Corrections

_chapters/01_set.md

@@ -16,7 +16,7 @@ Preface: What is an Abstract Theory
 
 Most scientific and mathematical theories have a specific domain, to which they are related, and in which they are valid. They are created with this domain in mind and are not intended to be used outside of it. For example, Darwin's theory of evolution is created so it explains how different biological species came to be. And quantum mechanics is a description of what particles are at the low scale. Even the work of most mathematicians, although it is not bound to a specific domain, is strongly related to it, as differential equations are linked to the description of events that change over time. 
 
-Set theory and category theory are different. They are not created for to provide a rigorous explanation of how a particular phenonium works, but to try to provide a more general framework for explaining all kinds of phenomena. Theories that are like that are called *abstract* theories. All theories *use abstraction*, else they would be pretty useless (without it Darwin would have to speak about specific animal species or even individual animals) but few are inherently abstract, so some of their core concepts are left unspecified. Or in other words, all theories are applicable outside of their domains, but set theory and category theory do not have a domain, to begin with.
+Set theory and category theory are different. They are not created to provide a rigorous explanation of how a particular phenomenon works, but to try to provide a more general framework for explaining all kinds of phenomena. Theories that are like that are called *abstract* theories. All theories *use abstraction*, else they would be pretty useless (without it Darwin would have to speak about specific animal species or even individual animals) but few are inherently abstract, so some of their core concepts are left unspecified. Or in other words, all theories are applicable outside of their domains, but set theory and category theory do not have a domain, to begin with.
 
 People have tried to be precise and at the same time down to Earth for centuries, and only recently discovered that "precise and down to Earth" is an oxymoron. Let's take Euclidian geometry as an example. Yes, Euclidian geometry is precise, because it is valid for all sets of objects, called ("point" "line" "angle" and "circle" etc.), which have relationships, as defined by the five famous axioms. Yes, geometry does, in many instances, describe the natural world, because there are many sets of objects which have these relations. However, its "precise" part and it's "down to Earth" part have nothing to do with each other. We can, for example, define a point as any stain on the floor of your room and that a line as a piece of duct tape, put on the same floor - that will be a completely valid application of the Euclidian laws, albeit not very useful one. Or we can try to use geometry to reason about points on the surface of the Earth, which is a very useful application, of geometry, however not of Euclidian geometry, because Euclidian geometry only describes points on a flat plane, and the Earth is not flat. You can argue that these are actually two separate theories there, which just happen to be perceived as one. You have the axioms, or the postulates on one hand, which are not useful for anything on their own, and you have applications in science and engineering which are somewhat based on them, but not quite. 
 
@@ -27,7 +27,7 @@ Everything in set theory is defined in terms of sets. A set is a collection of t
 
 ![Balls](elements.svg)
 
-For example, let's construct a set, call it **G** (as grey), consisting of *all* of them. This is how we can mark it:
+For example, let's construct a set, call it **G** (as gray), consisting of *all* of them. This is how we can mark it:
 
 ![The set of all balls](all.svg)
 
@@ -41,9 +41,9 @@ The key insight about what a set it is that it enables you to reason about sever
 Subsets 
 ---
 
-Let's construct one more set. The set of *all balls that are warm colour*. I will call it **Y**, because in the diagram is coloured in **y**ellow.
+Let's construct one more set. The set of *all balls that are warm color*. I will call it **Y**, because in the diagram is colored in **y**ellow.
 
-![Y - the set of all balls of warm colours](subset.svg)
+![Y - the set of all balls of warm colors](subset.svg)
 
 Notice that **Y** contains just elements that are also present in **G**. That is, every element of the set of **Y** is also an element in the set **G**. When two sets have this relation, we may say that **Y** is a *subset* of **G** (or **Y ⊆ G**).
 
@@ -56,7 +56,7 @@ The set of all *red balls* contains just one ball.
 
 ![The singleton set of red balls](singleton.svg)
 
-Like we said above, sets are all about for summarising *several* elements into one. Still, sets that contain just one element are perfectly valid. Simply, there are things which are one of a kind. Furthermore, if I have a function which expects a set of given items, here shouldn't be any issue if the "items" are just one item. Or to take a real-life example, the set of queens of England is a singleton set. The set of books written by the American writer Harper Lee and published during her lifetime is a singleton set - she has published just one novel.
+Like we said above, sets are all about for summarizing *several* elements into one. Still, sets that contain just one element are perfectly valid. Simply, there are things which are one of a kind. Furthermore, if I have a function which expects a set of given items, here shouldn't be any issue if the "items" are just one item. Or to take a real-life example, the set of queens of England is a singleton set. The set of books written by the American writer Harper Lee and published during her lifetime is a singleton set - she has published just one novel.
 
