Lecture: Richard Southwell - Category Theory For Beginners: Products

19 May 2019

In this video we explain how to define the categorical product, by starting with the Cartesian product in set theory. We see the categorical product in action within preordered sets, and see how the greatest common divisor emerges as an example. We also define the product of monoids, and the product of categories. We also show that any two categorical products of a given pair of objects will be isomorphic to each other.

Introduction

0:04

when I had the idea about making these videos it was this topic I'm going to

0:12

talk about today that I was most looking forward to basically what we're going to

0:18

do is look at something in set theory which seems like it's totally

0:23

fundamental ok this idea of taking the product of two sets I mean this is such

0:30

a basic thing it's like in science if we didn't know how to multiply numbers

0:36

together the science would be completely ruined and in a similar way this notion

0:42

has been able to take the product of two sets is really really basic and it seems

0:48

like it's a really really fundamental thing but it's not so what we're going

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to do today is to sort of peek behind this curtain of category theory and see

1:01

how this idea of products are really defined in terms of category theory and

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we're going to be rewarded for this my gosh we're going to be rewarded so there's three different special things

1:16

that we get out of this one is that we're going to get a new notion of

1:22

products for so called categorical product of two objects and this is a

1:28

notion which can be applied to all sorts of different mathematical objects we're

1:35

going to think about the product of models we're even going to bear to think

1:40

about the product of two categories and then the second great thing is that the

1:47

way we're going to approach this is just such a generalizable method I mean we're

1:56

going to look at this idea in a very basic kind of setting this idea of sets

2:03

and we're going to think hmm how can we generalize what's going on into terms of

2:11

category theory and simply thinking about that approach is so beneficial because

2:17

of these glorious concepts from category theory come from just taking that approach from just thinking well there's

2:24

something I'm interested in and how can I write that in terms of category theory but the third thing we're going to get

2:32

out of this is in my opinion definitely the best and that simply the definition

2:37

of this categorical product itself because the definition is just so deep I

2:44

mean it's based around the notion really of perfection so the way that we're

2:52

going to define this categorical product is essentially as a sort of perfect

2:57

object with respect to a particular property so this idea of things being

3:05

perfect has really been a more of more of a philosophical idea than a

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mathematical one for most of history but in these days with category theory we

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can really start to see how to formalize that kind of notion and I think this is

3:25

very great I mean I actually think that there's going to be quite a few really

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important implications of this idea of taking the product of things in category

3:39

theory terms and and the sort of underlying ideas I believe that they're

3:45

going to have a lot more impact and lead to a lot of new developments that we haven't even seen yet now to explain a

Cartesian Products

3:53

little bit about what it is this is the context that we are going to think about

4:00

products initially so if you have a set which has elements a and B and we have

4:07

another set which has elements 1 2 & 3 then there is something called the

4:13

Cartesian product of those two sets and this is something that's normally just

4:19

defined in terms of set theory so we'd say it's two sets of pairs with the first entry in the pair belonging to the

4:26

first set and the second entry in the pair born into the second sets and this is a very nice idea because if

4:35

we have a wine we could represent that by the real numbers okay so on this

4:41

number line we have 0 1 2 minus 1 and so on and we also have these numbers

4:48

between and the real numbers are a sort of model of all of the numbers on this

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number line and then if we take the Cartesian products of this line with

5:02

itself then we basically get the plane

5:08

okay because is they can't realized you can refer to any points on the plane

5:16

with a couple of numbers X and Y X being the amount we have to move right of the

5:23

origin and Y the amount you have to move up from the origin so we can think of

5:29

the plane as the Cartesian product of the line with itself this is also

5:35

written as R squared similarly R cubed

5:43

would be the Cartesian product of the plane with a line which gives you all the three-dimensional space it's a very

5:52

interesting idea because we can then think about things like after the power of 4 and that would represent four

5:59

dimensional space ok so we're going to focus on this idea of the Cartesian

6:05

product of two sets and we're going to look at this idea and I want you to pay

6:12

close attention to how we actually arrive at this definition because that's

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really the it's really a very important aspect of it because the way that we

