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title: About
---
{: style="text-align: right"}
*"Try as you may,*
{: style="text-align: right"}
*you just can't get away,*
{: style="text-align: right"}
*from mathematics"*
{: style="text-align: right" }
Tom Lehrer
About this book
===
This book is a product of my own endeavour of understanding category theory. 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.
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.
In this book, we will visit those formalisms and along the way, we would see all other kinds of mathematical objects, viewed under the prism of categories.
Acknowledgements
===
Thanks to my wife Dimitrina, who has no issue with taking after our daughter while I sit here and write my book.
Thanks to my high-school arts teacher, Mrs Georgieva who told me that I have some tallent, but I have to actually work to make it happen.
Thanks to Prathyush Pramod who encouraged me to finish the book and is also helping me out with it.
And also to everyone else who submitted feedback and helped me fix some of the numerous errors that I made - knowing myself, I know that there are a lot more.

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_chapters/02_category.md
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@ -17,7 +17,7 @@ The simplest composite type, of the sets **B**, which contains **b**'s and the s
![Product parts](product_parts.svg)
It is the set of *pairs* of **b**'s and **y**'s.
The product is the set of *ordered pairs* of **b**'s and **y**'s. It is denoted **B x Y** and it comes equipped with two functions for retrieving the **b** and the **y** from each **(b, y)**.
![Product](product.svg)
@ -25,6 +25,8 @@ The product is denoted **B x Y**, and it comes equipped with two functions for r
**Question**: Why is this called a product? Hint: How many elements does it have?
The Cartesian product was first defined by the mathematician and philosopher René Descartes as a basis for the Cartesian coordinate system. But as we shall see, it has many other use-cases.
Products as Objects
---
@ -66,7 +68,6 @@ Discovered in 1921 Kazimierz Kuratowski, this one uses just the component of the
![A pair, represented by sets](pair_as_set_1.svg)
Defining products in Terms of Functions
---
@ -88,7 +89,7 @@ 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.
@ -96,7 +97,9 @@ 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 represent 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
===
@ -151,6 +154,7 @@ 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
===
@ -159,10 +163,9 @@ If we compare the concepts of *sum* and *product*, we will find out that they ar
- 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 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:
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)
@ -184,7 +187,7 @@ What other categories, or applications of category theory are there, other than
This table illustrates how category theory allows us to see the big picture when it comes to sets and similar structures - when we are at the realm of sets we can view the set as a collection of individual elements. In category theory we don't have such notion, but we saw how taking this notion away allows us to define concepts such as the sum and product sets in a whole different and more general way.
**NB: Do note how the world "Object" is used in both programming languages and in category theory, but for completely different things. A categorical object is equivalent to a class in programming language.**
**NB: Do note how the word "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 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:
@ -227,6 +230,8 @@ Why that is true is pretty obvious if we go back to set theory for a second. (O
![Two sets connected with multiple functions](set_arrows.svg)
There are some types of categories in which only one morphism between two objects is allowed (or one in each direction), but we will talk about them later (they are also called orders).
Composition
---
@ -234,7 +239,6 @@ One of the few or maybe even the only requirement for a structure to be called a
![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))**.)

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@ -137,20 +137,20 @@ The chains in an order don't have to be completely disconnected from each other
The above set is not linearly-ordered - although the connection establishes the relationship between **D** and **G** (**D ≤ G**) and although the relationship between **F** and **G** is known as well (**F ≤ G**), the relationship between **D** and **F** is *not* known. Any element can be bigger than the other one.
Maximum and minimum
Greatest and least
---
Although posets don't give us definitive answer to who is better than who, some of them still can give us an answer to the more important question (in sports, as well as in other domains), namely *who is number one*, who is the champion, the player who is better than anyone else, or more generally the element that is bigger than any other element.
We call such elements the *maximum element* and some (not all) partial orders do have such element - in our last diagram **M** is the maximum element, in this diagram, the green element is the biggest one.
We call such elements the *greatest element* and some (not all) partial orders do have such element - in our last diagram **M** is the greatest element, in this diagram, the green element is the biggest one.
![Join diagram with one more element](join_additional_element.svg)
Sometimes we have more than one elements that are bigger than all other elements, in this case none of them are maximum.
Sometimes we have more than one elements that are bigger than all other elements, in this case none of them is the greatest.
![A diagram with no maximum element](non_maximal_element.svg)
![A diagram with no greatest element](non_maximal_element.svg)
In addition to the maximum element, a partial order may also have a minimum (smallest) element, which is defined in the same way.
In addition to the greatest element, a partial order may also have a least (smallest) element, which is defined in the same way.
Joins
---
@ -165,9 +165,9 @@ There can be multiple elements bigger than **A** and **B** (all elements that ar
Given any two elements in which one is bigger than the other (e.g. **A ≤ B**), the join is the bigger element (in this case **B**).
In a totally ordered set, the *join* of any subset of elements is just their the *maximum* element.
In a totally ordered set, the *join* of any subset of elements is just their the *greatest* element.
Like with the maximum element, if two elements have several upper bounds that are equally big, then none of them is a *join* (a join must be unique).
Like with the greatest element, if two elements have several upper bounds that are equally big, then none of them is a *join* (a join must be unique).
![A non-join diagram](non_join.svg)
@ -247,24 +247,34 @@ The poset example with the number dividers is also isomorphic to an inclusion or
![Divides poset](divides_poset_inclusion.svg)
Let's elaborate a bit about the isomorphism between the number poset and the prime inclusion order. It is comprised of one function from the prime inclusion order, to the number poset (which in this case is just the *multiplication* of all the elements in the set) and one function from the number poset to the prime inclusion order (which is an operation called *prime factorization* of a number, consisting (unsurprisingly) of finding the set of prime numbers that result in that number when multiplied with one another (as we said, this set is unique, and hence these functions really do comprise an isomorphism)).
