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Clean xhtml, convert to html5

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Thierry 4 years ago
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index.html View File

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<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="utf-8" />
<meta charset="utf-8">
<title>Categories Bartosz Milewski Youtube notes</title>
<link rel="shortcut icon" href="favicon.png" type="image/x-icon">
<link rel="copyright" href="http://www.gnu.org/copyleft/gpl.html"/>


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younotes1/1.1-motivation-philosophy.html View File

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<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="utf-8" />
<meta charset="utf-8">
<title>1.1: Motivation and Philosophy | Categories Bartosz Milewski Youtube notes</title>
<link rel="shortcut icon" href="favicon.png" type="image/x-icon">
<link rel="copyright" href="http://www.gnu.org/copyleft/gpl.html"/>


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<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="utf-8" />
<meta charset="utf-8">
<title>1.2: What is a category | Categories Bartosz Milewski Youtube notes</title>
<link rel="shortcut icon" href="favicon.png" type="image/x-icon">
<link rel="copyright" href="http://www.gnu.org/copyleft/gpl.html"/>


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younotes1/10.1-monads.html View File

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<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="utf-8" />
<meta charset="utf-8">
<title>10.1: Monads | Categories Bartosz Milewski Youtube notes</title>
<link rel="shortcut icon" href="favicon.png" type="image/x-icon">
<link rel="copyright" href="http://www.gnu.org/copyleft/gpl.html"/>


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younotes1/10.2-monoid-in-the-category-of-endofunctors.html View File

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<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="utf-8" />
<meta charset="utf-8">
<title>10.2: Monoid in the category of endofunctors | Categories Bartosz Milewski Youtube notes</title>
<link rel="shortcut icon" href="favicon.png" type="image/x-icon">
<link rel="copyright" href="http://www.gnu.org/copyleft/gpl.html"/>


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younotes1/2.1-functions-epimorphisms.html View File

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<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="utf-8" />
<meta charset="utf-8">
<title>2.1: Functions, epimorphisms | Categories Bartosz Milewski Youtube notes</title>
<link rel="shortcut icon" href="favicon.png" type="image/x-icon">
<link rel="copyright" href="http://www.gnu.org/copyleft/gpl.html"/>


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younotes1/2.2-monomorphisms-simple-types.html View File

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<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="utf-8" />
<meta charset="utf-8">
<title>2.2: Monomorphisms, simple types | Categories Bartosz Milewski Youtube notes</title>
<link rel="shortcut icon" href="favicon.png" type="image/x-icon">
<link rel="copyright" href="http://www.gnu.org/copyleft/gpl.html"/>


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younotes1/3.1-examples-orders-monoids.html View File

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<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="utf-8" />
<meta charset="utf-8">
<title>3.1: Examples of categories, orders, monoids | Categories Bartosz Milewski Youtube notes</title>
<link rel="shortcut icon" href="favicon.png" type="image/x-icon">
<link rel="copyright" href="http://www.gnu.org/copyleft/gpl.html"/>


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younotes1/3.2-kleisli-category.html View File

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<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="utf-8" />
<meta charset="utf-8">
<title>3.2: Kleisli category | Categories Bartosz Milewski Youtube notes</title>
<link rel="shortcut icon" href="favicon.png" type="image/x-icon">
<link rel="copyright" href="http://www.gnu.org/copyleft/gpl.html"/>
@@ -40,8 +40,8 @@ This deals with an interesting case to study.
<br>This is not a total order because there may exist disjoint sets or sets that partially overlap.
<br>We can have a diamond situation :
<div class="flex-wrap">
<img class="margin border" src="img/diamond1.jpg" alt="" />
<img class="margin border" src="img/diamond2.jpg" alt="" />
<img class="margin border" src="img/diamond1.jpg" alt="">
<img class="margin border" src="img/diamond2.jpg" alt="">
</div>

We have a category based on sets where the arrows are not functions, but relations. Some versions of this category is used in modeling topology ; open sets have this property of inclusion.


