Lecture 1: Richard Southwell - Category Theory For Beginners: Introduction

13 May 2019

Category theory is close to the perfect language. It can be used to describe many mathematical ideas, and see the relations between them, and their deeper structure. This is the first video in a course where we will carefully introduce the main ideas of category theory, and motivate them with lots of applications. This first video gives the definition of a category and give some motivation for studying the subject. In particular we define a category, give the definition of a monoid, we define a pre-ordered set (preorder), we define the category Set, of sets and functions. We introduce functional programming. We define the final object (terminal object), and explain how the ideas can be used to do set theory without considering the elements within sets directly.

Intro

0:05

okay so I want to say something about where the power of category theory comes

0:10

from and I think it basically comes from

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category theory being a sort of minimalistic language for describing

0:22

composition I mean so many things that people think about can be put in terms

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of composing things together like when we think about moving around in space

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you think well if I move from A to B and then B to C I can compose those two

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journeys to make a bigger journey from A to C or if you're going to describe to

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someone something like say how to make a machine it'll be a series of instructions a series of actions that

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they need to take they need to carry out and if you compose all those actions together you get the process of building

0:59

the machine so I think it's just something about the way that people

1:05

think we like the sort of break problems down into small chunks and for that

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reason category theory applies to so many different things we can think about and because it's a minimalistic language

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because it sort of represents this notion of composition in such a visual

1:28

and simple way we can see it being applied in all of these different

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systems and that's why when you look all these different areas of endeavor

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different areas of mathematics like set theory and logic and geometry and so on

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we can see that category theory can be used to describe all of those different

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areas or other areas of human thought like programming and computation and

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natural language again we can express these things in terms of category theory

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now there's a couple of important things that you get out of that one is that you

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can see all the sorts of commonality between these different structures and you see some amazing

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commonality because you can see things like for example there's a certain viewpoint by which logic where you're

2:24

proving theorems looks pretty much exactly the same as set theory where

2:30

you're doing function sending sets to other sets and so on and it's just

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remarkable to see these kind of relationships between things but what's even better is that you start to get an

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appreciation for what's fundamental so you can have these ideas like saying set

2:53

theory which we kind of have enshrined in mathematics as being the sort of

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foundations of the thing where all the other ideas come from it turns out that

3:04

actually those ideas from set theory we

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can start to see with category theory that hang on these notions are not fundamental they are they can be

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expressed in terms of category theory and generalized to lots of other categories so for me this is what really

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got me hooked on the subject is that you start to get a wider appreciation for

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what kinds of things are really fundamental in mathematics and so we

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then see that there are other object other ideas from category theory which apply all over the place and because

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we're sort of working in this language which is so fundamental everything fits

3:52

together very nicely and it's it's really a very satisfying thing to learn about so that's part of a reason I think

4:01

it's such a great language another reason is because we deal with the sort

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of we can deal with mathematics with the sort of large-scale structures of

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mathematics so category theory has been described as the mathematics of

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mathematics because what you can do is you can define a kind of mathematical

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objects like say assets or something and you can then say well is there a

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category that this object lives in naturally and there often tends to be and these categories are sort of

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infinite structures very nice to think about and we can then compare them with

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each other and so we can say for example what are the similarities between set

4:56

theory and topology and we'd have one category representing what's going on in

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topology and another one representing what's going on in set theory and we can just show relationships between them and

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this can be done remarkably easily considering these immense kind of

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structures that we're comparing so it's really sort of like going in a

5:20

helicopter very high above the landscape and being able to see all the details of the things that are going on so what's

What is category theory

5:28

category theory about categories so a category is a collection of objects and

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these objects we can draw as points because they're sort of atomic entities

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they don't really have any structure in and of themselves but between these objects we have arrows so these arrows

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could represent relations they could represent actions and the core notion in

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category theory is that we can compose arrows to get other arrows just like if

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you go on a journey from A to B and then a journey from B to C you can compose

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those to make a journey from the A to C okay so this is my first little category

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don't draw anymore parts of it and it has these objects a B and C and let's

6:36

label these arrows let's call them F and G so I say again a category is a

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collection of objects we can draw points with some arrows between them so at this

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point you might be thinking well isn't that just a network and it's true that networks do have a lot of similarity to

6:58

categories but there's a bit more to a category than a network because we can

7:06

compose these arrows so we have these two arrows which you know the head of

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this one is is the tail of this one and that means we can compose them yeah to

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get to another arrow and we'll call this our own so this is the area you get by going on F and then going wrong G it