Order isomorphisms
---
We mentioned order isomorphisms several times already so this is about time to elaborate a bit about what they are. Take the isomorphism between the number poset and the prime inclusion order as an example. Like an isomorphism between any two sets, it is comprised of two functions:
- One function from the prime inclusion order, to the number poset (which in this case is just the *multiplication* of all the elements in the set)
- One function from the number poset to the prime inclusion order (which is an operation called *prime factorization* of a number, consisting of finding the set of prime numbers that result in that number when multiplied with one another).
![Divides poset](divides_poset_isomorphism.svg)
For sets, an isomorphism means just that the functions are inverse of each other. But as orders, besides being sets, have the arrows that connect them there is an additional requirement for a pair of functions to form an isomorphism - to be an isomorphism, a function has *to respect those arrows, in other words should be *order preserving*. More specifically, applying the function (let's call it F) to any two elements in one set (**a** and **b**) should result in two elements that have the same corresponding order in the other set (so **a ≤ b** iff **F(a) ≤ F(b)**.
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.
<!-- TODO in a Power set P(X), the meet of a collection of subsets is their intersection, and the join is their union -->
Lattices
===
@ -277,7 +287,7 @@ Notice that when drawing our color-mixing lattice, we added the black ball at th
Bounded lattices
---
Our color-mixing lattice, has a *maximum element* (the black ball) and a *minimum element* (the white one). Lattices that have a minimum and maximum elements are called *bounded lattices*. It isn't hard to see that all finite lattices are also bounded.
Our color-mixing lattice, has a *greatest element* (the black ball) and a *least element* (the white one). Lattices that have a least and greatest elements are called *bounded lattices*. It isn't hard to see that all finite lattices are also bounded.
**Task:** Prove that all finite lattices are bounded.
@ -334,6 +344,12 @@ All of that structure arises naturally from the simple law of transitivity.
![Transitivity](transitivity.svg)
<!--
TODO: add the example of preorders as models for routes from/to a given set of destination.
Also the state machine example.
-->
Orders as categories
===
@ -347,7 +363,7 @@ What about that other law that was required in order to be a category - the iden
![Reflexivity](reflexivity.svg)
So it's official - preorders are categories (sounds kinda obvious, especially after we also saw that orders can be reduced to sets and functions using (through the inclusion order) and sets and functions form a category in their own right).
So it's official - preorders are categories (sounds kinda obvious, especially after we also saw that orders can be reduced to sets and functions (the inclusion order) and sets and functions form a category in their own right).
And since partial orders and total orders are preorders too (as they obey those two laws), they are categories as well.
@ -359,10 +375,12 @@ That is in the contrast with the category of sets where there are potentially in
![Orders compared to other categories](order_category.svg)
Note that although two objects in an order might be directly connected by just one arrow, they might still be be indirectly connected by more than one arrow. So when we define an order in categorical way it's crucial to specify that *these ways are equivalent* i.e. that all diagrams that show orders commute.
Products and sums
---
While we are rehashing diagrams from the previous chapters, let's look at the diagram defining the *sum* of two objects in a category.
While we are rehashing diagrams from the previous chapters, let's look at the diagram defining the *coproduct* of two objects in a category.
![Joins as coproduct](coproduct_join.svg)
@ -375,17 +393,17 @@ But wait, wasn't there something else that corresponded to set inclusion - oh ye
In category theory, an object **G** is the coproduct of objects **Y** and **B** if the following two conditions are met:
1. We have a morphism from any of the elements of the coproduct to the coproduct, so **Y → G** and **B → G**.
2. For any other object **P** that also has those morphisms, so for any **P** such that **P ≤ G** and **P ≤ B**, we would have morphism **G → P**.
2. For any other object **P** that also has those morphisms (so **Y → P** and **B → P**) we would have morphism **G → P**.
![Joins as coproduct](coproduct_morphisms.svg)
In the realm of orders, we say that **G** is the *join* of objects **Y** and **B** if:
1. It is bigger than both of these objects, so **Y ≤ G** and **P ≤ G**.
1. It is bigger than both of these objects, so **Y ≤ G** and **B ≤ G**.
2. It is smaller than any other object that is bigger than them, so for any other object **P** such that **P ≤ G** and **P ≤ B** then we should also have **G ≤ P**.
![Joins as coproduct](coproduct_join_morphisms.svg)
We can see that the two definitions (and the diagrams) are the same. So, speaking in category theoretic terms, we can say that the *categorical coproduct* in the category of orders is the *join* operation.
We can see that the two definitions and their diagrams are the same. So, speaking in category theoretic terms, we can say that the *categorical coproduct* in the category of orders is the *join* operation.

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---
layout: default
title: Logic
---
Logic
===
Here we go with this little *leit motif* of mine where I begin talking about something completely different and then it turns out 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 - I don't even have to start another chapter for it (althought it will make this one a bit lenghty).
Now let's talk about one more *seemingly* unrelated topic just so we can "surprise" ourselves when we realize it's category theory. By the way, in this chapter there will be another surprise in addition to that, so don't fall asleep.
Logic aims to study the *rules* by which knowing one thing leads you to conclude that some other thing is also true, and (most importantly) to do this in a formal way i.e. without regard of what these things are, or how you learned them.
Also, I will not merely transport you to a different branch of mathematics, but to an entirely different discipline - *logic*.
Not only that, but logicians try to organize those rules in what are called *systems*, or *formal systems* - containing selections of rules that have the expressive ability to prove everything that you can see by intuition.
What is logic
===
What does "prove" mean in this context? Simple, when we are able, using the rules of a given logical system, to transform one set of assertions **A** to another one **B** we say that we have proven that **A → B** in that logical system.
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. that which really exists. For example, if science explains how our universe works then logic is the part of the description which is also applicable to any other universe that is *possible to exist*.
So let's get our balls again.
Logic studies the *rules* by which knowing one thing leads you to conclude (or *prove*) that some other thing is also true, regardless of the things' domain (e.g. scientific discipline) and by only referring to their form.
