I mean, practically in the sense of mathematical stuff not so much philosophy

But from all this talk about how our brains work

The important part is that we want to be able to – the major tools in our

Arsenal, right, are: Abstraction, Composition, and i'm going to add one more thing to this, Identity

So as I said "abstraction" means we want to get rid of the details we want to

like forget about the assembly language or machine language of what we are

doing and that's not only in programming it's also in mathematics or physics we

want to get rid of unnecessary details

ok so once we get rid of unnecessary details then suddenly what happens is

that things that were different, but they were different because of unnecessary

details, they suddenly become identical. Like if you have let's say two

billiard balls of the same colour, they are not really identical if

you look under the microscope, they have different maybe scratches maybe they

have different configuration of atoms, right but you can replace one

with another when you're playing billiards

So once you abstract, things that used to be different now become

identical. this is why we have this notion of identity and this notion of

identity because of abstraction is always nontrivial. So there are

things that are strictly identical, so like you can replace one with another

and you won't notice any difference but then there are things that are identical

for all intents and purposes

ok and in mathematics is

this is like a very very important thing, that there is this distinction between

"it's really the same" or "it's not really the same, but we will look at it

as if it were the same". And there is even a whole foundation theory that's being now developed

that's based on distinguishing between what's identical and what's not identical

it's called Homotopy Type Theory, it's a very hot topic right now in mathematics

that just tries to solve this, just this one problem of what things, what are things

that are equal and what are things that are almost equal, or as they say "isomorphic"

is isomorphism and equality the same thing? Or not? Should it be treated the same way or not?

So composition and identity, these two things, they just define category theory

This is all there is in category theory, it just encompasses composition and identity

So you are now ready for the first definition: the definition of category. ok and

I wish I could define category – what a category is very precisely but I can't

Ok so I will be using terms like a bunch

Ok, a category is a bunch of objects

and like, you would want me to say it's a set of objects, right? Because a set

has a precise mathematical meaning, right? But it's not! Okay so you have categories

that have sets of objects and it's fine, but not all categories have sets of objects.

It turns out that there are things that are bigger than sets.

and you might think like "What can be bigger than a set?" I mean what's a set?

A set is something that has a bunch of elements, right?

A set is defined by membership. "Is this element a member of the set or not?"

But then you know, you can build sets with sets, you can say "I have a set of sets

that are two elements sets" and "I have a set of all sets that are of this

particular shape" right, so elements of a set can also be sets, right? It's like, from

a set theorist's point of view everything is a set right? They have a set-hammer

and everything is a set-nail.

So sets are built from sets and so on and and then you can say "So how do i

define a set? Ok I will specify what kind of elements it has." right, so I can

say okay "I have a set of sets", and there are some sets that are members of

themselves and there are some sets that are not, like a set of dogs is not a dog,

so that's one. Well let's think of an example of a set that is a member of itself

Yeah?

A set that only contains itself. Okay yeah. So now if you can define a set

that – so you can also define a set of sets that are not members of itself

and then you can ask "is this set a member of itself, or not?"

And if it is a member of itself then should not belong to the set

and if it's not, then it should belong. It's like, you know, the Barber's Paradox

if he shaves himself then he should not shave himself he doesn't shave himself-

so you get into these paradoxes in set theory, right? So if I say a

category is a set of objects, then I immediately get into this problem and in

particular you cannot define a set of all sets for some other reason, right?

Because it's too big, it's like a power set, it would have to contain all its subsets as

well so it's a set that contains all sets including its subsets and so on and so forth

these are the mind-blowing things that set theorists are thinking about

and they cannot sleep at night. So category theorists say "let's not to worry about this"

well, they do worry about it but they have ways of doing this.

So I will say it's a bunch of objects, ok?

So a category consists of objects

without specifying further whether they form a set or a bunch or a class – sometimes it's called a class

which is like less precise than a set.

And these objects I will just draw as dots.

And these arrows are called morphisms. I'll sometimes be saying "morphisms" sometimes I'll be saying "arrows"

So a morphism or an arrow is something that goes between two objects

So you have an object a, then you have an object b,

then you have an arrow between them and you call it f

And now you might want to ask "but what's an object?"

And I can't tell you!

An object is a primitive in this theory, it has no properties it has no

structure, internal structure, it's nothing. It's like the atom, it's a

point, it has no properties

Ok, what's a morphism? Well a morphism is also a primitive. It has no properties

well except that every arrow has a beginning and an end

ok so that's an important thing

so in fact the reason for having objects is so that you can mark the ends of arrows

They don't serve any other purpose as just being names for the ends of arrows.

We're identifying the ends of arrows

and notice that this is really funny that we are using

arrows here, it's like going back to what I talked about the primitive

human beings hunting mammoths with bows and arrows, right? This is really

a very interesting thing that we perceive the universe through these

notions that were developed by hunter-gatherers, right?

They described the world in terms of spatial relationships, right?