 The Empty set
 ---
@@ -80,7 +80,7 @@ A function is a relationship between two sets which matches each element of one
 > By function I mean the unity of the act of arranging various representations under one common representation.
 > Immanuel Kant, from Critique of Pure Reason
 
-Here is a function, **f** which maps each ball from the set **R** to the ball with the opposite colour in another set **G** ( in mathematics a function's name is often accompanied by the names of its domain and codomain, like this: **f: R → G**)
+Here is a function, **f** which maps each ball from the set **R** to the ball with the opposite color in another set **G** ( in mathematics a function's name is often accompanied by the names of its domain and codomain, like this: **f: R → G**)
 
 ![Opposite colors](function_one_one.svg)
 
@@ -115,11 +115,11 @@ Note that the question "Who is my child?" is *NOT* a straightforward function, b
 The Identity Function
 ---
 
-For every set **G**, no matter what it represents, we can define the function that does nothing, or in other words, a function which maps every element G to itself. It is called the *the identity function* of **G** or **idG: G → G**.
+For every set **G**, no matter what it represents, we can define the function that does nothing, or in other words, a function which maps every element G to itself. It is called the *the identity function* of **G** or **id G: G → G**.
 
 ![The identity function](function_identity.svg)
 
-You can think of **idG** as a function which represents the set **G** in the realm of functions. Its existence allows us to prove many theorems, that we "know" by intuition, formally.
+You can think of **id G** as a function which represents the set **G** in the realm of functions. Its existence allows us to prove many theorems, that we "know" by intuition, formally.
 
 Functions and Subsets
 ---
@@ -177,7 +177,7 @@ I will use the occasion to reiterate some of the more important characteristics
 
 Overall everything is OK, as long as you can always provide exactly one result (also known as *The result™*) per value, and in mathematics almost always do. Actually, math is designed in a way so its operations are valid functions:
 
-> Every generalisation of number has first presented itself as needed for some simple problem: negative numbers were needed in order that subtraction might be always possible, since otherwise a − b would be meaningless if a were less than b; fractions were needed in order that division might be always possible; and complex numbers are needed in order that extraction of roots and solution of equations may be always possible.
+> Every generalization of number has first presented itself as needed for some simple problem: negative numbers were needed in order that subtraction might be always possible, since otherwise a − b would be meaningless if a were less than b; fractions were needed in order that division might be always possible; and complex numbers are needed in order that extraction of roots and solution of equations may be always possible.
 > Bertrand Russell, from Introduction to Mathematical Philosophy
 
 Note that most mathematical operations, such as addition, multiplication etc. require two numbers in order to produce a result. This does not mean that they are not functions, it means that they are just a little more fancy ones. Depending on what we need, we may present those operations as functions from the sets of *tuples* of numbers to the set of numbers, or we may say that they take a number and return a function. More on that later.
@@ -201,7 +201,7 @@ Most of the types of programming are composite types - they are a combination of
 
 **Question:** What is the type equivalent of subsets in programming?
 
-**Question:** Do you recognise some of the basic functions we defined in programming languages you know?
+**Question:** Do you recognize some of the basic functions we defined in programming languages you know?
 
 Some functions in programming (also called methods, subroutines, etc.) kinda resemble mathematical functions - they sometimes take one value of a given type (or in other words, an element that belongs to a given set) and always return exactly one element which belongs to another type (or set). For example here is a function which that takes an argument of type `Char` and returns a `Boolean`, depending on whether the character is a letter.
 
@@ -232,11 +232,11 @@ Unlike the set above, most sets that we discussed (like the empty set and single
 
 ![Sets that don't contains themselves](sets_dont_contain_themselves.svg)
 
-In order to understand Russell's paradox we will try to visualise **the set all sets that do not contain themselves**. In the original set notation we can define this set as *Let R = { x => x ∉ x }* (let R be such that it contains all sets *x* such that *x* is not a member of *x*). 
+In order to understand Russell's paradox we will try to visualize **the set all sets that do not contain themselves**. In the original set notation we can define this set as *Let R = { x => x ∉ x }* (let R be such that it contains all sets *x* such that *x* is not a member of *x*). 
 
 ![Russel's paradox - option one](russells_paradox.svg)
 
-If we look at the definition, we recognise that the set that we just defined - *R* does not contain itself and therefore it belongs there as well.
+If we look at the definition, we recognize that the set that we just defined - *R* does not contain itself and therefore it belongs there as well.
 