On top of that, it (logic) tries to organize those rules in *logical systems* (or *formal systems* as they are also called).
Logic and mathematics
---
Seeing this description, we might think that the subject of logic is quite similar to the subject of set theory and category theory, as we described it in the first chapter - instead of the word "formal" we used another similar word, namely "abstract", and instead of "logical system" we said "theory". This observation would be quite correct - today most people agree that every mathematical theory is actually logic plus some additional definitions added to it. For example, part of the reason why *set theory* is so popular as a theory for the foundations of mathematics is that it adds just one single primitive to the standard axioms of logic which we will see shortly - the binary relation that indicates *set membership*. Category theory is close to logic too, but in a quite different way.
Primary propositions
---
A consequence of logic being the science of the possible is that in order to do anything at all in it, we should have an initial set of propositions that we accept as true or false. These are also called "premises", "primary propositions" or "atomic propositions" as Wittgenstein dubbed them.
![Balls](balls.svg)
Follows
In the context of logic itself, these propositions are abstracted away (i.e. we are not concerned about them directly) and so they can be represented with the colorful balls that you are familiar with.
Composing propositions
---
At the heart of logic, as in category theory, is the concept of *composition* - if we have two or more propositions that are somehow related to one another, we can combine them into one using a logical operators, like "and", "or" "follows" etc. The result would be a new proposition, not unlike the way in which two monoid objects are combined into one using the monoid operation. And actually some logical operations do form monoids, like for example the operation **and**, with the proposition **true** serving as the identity element.
Or and And
![Logical operations that form monoids](logic_monoid.svg)
However, unlike monoids/groups, logics have not one but *many* logical operations and logic studies *the ways in which they relate to one another*, for example, in logic we might be interested in the law of distributivity of **and** and **or** operations and what it entails.
![The distributivity operation of "and" and "or"](logic_distributivity.svg)
Important to note that **∧** is the symbol for **and** and **∨** is the symbol for **or** (although the law above is actually valid even if **and** and **or** are flipped).
The equivalence of primary and composite propositions
---
When looking at the last diagram, it is important to stress that, lthough in the leftmost proposition the green ball is wrapped in a gray ball to make the diagram prettier, propositions that are composed of several premises (symbolized by gray balls, containing some other balls) are not in any way different from "primary" propositions (single-color balls) and that they compose in the same way.
False and true
---
Notice we didn't say anything about true and false. Logic is not about truth and falsity.
![Balls as propositions](balls_propositions.svg)
Heyting algebra overview
Modus ponens
---
As an example of a proposition that contains multiple levels of nesting (and also as a great introduction of the subject of logic in its own right), consider one of the oldest (it was already known by Stoics at 3rd century B.C.) and most famous propositions ever, namely the *modus ponens*.
and say that it is an "atomic fact" as Wittgenstein calls it - it is not composed of any "more basic" facts, it is just the simplest thing that you can know.
Modus ponens is a proposition that states that if proposition **A** is true and also if proposition **(A → B)** is true (that is if **A** implies **B**), then **B** is true as well. For example, if we know that "Socrates is a human" and that "humans are mortal" (or "being human implies being mortal"), we also know that "Socrates is mortal."
![Modus ponens](modus_ponens.svg)
Lattices
Let's dive into it. The proposition is composed of two other propositions in a **follows** relation, where the proposition that follows (**B**) is primary, but the proposition from which **B** follows is not primary (let's call that one **C** - so the whole proposition becomes **C → B**.)
Going one more level down, we notice that the **C** propositions is itself composed of two propositions in an **and**, relationship - **A** and let's call the other one **D** (so **A ∧ D**), where **D** is itself composed of two propositions, this time in a **follows** relationship - **A → B**. But all of this is better visualized in the diagram.
Tautologies
---
Lattices and trees
We often cannot tell whether a given composite proposition is true or false without knowing the values of the propositions that compose it. However, with propositions such as *modus ponens* we can: modus ponens is *always true*, regardless of whether the propositions that form it are true or false. If we want to be fancy, we can also say that it is *true in all models of the logical system*, a model being a set of real-world premises are taken to be signified by our propositions.
For example, our previous example will not stop being true if we substitute "Socrates" with any other name, nor if we substitute "mortal" for any other quality that humans possess.
![Variation of modus ponens](modus_ponens_variations.svg)
Propositions that are always true are called *tautologies*. And their more-famous counterparts that are always false are called *contradictions*. You can turn each tautology into contradiction or the other way around by adding a "not".
The simplest tautology is the statement that each proposition implies itself (e.g. "All bachelors are unmarried"). It may remind you of something.
![Identity tautology](tautology_identity.svg)
Here are some more complex (less boring) tautologies (the symbol **¬** means "not"/negation.
![Tautologies](tautology_list.svg)
We will learn how to determine which propositions are a tautologies shortly, but first let's see why is this important at all i.e. what are tautologies good for: tautologies are useful because they are the basis of *axiom schemas*/*rules of inference*. And *axiom schemas* or *rules of inference* serve as starting point from which we can generate other true logical statements by means of substitution.
Axiom schemas/Rules of inferene
---
Realizing that the colors of the balls in modus ponens are superficial, we may want to represent the general structure of modus ponens that all of its variations share.
![Modus ponens](modus_ponens_schema.svg)
This structure (the one that looks like a coloring book in our example) is called *axiom schema*. And the propositions that are produced by it are *axioms*.
Note that the propositions that we plug into the schema don't have to be primary. For example, having the proposition **a** (that is symbolized below by the orange ball) and the proposition stating that **a** implies **a or b** (which is one of the tautologies that we saw above), we can plug those propositions into the *modus ponens* and prove that **a or b** is true.
![Using modus ponens for rule of inference](modus_ponens_composite.svg)
This is it about *axiom schemas* and *rules of inference are almost the same thing except they allow us to actually distill the conclusion from the premises. For example in the case above, we can use modus ponens as a rule of inference to proves that **a or b** is true.
