Introduction

In the previous post we were exposed to the notion of a category and we started exploring it a bit. We also explored the concept of Nothing a viewed it as a generative concept: a concept on which we can build other (perhaps richer) concepts.

Another (even more important) lesson from the previous post is that, concepts and theories about them are best used when we think about them relatively and that there is no such thing as The Geometry, The Logic, The Physics, … a universal theory of a subject that encompasses all the knowledge of the subject.

In the Theory of Everything, every concept we will define will be defined using relative means. Category theory embraces this way of working totally. It actually defines all its concepts using the notion of a Universal Property.

Universal properties

To explain you what is a universal property I will return again to Nothing. Imagine you are in a given category. There are many objects floating around, there are arrows between some of them (morphisms)… This is a very abstract setup, because like I explained previously, categories come in very different flavors, the objects (of concern) could be words, numbers, algebraic structures, theories, computer programs,… basically anything you wish to study the properties. The morphisms could also express rich relations between the objects.

In this very abstract setup we can define concepts using a universal property. Instead of trying to define you the notion of a universal property I will give an example. We will express this as a definition:

Definition 0: A category $\mathbb{C}$ has an initial object $\mathbb{0}\in\mathbb{C}$ if and only if for any object $X\in\mathbb{C}$ we must have a unique morphism
$\xymatrix@1{ \mathbb{0} \ar[r]^{!} & X}$

So imagine an arbitrary category, it may or may not have that initial object inside that satisfies. Until now we only gave 3 examples of a category: the category $\mathbb{T}$ of known living things, the category $\mathbb{N}$ of natural numbers and $\mathbb{Set}$ the category of sets.

Example 1: (counter-example) In the category $\mathbb{T}$ of known living things there is no object $\mathbb{0}$ that would satisfy the above definition most probably! Remember the morphisms in this category:
$\xymatrix{ Dogs \ar[r]^{are} & Mammals}$
This living thing $\mathbb{0}$ would need to have a morphism from it to any other living thing $T$. In other words, for this specie $\mathbb{0}$, we can say that $\mathbb{0}$’s are $T$’s for any $T$! This this special specie would be pretty weird since it would need to be at the same time a dog, a cat, an alligator, … a specie that would embody all species at once… The probability that one finds such a beast is almost 0 no? So $\mathbb{0}$ (an initial object) doesn’t exist in that category $\mathbb{T}$ of living things.

Example 2: Now let’s consider the second example of categories we gave: the category $\mathbb{N}$ of natural numbers. This category does have an initial object: it’s the number $0$ itself. For any natural number $n\in\mathbb{N}$ we do have a unique morphism
$\xymatrix{ 0 \ar[r]^{!} & n}$
because we definitely have $0\leq n$ for any $n\in\mathbb{N}$ (and this is the definition of the morphisms in this category). So $0$ (the number) viewed as an object in the category $\mathbb{N}$ is a special object. It satisfies this universal property of being an initial object.

Example 3: Now let’s consider the third example of categories we gave: the category $\mathbb{Set}$ of sets and functions. This category also has an initial object: the empty set $\emptyset$. Because there is trivially exactly one function from the empty set to any other set $X$ (the empty function).
$\xymatrix{ \emptyset \ar[r]^{!} & X}$

The essential things to remember here is that:

1. the notion of an initial object $\mathbb{0}$ may or may not exist in a given category
2. we did not define $\mathbb{0}$ as an intrinsic property but instead $\mathbb{0}$ is defined by the particular relations it has with all the other objects of the category.
3. We call this type of definition for $\mathbb{0}$ the universal property of $\mathbb{0}$ (an initial object).

From an epistemological point of view we have considered a first universal property that led us to understanding the true nature of $\mathbb{0}$ (Nothing). This universal property defines the true essence of $\mathbb{0}$ in any possible universe (category) where it exists. This definition of $\mathbb{0}$ is also very elegant and economic and corresponds to all the intuition we have on $\mathbb{0}$ viewed in all those universes.

In $\mathbb{N}$, $\mathbb{0}$ is indeed the integer $0$. In $\mathbb{Set}$, $\mathbb{0}$ is the empty set: $\emptyset$. In all the following examples of categories we will look at, $\mathbb{0}$, if it exists will correspond to the natural $\mathbb{0}$ of that universe of study, the category we study.

Maybe it’s not very clear for you how important it may be to have a universal definition of $\mathbb{0}$. Most of you probably never even considered that $\mathbb{0}$ needed to be defined in the first place. $0$, nada, … so obvious no? Why would you define that?

Well I told you why in the previous post: because from this $\mathbb{0}$ we can construct all the other concepts for all the things we are interested to study: mathematics, programming, physics, psychology,… All these will be formed using just the two definitions we gave until now somehow: the definition of a category and what is $\mathbb{0}$ in a category.