7:25

makes this big arrow when we call this G after 1/2 so this little circle here we

7:37

can read this as after ok so I think at

7:45

this point we're ready to start writing down a definition of what the category

What is a category

7:51

is

8:07

so it's a collection of objects and

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arrows between objects so I should say

8:30

at this point we can't have multiple arrows between a pair of objects we can

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of course have arrows going in the opposite direction we can have arrows

8:41

going from an object to itself in fact we must have objects going arrows going

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from objects in themselves I'll talk about those in a moment so yes we can

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have lots of arrows but now let me write

9:03

down this idea of arrow composition

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okay so I'm writing this here there's notation they're just things that f is

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an arrow that goes from object to object B so it's it's similar to the sort of

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notation used similar in meaning to the notation used over here it's just a little bit easier for sort of linear

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text so if F is an arrow from A to B G

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is an arrow from B to C then we have to

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have this arrow G after F which is an

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arrow from A to C so this is the core

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notion of arrow composition which is really at the core of category theory I

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mean often when people draw categories they draw these sorts of network pictures but not all the information is

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really shown there because to really know what to cut at what's going on in a category you need to know which are own

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compositions make which other arrows so we have a pair of arrows like this you

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know that the destination of this one is the source of the next and that means we

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can compose those arrows and that means their composition should be another

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arrow in the category an arrow that starts at a and ends at C and we see in

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this picture there are two such arrows so when we're defining the category we'd have to see which one of these

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corresponds to the composition of these two you so we're getting close to having

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a full definition of what a category is there's just two more two more

11:31

properties that the structure has to satisfy in order to be a category one of them is identity

11:38

arrows so every object has to have a arrow going from itself to itself a

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special kind of our own calm identity

11:52

arrow and so we call the identity arrow

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associate more object a IV a this is ID B and here's IBC and these identity

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errors have the property that if we compose them with an over arrow we just

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get the other arrow back so they sort of represent do-nothing operations a bit

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like when you add zero to a number you just get that number back so let's write

Associativity

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that down

12:47

so for every object B we have this identity ro of B which is an arrow which

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is not a from B to B and it's got the

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property there

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ivb after F is F and G after ID B equals

13:29

G okay so what I mean here is that if we

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go along this arrow and then we go around this identity so you know we go

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on F and then we go along the identity arrow that gives us there's a competition just F back again and

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similarly if we go along the identity arrow first and then we go along G then

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that just gives us G back and this of course has to hold for any functions any

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sort of incoming arrow I shouldn't call them functions at the moment any

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incoming arrow F and any outgoing arrow

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G right them so okay the final part of

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the definition of a category is associativity we want these arrow

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composition of this arrow composition to satisfy this extra more extra condition

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called associativity so what is it basically let's say we have three arrows

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like this F G and H and we want to

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compose these arrows together so I've just told you that we can compose pairs

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of our arrows together to get the third ones well here we have three so how do

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we compose them what order do we compose them together we could compose these two

15:22

together and then this one or we could compose

15:30

this one and then these two together and the answer is it doesn't matter because

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we're assuming that arrow composition is associative so what that boils down to

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is saying that if we compost H together

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with the thing that we get by composing

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G and F so if we compose this green arrow and then this orange are on the

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right that'll give us the same thing as if we compose this orange arrow on the

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left and this per player on the right so that would be so what we basically say I

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mean the way I'm writing this I'm using brackets to represent it in this case

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here we're composing these two hours first to get to get this green one and

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then we're pre-con we're composing with

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H after that on in this case here we're evaluating this composition first and

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then were composing with that but I'm saying it doesn't matter which one of

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these you do so essentially that just means that we can just write H after G

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after F and we don't have to use these brackets because we don't have to worry

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about which order where we're composing these pairs of arrows and this is

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similar to what we have in arithmetic in arithmetic we have of course 1 plus 2

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plus 3 it is the same thing as one plus two

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plus three okay so addition with whole numbers with

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any numbers is a associative operation and we're assuming that this arrow composition is also associative and

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really this is just to make category theory easier you know you could you

18:03

could play around with systems where this doesn't hold you might want to think about mathematical operations that

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are not associative can you think of one okay I'll tell you one is division so

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that's not always an associative thing but anyway that's basically the

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definition of a category so let's just reiterate a category is a collection of

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objects arrows between objects and we can compose arrows so whenever we have

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two arrows that the source of the the

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destination of the first is the source of the second then we can compose them

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and there'll be another arrow that's that composition that has the same