All axiom schemas can be easily applied as rules of inference and the other way around.
Logical systems
---
Knowing that we can use axiom schemas/rules of inference to generate new propositions, we might ask whether it is possible to create a small collection of such schemas/rules that is curated in such a way that it enables us to generate *all* other propositions. You would be happy (although a little annoyed, I imagine) to learn that there exist not only one, but many such collections. And yes, collections of this sort are what we call *logical systems*.
Here is one such collection which consists of the following five axiom schemes **in addition to the inference rule modus ponens** (These are axiom schemes, even though we use colors).
![A minimal collection of Hilbert axioms](min_hilbert.svg)
Proving that this and other similar logical systems are complete (can really generate all other propositions) is due to Gödel and is known as "Gödel's completeness theorem" (Gödel is so important that I specifically searched for the **ö** letter so I can spell his hame right.)
Conclusion
---
We now have an idea about how do some of the main logical constructs (axioms, rules of inference) work. But in order to prove that they are true, and to understand *what they are*, we need to do so through a specific *interpretation* of those constructs.
We will look into two interpretations - one very old and the other, relatively recent. This would be a slight detour from our usual subject matter of points and arrows, but I assure you that it would be worth it. So let's start.
Classical logic. The truth-functional interpretation
===
> Beyond the world that we inhabit and perceive every day, there exist the *world of forms* where reside all ideas and concepts that manifest themselves in the objects that we perceive e.g. beyond all the people that have ever lived, there lies the prototypical person, and we are people only insofar as we resemble that person, beyond all the things in the world that are strong, lies the ultimate concepts of strength, from which all of them borrow etc. And although, as mere mortals, we live in the world of appearances and cannot perceive the world of forms, we can, through philosophy, "recollect" with it and know some of its features.
The above is a summary of a worldview that is due to the Greek philosopher Plato and is sometimes called Plato's *theory of forms*. Originally, the discipline of logic represents an effort to think and structure our thoughts in a way that they apply to this world of forms i.e. in a "formal" way. Today, this original paradigm of logic is known as "classical logic". Although it all started with Plato, most of it is due to the 20th century mathematician David Hilbert.
The existence of the world of forms implies that even if there are many things that we people don't know, at least *somewhere out there* there exists an answer to every question. In logic, this translates to *the principle of bivalence* that states that *each proposition is either true or false*. Due to this principle, propositions in classical logic can be aptly represented using set theory by boolean set, which contains those two values.
![The set of boolean values](boolean_set.svg)
According to the classical interpretation, you can think of *primary propositions* as just a bunch of boolean values, *logical operators* are functions that take a one or several boolean values and return another boolean value (and *composite propositions* are, just the results of the application of these functions).
Let's review all logical operators in this context.
The *negation* operation
---
Let's begin with the negation operation. Negation is a unary operation, which means that it is a function that takes just *one* argument and (like all other logical operators) returns one value, where both the arguments and the return type are boolean values.
![negation](negation.svg)
The same function can also be expressed in a slightly less-fancy way by this table.
| p | ¬p |
|---| --- |
| True | False |
| False | True |
Tables like this one are called *truth tables* and they are ubiquitous in classical logic. They can be used not only for defining operators but for proving results as well.
Interlude: Proving results by truth tables
---
Having defined the negation operator, we are in position to prove the first of the axioms of the logical system we saw, namely the *double negation elimination*. In natural language, this axiom is equivalent to the observation that saying "I am *not unable* to do X" is the same as saying "I am *able* to do it".
![Double negation elimination formula](double_negation_formula.svg)
(despite its triviality, the double negation axiom is probably the most controversial result in logic, we will see why later.)
If we view logical operators as functions from and to the set of boolean values, than proving axioms involves composing several of those functions into one function and observing its output. More specifically, the proof of the formula above involves just composing the negation function with itself and verifying that it leaves us in the same place from which we started.
![Double negation elimination](double_negation_proof.svg)
If we want to be formal about it, we might say that applying negation two times is equivalent to applying the *identity* function.
![The identity function for boolean values](boolean_identity.svg)
If we are tired of diagrams, we can represent the composition diagram above as table as well.
| p | ¬p | ¬¬p |
|---| --- | --- |
| True | False | True |
| False | True | False |
Each proposition in classical logic can be proved with such diagrams/tables.
The **and** and **or** operations
---
OK, *you* know what **and** means and *I* know what it means, but what about those annoying people that want everything to be formally specified (nudge, nudge). Well we already know how we can satisfy them - we just have to construct the boolean function that represents **and**.
Because **and** is a *binary* operator, instead of a single value the function would accept a *pair* of boolean values.
![And](and.svg)
Here is the equivalent truth-table (in which **∧** is the symbol for **and**.)
| p | q | p ∧ q |
|---| --- | --- |
| True | True | True |
| True | False | False |
| False | True | False |
| False | False | False |
We can do the same for **or**, here is the table.
| p | q | p ∨ q |
|---| --- | --- |
| True | True | True |
| True | False | True |
| False | True | True |
| False | False | False |
**Task:** Draw the diagram for **or**.
Using those tables, we can also prove some axiom schemas we can use later:
- For **and**: **p ∧ q → p** and **p ∧ q → q** "If I am tired and hungry, this means that I am hungry".
- For **or**: **p → p ∨ q** and **p → p ∨ q** "If I have a pen this means that I am either have a pen or a ruler".
The *implies* operation
---
Let's now look into something less trivial: the *implies* operation, (also known as *entailment*). This operation binds two propositions in a way that the truth of the first one implies the truth of the second one. You can read **p → q** as "if **p** is true, then **q** must also be true.
Entailment is also a binary function - it is represented by a function from an ordered pair of boolean values, to a boolean value.
| p | q | p → q |
|---| --- | --- |
| True | True | True |
| True | False | False |
| False | True | True |
| False | False | True |
Now there are some aspects of this which are non-obvious so let's go through every case.
