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_chapters/02_category.md
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@ -87,7 +87,7 @@ Depending on our specific case there can be many other such objects.
![Product, external diagram](product_candidates.svg)
So how do we set apart all those "imposter" products from the one true product? Simple - they all can be converted to it. This is true, because:
So how do we set apart the true product from all those "imposter" products? Simple - by using the observation that *they all can be converted to it*, This observation is true, because:
1. By definition, each "impostor" can be converted to both elements of the pair.
2. The pair is nothing more than the sum of its elements.
@ -95,7 +95,9 @@ More formally, in order for a set **I** to serve as an impostor for the product
![Product, external diagram](products_morphisms.svg)
Notice that this definition does not rule out the sets which are isomorphic to the product - when we represents things using functions, an isomorphism is the same as equality.
In category theory, this type of property that a given object might posess (participating in a structure such that all similar objects can be converted to/from it) is called a *universal property*.
Notice that this definition does not rule out the sets which are isomorphic to the product - when we represents things using universal properties, an isomorphism is the same as equality.
Sums
===
@ -142,7 +144,7 @@ You might already notice that this definition is pretty similar to the previous
![Coproduct, external diagram](coproduct_candidates.svg)
All these sets are, express relationships which are more vague than the simple sum and therefore given any such set there would exist a function from the sum to it.
All these sets are, express relationships which are more vague than the simple sum and therefore given any such set there would exist a function from the sum to it.
For example, there exist a trivial function between the set **Y \| B** and the set **Y \| B \| R**.
@ -150,6 +152,7 @@ For example, there exist a trivial function between the set **Y \| B** and the
This diagram captures the **OR** relation in the same way as the previous one captures the essence of **AND**.
Duality
===
@ -158,10 +161,9 @@ If we have to compare the concepts of sum or and product we will find out that t
- The *product* of two sets is related to an element of the first one *and* one element of the second one.
- A *sum* of two sets is related to an element of the first one *or* one element of the second one.
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.
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**).
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:
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:
![Coproduct and product](coproduct_product_duality.svg)
@ -263,7 +265,7 @@ Functional composition is special not only because you can take any two morphism
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)**.
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).

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6
_chapters/04_order.md
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@ -263,14 +263,14 @@ For sets, an isomorphism means just that the functions are inverse of each other
Birkhoff's representation theorem
---
So far, we saw two different partial orders, one based on color mixing, and one based on number division, which can be represented by the inclusion orders of all possible combinations of sets of some *basic elements* (the primary colors in the first case, and the prime numbers (or prime powers) in the second one). Many other partial orders can be defined in this way. Which ones exactly is a question that is answered by an amazing result called *Birkhoff's representation theorem*. They are the partial orders that meet the following two criteria:
So far, we saw two different partial orders, one based on color mixing, and one based on number division, which can be represented by the inclusion orders of all possible combinations of sets of some *basic elements* (the primary colors in the first case, and the prime numbers (or prime powers) in the second one.) Many other partial orders can be defined in this way. Which ones exactly is a question that is answered by an amazing result called *Birkhoff's representation theorem*. They are the partial orders that meet the following two criteria:
1. All elements have *joins* and *meets* (those partial orders are called *lattices*, by the way)
2. Those *meet* and *join* operations *distribute* over one another, that is **x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z)**.
2. Those *meet* and *join* operations *distribute* over one another, that is if we denote joins as meets as **∨** or **∧**, then **x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z)** (those are called *distributive lattices*.)
(Just to note that this result is only proven for *finite* lattices, so it might not be valid for the numbers all the way to infinity. But it would be valid for any subset of them.)
I won't go into details about this result, I would only mention that the "prime" elements with which we can construct the inclusion order are the ones that are not the *join* of any other elements (for that reason, they are also called *join-irreducible* elements).
I won't go into details about this result, I would only mention that the "prime" elements with which we can construct the inclusion order are the ones that are not the *join* of any other elements (for that reason, they are also called *join-irreducible* elements.)
By the way, the partial orders that are *NOT* distributive lattices are also isomorphic to inclusion orders, it is just that they are isomorphic to inclusion orders that do not contain all possible combinations of elements.