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source as the first one the same destination as the last one every object

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has to have an identity arrow that has a property that if we compose any other

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amaura with it we just get that other arrow back and arrow composition has to

19:17

be associative anything that satisfies those conditions is a category and that's what category theory is all about

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it just turns out that this type of structure is really really useful for

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expressing loads of things okay so now I'm going to talk about some examples of categories and this is going to get very

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interesting very fast because there's some really mind-blowing kinds of

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infinite categories which are really fundamental to mathematics very very interesting but before that

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let's start with the simplest categories so the very simplest category is shown

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on this blackboard behind me it's the empty category it has no objects and no

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arrows and as fascinating object is that is let's look at something a bit more

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interesting which is the category which just has one object and as few our

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arrows as possible so are we allowed to have no arrows no we're not because we

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have to have an identity arrow for every object so moving up slightly we have

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these categories which have multiple objects but only these identity arrows I

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should call that I da excuse me and then

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this will have an identity arrow IDB and

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this will have an identity arrow are you see so we have categories like this and

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you can think about it's very nice exercise to see if you

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can draw out some of the very simplest categories which we have so here's our

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sort of non trivial one still very simple

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we just have one hour between those and

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you can carry on you can think about okay there's a category route to our Aires there's another one which occurs a

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lot and so on now let's talk about a

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type of category which is very important in mathematics and that's a category

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which has just one object but potentially many arrows so of course it

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has let's call its object to a

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and of course it has to have this identity error and it can have other

Mono age

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errors as well okay so potentially many other arrows going to itself and that type of

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structure is very important I will tell you about a couple of examples of it and

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it's something called a mono age okay so I don't want to overload you with

23:02

definitions at the beginning we'll go into what ma nodes are later but I mean

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I've already basically told you the definition of amyloid because I've said it's a category of one object and

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basically we have all these arrows going from the object to itself and I've told

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you about how you can compose these arrows of course if you compose a couple

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of arrows going from the object to itself you get another arrow going from

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your gift to itself so you can think of these sorts of self loops as elements

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and you can think of there's some sort of an operation where you take a couple

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of elements and you get an element out that's encoded in the arrow composition

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so an example is in this case let's say

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we call this identity elements let's call this one and let's just call this

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arrow zero okay so if we imagine that

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the arrow composition is multiplication then it works out quite nicely because

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we have 0 times 0 is 0 so in other words

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we're imagining the composition of this arrow with itself is is itself if we

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compose this with itself we get that arrow again and then this does

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indeed acts like the identity element because if we pre compose it with 0 we

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get 0 if we post compose it with 0 we get 0 and if we compose it with itself

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we get itself so this category where we

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just have one where we just have one

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object and we have we think of it and we

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think of these two arrows is 1 and 0 and we think of composition as mult multiplying those that gives us a

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bonafide category I've shown that the identity element is well behaved you can

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think about you know does this satisfy associativity as well which it would have to do for this to be a category and

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indeed it does so this is a nice simple idea of a category a more useful one

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perhaps is this time I'll have our we'll

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call our identity element 0 and we're going to have lots of arrows so there's

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one and we'll have another arrow which we'll call two and another arrow which

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we call three we carry on let's say we have an our own for every every every

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whole number okay every integer so in this case we can say that arrow

26:26

composition corresponds to addition and again we get a category so these are all

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examples of things called mono it's basically a mono it is just where you

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have you have a lot of elements you have an operation that takes any pair of them

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and makes another one and it is an associative operation

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and it has sort of identity element so it's just what I wrote on the board

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before for just applying to the arrows going from this single object to itself

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the arrow is being thought of as elements so mala weeds are very

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interesting and mmm they're well worth studying indeed if you want to

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understand category theory moloids are a kind of good place to start because like

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one of the things that it's kind of hard to wrap one's mind around at the

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beginning is this idea of composition and what type of operations you can you

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can what kind of things you can do with composition and modes are a great place

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to study that in miniature because in models we can compose it anything with

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anything whereas in categories they're a bit more intricate because we can only compose

27:57

arrows which are a head to tail okay so

Preorder

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the other example is something called a

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pre-order okay this is the other sort of so amyloid is a great way to get your

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head around composition because it's it's a category which just has one

28:34

object but lots of arrows potentially and the other extreme is a sort of pre

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order or a pre ordered set it's sometimes called and that's the kind of category that has lots of objects but it

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doesn't have that many arrows in particular it doesn't have more than one

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arrow going in a particular direction between two objects so