1. If *p* is true and *q* is also true, then **p** does imply **q** - obviously.
2. If *p* is true but *q* is false then **q** does not follow from **p** - cause *q* would have been true if it did.
3. If *p* is false but *q* is true, then **p** still does imply **q**. What the hell? Consider that by saying that *p* implies *q* we don't say that the two are 100% interdependent e.g. the claim that "drinking alcohol causes headache" does not mean that drinking is the only source of headaches.
4. And finally if **p** is false but **q** is false too, then **p** still does imply **q** (just some other day).
It might help you to remember that in classical logic **p → q** (**p** implies **q**) is true when **-p ∨ q** (either **p** is false or **q** is true.)
Proving results by axioms/rules of inference
---
Let's examine the above formula, stating that **p → q** is the same as **¬p ∨ q**.
![Hilbert formula](hilbert_formula.svg)
We can easily prove this by using truth tables.
| p | q | p → q | ¬p | q | ¬p ∨ q |
|---| --- | --- | --- | --- | --- |
| True | True | **True** | False | True | **True** |
| True | False | **False** | False | False | **False** |
| False | True | **True** | True | True | **True** |
| False | False | **True** | True | False | **True** |
But it would be much more intuitive if we do it using axioms and rules of inference. To do so, we have to start with the formula we have (**p → q**) plus the axiom schemas, and arrive at the formula we want to prove (**¬p ∨ q**).
Here is one way to do it. The formulas that are used at each step are specified at the right-hand side, the rule of inference is modus ponens.
![Hilbert proof](hilbert_proof.svg)
Note that to really prove that the two formulas are equivalen we have to also do it the other way around (start with (**¬p ∨ q**) and (**p → q**)).
Intuitionistic logic. The BHK interpretation
===
Although the classical truth-functional interpretation of logic works and is correct in its own right, it doesn't fit well the categorical framework that we are using here: It is too "low-level", it relies on manipulating the values of the propositions. According to it, the operations **and**, **or** and **implies** are just 3 of the 16 possible binary logical operations and they are not really connected to each other.
For these and other reasons (mostly other, probably), in the 20th century a whole new school of logic was founded, called *intuitionistic logic*. If classical logic is based on *set theory*, intuitionistic logic is based on *category theory* and its related theories. If *classical logic* is based on Plato's theory of forms, then intuinism began with a philosophical idea originating from Kant and Schopenhauer: the idea that the world as we experience it is largely predetermined of out perceptions of it. As the mathematician L.E.J. Brouwer puts it.
> [...] logic is life in the human brain; it may accompany life outside the brain but it can never guide it by virtue of its own power.
Classical and intuitionistic logic diverge from one another right from the start: because according to intuitionistic logic we are *constructing* proofs rather than *discovering* them or *unveiling* a universal truth, we are *off with the principle of bivalence*, that is, we have no basis to claim that each statements is necessarily *true or false*. For example, there might be a statements that might not be provable not because they are false, but simply because they fall outside of the domain of a given logical system (the twin-prime conjecture is often given as an example for this.)
So, intuitionistic logic is not bivalent, i.e. we cannot have all propositions reduced to true and false.
![The True/False dichotomy](true_false.svg)
One thing that we still do have there are 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 bivalence between true and false proposition might be thought out as similar to the bivalence between the existence or absense of a proof for a given proposition - there either is a proof of it or there isn't.
![The proved/unproved dichotomy](proved_unproved.svg)
This bivalence is at the heart of what is called the Brouwer–Heyting–Kolmogorov (BHK) interpretation of logic, something that we will look into next.
The original formulation of the BHK interpretation is not based on any particular mathematical theory. Here, we will first illustrate it using the language of set theory (just so we can abandon it a little later).
The **and** and **or** operations
---
As the existence of a proof of a proposition is taken to mean that the proposition is true, the definitions of **and** is rather simple - the proof of **A ∧ B** is just *a pair* containing a proof of **A**, and a proof of **B** i.e. *a set-theoretic product* of the two (see chapter 2). The principle for determining whether the proposition is true or false is similar to that of primary propositions - if the pair of proofs of **A** and **B** exist (i.e. if both proofs exist) then the proof of **A ∧ B** can be constructed (and so **A ∧ B** is "true").
![And in the BHK interpretation](bhk_and.svg)
**Question:** what would be the **or** operation in this case?
The *implies* operation
---
Saying that **A** implies **B** (**A → B**) would just mean that there exist a function which can convert a proof of **A** to a proof of **B**.
![Implies in the BHK interpretation](bhk_implies.svg)
And the *modus ponens* rule of inference is expressed by the fact that if we have a proof of **A** we can call this function (**A → B**) to obtain a proof of **B**.
(Note that in order for this to work, we also need to define the functions in terms of sets i.e. we need to have a set representing **A → B** for each **A** and **B**. We will come back to this later.)
The *negation* operation
---
So according to BHK interpretation saying that **A** is true, means that that we possess a proof of **A** - simple enough. But it's a bit harder to express the fact that **A** is false: it is not enough to say that we *don't have a proof* of **A** (the fact that don't have it, doesn't mean it doesn't exist). Instead, we must show that claiming that **A** is true leads to a *contradiction*.
To express this, intuitionistic logic defines the constant **⊥** which plays the role of *False* (and is also known as "absurdity" or "bottom value"). **⊥** is defined as the proof of a formula that does not have any proofs. And the equivalent of false propositions are the ones that imply that the bottom value is provable (which is a contradiction). So **¬A** is **A → ⊥**.
In set theory, the **⊥** constant is expressed by the empty set.
![False in the BHK interpretation](bhk_false.svg)
And the observation that propositions that are connected to the bottom value are false is expressed by the fact that if a proposition is true, i.e. there exists a proof of it, then there can be no function from it to the empty set.