6:25

arrive at this definition of a categorical product is basically to see what's going on in in sets and then

6:34

think well how do we express that in terms of category theory and so whenever

6:39

you see some process that you're interested in you can think well how

6:44

express that in terms of category theory and if you're able to do so you get two

6:50

kind of categorical definition of that thing and then you can apply it to

6:56

innumerable other cases and find interesting things so it turns out for

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example that this definition of a categorical problem that we're going to cook up we can apply it to other

7:08

categories as well so here I have another category where the objects are

7:15

positive integers positive whole numbers and we'll draw an arrow from one object

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to another if the first number divides

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the second number so to divide 6 because 6 is equal to 3 times 2 so we draw an

7:33

arrow from 2 to 6 and then we can ask for example what happens if we apply

7:41

this definition that I'm going to explain and we say well what's the categorical product of this object here

7:49

and this object here and it turns out that this is going to be the greatest

7:56

common divisor of these two of these two numbers the the biggest number which

8:03

divides into both of them which is 2 so

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this hopefully gives you a flavor about how this definition that we're going to

8:15

arrive at can be applied in lots of different places okay so here comes this

Categorical Products

8:21

definition of the categorical products and the way we're going to get there is

8:28

to look at what's happening in this category set and we're going to think

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about how we get the Cartesian products of this set a and this set B so this is

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just the set of all pairs with the first entry from a and the second entry from B

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and what we're going to do is we're going to ask ourselves how can we pick

8:56

out this objects by referring to things in

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terms of category theory so in category theory this set here is just an object

9:09

let's think of it like a dot and this is another object and this is another

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object so how can we identify this

9:22

object in terms of the arrows in this category sets now these arrows in the

9:29

category set are functions so let's think about the functions involved with these things and see if we can pick out

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the nature of this thing in terms of functions so what kind of functions do

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we know about this well we know that there is a sort of special function

9:49

which goes from this set here this product to this set a and that's the one

9:55

that just takes the first entry in a pair

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okay so we can call this function pi-1 and it's basically sort of like a

10:18

projection we're sort of projecting down onto the first entry in the path so we

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can draw that in our category theory

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picture here so there's a times B and

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there's a and there's B now similarly

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we're going to have a projection from this set here into be that just takes

10:53

four second entry in the path

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and that's pi/2 so there it is pi/2 so

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we're getting towards this definition of a categorical product we know it's

11:29

something that has a full we know it's some object that has an arrow into a and

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an arrow into B now that's obviously not enough so uniquely identify this thing

11:41

but it's a start and so we're going to use that to define

11:46

something we're going to call it a

11:54

product candidate okay so it's something that might be a problem so this is our

Product Candidates

12:06

definition of what we mean when we say something is a candidate for being a

12:12

product of a and B and we just mean that it has some arrow F into a and some ROG

12:21

into B in my case here I'm calling arrow f PI 1 and the arrow G hi - okay

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so clearly this is a product candidates and it's represented in category theory

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terms here but that's not enough obviously to pick out this as actually

12:41

being the Cartesian product of this set here and this head here the Cartesian

12:48

product being the collection of all these pairs so what else do we know well

12:54

here is the very interesting idea what we do we say well what if there is

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another candidate so let's pick out another candidates

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so there is another calendar there acts and it does have a function into a let's

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make one off and it does have a function

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into B so there's another candidate now

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what we're going to do so I'll just draw that in terms of category theory as our

13:59

other candidate now there's one more

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important thing that we know and that is that whatever this candidate can do I

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mean what I'm saying it's doing is it's doing these functions here into a and B and the idea is that whatever it can do

14:29

this sorts of perfect objects can do better what do I mean by that well what

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I mean is that say we want to apply this

14:42

function f from X to a well there's a way that we can instead do a function

14:50

into this object here and then apply

14:55

Paiwan to obtain the same thing and similarly

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there's a way to sort of simulate what's happening with this function G and

