new layout

03_categories/text.md

@@ -1,124 +0,0 @@
-Defining Categories
-===
-
-Every category theory guide (including mine) starts by talking about set theory. However, looking back, I really don't know why that is the case - surely most other books that focus around a specific subject don't start by introducing an entirely different subject. Perhaps the set-first approach is the best way to introduce people to categories. or perhaps using sets to introduce categories is just one of the things that people just do, simply because everyone else does it. But one thing is for sure - we don't need to study sets in order to understand categories. So in this chapter I would like to start over and talk about categories as a first concept. So pretend like it's a new book:
-
-Starting over with categories
-===
-
-So, a category is a collection of objects (things) where the "things" can be anything you want. Consider, for example, these ~~colourful~~ grey balls:
-
-![Balls](elements.svg)
-
-A category consists of a collection of objects as well as some arrows connecting some of them to one another, that we call (the arrows) morphisms.
-
-![A category](category.svg)
-
-Wait a minute - we said that all sets form a category, but at the same time any one set can be seen as a category on its own right (just one which has no morphisms). This is true and an example of a phenomenon that is very characteristic of category theory - one structure can be examined from many different angles and may play many different roles, often in a recursive fashion.
-
-This particular analogy (a set as a category with no morphisms) is, however, not very useful. Not because it's in any way incorrect, but because category theory is *all about the morphisms*. If in set theory arrows are nothing but a connection between a source and a destination, in category theory it's the *objects* that are nothing but a source and destination for the arrows that connect them to other objects. This is why, in the diagram above, the arrows, and not the objects, are coloured: the category of sets should really be called the category of set functions.
-
-Speaking of which, note that objects in a category can be connected by multiple arrows and that arrows having the same domain and codomain does not in any way make them equivalent.
-
-![Two objects connected with multiple arrows](arrows.svg)
-
-Why is that is pretty obvious if we go back to set theory for a second (OK, maybe we really *have* to do this from time to time) - there are, for example, an infinite number of functions that go from number to boolean and the fact that they have the same input type and the same output type (or the same *type signature*, as we like to say) does not in any way make them equivalent to one another.
-
-![Two sets connected with multiple functions](set_arrows.svg)
-
-Composition
----
-
-One of the few, or maybe even the only, requirement for a structure to be called a category is that two morphisms can make a third, or in other words that morphisms are *composable* - given two successive arrows with appropriate type signature, we can draw a third one that is equivalent to the consecutive application of the other two.
-
-![Composition of morphisms](composition.svg)
-
-
-Formally, this requirement says that there should exist an operation (denoted with the symbol **•**) such that for each two functions **g: A → B** and **f: B → C**, there exists exactly one function **(f • g): A → C**.
-
-**NB:** Note that functional composition is written from right to left. e.g. applying **g** and then applying **f** is written **f • g** and not the other way around (you can think of it as a shortcut to **f(g(a))**).
-
-Again, note that this criteria is not met by just *any* morphism with this type signature. There is *exactly one* morphism that fits these criteria and there may be some which don't.
-
-
-![Composition of morphisms in the context of additional morphism](composition_arrows.svg)
-
-Commuting diagrams
----
-
-The diagram above, uses colors to illustrate the fact that the green morphism is equivalent to the other two (and not just some unrelated morphism), but in practice this notation is a little redundant - the only reason to draw diagrams with arrows in the first place is represent paths between objects that are equivalent to each other, all other paths just belong in different diagrams. For this reason, we can drop the colors from our diagrams.
-
-![Composition of morphisms - a commuting diagram](composition_commuting_diagram.svg)
-
-Diagrams that are like that (ones in which any two paths between two objects are equivalent to one another) are called *commutative diagrams* (or diagrams that *commute*). All diagrams that we have here (except the wrong ones) commute.
-
-
-The law of associativity
----
-