![False in the BHK interpretation](bhk_false_function.svg)
The only way for there to be such function is if the set of proofs of the proposition is empty as well.
![False in the BHK interpretation](bhk_false_function_2.svg)
**Task:** Look up the definition of function and verify that there cannot exist a function from any set *to the empty set*
**Task** Look up the definition of function and verify that there does exist a function *from the empty set* to itself (in fact there exist a function from the empty set to any other set.
Classical VS intuitionistic logic
---
Although from first glance intuitionistic logic seems to differ a lot from classical logic, it actually doesn't - if we try to deduce the axiom schemas/rules of inference that correspond to the definitions of the structures outlined above, we would see that they are virtually the same as the ones that define classical logic. With one exception concerning the *double negation elimination axiom* that we saw earlier, a version of which is known as *the law of excluded middle*.
![The formula of the principle of the excluded middle](excluded_middle_formula.svg)
This law is valid in classical logic and is true when we look at it in terms of truth tables, but there is no justification for it in the BHK interpretation - a fact that spawned a heated debate between the inventor of classical logic David Hilbert and the inventor of intuitionistic logic L.E.J. Brouwer, known as *the Brouwer–Hilbert controversy*.
Logics as categories
===
Leaving the differences between intuitionistic and classical logics aside, the BHK interpretation is interesting because it provides that higher-level view of logic, that we need in order to construct a interpretation of it based on category theory.
Such higher-level interpretations of logic are sometimes called *algebraic* interpretations, *algebraic* being an umbrella term describing all structures that can be represented using category theory, like groups and orders.
The Curry-Howard isomorphism
---
Programmers might find the definition of the BHK interpretation interesting for other reason - it is very similar to a definition of a programming language: propositions are *types*, the **implies** operations are *functions*, **and** operations are composite types (objects), and **or** operations are *sum types* (which are currently not supported in most programming languages, but that's a separate topic.) Finally a proof of a given proposition is represented by a value of the corresponding type.
![Logic as a programming language](logic_curry.svg)
This similarity is known as the *Curry-Howard isomorphism*.
**Task:** The Curry-Howard isomorphism is also the basis of special types of programming languages called "proof assistants" which help you verify logical proofs. Install a proof assistant and try to see how it works (I recommend the Coq Tutorial by Mike Nahas).
Cartesian closed categories
---
Knowing about the Curry-Howard isomorphism and knowing also that programming languages can be described by category theory may lead us to think that *category theory is part of this isomorphism as well*. And we would be quite correct, this is why it is sometimes known as the Curry-Howard-*Lambek* isomorphism, Lambek being the person who discovered the categorical side.
Let's examine this isomorphism (without being too formal about it). As all other isomorphisms, it comes in two parts. The first part is finding a way to convert a logical system into a category - this would not be hard for us, as sets form a category and the flavor of the BHK interpretation that we saw is based on sets.
![Logic as a category](category_curry_logic.svg)
**Task:** See whether you can prove that logic propositions and entailments forms a category. What is missing?
The second part involves converting a category into a logical system - this is much harder. To do 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.
![Logic as a category](logic_curry_category.svg)
Categories that adhere to these criteria are called *cartesian closed categories*. We won't describe them here directly, but instead we would start with a similar but simpler structures that are instance of them and that we already examined - orders.
Logics as orders
---
We will now do something that is quite characteristic of category theory - examining a concept in a more limited version of the theory, in order to make things simpler for ourselves.
So we already saw that a logical system along with a set of primary propositions forms a category.
![Logic as a preorder](logic_category.svg)
If we assume that there is only one way to go from proposition **A**, to proposition **B** (or there are many ways, but we are not interested in the difference between them), then logic is not only a category, but a *preorder* 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 are proven by the same proof) as equivalent, then logic is a proper *partial order*.
![Logic as an order](logic_order.svg)
And so it can be represented by a Hasse diagram, yey.
![Logic as an order](logic_hasse.svg)
Now let's examine the question that we asked before - exactly which ~~categories~~ orders represent logic and what laws does an order have to obey so it is isomorphic to a logical system? We will attempt to answer this question as we examine the elements of logic again, this time in the context of orders.
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). As we saw, in the BHK interpretation those are represented by set *products* and *sums*. The equivalent constructs in the realm of order theory are *meets* and *joins* (in category-theoretic terms *products* and *coproducts*.)
![Order meet and joing](lattice_meet_join.svg)
Here comes the first criteria for an order to represent a logical system accurately - *it has to have **meet** and **join** operations for all elements*. Having two elements without a meet would mean that you would have a logical system where there are propositions for which you cannot say that one or the other is true. And this not how logic works, so our order has to have meets and joins for all elements. Incidentally we already know how such orders are called - they are called *lattices*.
One important law of 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 they distribute, over one another.
![The distributivity operation of "and" and "or"](logic_distributivity.svg)
Lattices that obey this law are called *distributive lattices*.
Wait, where have we heard about distributive lattices before? In the previous chapter we said that they are isomorphic to *inclusion orders* i.e. orders which contain all combinations of sets of a given number of elements. The fact that they popped up again is not coincidental - "logical" orders are isomorphic to inclusion orders. To understand why, you only need to think about the BHK interpretation - the elements which participate in the inclusion are our prime propositions. 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: For historical reasons, the symbols for *and* and *or* logical operations are flipped when compared to arrows in the diagrams ∧ is *and* and ∨ is *or*.**
The *negation* operation
---
In order for a distributive lattice to represent a logical system, it has to also have objects that correspond to the values **True** and **False**. But to mandate that these objects exist, we must first find a way to specify what they are in order/category-theoretic terms.
A well-known 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 of classical logic), then any and every other statement can be proven. And we also know that no true statement implies **False** (in fact in intuinistic logic this is the definition of a true statement). Based on these criteria we know that the **False** object would look like this when compared to other objects:
![False, represented as a Hasse diagram](lattice_false.svg)
Circling back to the 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 *greatest* and *least* objects of our order (in category-theoretic terms *terminal* and *initial* object.)