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_chapters/05_logic.md
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@ -6,14 +6,14 @@ title: Logic
Logic
===
Now let's talk about one more *seemingly* unrelated topic, just so we can surprise ourselves that it is all connected. This time I will not merely transport you to a different branch of mathematics, but an entirely different discipline, namely *logic*. Or, to be more precise, intuitionistic logic. This discipline may seem to you as detached from what we have been talking about as it possibly can, but it is actually very close.
Now let's talk about one more *seemingly* unrelated topic, just so we can surprise ourselves that it is all connected. This time I will not merely transport you to a different branch of mathematics, but an entirely different discipline, namely *logic*. This discipline may seem to you as detached from what we have been talking about as it possibly can, but it is actually very close.
What is logic
===
Logic is the science of the *possible*. As such, it is at the root of all other sciences, all of which are sciences of the *actual*, i.e. of that which really exists. For example, if the laws of physics show how particles behave in our universe (or multiverse), we might use logic to deduce how would they behave in any universe that is possible to exist (under a given set of postulates, real or made up). The key is that everything that is actual is also possible, and so all sciences are (or should be) be based on logic. But at the same time (and that's sometimes overlooked) nothing real is purely logical.
Logic is the science of the *possible*. As such, it is at the root of all other sciences, all of which are sciences of the *actual*, i.e. of that which really exists. For example, if the laws of physics show how particles behave in our universe (or multiverse), we might use logic to deduce how would they behave in any universe that is possible to exist (under a given set of postulates, whether real or made up). The key is that everything that is actual is also possible, and so all sciences are (or should be) be based on logic. But at the same time (and that's sometimes overlooked) nothing real is purely logical.
Logical proofs
Proofs
---
OK, let's be more specific. Logic aims to study the *rules* by which knowing one thing leads you to conclude or (*prove*) that some other thing is also true, regardless of the things's domain (e.g. scientific discipline) i.e. by only referring to their form.
@ -66,7 +66,8 @@ As a result of this is that we can compose propositions with multiple levels of
Modus ponens
---
As an example of a proposition that contains multiple levels of nesting (and a great introduction of the subject in its own right), consider one of the most famous propositions ever, namely *modus ponens*.
As an example of a proposition that contains multiple levels of nesting (and a great introduction of the subject in its own right), consider one of the oldest (it was
alredy known by Stoics at 3rd century B.C.) and most famous propositions ever, namely *modus ponens*.
Modus ponens is a proposition that states that if **A** is true and if also **A → B** is true (if **A** implies **B**), then **B** is true as well. For example, if we know that "Socrates is a human" and that "Being human implies being mortal", we also know that "Socrates is mortal".
@ -254,17 +255,20 @@ It might help you to remember that **p → q** (**p** implies **q**) is true whe
| False | True | **True** | True | True | **True** |
| False | False | **True** | True | False | **True** |
You can see that truth tables don't scale well for longer problems.
Proving results by axioms/rules of inference
---
p → q
Here is a way to derive the above result using axioms and rules of inference.
if p is true
p ∨ -p ( a ∨ -a)
p ∨ -q
q ∨ -p (p → q)
if p is false
(-p ∨ q) ∨ -p ( a → a ∨ b)
-p ∨ q
(-p ∨ q) ∨ (-p ∨ q) ( a → a ∨ b)
-p ∨ q (a ∨ a → a)
Intuinistic logic
@ -287,11 +291,13 @@ So, due to the reasons outlined above, intuinistic logic is not bivalent, we can
But one thing that we still do have propositions that are "true" in the sense that a proof for them is given - the primary propositions.
So with some caveats (which we will see later) the dichotomy between the existence or non-existence of a proof for a given proposition may be taken as similar to the proposition being true or false - there either is a proof of a given proposition or there isn't.
So with some caveats (which we will see later) the bivalence of the existence or non-existence of a proof for a given proposition may be taken as similar to the proposition being true or false - there either is a proof of a given proposition or there isn't.
![The proved/unproved dichotomy](proved_unproved.svg)
This is known as the as the Brouwer–Heyting–Kolmogorov (BHK) interpretations of intuinistic logic
This is known as the as the Brouwer–Heyting–Kolmogorov (BHK) interpretations of intuinistic logic.
The original formulation of the BHK interpretation does not depend on any particular mathematical theory, but here we chose to illustrate it using the language of set theory.