-Functional composition is special not only because you can take any two morphisms with appropriate signatures and make a third, but because you can do so indefinitely, i.e. given **n** successive arrows, each of which starts from the object that the other one finishes, we can draw one (exactly one) arrow that is equivalent to the consecutive application of all **n** arrows.
-
-![Composition of morphisms with many objects](composition_n_objects.svg)
-
-This approach for building stuff is often used in programming. To see some examples, you don't need to look further than the way the pipe operator in bash (`|`), that feeds the standard output of a program with the standard input of another program, is (ab)used (if you *want* to look further, note that there is a whole programming paradigm based on functional composition, called "concatenative programming").
-
-But let's get back to the math. If we carefully review the definition above can see that it can be reduced to multiple applications of the following formula: given 4 objects and 3 morphisms between them **f** **g** **h**, combining **h** and **g** and then combining the end result with f** should be the same as combining **h** to the result of **g** and **f** (or simply **(h • g) • f = h • (g • f)**. 
-
-**Task:** show how the definition can be reduced to the formula (the approach resembles mathematical induction).
-
-This formula is true if and only if this diagram commutes. 
-
-![Composition of morphisms with many objects](composition_associativity.svg)
-
-Given that all our category-theoretic diagrams commute, we can say, in such cases, that the formula and the diagram are equivalent.
-
-This formula (and the diagram) is the definition of a property called **associativity**. Being associative is required for functional composition to really be called functional composition (and for a category to really be called category), if only because it is required for us to be able to draw diagrams of a category's objects and morphisms, as diagrams can only represent associative structures (if the diagram above does not commute it would be super weird).
-
-Associativity is not just about diagrams, for example when we express relations using formulas, associativity just means that brackets don't matter, in our formulas (as evidenced by the definition **(h • g) • f = h • (g • f)**). And it is not only about categories either, it is a property of many other operations on other types of objects as well e.g. if we look at numbers, we can see that the multiplication operation is associative e.g. **(1 . 2) . 3 = 1 . (2 . 3)**. While division is not **(1 / 2) / 3 = 1 / (2 / 3)**.
-
-
-Identity
----
-
-Ancient mathematicians invented the number zero that, although useless by itself, allowed them to define many useful numbers. In order to be able to define more stuff using morphisms in category theory, we too would want to define what we call the "identity morphism" for each object. In short, this is a morphism, that doesn't do anything.
-
-![The identity morphism (but can also be any other morphism)](identity.svg)
-
-It's important to mark this morphism, because there can be (let's add the very important (and also very boring) reminder) many morphisms that go from one object to the same object, many of which actually do stuff. For example mathematics deals with a multitude of functions that have the set of numbers as domain and codomain, such as **negate**, **square**, **add one**, and are not at all the identity morphism.
-
-**Question:** What is the identity morphism in the category of sets?
-
-Why do we need to define a morphism that does nothing? 
-
-Isomorphisms
----
-
-Once we have the concept of identity morphism defined we can have a category-theoretic definition of an isomoprhism (which is important, because the concept of an isomorphism is very important for cathegory theory): An isomorphism between two objects (**A** and **B**) consists of two morphisms - (**A → B**.  and **B → A**) such that their compositions are equivalent to the identity functions of the respective objects. 
-
-Here is how this looks when expressed using a formulas:
-
-Objects **A** and **B** are isomorphic 
-iff there exist mophisms 
-**f: A → B**
-**g: B → A** 
-such that  
-**f • g = idB** 
-**g • f = idA** 
-
-And here is the same thing expressed with a commuting diagram.
-
-![Isomorphism](isomorphism.svg)
-
-Like the example with the law of associativity, the diagram expresses the same (simple) fact as the formula, namely that going from the one of objects (**A** and **B**) to the other one and then back again is the same as applying the identity morphism i.e. doing nothing. 
-
-A summary
----
-
-For future reference, let's repeat what a category is.
-
-A category is a collection of **objects** (we can think of them as points) and **morphisms** (arrows) that go from one object to another, where:
-1. There should be a way to compose two morphisms with an appropriate type signature into a third one in a way that is associative.
-2. Each object has to have the identity mophism.
-
-This is it.
-