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there's a unique way to do that and that's the definition of the product so

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what does this unique way look like well

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it's going to be a function H

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and the way it's going to be defined is it's going to send an element of X to

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the elements f of X G of X that way this

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thing he is going to get sent to f of X G of X when we projects its first components we're going to get the same

15:54

results as doing f and similarly with G so let's draw that so P here get sent to

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one onto F and it gets into a under G so

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let's send P here it's a 1 comma a and Q

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here get sent to 1 under F and B under G

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so let's send one so let's send q to 1

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comma B and that does the job ok so what

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we're going to do now is to formalize this in terms of category theory and

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we're going to have our definition of a categorical product so all we have to

16:56

say is that from this candidate with its with its functions into a and B there's

17:10

going to be this unique roh that makes this diagram commute what I mean by it

17:16

makes it commute is just that when we have two paths from one object to

17:24

another then those two paths give the same results when we compose all the arrows within them so this is our

17:32

definition then of a cat Oracle products we're saying that it's

17:40

an object that has an arrow into a and an arrow into B and for any other such

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objects for any other sorts of candidates with its arrows F and G

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there's going to be a unique arrow H which has the property that it makes

18:05

this diagram commute okay so this is the

Categorical Product

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proper definition then of the categorical problems we're saying that

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it's a candidate so that just means it's an object which has an arrow into a and

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an arrow into B and it has the property that whatever any other candidate can do

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it can do better in some sense so what

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I'm exactly saying is that if we have any other candidate X any other thing which has an arrow into a and an arrow

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into B there's a way that we can simulate the effects of this arrow and

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this arrow G and there's a unique way of doing it this is very important okay so

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we're saying that there exists a unique arrow in a unique sort of intermediary

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arrow H with the property that PI one

19:10

after H is f and PI 2 after H is G so

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can we see that this is indeed the categorical product of this object here

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and this object here well yes it has an arrow into a yes it has an arrow into B

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so it is a it is a product candidates and also for any other products

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candidates we can make an arrow from that candidates into our product object

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here and our like this which sends an element X into

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f of X comma G of X and we also need

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this is the only function which which sort of makes this diagram commutes and

20:05

is it well I think it's you can see that

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it is okay because we'd have to send an elephant acts to f of X comma G of X if

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we didn't saying we set the first entry to something different well then when we

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do PI 1 and just grab that first entry it wouldn't match up with the way F Maps

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it and similarly if we didn't send the

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second if we didn't send X to something with a second entry which was G of X

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then when we did PI 2 on that path we wouldn't end up with G of X but we have

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to in order for this for these conditions here to be satisfied so

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hopefully you can see now that this is a categorical product hopefully you can

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see that the Cartesian product of two sets is the categorical product of those

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two objects of the two sets of those two

21:16

you well the two objects representing those sets in this category set but you

21:25

may still be wondering well ok so I can see that the Cartesian product of two

21:35

sets satisfies this definition but how do I know that this definition is

21:40

correct I mean maybe there are some other things which satisfy this

21:46

condition which look nothing like the product of two sets okay so I have to

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then try and argue the converse so have to try and argue that something that satisfies this condition in the category

22:00

set is actually Cartesian products okay and so how to do

22:09

that well the way that that works is very very interesting and it's a lot easier than you might expect because it

22:18

turns out that in general in a general category any two objects which satisfy

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this condition you know only two things that we could call a times B under this

22:32

condition are actually going to be isomorphic to each other okay so this

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basically says that you know if you have some category and you're saying okay

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what's the categorical product of these two objects and you think well something that satisfies this condition

22:52

well the many maybe many sorts on there may be many such objects which satisfy

22:57

this condition so which one do you pick well they're all isomorphic so in some

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sense it doesn't matter and we're going to demonstrate that later and I'm also going to recap the definition of what an

23:11

isomorphism is so you can get so you can

23:16

recall that so let me just reiterate

23:27

this definition of a categorical product so in a general category if we have a

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pair of objects then a categorical