![The whole logical system, represented as a Hasse diagram](lattice_true_false.svg)
This is another example of the categorical concept of duality - ***True** and **False** are dual to each other*, (which makes a lot of sense if you think about it).
So in order to represent logic, our distributive lattice has to also be *bounded* i.e. it has to have greatest and least elements (which play the roles of **True** and **False**).
The *implies* operation
---
Finally, if a lattice really represents a logical system (that is, it is isomorphic to a set of propositions) 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 logic are followed.
How would this object be described? You guessed it, using categorical language i.e. 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 favorite rule of inference, the *modus ponens* (**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 is related to objects **A** and **B** in such a way that such that **A ∧ (A → B) → 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 of it's equivalent for orders, the *meet* operation) you would already know where this is going: we define the function object using a *universal property*, by recognizing that all other formulas that can be in the place of **X** in **A ∧ X → B** point to **(A → B)** i.e. they are below **(A → B)** in a Hasse diagram.
![Implies operation with universal property](implies_universal_property.svg)
Or, using the logic terminology, we say that **A → B ∧ C** and **A → B ∧ C ∧ D** etc. are all "stronger" results than (**A → B**) and so (**A → B**) is the weakest result that fits the formula (stronger results lay lower in the diagram).
So this is the final condition for an order/lattice 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*.
Without being too formal, 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, (**A → B**) (or (**A → A**) if you want to be pedantic) would be the topmost object **X** for which the criteria given by the formula **A ∧ X → A** is satisfied. But in this case the formula is *always satisfied* as the *meet* of **A** and any other object would always be below **A**. So this formula is always for all **X**. The topmost object that fits it is, then, the topmost object out there 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** if **A** implies **B**, then **(A → B)** is always true.
And what if **B** is lower than **A**. In this case the topmost object that fits the formula **A ∧ X → B** is **B** itself: **B** fits the formula because the meet of two objects is always below those same objects, so **A ∧ B → B** for all **A** and **B**. And **B** is definitely the topmost object that can possibly fit it, as it literary sets its upper bound.
![Implies when B follows from A](implies_a_follows.svg)
Translated to logical language, says that if **B → A**, then the proof of **(A → B)** is just the proof of **B**.
Note that this definition does not follow the one from the truth tables exactly. This is because this definition is valid specifically for intuinistic logic. For classical logic, the definition of **(A → B)** is simpler - it is just equivalent to (**-A ∨ B**).
By the way, the law of distributivity follows from this criteria, so the only criteria that are left for an lattice to follow the laws of intuinistic logic is for it to be *bounded* i.e. to have greatest and least objects (**True** and **False**) and to have a function object as described above. Lattices that follow these criteria are called *Heyting algrebras*.
And for a lattice to follow the laws of classical logic it has to be *bounded* and *distributive* and to be *complemented* which is to say that each proposition **A** should be complemented with a unique proposition **-A** (such that **A ∨ -A = 1** and **A ∧ -A = 0**). These lattices are called *Boolean algebras*.

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Functors
===
With this chapter, we will change the tactic a bit, (as I am sure you are a bit tired of jumping through different subjects) and we will examine some purely categorical concepts, using the structures that we saw so far as context. We will also generalize some of the concepts that we saw in these structures so they are valid for all categories.
Categories we saw so far
===
We already saw a many different types of categories, like categories that have just one *object* (monoids, groups) and categories that have only one *morphism* between any two objects (preorders, partial orders).
![Types of categories](category_types.svg)
We also defined a lot of categories, most notably the ones based on logics/programming languages, but also less serious categories, as for example the color-mixing one
![Types of categories](category_types.svg)
We saw that we can make a category out of everything, as for example our city's road system, or a soccer ranglist starring me and my grandmother.
While examining all of this, we also saw the mother of all categories - the category of sets, as well as various *subcategories* to that category, as for example the set inclusion orders. We saw that many other categories are similar to subcategories of the category of sets.
![Set category](category_set.svg)
But how can we utilize the insights that we get from the fact that so many different things are actually one and the same thing in disquise? To do so we must specify ways to connect categories to one another. This is the topic of this chapter.
Categorical Isomorphisms
===
In chapter 1 we talked about set isomorphisms, which establish the equivalence between two sets. If you remember a set isomorphism consisted of two functions between the sets that when composed are equivalent to the identity function. Or more simply put, of one invertible function.
![Set isomorphism](set_isomorphism.svg)
![Set isomorphism](set_isomorphism_invertable.svg)
Then in chapter 4 we mentioned order isomorphisms, which were just set isomorphisms, with the extra condition that not only the objects, but the arrows should be the same, so for any two elements **a** and **b**, *a ≤ b** iff **F(a) ≤ F(b)**.
![Order isomorphism](order_isomorphism.svg)
It's a little (but not a lot) more complex to extend this definition to work for categories as well. Given two categories, an isomorphism between them is an invertible function between the underlying sets of objects, plus an isomorphism between the morphisms that connect them.
![Category isomorphism](category_isomorphism.svg)
Although longer, this definition is the same that we have for orders - it is just that orders can have at most one morphism between two objects, so it's obvious from the signature which corresponds to which - all we have to do is to verify that the morphisms exist, which is taken care of by the additional condition.
So those are categorical isomorphisms. But isomorphisms are actually almost useless - honestly, if two categories are isomorphic, then they are so similar that you can hardly see the difference between one and the other. It will be non-essential.
[Category isomorphism squares](category_isomorphism_squares.svg)
Functors
===
Unlike two-way relations between two categories may be established only in a very limited set of cases, the one-way relations are very common.