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how do we get a pre ordered set well a

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nice easy way to think about it is if we

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start with any we can sort of any

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directed Network okay so a directed network is a kind of simpler idea than a

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category it's just where you have a load of dots you can call nodes or vertices or objects whenever you want to call

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them and then you have some arrows between them

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so there's an example of a directed Network it's it's not a category because

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for example we have an arrow from here to here and from here to here but we

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don't have an arrow from there to there so the composition is missing we also

30:19

don't have any self loops so it's not a category it's just a directed network

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but we can turn it into a so-called pre ordered set I mean we have this set of

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set of elements nodes if you like and we

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can basically yeah we can put we can turn it into a pre ordered set so how do we do that well basically we we think

30:55

about walking around on this network and what we say is we're going to draw an arrow whenever we can walk from one

31:05

element to another using these arrows so we already have an arrow from here to

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here so we don't have to draw one in but we can indeed walk from here to here to here so we should add an extra arrow

31:21

from here to here because that's basically the arrows are representing

31:28

the presence of paths similarly we can we can go from here and we can go from

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here to here and we can also do a sort of no distance walk from an object to

31:47

itself

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so if you started with a directed

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network which didn't have any multi edges okay so there wasn't any of this business where we have multiple arrows

32:12

between an object and the thing we started with like that none of this initially then if we add

32:22

these extra arrows to represent the possible paths then we end up with

32:28

something basically called a pre-order I mean usually in mathematics and also I

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should say any pre-order can be created like this if you look at the definition of a pre-ordered sets

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it'll be defined somewhat differently probably but this is an equivalent

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definition of it and usually you would write so if there's an arrow from let's

33:00

say a to see is there is you'd write a less than or equal to see okay so now I

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want to talk about one of my favorite ideas in Category and it's a sort of

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large-scale infinite category which is amazingly important for mathematics and

33:22

it's really a very fundamental and rather beautiful notion it's called the

33:30

catenary set capital S et it's a

33:37

category and the objects in this category represents sets so we can think

33:43

of a set as a load of elements

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so for example this is for sets which has one two and three and the one of the

Set

34:04

remarkable things about this category called set is that its objects each of

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its objects represents a set and we imagine that all sets are represented so

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this is a pretty mind-blowing idea already I mean actually it goes a bit

34:27

too far because if you start thinking about the collection of all sets you

34:32

you're in paradox country okay the things can go a bit wrong with the logic

34:38

because that's just - there's so many of these things that things can go wrong so

34:45

there is a way to set up this kind of thing rigorously you can essentially

34:51

define the sort of universe of sets which holds all of the sets which you

34:59

may encounter by doing the familiar operations and it's an immense it's an

35:05

immense kind of collection of different sets and everything we do in mathematics

35:10

every set that we refer to is an element of that and we can just say that each of

35:16

the members of that universe each set in that universe is one of our objects and those are our objects but I'm not going

35:24

to get too hung up on these foundational issues I just want to introduce this amazing idea so each object represents a

35:31

set so for example this is a set holding one two and three and then there'll be

35:37

another object over here which is a set

35:43

holding true and false

35:49

and yeah so these are the objects and

35:54

then what are the arrows well the arrows are functions so a function let's say

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we'll call this capital A and this one capital be a function is something that

36:08

sends each element in the function from A to B is something that sends each element in a to an element in B so

36:22

there's a function f from act opinion this various or the ones so our arrows

Composition

36:29

are functions and all functions between these sets are arrows in this category

36:37

so this category set has objects as sets

36:50

and arrows as functions okay so I still

37:06

need to tell you how we can compose these functions how can we compose these

37:12

arrows together I mean remembering the definition of a category if you have two arrows and the

37:18

head of one is the tail of the next then you can compose them so that composition

37:24

must exist so we have to define how we're going to compose these arrows together so let's say we have another

37:31

set over here let's say this has elements little a little B little C

37:41

little D and say we have another function

37:49

gee so how do we compose F and G that's

38:00

something that I'm going to have to define I mean basically it's pretty straightforward we just go along a