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product is going to be an object which has an arrow into each of them and it

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has the property that for any other object with an arrow into each of them

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there's going to be a unique arrow which makes this diagram commute in the sense

23:58

that these paths like along here and here this composition gives the same as this and this composition here pi/2

24:06

after H like this gives the same as G ok so the many there may be many arrows

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from this candidate here into our actual products but we're just

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requiring that there's exactly one which makes this diagram commute in this sense

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okay so this is the definition of a categorical product and there may be

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many categorical products of an object and there may not be any okay we're not

24:38

guaranteed in an arbitrary categories are there defined something like this but in many nice categories we're

24:47

looking enough to bear to find these things and when we can we have a natural

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sort of notion of what products are of objects from that category and if there

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are many such objects which satisfy this condition they're all going to be

25:08

isomorphic to each other and I'm going to demonstrate that to you later

Preordered Sets

25:19

okay so a particularly easy situation to

25:24

understand what the categorical products is is in pre ordered sets so a pre

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ordered sets is just a category which doesn't have any parallel arrows so

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there's at least most one arrow from one object to another and in situations like

25:46

that we don't have to worry so much about arrow composition because whenever

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we have two arrows head to tail we know that composition has to be in

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the category and there's at most one arrow which is the composition of this

26:12

one and this one so we don't actually have to go to the effort of labeling

26:20

which arrows are the compositions of which other ones because there's only one possibility and that makes the

26:29

definition of a categorical products much simpler because those parts of the

26:35

definition which involve thinking about what the actual composition of arrows is

26:40

and checking that the composition of these two is indeed that one and so on we don't need to worry about that the

26:49

definition of a categorical products for a pre ordered sets is much more sort of

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visual geometric if you like and basically for a pre ordered sets we have

27:05

the the an objects which are pre-emptive

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recall a times B is the products

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if a and B when

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for any X with an arrow from X to a and

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an arrow from the extra fee we have

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there's an arrow from X to a times B the

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products okay so that's definitely Stephon ition of a categorical call

28:24

product looks like in a pre ordered set that's in general but in a pre ordered

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set in a pre order and that's a category

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we don't have any parallel arrows okay

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so what does this look like well here is a couple of objects a and B in this pre

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ordered set I haven't drawn the identity arrows but you know imagine they're

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there as well so we want to work out what's the categorical product Avista

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now there won't always be a categorical product even in a pre ordered set I mean

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there may not be any objects which have arrows to both of these things for

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example but in this case there are and

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so what are the possible candidate products well they're going to be objects which have arrows into both a

29:30

and B so there's going to be three of them this one here this one here and

29:36

this one here but this one here has the

29:43

property that there's also an arrow to its from any of the candidates and

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because this is a pre there's only one arrow into it from any

29:56

of these of the candidates because there's no parallel arrows but anyway this gives us a rather nice sort of

30:05

visual way to think about what a what the product is at least in this context

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so if you think about rivers you can think about sorts of watersheds okay so

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imagine this river is flowing and then it sort of Forks into two and if you

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have one object so here let's say a and one object here let's say B then both of

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these objects are going to be accessible by any objects over here but the product

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is going to sit right at these sorts of watershed of these two rivers because it

30:54

can you can reach a and B from it but anywhere else or further this way along

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this stream also has the property that you can reach a and B from it but you

31:10

can also always reach a am a bit this problems from anywhere further along

31:16

this stream okay then so I want you to

Isomorphism

31:29

remind you about the definition of an isomorphism in category theory so recall

31:38

that we started thinking about isomorphisms and what they meant in this category of sets where the objects

31:45

represented sets and the arrow is represented functions and in that

31:51

context an isomorphism is a function which has a sort of inverse it's one of

31:59

these one-to-one on two mappings which just pairs things up in a

32:06

in a sort of one-to-one correspondence so basically the definition in okay so

32:13

in in terms of category theory and isomorphism is going to be an arrow from

32:20

one object to another such that there's an arrow in the opposite direction and

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if we compose these two arrows starting from a we go to B and then we go back