The **and** and **or** operations
---
@ -311,7 +317,7 @@ In this case, saying that **A** implies **B** (**A → B**) would just mean that
And the *modus ponens* rule of inference is just the fact that if we have a proof of **A** we can call this function (**A → B**) to obtain a proof of **B**.
**Task:** In order for this to work, you need to define the function in terms of sets. Like with *and* you can do this by using the concept of a pair, work out the details.
Note that in order for this to work, we we need to define the function itself in terms of sets i.e. we need to have a set representing **A → B** for each **A** and **B**. This is possible (using the concept of a *pair*) but we won't do it now.
The *negation* operation
---
@ -346,7 +352,9 @@ This law is valid in classical logic and is true when we look at it in terms of
Logics as categories
===
Leaving the differences between intuinistic and classical logics aside, the BHK interpretation is interesting because it provides a bit of the higher-level view of logic, that we need in order to define it in terms of category theory. And this is just what we will attempt here.
Leaving the differences between intuinistic and classical logics aside, the BHK interpretation is interesting because it provides a bit of the higher-level view of logic, that we need in order to represent it in terms of category theory.
This representation of logic, the one that does nor rely on formulas and propositions, but on objects and operations which obey is sometimes called an *algebraic* representation, *algebraic* being an umbrella term describing all structures that can be represented using category theory, like groups and orders
The Curry-Howard correspondence
---
@ -372,18 +380,18 @@ Defining logic in a categorical language, however, is a more complex question.
![Logic as a category](logic_curry_category.svg)
In order to answer it, we have to ennumerate the criteria that a given category has to adhere to in order for it to be "logical". These criteria have to guarantee that the category has objects that correspond to all valid logical propositions and no objects that correspond to invalid ones. We won't describe these categories directly (by the way, they are called *Cartesian closed* categories). Instead we would start with a simpler structures that we already examined - orders.
In order to answer it, we have to enumerate the criteria that a given category has to adhere to in order for it to be "logical". These criteria have to guarantee that the category has objects that correspond to all valid logical propositions and no objects that correspond to invalid ones. We won't describe these categories directly (by the way, they are called *Cartesian closed* categories). Instead we would start with a similar but simpler structures that we already examined - orders.
Logics as orders
---
Once you know that logics form categories, you might think you've seen it all. You might think that can already skip the rest of the chapter, or throw the whole book away, while putting on your sunglasses and jumping in your convertible car. But you would be missing out, as there is an even simple structure that captures all of the concepts that we saw, while providing some interesting insights on what logic is.
Order theory captures all of the concepts that we saw, while providing some interesting insights on what logic is. This is why it is the usual default algebraic representation of simpler logics such as the ones we are introducing here.
If we assume that there is only one way to go from proposition **A**, to proposition **B** (or there are many ways, but they are equivalent), then logic is not only a category, but a *preorder* (a category that has just one morphism between any two objects).
If we assume that there is only one way to go from proposition **A**, to proposition **B** (or there are many ways, but they are equivalent), then logic is not only a category, but a *preorder* (which as we said is a category that has just one morphism between any two objects) in which the relationship "bigger than" is taken to mean "implies".
![Logic as a preorder](logic_preorder.svg)
Furthermore, if we count propositions that follow from each other (or sets of propositions that have the same truth value and can be proven by the same proof) as equivalent, then logic is a *partial order*.
Furthermore, if we count propositions that follow from each other (or sets of propositions that have the same truth value and can be proven by the same proof) as equivalent, then logic is a proper *partial order*.
![Logic as an order](logic_order.svg)
@ -391,18 +399,97 @@ And so it can be represented by a Hasse diagram, yey.
![Logic as an order](logic_hasse.svg)
By the way, this representation of logic is sometimes called an *algebraic* representation.
Now back to the question we asked before: exactly which orders represent logic and what laws does an order have to obey so it is isomorphic to a logic? We will answer this question as we examine the main logical constructs again, this time in the context of logic.
The **and** and **or** operations
---
By now you probably realized that the **and** and **or** operations are the bread and butter of logic (although it's not clear which is which). In orders, these operations are represented by the **meet** and **join** operations, correspondingly.
In logic you can use the **and** and **or** operations for all propositions and here comes the first criteria for an order to be logical - it has to have **meet** and **join** operations for all elements. Incidentally we already know that such orders are called *lattices*.