Functors in programming
===
Functors as maps
===
Diagrams
===
Natural transformations
===
Limits
===

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Coproduct
morphism
coproducts
morphism
morphisms
preorders
Preorders
@ -51,6 +52,7 @@ non-abelian
Dih3
composable
forall
<<<<<<< HEAD
codomain
Euclidian
subtypes
@ -61,3 +63,34 @@ structs
DeMorgan's
getter
OOP
=======
BHK
yey
intuitionistic
intuinism
Brouwer
schemas
Nahas's
Coq
Kolmogorov
entailments
Logics
logics
bivalence
Zermelo
Fraenkel
distributivity
CTI
monads
monad
hom-set
Curry-Howard-Lambek
Curry-Howard
Lambek
cartesian
functor
functors
Functor
Functors
Endofunctors
invertible

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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 is used in other scientific disciplines e.g. quantum mechanics:
> [...] The categorical formalization 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)
- It may also be used as an axiomatic foundation for mathematics, as an alternative to set theory and other proposed foundations.
- Last but not least (perhaps even most importantly) category theory has a very good potential as a *teaching tool* for a variety of other subjects.
About the book
---
I am writing a primer in category theory and 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 serves as chapter 0 going through the gist of the material covered by other similar introductory books, but doing so in a way that would enable non-mathematicians to swift through with ease.
Reading it would enable my readers to effortlessly go through any academic introduction to category theory, as well as to start tacking resources that use category theory as a tool to threat other subjects.
About the author
---
I am a nerd who works in web programming with background in design and technical communication. I first discovered category theory through programming, but then started pursuing it in its own right. Finding math more and more fascinating.
Currently living in Sofia with my wife and kid.
Similar resources
===
Here is [a good list of category theory introductions](https://github.com/prathyvsh/category-theory-resources)
"Category Theory Illustrated" (which I will call CTI from now on), however possesses some unique features that make it different from the ones that I looked into. Here I will examine those features of CTI when compared specifically to my two favorite introductions, which also target the two biggest target audiences for CTI - programmers (nerds) and scientists (academics).
- 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](http://math.mit.edu/~dspivak/CT4S.pdf)
I am attaching excerpts from both of them, as well as one from CTI, that explain a similar concept - 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)
The categorical aspect of orders is described 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.
[Source](/category-theory-illustrated/04_order)
Features
===
Diagrams
---
Category theory is very *visual* in its nature. Category-theoretic diagrams are not merely supplemental illustrations for its concepts, 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.
However, books on category theory typically have as many diagrams as other math textbooks. I consider this a huge missed opportunity for making the subject more approachable. This was the original motivation for the creation of CTI.
Besides being more in quantity, the diagrams of CTI use many techniques from traditional communication design (such as the use of color for example) in order to illuminate the different subjects and abstractions that I am covering.
Both books that I am reviewing have less diagrams than CTI. From the resources that I have examined, the only author who stresses on diagrams as much as I do is Tai-Danae Bradley (her blog is [https://www.math3ma.com/](https://www.math3ma.com/).)
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) ([Lecture slides](https://raw.githubusercontent.com/compas-teaching/COMPAS-II-FS2021/main/lecture_06/lecture_06.pdf).)
Verbosity of language
---
Leaving the diagrams aside, we can see that the descriptions in CTI are much more verbose than equivalent descriptions in other books. This is clearly visible in the excerpts attached: although the authors of the books that I am reviewing have quite different styles of presentation, they both move much faster with the exposition, which would be OK for readers who have prior experience with math and computer science, but would be very hard to follow by lay people (I know that from my own experience).
In CTI, I try above all to be gentle in my exposition, taking the time to stress out the important parts of the descriptions and the places where misunderstanding might occur, such as the fact that the term "object" has entirely different meanings in computer science and in category theory.
Chapter on logic
---
Category theory is deeply bound with mathematical logic. But although the connection is mentioned in many books, it isn't fully examined by any introductory texts on category theory that I have seen.
CTI has a dedicated chapter on logic (the longest one in the book), which introduces the connection and makes use of it to give context to some categorical phenomena, such as limits.
Target audiences
===
The types of audiences of CTI.
Programmers
---
### who are curious about category theory, because they are into functional programming.
Functional programming is on the rise with both new languages and technologies that are gaining traction (Elixir, Clojure), and mainstream languages, trying to catch up (Java, JavaScript) and so there are many programmers who are interested in functional programming and category theory, but lack the math background to tackle the existing resources. CTI is a way for those people to understand "what the fuss is about" before diving into such technologies.
Students
---
### who are studying category-theory-related disciplines
CTI would be of much help for people who want to study category theory or any of the growing number of resources that use category theory and diagrammatic language to introduce other subjects:
- Picturing Quantum Processes by Bob Coecke and Aleks Kissinger (Cambridge University Press - 2017) that uses it to introduce quantum mechanics
- 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
- "Topology. A Categorical Approach" by Tai-Danae Bradley, Tyler Bryson and John Terilla (MIT press - 2020)
As David Spivak puts it in his review of "Topology. A Categorical Approach":
> This book is at the leading edge of what will **likely become a major pedagogical trend in mathematics: teaching the fundamentals from a categorical perspective**.
CTI aids students who are studying these disciplines by communicating what is considered as a "common knowledge" by category theorists.
Any other kind of nerds
---
### who want to learn some mathematics just for fun
Aside from being informative, CTI is easy and fun to read so many people read it just as a form of recreation.
Feedback
===
My book received a lot of positive feedback and a lot of people shared it on social media.
Comments
---
> This is the most beautiful and clearly written introduction to categories I’ve ever seen. Highly recommended. [Source](https://twitter.com/y0b1byte/status/1417567589241339912)
> Great stuff man! Internet needs more accessible Category Theory expositions. [Source](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! [Source](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. [Source](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! [Source](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
---
[Lobste.rs](https://lobste.rs/s/bc11fo/category_theory_illustrated_monoids)
[HackerNews](https://news.ycombinator.com/item?id=26658111)
[Twitter](https://twitter.com/search?q=category-theory-illustrated&src=typed_query&f=live)

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