_chapters/00_about.md

@@ -0,0 +1,13 @@
+---
+layout: default
+title: About this book
+---
+
+About
+===
+
+This book is a product of my own endeavour of understanding category theory from a more non-mathematical viewpoint - it is just that as I am explaining something, I am understanding it better.
+
+It is aimed at programmers, as well as anyone else who is interested in this stuff. 
+
+The main reason I am interested in Category Theory is that it allows us to formalise some common concepts that we use in our daily (intellectual) lives. Much of our language is based on intuition and rightfully so: relying on intuition is a very easy way to get your point across so it is understood by other human beings. However, that is part of the problem: sometimes intuition makes it too easy to communicate with someone. So easy that he might, in fact, understand things that you haven't actually said. For example, when I say that two things are equal, it would seem obvious to you what I mean, although it isn't obvious at all (how are they equal, at what context etc). That is the place when we might want to provide a more rigorous definition of what am I saying (even if I did not have one, to begin with). But providing such definition in natural language, which is designed to use intuition as a means of communication, is no easy task. It is in these situations that people often resort to diagrams to explain their thoughts. Diagrams are ubiquitous in science and mathematics because they are an understandable way to communicate a formal concept clearly. Category theory formalises the concept of a diagram and their components - arrows and objects and creates a language for presenting all kinds of ideas. Along the way, we would see all other kinds of mathematical objects, viewed under the primsm of categories.

_chapters/01_set.md

@@ -1,3 +1,8 @@
+---
+layout: default
+title: Sets
+---
+
 Sets
 ===
 

_chapters/02_category.md

@@ -1,3 +1,8 @@
+---
+layout: default
+title: From Sets to Categories
+---
+
 From Sets to Categories
 ===
 
@@ -195,3 +200,127 @@ One remark before we go - the last paragraphs I sound as if I'm *comparing* cate
 
 Perhaps that notion would be somewhat correct if category and set theory were meant to describe *concrete* phenomena, in the way that the theory of relativity and the theory of quantum mechanics in physics. Concrete theories are conceived mainly as *descriptions* of the world, and as such it makes sense for them to be connected to one another in some sort of hierarchy. Abstract theories, like category theory and set theory, on the other hand, are more like languages for expressing such descriptions. They still can be connected, and are connected in more than one way, but there is no inherent hierarchy between the two and therefore arguing over which of the two is more basic, or more general, is just a chicken-and-egg problem, as you would see in the next chapter.
 