One more important law concerning the **and** and **or** operations that is not always present in the **meet**-s and **join**-s concerns the connection between the two, i.e. way that the **and** and **or** operations distribute, over one another.
![The distributivity operation of "and" and "or"](logic_distributivity.svg)
Lattices that obey this law are called *distributive lattices*, so logical systems are represented by distributive lattices.
In the previous chapter we said that *distributive lattices* are isomorphic to *inclusion orders* i.e. orders which contain all combinations of sets of a given number of elements and, circling back to the set-theoretic BHK interpretation, we see that the two theorems are really saying the same thing - the elements which participate in the inclusion are our prime propositions and are the the and the inclusions are all combinations of these elements, in an **or** relationship (for simplicity's sake, we are ignoring the **and** operation.)
![A color mixing poset, ordered by inclusion](logic_poset_inclusion.svg)
**NB: The symbols that we are using for *and* and *or* logical operations are flipped when compared tothe arrows of the arrows they don't ∧ is *and* and ∨ is *or*. That is because the logic order is drawn upside-down at some places.**
In categorical logic, the **and** and **or** operations are represented (unsurprisingly) by *products* and *coproducts*.
The *negation* operation
---
In order to prove that our color-mixing lattice represents logic completely, we have to identify which objects correspond to the values **True** and **False**.
A trivial result in logic, called *the principle of explosion*, states that if we have a proof of **False** (or if "**False** is true" if we use the terminology from classical logic), than any and every statement can be proven. And it is also obvious that no true statement implies False. So here is it.
![False, represented as a Hasse diagram](lattice_false.svg)
Circling back to the set-theoretic BHK interpretation, we see that the empty set fits both conditions.
![False, represented as a Hasse diagram](lattice_false_bhk.svg)
Conversely, the proof of **True** (or the statement that "**True** is true") is trivial and doesn't say anything, so *nothing follows from it*, but at the same time it follows from every other statement.
![True, represented as a Hasse diagram](lattice_true.svg)
So **True** and **False** are just the *maximum* and *minimum* objects of our order.
![The whole logical system, represented as a Hasse diagram](lattice_true_false.svg)
And in categorical logic, the **True** and **False** objects are represented by what we call *terminal* and *initial* objects. This is another example of the categorical concept of duality - **True** and **False** are dual to each other, just like **and** and **or**.
The *implies* operation
---
Finally, if a lattice really is isomorphic to a set of propositions, we it also has to have *function objects* i.e. there needs to be a rule that identifies a unique object **A → B**, for each pair of objects **A** and **B**, such that all axioms of intuinistic logic are followed.
The *negation* operation
And we would define the rule using categorical language - by recognizing a structure, that consists of set of relations between objects in which (A → B) plays a part.
![Implies operation](implies.svg)
This structure is actually a categorical reincarnation our rule of inference, called *modus ponens*, stating that **A ∧ (A → B) → B**. This rule is the essence of the **implies** operation and because we already know how the operations that it contains (**and** and **implies**) are represented in our lattice, we can directly "categorize" it and use it as a definition, saying that **(A → B)** is the object which has the following relations to objects **A** and **B**.
![Implies operation with impostors](implies_modus_ponens.svg)
This definition is not complete, however, because **(A → B)** is *not the only object* that fits in this formula. For example, the set **A → B ∧ C** is also one such object, as is **A → B ∧ C ∧ D**. So how do we set apart the real formula from all those "imposter" formulas? If you remember the definitions of the *categorical product* or the definiton of the *meet* operation in orders (as *meet* is an instance of a categorical product) you would already know where this is going: we define the function object using a *universal property* by saying that all other formulas that can be in **A ∧ X → B** point to **(A → B)**. And are below **(A → B)** in a Hasse diagram.
![Implies operation with universal property](implies_universal_property.svg)
Or, we use the logic terminology, we say that **A → B ∧ C** and **A → B ∧ C ∧ D** etc. are all "stronger" results than **A → B**.
Let's try to test if this definition captures the concept correctly by examining a few special cases.
For example, let's take **A** and **B** to be the same object. In this case we have the formula **A ∧ X → A** (where **X** is **A → A**). But the *meet* of **A** and any other object would always be below **A**. So this formula is always for all **X**. So the biggest object that fits the description is just the biggest object there is i.e. **True**.
![Implies identity](implies_identity.svg)
This corresponds to the identity axiom in logic, that states that everything follows from itself.