38:07

narrow F see what element were at and then follow the arrow out of that

38:12

element with G and see where it ends up in this set capital C so the composition

38:21

of these two arrows these two functions

38:27

looks like this so two goes to F goes to

38:32

B so in the composition to just go to straight to B 3 goes to F goes to be

38:39

something the composition 3 just going

38:44

straight to B so this here is the composition and we can call it G up

38:50

there ok so let me just introduce a bit

38:57

more a bit more sort of background for you to think about these functions one

39:04

view of a function is it something like a black box and if we feed in an element

39:12

so this is a func misses this green function from A to B if we feed in and

39:18

elements from a then we get out an

Definition of Composition

39:24

element from beef and the way that we

39:29

write this normally is F of the inputs

39:35

is equal to the output okay so this is

39:41

basically a way we'd sort of write about

39:46

these things and how about the composition how do we define that well the way it will be defined as we'd say

39:54

that G after F

39:59

is a function and if it's operating on an element X what we get is what we get

40:10

if we acquired G - what we get if we apply F to X so if we start with one we

40:21

apply F to it we get T then we apply G to it we get a therefore G after F of 1

40:29

is equal to what way okay so we can compose our arrows we have to or we wish

Categories

40:37

to prove that this so we want to now show that this is actually a category okay so this means that this arrow

40:47

composition has to satisfy the different laws of the category and remember

40:54

there's two of them that we have to satisfy one is that we have to have these identity arrows which just are

41:03

sort of four arrows which go from objects to themselves so in our case these objects are sets and so we need

41:11

some kind of function that goes from a set to itself and it sort of has to be

41:17

like a do-nothing kind of function because it has to have the property that

41:22

if we compose it with other functions we just get the other function back and there are such functions and they're

41:31

basically called identity functions or identity maps so it's just a function

41:41

which sends every element to itself

41:51

and I think quite clearly there's going to be a function like that for every set

41:56

so that's good we have our identity arrows the other thing we need is

42:02

associativity so we need to show this

42:07

for these composable functions we have

42:25

this anyhow remember though so I've

42:32

drawn some of the sort of goings-on in this category set up here but this isn't

42:40

really how we represent it in category theory because remember I told you this

Objects vs Sets

42:47

in category theory our objects which in

42:52

this case representing sets while our objects are just thought of this sort of

42:57

atomic things they don't really have any internal structure so this is a bit

43:06

strange where is all our information gone how do we know how many elements

43:11

there are in these sets so what do we know in category theory well we have the

43:17

the objects that represent the sets and we have these arrows between them which

43:25

represent the functions

43:54

and so how about we just lost some of the information here because we're no

44:01

longer depicting the elements within the sets and the answer which is quite

44:09

remarkable is no we haven't really lost any information it turns out that we can

44:14

recover the information about these particular sets you know how many

44:21

elements they have in them we can recover that information just by having

44:27

this category so in this category here we have the objects and we have the

44:32

arrows and we also know which arrows compose together to make other arrows

44:37

and it's a remarkable fact that in fact that's enough for us to be able to

44:44

identify particular sets and particular

44:50

mappings and so on so how does that work I'm going to talk a little bit about

44:56

that in a bit let me just talk something more about why this category set is so

45:04

useful in another way and that is a sort of application that it has in

45:09

programming in particular in functional programming so there's various different

45:16

ways to do programming one of them is by writing loops and mutating values and so

45:22

on there's another very elegant way to do computer programming which is functional computer programming and in

45:30

that kind of scenario what you're doing what your code is basically is a load of

45:36

functions which you compose together and you end up getting an output at the end

45:42

and you're finished so this is a very nice idea and it's called

45:49

related to this category set so just to give you a very small flavor of this

45:54

relationship and what functional programming is about let's say you have

46:00

a programming problem which is that you want to write a code which is going to

46:06

tell you if a particular word has three or less letters so here's how you can

46:13

think about that in this context so you could say okay I have a set of words and

46:33

then I also have this here which is the

46:40

set of all non negative whole numbers so 0 1 2 3 4 and so on

46:46

and we'll also have this here which just

46:53

has two elements called true and false

46:59

so how are we going to write this program well we're going to use function

47:05

composition okay so let's make a function which goes from a set of words

47:10

to the set of non-negative whole numbers and we're going to call this function

47:17

once and it's going to send a particular

47:24

word to the numbers of letters in that word so it would send cat to 3 for

47:31

example now let's have another function which goes from these non negative whole

47:39

numbers to true and false and let's call that function less than or equal to 3

47:51

okay so the way this function works is that if

47:58

if you if your number is less than or

48:03

equal to three then it's sent to true otherwise it's sent to false okay

48:11

and so now if we compose these two functions together we'll get another

48:21

function which basically sends a word to

48:26

true if it has less than or equal to three letters otherwise it sends it to

48:31

false okay so that's something about how you can use how function composition is

48:38

used in in functional programming and this is quite remarkable because it

48:43

basically boils down to saying that when you're doing functional programming

48:48

you're sort of just walking around on this category set I mean there are details it might not be exactly the same