+Defining Categories
+===
+
+Every category theory guide (including mine) starts by talking about set theory. However, looking back, I really don't know why that is the case - surely most other books that focus around a specific subject don't start by introducing an entirely different subject. Perhaps the set-first approach is the best way to introduce people to categories. or perhaps using sets to introduce categories is just one of the things that people just do, simply because everyone else does it. But one thing is for sure - we don't need to study sets in order to understand categories. So in this chapter I would like to start over and talk about categories as a first concept. So pretend like it's a new book:
+
+Starting over with categories
+===
+
+So, a category is a collection of objects (things) where the "things" can be anything you want. Consider, for example, these ~~colourful~~ grey balls:
+
+![Balls](elements.svg)
+
+A category consists of a collection of objects as well as some arrows connecting some of them to one another, that we call (the arrows) morphisms.
+
+![A category](category.svg)
+
+Wait a minute - we said that all sets form a category, but at the same time any one set can be seen as a category on its own right (just one which has no morphisms). This is true and an example of a phenomenon that is very characteristic of category theory - one structure can be examined from many different angles and may play many different roles, often in a recursive fashion.
+
+This particular analogy (a set as a category with no morphisms) is, however, not very useful. Not because it's in any way incorrect, but because category theory is *all about the morphisms*. If in set theory arrows are nothing but a connection between a source and a destination, in category theory it's the *objects* that are nothing but a source and destination for the arrows that connect them to other objects. This is why, in the diagram above, the arrows, and not the objects, are coloured: the category of sets should really be called the category of set functions.
+
+Speaking of which, note that objects in a category can be connected by multiple arrows and that arrows having the same domain and codomain does not in any way make them equivalent.
+
+![Two objects connected with multiple arrows](arrows.svg)
+
+Why is that is pretty obvious if we go back to set theory for a second (OK, maybe we really *have* to do this from time to time) - there are, for example, an infinite number of functions that go from number to boolean and the fact that they have the same input type and the same output type (or the same *type signature*, as we like to say) does not in any way make them equivalent to one another.
+
+![Two sets connected with multiple functions](set_arrows.svg)
+
+Composition
+---
+
+One of the few, or maybe even the only, requirement for a structure to be called a category is that two morphisms can make a third, or in other words that morphisms are *composable* - given two successive arrows with appropriate type signature, we can draw a third one that is equivalent to the consecutive application of the other two.
+
+![Composition of morphisms](composition.svg)
+
+
+Formally, this requirement says that there should exist an operation (denoted with the symbol **•**) such that for each two functions **g: A → B** and **f: B → C**, there exists exactly one function **(f • g): A → C**.
+
+**NB:** Note that functional composition is written from right to left. e.g. applying **g** and then applying **f** is written **f • g** and not the other way around (you can think of it as a shortcut to **f(g(a))**).
+
+Again, note that this criteria is not met by just *any* morphism with this type signature. There is *exactly one* morphism that fits these criteria and there may be some which don't.
+
+
+![Composition of morphisms in the context of additional morphism](composition_arrows.svg)
+
+Commuting diagrams
+---
+
+The diagram above, uses colors to illustrate the fact that the green morphism is equivalent to the other two (and not just some unrelated morphism), but in practice this notation is a little redundant - the only reason to draw diagrams with arrows in the first place is represent paths between objects that are equivalent to each other, all other paths just belong in different diagrams. For this reason, we can drop the colors from our diagrams.
+
+![Composition of morphisms - a commuting diagram](composition_commuting_diagram.svg)
+
+Diagrams that are like that (ones in which any two paths between two objects are equivalent to one another) are called *commutative diagrams* (or diagrams that *commute*). All diagrams that we have here (except the wrong ones) commute.
+
+
+The law of associativity
+---
+