 ![Russel's paradox - option one](russells_paradox_2.svg)
 
@@ -303,9 +303,9 @@ Notice that the function is invertible, that is if you flip its arrows you get a
 
 ![Opposite colors](isomorphism_one_one.svg)
 
-Invertible functions are called *isomorphisms*. When there is an invertible function between two sets we can say that the sets are *isomorphic*. For example, the temperature measured in Celcius is isomorphic to the temperature, measured in Fahrenheit.
+Invertible functions are called *isomorphisms*. When there is an invertible function between two sets we can say that the sets are *isomorphic*. For example, the temperature measured in Celsius is isomorphic to the temperature, measured in Fahrenheit.
 
-More formally, two sets **R** and **G** are isomorphic, or **R ≅ G** if there exist functions **f: G → R** and its reverse **g R → G**, such that **f ∘ g = idR** and **g ∘ f = idG** (notice how the identity function comes in handy).
+More formally, two sets **R** and **G** are isomorphic, or **R ≅ G** if there exist functions **f: G → R** and its reverse **g R → G**, such that **f ∘ g = id R** and **g ∘ f = id G** (notice how the identity function comes in handy).
 
 Isomorphism and equality
 ---
@@ -327,7 +327,7 @@ Isomorphisms Between Singleton Sets
 
 Between any two singleton sets, we may define the only possible function.
 
-![The only possible function between sungletons](singleton_function.svg)
+![The only possible function between singletons](singleton_function.svg)
 
 The function is invertible, which means that all singleton sets are isomorphic to one another.
 

_chapters/02_category.md

@@ -13,7 +13,7 @@ Products
 
 In the previous chapter, we needed a way to make a set that is a composite of other sets that we already have. For example, when we discussed mathematical functions, we couldn't define **+** and **-** functions, because we only knew about functions that take one argument. When we talked about programming, we talked about the primitive types, `Char` and `Number`, and we mentioned that most of the types are composite types. So how do we construct those?
 
-The simplest composite type, of the sets **B**, which contains **b**'s and the set**Y**, which contains **y**'s is the the *product* or **B** and **Y**.
+The simplest composite type, of the sets **B**, which contains **b**'s and the set**Y**, which contains **y**'s is the *product* or **B** and **Y**.
 
 ![Product parts](product_parts.svg)
 
@@ -30,7 +30,7 @@ Products as Objects
 
 We established that in programming sets resemble types and functions resemble functions. Products, in this case, are like stripped-out *classes* (also called *records* or *structs*). The composite sets (the ones which form the product) are just the class's fields (also called *members*). The functions for accessing them are like what programmers call *getter methods*.
 
-For example, the famous OOP example of `Person` class with `name` and `age` fields is nothing more than a product of the set of strings, and the sets of numbers (we still haven't defined strings and lists in terms of set theory but we will get to that). Objects with more than two values can be expressed as products the composites of which are themselves products.
+The famous example of object-oriented programming of a `Person` class with `name` and `age` fields is nothing more than a product of the set of strings, and the sets of numbers (we still haven't defined strings and lists in terms of set theory but we will get to that). Objects with more than two values can be expressed as products the composites of which are themselves products.
 
 Using Products to Define Numeric Operations
 ---
@@ -41,12 +41,12 @@ Joking, here it is.
 
 ![The plus function](plus.svg)
 
-Note that there are languages where the *pair* datastructure (also called a *tuple*) is a first-level construct, and multi-argument functions are really implemented in this way. 
+Note that there are languages where the *pair* data structure (also called a *tuple*) is a first-level construct, and multi-argument functions are really implemented in this way. 
 
 Defining products in Terms of Sets 
 ---
 
-Now we will define the abstract concept of a product of two sets sets in terms of sets themselves. It is not hard: the product of two sets **Y** and **B** is just the set of all possible *ordered pairs*, which contain one element of the set **Y** and one element of the set **B**. Or formally speaking **Y x B = { (y, b) }** where **y ∈ Y, b ∈ B** (**∈** means "is an element of").
+Now we will define the abstract concept of a product of two sets in terms of sets themselves. It is not hard: the product of two sets **Y** and **B** is just the set of all possible *ordered pairs*, which contain one element of the set **Y** and one element of the set **B**. Or formally speaking **Y x B = { (y, b) }** where **y ∈ Y, b ∈ B** (**∈** means "is an element of").
 
 ![A pair](pair.svg)
 
@@ -122,7 +122,7 @@ Like with the product, there is a low-level way to express a sum using sets alon
 
 ![A member of a coproduct, examined](coproduct_member_set.svg)
 
-But again, this distinction is only rellevant only when the two sets have common elements.
+But again, this distinction is only relevant only when the two sets have common elements.
 
 
 Defining Sums in Terms of Functions
@@ -162,7 +162,7 @@ If we have to compare the concepts of sum or and product we will find out that t
 Actually, the two concepts are captured by one and the same external diagram, just the arrows are flipped - many-to-one relationships become one-to-many and the other way around.
 