And by the similar logic we can see easily that if we take **A** to be any object that is below **B**, then **(A → B)** will also correspond to the True object.
![Implies when A follows from B](implies_b_follows.svg)
So if we have **A → B** then **(A → B)** is true.
Let's take one more example - what if **B** is lower than **A**. In this case the highest object that fits the formula **A ∧ X → B** is... **B** itself. **B** fits the formula (because **A ∧ B → B**) and is definitely the highest object that does so.
![Implies when B follows from A](implies_a_follows.svg)
This translated to logical language, says that if **A → B** and **B → A**, then
Conclusion
---
So this is the final condition for an order to be a representation of logic - for each pair **A** and **B**, it has to have a unique object **X** which obey the formula **A ∧ X → B** and the universal property. In category theory this object is called the *exponential object*.
By the way, distributivity follows from this criteria, so we are left with just these two points: Logic is represented by an order that has with *meet and joins* and a *functional object*.
Which is the shortest definition of logic there is.

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About
===
About category theory
---
> Category Theory is considered by many to be an involved domain of study to get into. It becomes a ground for unification of mathematical ideas for a wide variety of domains. And the way it achieves this is by taking an abstract vantage point on the ideas, their properties, and processes in these disciplines. This viewpoint gives it the ability to reason about analogies happening in distinct domains and draw rigorous analogies across the patterns happening in them in a rigorous way.
Source: https://github.com/prathyvsh/category-theory-resources
Category theory has a variety of applications:
- It is used in **programming language theory**, for example the usage of monads in functional programming.
- It may also be used as an axiomatic foundation for mathematics, as an alternative to set theory and other proposed foundations.
- It is used in other scientific disciplines e.g. quantum mechanics:
> The conference series Quantum Physics and Logic (QPL), founded by Peter Selinger in 2003 under a different name (but with the same abbreviation!), was a particularly important forum for the development of the key results leading up to this book. In fact, the first paper about diagrammatic reasoning for novel quantum features (Coecke, 2003) was presented at the first QPL. The categorical formalisation of this result (Abramsky and Coecke, 2004), now referred to as categorical quantum mechanics, became a hit within the computer science semantics community, and ultimately allowed for several young people to establish research careers in this area. **Top computer science conferences (e.g. LiCS and ICALP) indeed regularly accept papers on categorical quantum mechanics, and more recently leading physics journals (e.g. PRL and NJP) have started to do so too**.
Source: Picturing Quantum Processes by Bob Coecke (Cambridge University Press - 2017)
About this book
---
I am writing a primer in category theory and of various related concepts in "higher" mathematics that is *really* accessible to people with no prior exposure to the subject without being dumbed down, by utilizing visual explanations.
My book will serve as chapter 0, going through the material that any other similar introductory book covers and enabling people to later go through any other introductory books effortlessly.
With the
Similar resources
===
To my knowledge, Category Theory Illustrated does not have any *direct* competition - although there are some books that are similar to it in terms of *goals*, no book is similar in terms of *execution*.
A good list of category theory introductions can be found here:
https://github.com/prathyvsh/category-theory-resources
In this document, I will concentrate on the following two books:
- Category Theory for Programmers by Bartosz Milewski (self-published- 2018) - Full text: https://github.com/hmemcpy/milewski-ctfp-pdf/
- Category Theory for the Sciences by David I. Spivak (MIT press - 2014) - Free version available at: http://math.mit.edu/~dspivak/CT4S.pdf
I chose those two because:
1. They are the closest to my book that I could find.
2. Their readers (and authors) represent the two biggest target audiences for my book - programmers (nerds) and scientists (academics).
I will continue with examining the points which make my work unique compared to those books.
Here I am attaching excerpts from both of them, as well as one from CTI, that explain the concept of an *order*
Category Theory for Programmers
---
> A preorder is a set with an ordering relation between its elements that’s traditionally written as <= (less than or equal). The “pre” in preorder is there because we’re only requiring the relation to be transitive and reflexive but not necessarily antisymmetric (so it’s possible to have cycles).
> A set with the preorder relation gives rise to a category. The objects are the elements of this set. A morphism from object a to b either doesn’t exist, if the objects cannot be compared or if it’s not true that a <= b; or it exists if a <= b, and it points from a to b. There is never more than one morphism from one object to another. Therefore any hom-set in such a category is either an empty set or a one-element set. Such a category is called thin.