+Functional composition is special not only because you can take any two morphisms with appropriate signatures and make a third, but because you can do so indefinitely, i.e. given **n** successive arrows, each of which starts from the object that the other one finishes, we can draw one (exactly one) arrow that is equivalent to the consecutive application of all **n** arrows.
+
+![Composition of morphisms with many objects](composition_n_objects.svg)
+
+This approach for building stuff is often used in programming. To see some examples, you don't need to look further than the way the pipe operator in bash (`|`), that feeds the standard output of a program with the standard input of another program, is (ab)used (if you *want* to look further, note that there is a whole programming paradigm based on functional composition, called "concatenative programming").
+
+But let's get back to the math. If we carefully review the definition above can see that it can be reduced to multiple applications of the following formula: given 4 objects and 3 morphisms between them **f** **g** **h**, combining **h** and **g** and then combining the end result with f** should be the same as combining **h** to the result of **g** and **f** (or simply **(h • g) • f = h • (g • f)**. 
+
+**Task:** show how the definition can be reduced to the formula (the approach resembles mathematical induction).
+
+This formula is true if and only if this diagram commutes. 
+
+![Composition of morphisms with many objects](composition_associativity.svg)
+
+Given that all our category-theoretic diagrams commute, we can say, in such cases, that the formula and the diagram are equivalent.
+
+This formula (and the diagram) is the definition of a property called **associativity**. Being associative is required for functional composition to really be called functional composition (and for a category to really be called category), if only because it is required for us to be able to draw diagrams of a category's objects and morphisms, as diagrams can only represent associative structures (if the diagram above does not commute it would be super weird).
+
+Associativity is not just about diagrams, for example when we express relations using formulas, associativity just means that brackets don't matter, in our formulas (as evidenced by the definition **(h • g) • f = h • (g • f)**). And it is not only about categories either, it is a property of many other operations on other types of objects as well e.g. if we look at numbers, we can see that the multiplication operation is associative e.g. **(1 . 2) . 3 = 1 . (2 . 3)**. While division is not **(1 / 2) / 3 = 1 / (2 / 3)**.
+
+
+Identity
+---
+
+Ancient mathematicians invented the number zero that, although useless by itself, allowed them to define many useful numbers. In order to be able to define more stuff using morphisms in category theory, we too would want to define what we call the "identity morphism" for each object. In short, this is a morphism, that doesn't do anything.
+
+![The identity morphism (but can also be any other morphism)](identity.svg)
+
+It's important to mark this morphism, because there can be (let's add the very important (and also very boring) reminder) many morphisms that go from one object to the same object, many of which actually do stuff. For example mathematics deals with a multitude of functions that have the set of numbers as domain and codomain, such as **negate**, **square**, **add one**, and are not at all the identity morphism.
+
+**Question:** What is the identity morphism in the category of sets?
+
+Why do we need to define a morphism that does nothing? 
+
+Isomorphisms
+---
+
+Once we have the concept of identity morphism defined we can have a category-theoretic definition of an isomoprhism (which is important, because the concept of an isomorphism is very important for cathegory theory): An isomorphism between two objects (**A** and **B**) consists of two morphisms - (**A → B**.  and **B → A**) such that their compositions are equivalent to the identity functions of the respective objects. 
+
+Here is how this looks when expressed using a formulas:
+
+Objects **A** and **B** are isomorphic 
+iff there exist mophisms 
+**f: A → B**
+**g: B → A** 
+such that  
+**f • g = idB** 
+**g • f = idA** 
+
+And here is the same thing expressed with a commuting diagram.
+
+![Isomorphism](isomorphism.svg)
+
+Like the example with the law of associativity, the diagram expresses the same (simple) fact as the formula, namely that going from the one of objects (**A** and **B**) to the other one and then back again is the same as applying the identity morphism i.e. doing nothing. 
+
+A summary
+---
+
+For future reference, let's repeat what a category is.
+
+A category is a collection of **objects** (we can think of them as points) and **morphisms** (arrows) that go from one object to another, where:
+1. There should be a way to compose two morphisms with an appropriate type signature into a third one in a way that is associative.
+2. Each object has to have the identity mophism.
+
+This is it.
+