32:32

again then their composition gives us the identity arrow of a and similarly if

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we start at B and we compose this arrow

32:43

into a and then the arrow back again then we get the identity arrow of B so

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this is the definition of when F is an isomorphism when there exists G switch

33:08

paths J after F is ID a and F after G is

33:18

ID B okay so speaking of cats I can

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identify that our cat has been walking around in this room because there are

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footprints of chalk all around and I think it's been stepping around then

33:35

this chalk dust here but anyway let's get back to talking about isomorphisms

33:41

so I want to show you that if we have two objects and they are both

33:48

categorical products of a and B then these two objects have to be isomorphic

33:55

so what we're going to do we're going to start by thinking about things from the

34:00

perspective of this product here a times B now it sees this other thing here as a

34:11

candidate products there it is with its arrow into a and its arrow into B and so

34:18

we have property that there's a unique herro a unique intermediary roh into a times b

34:26

with the property that if we go along h and then PI one we get PI one bar the

34:34

arrow from this thing and similarly we can get this arrow from going along here and then here okay so now let's flip

34:44

around and think about things from the perspective of this this is also a categorical product and so there's going

34:53

to be an arrow the other way which we get all h-bar and this has the property

35:02

sort of the opposite way around that if we think about this as a candidate there

35:08

it is with its arrow into a well we can simulate that by going on on h-bar and

35:18

then PI 1 bar and similarly PI 2 is equal to PI 2 bar after H bar okay so

35:29

let's write down what we know considering this so we know now that PI

35:39

1 bar is equal to

35:53

hye-won after h so we know that because

36:04

this is a product and we also know that

36:09

pi 1 is equal to PI 1 bar after H bar

36:18

and we know that because this is a product ok so now we're going to

36:23

consider we're just going to sidestep a little bit and we're going to think about the sort of degenerate case of

36:30

this candidate comparison type of job we have to do when we're defining a

36:36

products so we're going to say okay this is our object and can it do what itself

36:50

can do ok so there is our there is our

36:56

products with its projections into a and B and then we want to compare it with itself so this will now think of this as

37:04

a candidates and there it is with its projections PI 1 and PI 2 into a and B

37:14

and we're wanting to know can we replicate the effects of these

37:19

projections with some intermediary error and indeed we can if we just pick the

37:25

identity our own then yes we have the property that going on this arrow and

37:31

then projecting gives the same as this arrow and going along this our own projecting this way gives the same as

37:37

this our so indeed the our product does

37:44

work out when we compare it with itself now what we're going to do is to notice

37:52

that if we go along h-bar and then we go on H we're going to get an arrow which

37:59

goes from a times B so a times B so we're going to draw that as

38:06

a diagram that looks a bit like this so now we've redrawn the diagram we've sort

38:13

of tried to illustrate what's going on some of the stuff that's going on here okay so we've drawn another diagram

38:19

where we have two copies of a times B but now I've just for no reason which is

38:28

apparent yet I've just drawn this arrow H after H bar which goes from a times B

38:37

to a times B now I want to convince you that this is a intermediary arrow from

38:48

this object to this object so what I'm saying is that if we think of this as

38:53

our product and we think of this as a product candidates then this arrow has

39:01

the property that if we compose it with PI 1 we get PI 1 and if we compose it

39:09

with PI 2 we get PI 2 and we can see this from the algebra okay so if we just

39:18

substitute for this equation here into

39:24

here we get the PI 1 is equal to PI 1

39:35

after H after H bar and buy some more

39:41

logic PI 2 is equal to PI 2 after H

39:50

after H bar and that's what we wanted we

39:56

wanted that PI 1 is equal to PI 1 after

40:03

H after H bar and similarly with PI 2 so that means that this is an intermediary

40:11

arrow that makes this diagram commute but here's the part remember the definition over