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<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="utf-8" />
<meta charset="utf-8">
<title>4.1: Terminal and initial objects | Categories Bartosz Milewski Youtube notes</title>
<link rel="shortcut icon" href="favicon.png" type="image/x-icon">
<link rel="copyright" href="http://www.gnu.org/copyleft/gpl.html"/>


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<!DOCTYPE html>
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<title>4.2: Products | Categories Bartosz Milewski Youtube notes</title>
<link rel="shortcut icon" href="favicon.png" type="image/x-icon">
<link rel="copyright" href="http://www.gnu.org/copyleft/gpl.html"/>
@@ -119,7 +119,7 @@ We say that <code>c</code> is better than <code>c'</code> if there is a unique m
<br>The definition of product is :

<div class="flex-wrap">
<img class="margin border" src="img/product4.jpg" alt="Cartesian product 4" />
<img class="margin border" src="img/product4.jpg" alt="Cartesian product 4">
<div class="inline-block border margin padding">
A categorical product of two objects <code>a</code> and <code>b</code> is a third object <code>c</code> with two projections


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<!DOCTYPE html>
<html lang="en">
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<meta charset="utf-8" />
<meta charset="utf-8">
<title>5.1: Coproducts, sum types | Categories Bartosz Milewski Youtube notes</title>
<link rel="shortcut icon" href="favicon.png" type="image/x-icon">
<link rel="copyright" href="http://www.gnu.org/copyleft/gpl.html"/>


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younotes1/5.2-algebraic-data-types.html View File

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<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="utf-8" />
<meta charset="utf-8">
<title>5.2: Algebraic data types | Categories Bartosz Milewski Youtube notes</title>
<link rel="shortcut icon" href="favicon.png" type="image/x-icon">
<link rel="copyright" href="http://www.gnu.org/copyleft/gpl.html"/>


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younotes1/6.1-functors.html View File

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<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="utf-8" />
<meta charset="utf-8">
<title>6.1: Functors | Categories Bartosz Milewski Youtube notes</title>
<link rel="shortcut icon" href="favicon.png" type="image/x-icon">
<link rel="copyright" href="http://www.gnu.org/copyleft/gpl.html"/>


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<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="utf-8" />
<meta charset="utf-8">
<title>6.2: Functors in programming | Categories Bartosz Milewski Youtube notes</title>
<link rel="shortcut icon" href="favicon.png" type="image/x-icon">
<link rel="copyright" href="http://www.gnu.org/copyleft/gpl.html"/>
@@ -61,7 +61,7 @@ We need to prove that <pre>fmap id<sub>a</sub> = id<sub>Maybe a</sub></pre>
We need to prove that
<pre>fmap(g . f) = fmap g . fmap f</pre>
Corresponds to this diagram :
<div><img class="margin border" src="img/fmap-composition.jpg" alt="fmap composition" /><br>(TODO : add <code>fmap g . fmap f</code>)</div>
<div><img class="margin border" src="img/fmap-composition.jpg" alt="fmap composition"><br>(TODO : add <code>fmap g . fmap f</code>)</div>
This can be showed by equational reasoning using the same method as we did for identity (It's done on the <a href="https://bartoszmilewski.com/2015/01/20/functors/">blog page about functors</a>).
<br>Strictly speaking there is no need to prove it because as we use parametric polymorphism, this is a theorem for free. Once the id property is proven, this follows.



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<meta charset="utf-8">
<title>7.1: Functoriality, bifunctors | Categories Bartosz Milewski Youtube notes</title>
<link rel="shortcut icon" href="favicon.png" type="image/x-icon">
<link rel="copyright" href="http://www.gnu.org/copyleft/gpl.html"/>


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<head>
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<meta charset="utf-8">
<title>7.2: Monoidal Categories, Functoriality of ADTs, Profunctors | Categories Bartosz Milewski Youtube notes</title>
<link rel="shortcut icon" href="favicon.png" type="image/x-icon">
<link rel="copyright" href="http://www.gnu.org/copyleft/gpl.html"/>


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<!DOCTYPE html>
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<meta charset="utf-8">
<title>8.1: Function objects, exponentials | Categories Bartosz Milewski Youtube notes</title>
<link rel="shortcut icon" href="favicon.png" type="image/x-icon">
<link rel="copyright" href="http://www.gnu.org/copyleft/gpl.html"/>


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<!DOCTYPE html>
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<meta charset="utf-8">
<title>8.2: Type algebra, Curry-Howard-Lambek isomorphism | Categories Bartosz Milewski Youtube notes</title>
<link rel="shortcut icon" href="favicon.png" type="image/x-icon">
<link rel="copyright" href="http://www.gnu.org/copyleft/gpl.html"/>


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<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="utf-8" />
<meta charset="utf-8">
<title>9.1: Natural transformations | Categories Bartosz Milewski Youtube notes</title>
<link rel="shortcut icon" href="favicon.png" type="image/x-icon">
<link rel="copyright" href="http://www.gnu.org/copyleft/gpl.html"/>


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<meta charset="utf-8">
<title>9.2: bicategories | Categories Bartosz Milewski Youtube notes</title>
<link rel="shortcut icon" href="favicon.png" type="image/x-icon">
<link rel="copyright" href="http://www.gnu.org/copyleft/gpl.html"/>


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