|
||||
<pre><code class="haskell">data Tree a = Empty | Node a [Tree a]
|
||||
<pre><code class="haskell small">data Tree a = Empty | Node a [Tree a]
|
||||
toTree :: [a] -> Tree a
|
||||
toTree [] = Empty
|
||||
toTree (x:xs) = Node x [toTree xs]</pre></code>
|
||||
|
|
|
|||
|
|
@ -1,5 +1,5 @@
|
|||
<h2 id="natural-transformation-examples-24">Natural Transformation Examples (2/4)</h2>
|
||||
<pre><code class="haskell">data Tree a = Empty | Node a [Tree a]
|
||||
<pre><code class="haskell small">data Tree a = Empty | Node a [Tree a]
|
||||
toList :: Tree a -> [a]
|
||||
toList Empty = []
|
||||
toList (Node x l) = [x] ++ concat (map toList l)</pre>
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
Natural Transformation Examples (2/4)
|
||||
------------------------------------
|
||||
|
||||
<pre><code class="haskell">data Tree a = Empty | Node a [Tree a]
|
||||
<pre><code class="haskell small">data Tree a = Empty | Node a [Tree a]
|
||||
toList :: Tree a -> [a]
|
||||
toList Empty = []
|
||||
toList (Node x l) = [x] ++ concat (map toList l)</pre></code>
|
||||
|
|
|
|||
|
|
@ -1,5 +1,5 @@
|
|||
<h2 id="natural-transformation-examples-34">Natural Transformation Examples (3/4)</h2>
|
||||
<pre><code class="haskell">toMaybe :: [a] -> Maybe a
|
||||
<pre><code class="haskell small">toMaybe :: [a] -> Maybe a
|
||||
toMaybe [] = Nothing
|
||||
toMaybe (x:xs) = Just x</pre>
|
||||
</code>
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
Natural Transformation Examples (3/4)
|
||||
-------------------------------------
|
||||
|
||||
<pre><code class="haskell">toMaybe :: [a] -> Maybe a
|
||||
<pre><code class="haskell small">toMaybe :: [a] -> Maybe a
|
||||
toMaybe [] = Nothing
|
||||
toMaybe (x:xs) = Just x</pre></code>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,5 +1,5 @@
|
|||
<h2 id="natural-transformation-examples-44">Natural Transformation Examples (4/4)</h2>
|
||||
<pre><code class="haskell">mToList :: Maybe a -> [a]
|
||||
<pre><code class="haskell small">mToList :: Maybe a -> [a]
|
||||
mToList Nothing = []
|
||||
mToList Just x = [x]</pre>
|
||||
</code>
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
Natural Transformation Examples (4/4)
|
||||
-------------------------------------
|
||||
|
||||
<pre><code class="haskell">mToList :: Maybe a -> [a]
|
||||
<pre><code class="haskell small">mToList :: Maybe a -> [a]
|
||||
mToList Nothing = []
|
||||
mToList Just x = [x]</pre></code>
|
||||
|
||||
|
|
|
|||
|
|
@ -16,7 +16,7 @@
|
|||
<link rel="stylesheet" href="extensions/navigation/deck.navigation.css">
|
||||
<link rel="stylesheet" href="extensions/status/deck.status.css">
|
||||
<link rel="stylesheet" href="extensions/hash/deck.hash.css">
|
||||
<link rel="stylesheet" href="extensions/scale/deck.scale.css">
|
||||
<!-- <link rel="stylesheet" href="extensions/scale/deck.scale.css"> -->
|
||||
|
||||
<!-- Transition theme. More available in /themes/transition/ or create your own. -->
|
||||
|
||||
|
|
|
|||
|
|
@ -39,7 +39,7 @@
|
|||
<script src="extensions/goto/deck.goto.js"></script>
|
||||
<script src="extensions/status/deck.status.js"></script>
|
||||
<script src="extensions/navigation/deck.navigation.js"></script>
|
||||
<script src="extensions/scale/deck.scale.js"></script>
|
||||
<!-- <script src="extensions/scale/deck.scale.js"></script> -->
|
||||
|
||||
<!-- Initialize the deck. You can put this in an external file if desired. -->
|
||||
<script>
|
||||
|
|
|
|||
|
|
@ -915,8 +915,20 @@ body.deck-container {
|
|||
.deck-container {
|
||||
font-size: 3.5em; } }
|
||||
@media only screen and (max-width: 1380px) {
|
||||
body {
|
||||
width: 760px !important;
|
||||
min-width: 760px !important;
|
||||
max-width: 760px !important;
|
||||
height: 500px !important;
|
||||
min-height: 500px !important;
|
||||
max-height: 500px !important;
|
||||
padding: 0 20px !important;
|
||||
margin: 0 auto;
|
||||
box-shadow: 0 0 3px white;
|
||||
height: 50%; }
|
||||
|
||||
.deck-container {
|
||||
font-size: 2.5em; } }
|
||||
font-size: 24px; } }
|
||||
@media only screen and (max-width: 640px) {
|
||||
.deck-container {
|
||||
font-size: 1.5em; } }
|
||||
|
|
@ -930,11 +942,13 @@ body.deck-container {
|
|||
body.deck-container {
|
||||
overflow-y: hidden; }
|
||||
|
||||
html {
|
||||
box-shadow: 0 0 50px black inset; }
|
||||
|
||||
.deck-container {
|
||||
color: #839496;
|
||||
background-color: #002b36;
|
||||
text-shadow: 0 1px 5px #001122;
|
||||
box-shadow: 0 0 50px black inset; }
|
||||
text-shadow: 0 1px 5px #001122; }
|
||||
.deck-container h1 {
|
||||
font-size: 3em;
|
||||
line-height: 1.2em;
|
||||
|
|
|
|||
|
|
@ -1157,8 +1157,19 @@ body.deck-container
|
|||
.deck-container
|
||||
font-size: 3.5em
|
||||
@media only screen and (max-width: 1380px)
|
||||
body
|
||||
width: 760px !important
|
||||
min-width: 760px !important
|
||||
max-width: 760px !important
|
||||
height: 500px !important
|
||||
min-height: 500px !important
|
||||
max-height: 500px !important
|
||||
padding: 0 20px !important
|
||||
margin: 0 auto
|
||||
box-shadow: 0 0 3px #FFF
|
||||
height: 50%
|
||||
.deck-container
|
||||
font-size: 2.5em
|
||||
font-size: 24px
|
||||
@media only screen and (max-width: 640px)
|
||||
.deck-container
|
||||
font-size: 1.5em
|
||||
|
|
@ -1173,11 +1184,12 @@ body.deck-container
|
|||
body.deck-container
|
||||
overflow-y: hidden
|
||||
|
||||
html
|
||||
box-shadow: 0 0 50px #000 inset
|
||||
.deck-container
|
||||
color: $base0
|
||||
background-color: $base03
|
||||
text-shadow: 0 1px 5px #012
|
||||
box-shadow: 0 0 50px #000 inset
|
||||
h1
|
||||
font-size: 3em
|
||||
line-height: 1.2em
|
||||
|
|
|
|||
Loading…
Reference in a new issue