From nothing to something

Now let’s apply this way of defining things universally to create something. Until now, we have only talked about categories (a very abstract concept I grant you) and the concept of $\mathbb{0}$ (nothing) an initial object. Can we get something out of this? Yes! You can get something by looking at the definition of $\mathbb{0}$ (nothing) in a mirror.

Take a look at definition 0 above. If you would look at it in a mirror it would certainly get confusing… But if you just reverse the direction of the arrow you get a new definition: the definition of (let’s call it for now) $\mathbb{0}'$ for which for any object $X\in\mathbb{C}$ there must be a unique arrow:

$\xymatrix{ X \ar[r]^{!} & \mathbb{0}'}$

which is the mirror of

$\xymatrix{ \mathbb{0} \ar[r]^{!} & X}$

in the sense that it’s the same definition as $\mathbb{0}$ but with the arrow reversed.

In a category, if an object like that $\mathbb{0}'$ exists we call it a terminal object and we usually give it the name $\mathbb{1}$. Yes! $\mathbb{1}$ has the symmetric universal property that $\mathbb{0}$ has. We call $\mathbb{1}$ the dual universal property of $\mathbb{0}$. Therefore we say that $\mathbb{1}$ and $\mathbb{0}$ are dual to each other.

Let’s summarize this in the following definition 1 that we put side by side with definition 0 to better view the symmetry:

Definition 1: A category $\mathbb{C}$ has an terminal object $\mathbb{1}\in\mathbb{C}$ if and only if for any object $X\in\mathbb{C}$ we must have a unique morphism
$\xymatrix@1{ X \ar[r]^{!} & \mathbb{1}}$

---

Definition 0: A category $\mathbb{C}$ has an initial object $\mathbb{0}\in\mathbb{C}$ if and only if for any object $X\in\mathbb{C}$ we must have a unique morphism
$\xymatrix@1{ \mathbb{0} \ar[r]^{!} & X}$

So basically from Nothing we constructed Something: $\mathbb{1}$. So we are onto something!

If again we reconsider the three basic examples of categories we have on the table, we may ask ourselves if $\mathbb{1}$ exists in these universes.

Example 1: For $\mathbb{T}$ the category of known living species the concept of $\mathbb{1}$ exists: it would be the specie of $Organic Things$… the most general specie of all. For all known species of living thing $T\in\mathbb{T}$ we can definitely affirm that “All $T$’s are $Organic Things$” (all $Dogs$ are indeed $Organic Things$, all $Cats$,… ); therefore we have a unique arrow
$\xymatrix{ T \ar[r]^{!} & \mathbb{1}}$
for any $T$.

Example 2: For the category $\mathbb{N}$ of natural numbers the $\mathbb{1}$ is not what you would expect… like number $1$. In that category $\mathbb{1}$ doesn’t exist in fact. To exist, such a $\mathbb{1}$ would need to satisfy $n\leq\mathbb{1}$ for any $n\in\mathbb{N}$. There is no integer greater than equal than any other integer… To force it to exist, we could declare that $\infty$ is a new integer and that for any $n\in\mathbb{N}$ we have $n\leq\infty$. This is all possible but then this is a new category, with one more object than the original category. In this category $0$ is the dual of $\infty$ (it’s mirror image). This is again an instance of our initial motto: “From nothing comes everything”.

Example 3: Finally, in the category of $\mathbb{Set}$ the concept of $\mathbb{1}$ exists, any singleton set $\{*\}$ (a set with just one element) can play the role of $\mathbb{1}$. This is because for any set $X$ we always have a unique function $X\rightarrow \mathbb{1}$: the mapping sending each $x\in X$ to $*$ (the unique element of $\mathbb{1}$).

Duality

We see that from this general definition of a category we could define a universal notion of $\mathbb{0}$ and a universal notion of a $\mathbb{1}$. This is truly the essence of the genesis of it all. These two mirror concepts are key concepts that will be at the basis of all future ideas we will explore. They form the origin of duality itself: the yin/yang, truth/falsity, woman/man, beautiful/ugly, good/bad, rich/poor …

These are the pillars of what we can call the logic of the extremes (boolean classical logic). They are degenerate extreme cases of objects in a category. The degenerate cases can help us understand the more subtle non-degenerate cases that usually happen in our reality. Limiting our scope to just those degenerate cases would be great mistake, this is obvious to most of you no? Well this is the choice that some mathematicians took in the beginning of the 20th century.

This choice had a tremendous impact on our view of what is mathematics and what mathematics we were going to develop in the following century until now. And since science and social sciences rely on the language of mathematics to structure the knowledge in their domains of expertise, this choice was disastrous for the whole advancement of human knowledge. This will the subject of further more detailed discussions.