So like, when I'm talking about category theory and when

mathematicians talk about it, they will put things in space. Maybe with their

hands – manipulate it – so we know how to manipulate things, right? We know how to

position things in space. We talk about spatial relationships: something is

above, something is below. Higher level of abstraction, lower level of abstraction

these are all spatial relationships, right?

Hunter-gatherers also understand movement. It comes from here, it goes here.

This is movement this is an arrow from A to B, right? It has this little thing

here with barbs, so when you when you hit the animal it can't just pull it

And of course we have social language we are

hunter-gatherers, we're social animals so they talk about you know

relationships between things in terms of this guy points at this guy and

and by movement going from this place to that place through this

place and so on. So we'll be using this language all the time and I want you to

like realize what kind of language we are using and how it actually constrains us

Ok? But anyway, we have objects and objects we draw in some kind of spatial

relationship usually, because we are using the spatial part of our brain, we have

morphisms which are arrows so they suggest movement and relationship.

So, what kind of things can happen? I mean we can have, let's say,

multiple arrows going between objects so we could have object A and B, you can have zero or

more arrows going between them. So every time you define the category you specify

"what are the objects of this category?" and for each pair of

objects you specify the arrows that go between these objects.

Some objects are not connected with arrows, other objects are connected with one arrow,

other objects are connected with infinitely many arrows,

it could be an uncountable number of arrows going between two objects

So, like, if you have an idea that the category is sort of like a graph that's

a good idea, right, except that you have to be open-minded about what the graph is

it might have infinitely many nodes and can have infinitely many arrows between

between two nodes or you have zero or, you know.

But you have to expand your mind around this

And of course you can have arrows going from B to A as well, and you can have

arrows going from B back to B, or from A back to A, or you can have multiple arrows

going from A to A, and so on. So all these are possibilities – sometimes people

get stuck and I get questions like "How is it possible that you can

have more than one arrow? Aren't they all the same?" no, they are different you

know? You can have infinitely many or an uncountable number of arrows going from

A back to A, it's okay, you just give them different names

This is f, this is g, this is h. So this is what a category is .

Now that's not all, because that just tells us about what are the elements of a category.

We haven't talked about these two.

So composition is the property,

a very simple property that if you have an arrow from A to B, and you have

another arrow from B to C, so we have object A, object B and object C and you

have these two arrows, then there always must exist an arrow that's a

composition of these. So if I call this one f call this one g, I'm using

these names f and g that suggests functions, right? Because at some point this will be

one of our models for a category. So this is called "g after f".

"After" is this little circle. So this arrow is called "g after f". And here we have this idea that

going from A to B using f and going from B to C using g is identical to

going from A to C using this path called g after f. Ok, so this arrow is identical to

the composition of these arrows

ok? It's very important to understand that there might be multiple arrows going

from A to C, right? But this one is a composition of these two and it must exist.

So for every composable pair of arrows, composable means the end of one

is the same as the beginning of the other. And here it's important that we

have these objects to identify the ends and the beginnings, right? So the end of

this is B, the beginning of this B, so they are composable. And if they are

composable then there must be a composition, there must be an arrow going from here to here.

So this is called composition; check.

Now when we are defining a category, the category is defined by giving us

objects, saying what the objects and arrows are, and then defining composition

which is sort of like a multiplication table for arrows. So for every two arrows

you have to define what is their composition. It's a humongous

multi-dimensional multiplication table or infinitely dimensional multiplication table

for every three objects you have to define all possible

combinations in which you you can compose arrows going between these

objects and so on

right? so it's a humongous multiplication table and the whole information about

the category is in this multiplication table

So remember this is the multiplication table for a category, it's

a composition table - how you compose morphisms.

And different composition tables will give you different categories

Because since objects don't have structure, since arrows don't have structure

they don't contain any information. But the composition contains the information

and we just want to encode everything, everything, within this composition.

Now Identity. For every object – let's call it A – there is an identity arrow

Which we will call "id". And sometimes id with a subscript "a", meaning this an identity for the object A. So there is this

arrow that we call identity. There's one per object, for every object in a category

there is an identity for this object. Now why am I calling this identity?

Because of composition. So if i have an arrow going from, well let's call it A and let's call it B,

okay, if I compose this arrow f with id_b I get back arrow f. In this

sense this is an identity, right? If you think of this is a multiplication

table, f times id is again f. So this is like a 1 in terms of composition.

So I can write it using this notation that id_b after f (so first I go f, then I go id_b) is the same as f.

Equal. Ok? This is one morphism, this is another morphism. They are equal, it's the same morphism, ok?

And of course there is this symmetric thing where I have id_a and have some, lets call it g,

from A to B. So if I start with an id (id_a) and follow it with g, I will get g.

Ok so these two things, they are not the same, right? I mean this is the left identity,

this is the right identity. Sometimes they are well maybe I shouldn't say that

"sometimes they're the same" but its just that you have to have a left identity and you have to have a right identity

So this is one of the axioms of a category, or laws of the category, that in

every category – so if you think of the category as a graph, this graph has to