40:19

product we said there's only supposed to be one such intermediary arrow which

40:26

makes the diagram commutes and here we have one so if we only have one and we

40:34

see two expressions for its those expressions must be equal so we have the

40:42

H after H bar must equal ID event times

40:48

B and that's just the exact well by

40:55

similar logic sort of just flipping things around and looking things from

41:00

the perspective of a bar a times B bar we can get this H bar after H is equal

41:12

to the identity of a times B bar and

41:18

this exactly means that H and H bar are isomorphisms between these two products

41:25

so there we have it the reason why two categorical products of the same objects

41:33

have to be isomorphic to each other I think it's remarkable that it's possible

41:39

through all this stuff just by thinking about arrows so just as a bit of a tangent I noticed that there's a sort of

Tangent

41:48

vague relationship between this idea of a categorical product and a very

41:54

interesting idea by the physicist Alexander wizna gross who had this very

42:01

interesting notion that you could define intelligence I'm putting it very roughly

42:08

but he said that you can define intelligence as something that tries to maximize the number of possible actions

42:16

that it can do and if you think about the categorical product in those kind of

42:24

terms you could almost say that it's the most intelligent a

42:29

the product candidates because it can do whatever any of these other product

42:36

candidates can do and it does so with great economy by just using these

42:42

intermediates arrows so now we have this fantastic tool this notion in their

42:49

categorical products we can just point it at a category and ask is the

42:55

categorical product defined in this category and when it is we usually have

43:01

some kind of natural notion of what the product of objects within that category

43:09

might mean so for example if we consider this category of mono AIDS so remember

43:17

amyloid we can think of it as a category

43:26

which just has one object so I told you

43:33

in the last video about this idea of what we might consider to be arrows

43:39

between these objects in this category what we might consider to be an arrow

43:45

between models and so we have this category of models we know what the

43:51

objects are down all the models and we know what the arrows are they're

43:56

basically correspond to four motors from one model to another and now we can ask

44:05

is there a natural notion of the products of a pair of these objects in

44:13

this category of mono needs so can we think of how its natural to define the

44:19

products of one monoid with another and when we go through the work of it we

44:26

find that there is a very natural definition I'm going to tell you it's a

44:31

bit later but just do illustrate how it works we have this mono lead where so

44:39

this is the categorical view of a mono leaders that objects with these arrows the sort of normal view is that it's

44:46

some set with some operation and we can

44:54

then ask so it's sunset with some operation which takes a pair of elements

44:59

and outputs and other elements which is associative and which has an identity

45:06

elements so these elements here in this set would just correspond to the arrows

45:11

in our moloids so here's an example the kind of standard example which is just

45:20

the whole numbers with addition as our

45:26

operation that guess is a new element from the old ones and we can ask what's

45:31

the product of this mono eight with itself and the result is a very natural

45:52

one because we can think of this mono lead as the number line and an element

46:05

is a point on this number line fits into gym space maybe I'll use this one and

46:13

then we you know we can take a couple of elements and just literally add them together to get the new one and the

46:20

products it basically consists of the plain so an element in this new mono did

46:32

we get from product doing these once together will be a path with the first

46:39

element from the first mono and the second element from a second moloids and we can just add these pairs together

46:52

so for example 1 comma 2 plus 1 comma 3

47:00

yes it's going to be 2 comma 5 ok so we

47:07

get this very natural notion of taking the product of two monologues which is

47:15

exactly how people to find it before they knew about a category theory

47:20

and it just comes from the categorical

47:26

products notion and I'm going to tell you exactly how this problem of monolids

47:34

works in a bit I'm going to define

47:39

something first and then we'll have a very succinct definition of how we can

47:45

take the product of two modes ok then so now we have this tool over the

47:51

categorical product we can point it to any category that we like and then see

47:58

can we define the categorical product within that category we've already pointed it at set and