 
-That's normal right? After all, **AND** *is* the opposite of **OR**. The connection between the two has always been there, evidenced, for example, by the De Morgan's law, citing that **NOT (A AND B) ↔ (NOT A) OR (NOT B)** (or to put it in everyday language, "If either A or B is false, then (and only then) A *and* B is also false). But only with category theory, this connection can be expressed in such a concise way:
+That's normal right? After all, **AND** *is* the opposite of **OR**. The connection between the two has always been there, evidenced, for example, by the DeMorgan's law, citing that **NOT (A AND B) ↔ (NOT A) OR (NOT B)** (or to put it in everyday language, "If either A or B is false, then (and only then) A *and* B is also false). But only with category theory, this connection can be expressed in such a concise way:
 
 ![Coproduct and product](coproduct_product_duality.svg)
 
@@ -186,7 +186,7 @@ This diagram illustrates how category theory allows us to see the big picture wh
 
 **NB: Do note how the world "Object" is used in both programming languages and in category theory, but for completely different things. The equivalent a categorical object is equivalent to a class in programming language.**
 
-Looking at the table I cannot help but notice the somehow weird, but otherwise completely logical symmetry (or perhaps "reverse symetry") between the the world as viewed through the lense of set theory, and the way it is viewed through the (inverted) lens of cathegory theory:
+Looking at the table I cannot help but notice the somehow weird, but otherwise completely logical symmetry (or perhaps "reverse symmetry") between the world as viewed through the lenses of set theory, and the way it is viewed through the (inverted) lens of category theory:
 
 | Category Theory | Set theory | 
 | ---             | ---        |
@@ -194,7 +194,7 @@ Looking at the table I cannot help but notice the somehow weird, but otherwise c
 | Objects and  Morphisms        | Sets and functions |
 | **N/A**             | Element    | 
 
-By switching to external diagrams, we lose sight of the particular (the elements of our sets), but we have gained the ability to see the whole universe that we have been previously trapped in. The whole realm of sets, can be thought as one category, a programming language can also be thought as a category. The concept of a category allows us to find and analyse similarities between these and other structures.
+By switching to external diagrams, we lose sight of the particular (the elements of our sets), but we have gained the ability to see the whole universe that we have been previously trapped in. The whole realm of sets, can be thought as one category, a programming language can also be thought as a category. The concept of a category allows us to find and analyze similarities between these and other structures.
 
 ![Category theory and set theory compared](set_category.svg)
 
@@ -207,7 +207,7 @@ Defining Categories (again)
 
 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 - most books that focus around a given 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 because everyone else does it. But one thing is for sure - we don't need to study sets in order to understand categories. So now I would like to start over and talk about categories as a first concept. So pretend like it's a new book.
 
-So, a category is a collection of objects (things) where the "things" can be anything you want. Consider, for example, these ~~colourful~~ grey balls:
+So, a category is a collection of objects (things) where the "things" can be anything you want. Consider, for example, these ~~colorful~~ gray balls:
 
 ![Balls](elements.svg)
 
@@ -217,7 +217,7 @@ A category consists of a collection of objects as well as some arrows connecting
 
 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.
+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 colored: 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.
 
@@ -293,17 +293,17 @@ It's important to mark this morphism, because there can be (let's add the very i
 Isomorphisms
 ---
 
-Why do we need to define a morphism that does nothing? It's because morphisms are the basic building blocks of our language and we need this one to be able to speak properly. For example, 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. 
+Why do we need to define a morphism that does nothing? It's because morphisms are the basic building blocks of our language and we need this one to be able to speak properly. For example, once we have the concept of identity morphism defined we can have a category-theoretic definition of an *isomorphism* (which is important, because the concept of an isomorphism is very important for category 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 
+iff there exist morphisms 
 **f: A → B**
 **g: B → A** 
 such that  
-**f • g = idB** 
-**g • f = idA** 
+**f • g = id B** 
+**g • f = id A** 
 
 And here is the same thing expressed with a commuting diagram.
 
@@ -318,7 +318,7 @@ 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.
+2. Each object has to have the identity morphism.
 
 This is it.
 

dictionary.txt

@@ -1,4 +1,7 @@
+Invertible
+invertible
 coproduct
+Coproduct
 morphism
 coproducts
 morphisms
@@ -28,6 +31,7 @@ Monoid
 Monoids
 isomorphism
 isomorphisms
+Isomorphisms
 Hasse
 Antisymmetry
 connexity
@@ -47,3 +51,13 @@ non-abelian
 Dih3
 composable
 forall
+codomain
+Euclidian
+subtypes
+Immanuel 
+iff
+concatenative
+structs
+DeMorgan's
+getter
+OOP