_chapters/03_monoid.md

@@ -1,5 +1,9 @@
-Monoids etc.
+---
+layout: default
+title: Monoids etc
+---
 
+Monoids etc
 ===
 
 Since we are done with categories, let's look at some other structures that are also interesting - monoids. Like categories, monoids/groups are also abstract systems consisting of objects and rules for manipulating these objects. 
@@ -11,13 +15,13 @@ Monoids are simpler than categories. A monoid is defined by a collection (set) o
 
 Let's take our familiar colourful balls.
 
-[Balls](balls.svg)
+![Balls](balls.svg)
 
 In this case a monoid would be a rule (operation) for "combining" two balls into one. 
 
 An example of such rule would be blending the colors of the balls, as if we are mixing paint.
 
-[A rule for combining balls](balls_rule.svg)
+![A rule for combining balls](balls_rule.svg)
 
 You can probably think of other ways to define such a rule. This will help you realize that there can be many ways to create a monoid from a given set of items. The monoid is not the set itself, it is the set *together with the rule*.
 
@@ -26,18 +30,18 @@ Associativity
 
 The monoid rule should, like functional composition, be "associative" i.e. applying it on the same number of elements in a different order should make no difference.
 
-[Associativity in the color mixing operation](balls_associativity.svg)
+![Associativity in the color mixing operation](balls_associativity.svg)
 
 When a rule is associative, this means we can use all kinds of algebraic operations to any sequence of terms (or in other words to apply equation reasoning), like for example we can add or remove a term that is present at both sides of an equation and retaining the equality of the existing terms:
 
-[Associativity in the color mixing operation](balls_arithmetic.svg)
+![Associativity in the color mixing operation](balls_arithmetic.svg)
 
 The identity element
 ---
 
 Actually, not any (associative) rule for combining elements makes the balls form a monoid (it makes them form a "semigroup", which is also a thing, but that's a separate topic). In order to be a monoid, a set must feature what is called an "identity element" of a given rule (or a *zero* element, if you prefer) - one that, when combined with any other element gives back that same element not the identity but the other one. Or simply **x • i = x and i • x = x for any x**. In the case of our color-mixing monoid the identity element is the color white, represented by the white ball:
 
-[The identity element of the color-mixing monoid](balls_identity.svg) 
+![The identity element of the color-mixing monoid](balls_identity.svg) 
 
 As you probably remember from the last chapter, functional composition is also associative and it also contains an identity element, so you might start suspecting that it forms a monoid in some way. And it is really the case with one little caveat. 
 
@@ -52,13 +56,13 @@ Monoids from numbers
 
 Mathematics is not all about numbers, however numbers do tend to pop up in most of its areas and monoids are no exception. The set of natural numbers *N* form a monoid when combined with the all too familiar operation of addition (or to use the official terminology *N* *form* a monoid *under* addition).
 
-[The monoid of numbers under addition](numbers_addition.svg)
+![The monoid of numbers under addition](numbers_addition.svg)
 
 (if you see a **1 + 1 = 2** in your texbook you know you are working on math foundations (or you are in kindergarden)).
 
 The natural numbers also form a monoid under multiplication as well:
 
-[The monoid of numbers under multiplication](numbers_multiplication.svg)
+![The monoid of numbers under multiplication](numbers_multiplication.svg)
 
 **Task:** Which are the identity elements of those monoids?
 
@@ -72,13 +76,17 @@ Thinking about other operations that we covered (operation being a function whic
 We can prove that **AND** is associative by expanding the formula **(A AND B) AND C = A AND (B AND C)** in all possible ways:
 
 **(TRUE AND FALSE) AND TRUE = TRUE AND (FALSE AND TRUE)**
+
 **(TRUE AND FALSE) AND FALSE = TRUE AND (FALSE AND FALSE)**
+
 **(FALSE AND FALSE) AND TRUE = FALSE AND (FALSE AND TRUE)**
+
 ...
 
 And we can prove that **TRUE** is the identity element by expanding the other formulas that state that for all elements **A** **I AND A = A**
 
 **FALSE AND TRUE = FALSE**
+
 **TRUE AND TRUE = TRUE**
 
 ...and then do the same for **A AND I = A**.
@@ -102,13 +110,13 @@ Commutative monoids
 
 Looking at the monoid laws and the examples we gave so far, we observe that all of them obey one more rule (law) which we didn't specify, namely that the order in which the operations are applied is irrelevant to the end result.
 
-[Commutative monoid operation](monoid_commutative.svg)
+![Commutative monoid operation](monoid_commutative.svg)
 
 Such operations (ones for which combining a given set of objects yields the same result no matter which one is first and which one is second) are called *commutative* operations. Monoids with operations that are commutative are called *commutative monoids*. 
 
 As we said, addition is commutative as well - it does not matter whether if I have given you 1 apple and then 2 more, or if I have given you 2 first and then 1 more i.e. **x + y = y + x**.
 
-[Commutative monoid operation](addition_commutative.svg)
+![Commutative monoid operation](addition_commutative.svg)
 
 All monoids that we examined so far are also *commutative*, and we will see some non-commutative ones later. 
 
@@ -132,37 +140,3 @@ Here are the algebraic structures that we have seen so far and the laws that the
 |Associativity| X | X | X |
 |Identity| | X | X |
 |Invertability | |  | X |
-
-Symmetry groups
-===
-
-An interesting set of groups/monoids of *symmetries* of some object
-
-Group of rotations 
----
-
-Group of reflections
----
-
-Cancellation law
----
-
-
-
-Categories
-===
-
-Some more advanced examples
-===
-
-The trivial monoid
----
-Peano arithmetic
-
-
-Monoids as categories 
-===
-
-The free monoid.
-===
-