> It’s easy to convince yourself that this construction is indeed a category: The arrows are composable because, if a <= b and b <= c then a <= c; and the composition is associative. We also have the identity arrows because every element is (less than or) equal to itself (reflexivity of the underlying relation).
Source: https://bartoszmilewski.com/2015/10/28/yoneda-embedding/
Category Theory for the Sciences
---
> 3.4.1 Definitions of preorder, partial order, linear order
> Definition 3.4.1.1. Let S be a set and R Ď S ˆS a binary relation on S; if ps,s1q P R
we will write s ≤ s1. Then we say that R is a preorder if, for all s,s1,s2 P S we have
> Reflexivity: s ≤ s, and
> Transitivity: if s ≤ s1 and s1 ≤ s2, then s ≤ s2.
> We say that R is a partial order if it is a preorder and, in addition, for all s,s1 P S we
have
> Antisymmetry: If s ≤ s1 and s1 ≤ s, then s “ s1.
> We say that R is a linear order if it is a partial order and, in addition, for all s,s1 P S we have
> Comparability: Either s ≤ s1 or s1 ≤ s.
> We denote such a preorder (or partial order or linear order) by pS,≤ q.
> Exercise 3.4.1.2.
> a.) Decide whether the table to the left in Display (3.9) constitutes a linear order.
> b.) Show that neither of the other tables are even preorders.
Source: http://math.mit.edu/~dspivak/CT4S.pdf (page 93)
Orders also discussed on page 132 in the same link
Category Theory Illustrated
---
> The most straightforward type of order that you think about is linear order i.e. one in which every object has its place depending on every other object. In this case the ordering criteria is completely deterministic and leaves no room for ambiguity in terms of which element comes before which. For example, ordering the colors by the length of their waves (or by how they appear in the rainbow).
![Linear order](_chapters/04_order/linear_order.svg)
> In most programming languages, we can order objects linearly by providing a function which, given two objects, tells us which one of them is "bigger" (comes first) and which one is "smaller".
```
[1, 3, 2].sort((a, b) => {
if (a > b) {
return true
} else {
return false
}
})
```
> But in order for such a function to really define an order (e.g. give the same output every time, independent of how the objects were shuffled initially), it has to obey several rules.
> Incidentally, (or rather not incidentally at all), these rules are nearly equivalent to the mathematical laws that define the criteria of the relationship between elements in an order i.e. those are the rules that define which element can point to which. Let's review them.
> Reflexivity
> Let's get the most boring law out of the way - each object has to be bigger or equal to itself, or **a ≤ a** (the relationship between elements in an order is commonly denoted as **≤** in formulas, but it can also be represented with a simple arrow from first object to the second).
![Reflexivity](_chapters/04_order/reflexivity.svg)
> No special reason for this law to be so, except that the "base case" should be covered somehow.
> We can formulate it the opposite way too and say that each object should *not* have the relationship to itself, in which case we would have a relation than resembles *bigger than*, as opposed to *bigger or equal to* and a slightly different type of order, sometimes called a *strict* order.
> Transitivity
> The second law is maybe the least obvious, (but probably the most essential) - it states that if object **a** is bigger than object **b**, it is automatically bigger than all objects that are smaller than object **b** or **a ≤ b and b ≤ c ➞ a ≤ c**.
![Transitivity](_chapters/04_order/transitivity.svg)
> This is the law that to a large extend defines what an order is: if I am better at playing soccer than my grandmother, then I would also be better at it than my grandmother's friend, whom she beats, otherwise I wouldn't really be better than her.
> Antisymmetry
> The third law is called antisymmetry and it states that the function that defines the order should not give contradictory results (or in other words you have **x ≤ y** and **y ≤ x** only if **x = y**).
![antisymmetry](_chapters/04_order/antisymmetry.svg)
> It also means that no ties are permitted - either I am better than my grandmother at soccer or she is better at it than me.
> Totality
> The last law is called *totality* (or *connexity*) and it mandates that all elements that belong to the order should be comparable - **a ≤ b or b ≤ a**. That is, for any two elements, one would always be "bigger" than the other.
> By the way, this law makes the reflexivity law redundant, as reflexivity is just a special case of totality when **a** and **b** are one and the same object, but I still want to present it for reasons that will become apparent soon.