48:06

found that the categorical product yields this notion of the Cartesian

48:12

product which is a very natural way to take the product of sets so now let's

48:19

take the bull by the horns and see what happens when we points this categorical

48:26

product tool at the biggest and most important category that we know about so

48:31

far which is this category of all categories ok so what does that mean

48:39

basically what that means is we're wondering if there is a natural notion

48:45

of taking the products of a pair of categories is there a way we can get a

48:50

pair of categories and somehow make a new category that can be considered to

48:57

be their products and that's what I'm going to tell you about because it turns

49:03

out there is a very natural definition now we're not going to worry about how this

49:10

relates to the notion of categorical product which I've been discussing so

49:16

far I'm simply going to tell you the results I'm just going to tell you what this natural notion of taking the

49:23

product of two categories is okay so let's say we have a category C and a

49:32

category D or I've drawn the product of these two categories according to this

49:38

definitions I'm going to explain over here okay so what is it well it has

49:46

objects which are pairs the first element in the pair is an object from C

49:53

and the second element of the pair is an object from D and all such pairs are

49:58

appearing here as objects so if you like you could say that the set of objects of

50:06

this products thing are the Cartesian problem is the Cartesian product of the

50:14

set of objects of the C with a set of objects of deep so we're writing the

50:21

definition here the products C times D over category C and a category D it has

50:29

these objects a comma a bar for each a

50:34

which is an object of C in age a bar which is an object of D and we also have

50:41

arrows in this product category and we get those in a very natural way by

50:48

taking arrows from over here and arrows from over here so in this product category we're going

50:55

to have arrows F comma F bar which is going to be a arrow from a comma a bar to B comma B

51:04

bar and we're going to do that for every arrow F of C and every arrow F bar

51:16

f d okay so whenever we have a narrow F

51:23

from A to B in C and we have a narrow f

51:28

bar from a bar to B bar in D then that's

51:34

going to give us this kind of paired thing F comma F bar in this product and

51:42

that's going to go from a comma a bar to become a B bar now the only other thing

51:50

you need to know is how are we going to compose these arrows and it's very

51:56

straightforward actually remarkably straightforward so let's say we have an

52:02

arrow G comma G bar and an arrow F comma

52:07

f bar in this product and we want to compose those together so we want to

52:13

find out what is G comma G bar after F comma f bar well all we do is we look at

52:25

what's going on over here and in this category C we we compose G after F and

52:33

in this category D we compose G bar after F bar and then we just pair them

52:39

together so G comma G bar after F comma f bar is G comma is G after F comma G

52:52

bar after F bar okay so all we're basically doing is composing these

52:57

arrows in a natural way and so I'm explained to you now what the objects

53:04

are of this product and what the arrows

53:11

are and how to compose them you may want to satisfy yourself that this is indeed

53:17

a category think about for example does this thing

53:25

have identity arrows indeed the identity arrows over here going to be the problem so for example

53:33

ID a comma ID a bar is going to be the

53:39

identity arrow of the object a a bar okay so it will have will have identity

53:47

arrows and it's also associative because the arrow composition is just basically

53:56

working just like it works in C and D just with this comma here okay so it is

54:03

indeed a category it has identity arrows and it's associative if you're feeling

54:10

adventurous you might want to try and convince yourself that this actually is

54:15

what you get if you ask what is the categorical product of a pair of objects

54:23

in this category of categories so now we

54:28

know how to define the product of two categories it's very easy then for us to

54:35

return to this category of moments remember which has objects of as mono

54:41

eats and these arrows --is transformations from one mode to another and we can ask well what's the product

54:50

of a couple of model it's how we differ how do we define that and it is exactly

54:56

the notion of the product of categories which we've just been explaining we

55:03

simply think of these mono aids as single objects categories and we apply

55:10

that they're finishing we've just been discussing and you know we think though

55:16

this is a category of one object and this is a category of one object if we take the products of these two

55:22

categories we get another one with one object and it has these arrows which are

55:29

sorts of pairs of an arrow from here and an arrow from here and the way that

55:35

they're composed is as I discussed before so you know we're really because

55:42

these definitions so nicely related we don't have to do too much work to think to get a notion

55:49

of what a natural idea of products are for this category of Maoris