_config.yml

@@ -3,43 +3,21 @@
 #
 
 # Name of your site (displayed in the header)
-name: Category theory illustrated
+name: Category Theory Illustrated
 
 # Short bio or description (displayed in the header)
-description: Sets, categories, logic and other stuffs, represented visually
-
-# URL of your avatar or profile pic (you could use your GitHub profile pic)
-avatar: https://raw.githubusercontent.com/barryclark/jekyll-now/master/images/jekyll-logo.png
+description: A book about category theory
 
 #
 # Flags below are optional
 #
 
-# Includes an icon in the footer for each username you enter
-footer-links:
-  dribbble:
-  email:
-  facebook:
-  flickr:
-  github: barryclark/jekyll-now
-  instagram:
-  linkedin:
-  pinterest:
-  rss: # just type anything here for a working RSS icon, make sure you set the "url" above!
-  twitter: jekyllrb
-  stackoverflow: # your stackoverflow profile, e.g. "users/50476/bart-kiers"
-  youtube: # channel/<your_long_string> or user/<user-name>
-
-# Enter your Disqus shortname (not your username) to enable commenting on posts
-# You can find your shortname on the Settings page of your Disqus account
-disqus:
-
 # Enter your Google Analytics web tracking code (e.g. UA-2110908-2) to activate tracking
 google_analytics:
 
 # Your website URL (e.g. http://barryclark.github.io or http://www.barryclark.co)
 # Used for Sitemap.xml and your RSS feed
-url:
+url: boris-marinov.github.io/category-theory-illustrated
 
 excerpt_separator: <!--more-->  # you can specify your own separator, of course.
 
@@ -47,7 +25,13 @@ excerpt_separator: <!--more-->  # you can specify your own separator, of course.
 # (http://yourusername.github.io/repository-name)
 # and NOT your User repository (http://yourusername.github.io)
 # then add in the baseurl here, like this: "/repository-name"
-#baseurl: "/funktion"
+baseurl: "/category-theory-illustrated"
+
+collections:
+  chapters:
+    output: true
+    permalink: /:path/
+
 
 #
 # !! You don't need to change any of the configuration flags below !!
@@ -59,10 +43,6 @@ permalink: /:title/
 # The release of Jekyll Now that you're using
 version: v1.1.0
 
-# Set the Sass partials directory, as we're using @imports
-sass:
-  style: :expanded # You might prefer to minify using :compressed
-
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+<p>This book is a product of my own endeavour of understanding category theory from a more non-mathematical viewpoint - it is just that as I am explaining something, I am understanding it better.</p>
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+<p>It is aimed at programmers, as well as anyone else who is interested in this stuff.</p>
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+<p>The main reason I am interested in Category Theory is that it allows us to formalise some common concepts that we use in our daily (intellectual) lives. Much of our language is based on intuition and rightfully so: relying on intuition is a very easy way to get your point across so it is understood by other human beings. However, that is part of the problem: sometimes intuition makes it too easy to communicate with someone. So easy that he might, in fact, understand things that you haven’t actually said. For example, when I say that two things are equal, it would seem obvious to you what I mean, although it isn’t obvious at all (how are they equal, at what context etc). That is the place when we might want to provide a more rigorous definition of what am I saying (even if I did not have one, to begin with). But providing such definition in natural language, which is designed to use intuition as a means of communication, is no easy task. It is in these situations that people often resort to diagrams to explain their thoughts. Diagrams are ubiquitous in science and mathematics because they are an understandable way to communicate a formal concept clearly. Category theory formalises the concept of a diagram and their components - arrows and objects and creates a language for presenting all kinds of ideas. Along the way, we would see all other kinds of mathematical objects, viewed under the primsm of categories.</p>
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+            <p><a href="/category-theory-illustrated/00_about/">About this book</a></p>
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+            <p><a href="/category-theory-illustrated/01_set/">Sets</a></p>
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+            <p><a href="/category-theory-illustrated/02_category/">From Sets to Categories</a></p>
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+            <p><a href="/category-theory-illustrated/03_monoid/">Monoids etc</a></p>
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_site/01_set/function_singleton.svg

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