![connexity](_chapters/04_order/connexity.svg)
> You might say that this law is not as self-evident as the rest of them - if you think about different types of real-life objects that we typically order, you would probably think of some situations in which it does not apply. For example, if we aim to order all people based on soccer skills there are many ways in which we can rank a person compared to their friends their friend's friends etc. but there isn't a way to order groups of people who never played with one another.
Diagrams
---
Category theory is very *visual*. Diagrams are not merely an illustration of it's concept, but are often the very language that is used to define those concepts e.g. defining a more advanced concept such as *natural transformation* without diagrams is practically impossible. So having a lot of diagrams is essential for people who are inexperienced to understand the concepts (it is essential for people who are experienced as well, they just can draw the diagrams themselves).
However, books on category theory typically have as many diagrams as other math textbooks. I considered that (and still do) a huge missed opportunity for making the subject more approachable, which was the original motivation for the creation of CTI.
Besides being more in quantity, the diagrams of CTI are many different kinds combining and combine different prioms from traditional communication design in order, such as the use of color, for example, in order to illuminate the different subjects and abstractions that I am covering.
The diagrams in CTI had received universal praise from many audiences. The university professor Gonzalo Casas used some of them for his lectures on robotic fabrication at ETH Zurich.
https://github.com/compas-teaching/COMPAS-II-FS2021/tree/main/lecture_06
https://raw.githubusercontent.com/compas-teaching/COMPAS-II-FS2021/main/lecture_06/lecture_06.pdf
Although there are not any category theory introductions that make extensive use of diagrammatic language, there are some books that use it to introduce other subjects:
- Picturing Quantum Processes by Bob Coecke and Aleks Kissinger (Cambridge University Press - 2017) that uses it to introduce quantum mechanics
By the way, I showed Bob the first few chapters of CTI and he liked them :)
- Visual Group Theory by Nathan Carter (MATHEMATICAL ASSOCIATION OF AMERICA - 2009) sample: http://www.mathcs.emory.edu/~dzb/teaching/421Fall2014/VGT-Ch-1-2.pdf
- Also, there is the blog https://graphicallinearalgebra.net/ that introduces linear algebra visualy.
Competency of the readers
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Coverage of secondary topics
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Logic
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Target audience
===
- Programmers, who are curious about category theory, because they are into functional programming.
- Math undergraduates, who are struggling with reading academic texts.
- Any other kind of nerds who want to learn some mathematics just for fun.
- Children.
People with programming background
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People with academic background
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People who are generally curious
---
Feedback
===
I have reseived a lot of positive feedback on CTI, and a lot of people shared it on social media. Some comments and links follow:
Comments
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> This is the most beautiful and clearly written introduction to categories I’ve ever seen. Highly recommended.
https://twitter.com/y0b1byte/status/1417567589241339912
> Great stuff man! Internet needs more accessible Category Theory expositions.
https://twitter.com/prathyvsh/status/1253303971185221634
> What a wonderful resource this Illustrated Category Theory series is. It's an easy(ier), concise, on-ramp to the topic, that would make a nice introduction. A good resource for sharing. I look forward to the remaining topics. Thanks Boris!
https://www.reddit.com/r/haskell/comments/mhs3ov/category_theory_illustrated/gt9xbp9?utm_source=share&utm_medium=web2x&context=3
> Awesome website! Well written and crystal clear. It's truly a feat to explain simply such a complex topic.
https://news.ycombinator.com/item?id=26659190
> My God that site is beautiful. If only every "maths" site could looks like this, I'd have won a field medal!
https://news.ycombinator.com/item?id=26660369
> I saw your site on Hacker News. I just wanted to send a note saying it's beautiful! I have been writing Haskell (PureScript really) for years, and I've been wanting to get more into Category Theory. I'm excited to read through your site, and thanks for taking the time to create it!
(received by email)
> Thank you for creating Category Theory Illustrated, for me the book makes the concepts easier to understand and build an intuition.
(received by email)
Discussions:
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Lobsters: https://lobste.rs/s/bc11fo/category_theory_illustrated_monoids
HackerNews: https://news.ycombinator.com/item?id=26658111
Twitter shares:
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https://twitter.com/search?q=category-theory-illustrated&src=typed_query&f=live