48:57

because types and infinite procedures

49:03

and so on but I mean basically this gives you a very nice idea of what's

49:09

going on in functional programming and this is one of the reasons that category theory is often of interest to people

49:16

who are doing that okay so now I want to start to substantiate this claim I was

Final Objects

49:22

making which is that if we're looking at this category set and if we're taking a

49:31

categorical viewpoint on it so we're only looking at the objects as sort of

49:36

points and we have these arrows between them then I claim that we can actually identify the different sets which are

49:45

involved and the different and which functions these different arrows

49:51

represent and so on so how can we do that well the key is a general

49:58

definition there's something called a final object so final object

50:12

is an object search this for any object

50:30

hey there's a unique arrow just one

50:39

arrow from this object 82f okay so if

50:54

these are our objects in our category and this is a final object then there's

51:03

just going to be one arrow from every object to this one and just one or one

51:11

arrow from this object to itself the identity our okay so why is this

51:17

important for understanding about this category set well because there's a

51:23

particular object in this category sets that was a killer set if you like which

51:30

is a final object of this category and maybe you would like to pause the video and think about what kind of sets would

51:38

represent a final object in the category set because it's really the key to understanding this this issue well it

51:46

turns out that if we just pick a set which just has one element in it then

51:56

that object that corresponds to that set is actually a final object and it's

52:04

quite easy to see why if we have another set

52:18

so this will be represented by another object in our category and how many

52:24

arrows are with them from this object to this object how many functions are there from this set to this set well there's

52:31

just one just the function which kind of claps is all of the elements of this set

52:37

on the left into this set on the right it's kind of like a black hole so that's

52:43

great because that means that just by looking at the objects and the arrows we

52:49

can identify the singleton set the object representing the singleton set in

52:56

the category without having to actually look inside of these objects you know

53:02

what what elements they have and what what they're supposed to represent that we can just look at the arrows and

53:08

determine determine where the singleton

53:13

which object represents four singleton sets now that's not all it's very

53:21

fruitful to then ask well what are the arrows going from this singleton this object representing the singleton sets

53:28

or other objects so what functions are there going from this set here to this

53:35

set here well there are three of them there's one like that one like that and

53:49

one like that so this is really quite remarkable because we can then identify

53:57

a set width we can identify an object

54:04

representing a sets with three elements in this category as something that has

54:09

three arrows in coming upon it from the singleton sets so I mean this is how

54:18

this picture looks like when we're allowed to actually look at the element look at the elements in the

54:25

sets but from a categorical viewpoint we would just see something more like this

54:32

we have this sort of final object here final objects are also called terminal

54:40

objects in categories but anyway you would have this final object here and of

54:45

course there's just one arrow from this other object to it but there are three

54:53

arrows going from this final object to

55:01

this object on the left here

55:11

and so by just looking at the arrows we can then say hey this must be a this

55:17

must represent assets which has three elements and moreover each of the

55:23

elements in this set corresponds to a particular arrow so we no longer have to

55:29

talk about the elements of assets we can just talk about the the arrows coming

55:35

from this final object into them and this really gives us a way to sort of generalize lots of the things that we

55:43

talk about in set theory and there are various all the ways to generalize things as well just to make sure I've

55:49

got this idea clear I mean let's say we

55:56

have another set over here true and false how could we identify this the

56:05

object representing this set in our category sets well we just again look at

56:11

what are the arrows coming out of this final object here then we see there are

56:22

two of them so when we're just looking in our category set and we're not allowed to sort of peep inside these

56:28

objects and see what they represent we can still identify this as a two element set because there's just going to be two

56:35

arrows coming out of this final object into it so yeah and there's various

56:44

other ideas from set scenery that you can generalize in a similar kind of way

56:49

for example if we have two sets like this one here and this one here we can

56:56

form another set which has pairs one element from here and while then from

57:03

here this is the so-called Cartesian

57:10

product of this set here and this ax here it's another set and I always used

57:17

to think that the such a kind of product of two sets was a fundamental idea

57:24

mathematics turns out it's not turns out this idea can be expressed in terms of

57:29

category theory and generalize to lots of other categories as well not all

57:34

categories not all categories have final objects not all categories can have this

57:41

kind of notion to find in them but many important ones can and so we're going to

57:47

get some understanding of what's fundamental in mathematics from a kind of new more global perspective.