Category Theory For Beginners: Duality And Functors

3 Jun 2019

In this video we introduce the idea of duality, and the notion of the opposite category. We define the initial object of a category, and show that it can be viewed as the empty set, within the category of sets. We also introduce the idea of the coproduct and show that it corresponds to the discriminated union, within the category of sets. We also show how the Cartesian product can be viewed as a functor in the category of sets. More generally, we show that the categorical product can be viewed as a functor, when the categorical product of each pair of objects is defined. This also gives us a notion of the categorical product of a pair of arrows.

Transcript

0:04

okay so so far in this series we've encountered a lot of ideas we've

0:11

encountered the ideas of categories and functors and products and the notion of

0:20

taking the product of two categories and what we're going to do today is to reap

0:27

the rewards we're going to explore the

0:32

connections between these topics and that's really going to come together when we look at the product as a functor

0:39

okay so this gives us a much deeper appreciation for what this notion of the

0:46

categorical product really means and we're also going to do something else

0:51

which is essentially going to double the amount of insights that we get without

0:57

any extra work and that's simply to realize that every category has a kind

1:04

of dual called its opposite category that's just obtained by reversing all

1:11

the directions of the arrows so what that means is that almost every time which we make up some concepts in

1:18

category theory and we work out some implications of it we then get another

1:24

big idea for free just by thinking about the same situation with the arrows

1:30

reversed so we're going to do that for the notion of a terminal object and

1:38

that's going to give us the important idea of an initial object but the most

1:44

useful thing perhaps that we're going to do is to look at this idea of product

1:50

and the different things which we've learned about it and then we're just going to generalize them so we're just

1:58

going to say well what about if we're doing the same kind of constructions in

2:03

the opposite category and that's going to give us the notion of the co product

2:09

of two objects now Co products I would

2:15

say is equally important to the notion of product and between the two of them we

2:22

can construct lots and lots of very useful data structures okay so now I

2:28

want to talk about duality so basically there's this idea in category theory

2:37

which is on one hand seems quite basic but on the other hand it's very powerful

2:43

because it sort of doubles how much stuff you get out of the theory and

2:49

that's just the idea that if you have a category you can just reverse the

2:54

directions of all of the arrows and you get another category so this means

3:00

there's a natural source of duality in category theory and when you define some

3:08

kind of a object or often when you prove some sort of a result you can then sort

3:18

of for free you can just say well there's another corresponding construction when all the arrows are

3:25

reversed so often when you do reverse the arrows in a construction you get

3:31

something which has quite different properties so it can be quite surprising

3:37

how much economy you get out of this idea so one of the simple things that

3:44

we've talked about is this idea of a terminal object for example and remember

3:50

that a terminal objects has this property that was defined by this property that for any object there's

3:57

just going to be one arrow into the terminal object so we can then say well

4:05

what is the jewel of this concept and so

4:10

the jewel idea that we shall call an initial object we can say an initial

4:17

object is a terminal object in the opposite category so remember this

4:25

opposite category is this category where we reverse for directions of all of the

4:30

arrows so if you have a category and then you

4:36

take it and then you get the opposite category so you reverse the arrows and

4:42

then if you find a terminal object in there then you can you can say well what

4:52

does this correspond to it corresponds to an initial object and I think it's

4:57

fairly straightforward to see what this is going to be it's going to look like this

5:03

there's our initial object so let's write down a more direct definition of

5:11

what this initial object is so an initial object is defined by the feature

5:18

that for any object is just going to be one arrow from the initial object Tyrians and that includes itself okay so

5:26

this single identity arrow that it has is going to be the only arrow from

5:33

itself to itself okay so at the moment

5:38

we're just doubling definitions but there's quite a lot more that you get

5:45

out of this notion of duality okay in

5:50

general if we have a category C then the category that we get by reversing all

5:57

the arrows in C is often denoted as C up

6:02

for opposites so we can ask I mean we

6:08

have this notion of an initial object now and we can ask about the category

6:13

sets I mean we identified a terminal

6:18

object in the category sets that will be a singleton set with just one element and we can then ask well what is a

6:27

initial object in the category set and it's a very good question to think about

6:34

it's it's actually quite quite weird the idea of an initial object in the

6:41

category set there is one but it's maybe quite have to guess what it would be okay so

6:48

we have this conundrum then we have the category set that has these objects

6:55

representing sets and these arrows representing functions and we want to know what is an initial object in this

7:03

category set and the answer is the empty

7:11

set okay so this is perhaps peculiar if

7:17

it's the first time you've seen this kind of thing I mean maybe the empty set

7:22

didn't cross your mind this empty set this is for set that doesn't have any elements and it's very important in set

7:30

theory for example the empty sets is contained within every other set it's it

7:36

has very important properties much like zero does in arithmetic and so how can

7:43

we see that the empty set is an initial

7:48

object in this category set well if that's true then there must be a

7:57

function from the empty set to any other set

8:03

that seems a bit strange isn't it how can there be a function from

8:09

something that doesn't have any elements well there's different views that you

8:15

can take on this one is that this function would be a sort of bluff I mean

8:24

I tell you that i have a function from the empty set to this set here with a

8:30

and b and you say really and i say yeah

8:37

you give me an element of the empty set and I'll give you an element of this and you can't give me an element of the

8:43

empty set so there is a function from the empty set to this one and why is

8:52

okay so if you buy that then you maybe you'd say well okay why isn't there more

8:57

than one reason that there isn't more than one function from me have deceptive this set

9:02

is that we define two functions to be the same when they have the same source

9:12

set and the same destination set and they're equal for every argument and any

9:18

of these kind of no kind of functions from here to here would be the same that

9:26

all just not have any inputs and not have any output so there's exactly one from here to here maybe even stranger is

9:35

that there's a function from the empty set to the empty set but again we just

9:42

say well you give me an element with the empty set and as an input and I'll give

9:48

you an a member of the empty set as an output but I mean that is a little bit strange because if we have any other

9:56

sets and non-empty sets then clearly there are no functions from that into

10:02

this empty set because if you have an element in here how can you send it to

10:08

an element of the empty set when there aren't any okay but you know this

10:14

peculiar kind of reasoning does seem to make a bit of sense and perhaps a easier

10:22

way to get your head around this idea is to take an alternative view about what a

10:28

function is okay so here we have a function and this is the sort of

10:34

standard view it's something that you take an element of this set on the left

10:40

and you get an element of this set on the right another way to think of what a

10:45

function is is that the function is itself a set of pairs and these pairs

10:53

well the first entry in the pair comes from the source set and the second entry

10:59

in the pair comes from the target set and every elements of the source set

11:04

appears in exactly one pair and it's paired with the thing that it gets sent

11:11

to under the function so I mean this makes a lot of sense

11:16

we're essentially just making a sort of something like a sort of table except

11:23

the order of the entries don't matter but we're making a sort of table of all of the outputs that this function would

11:32

have for all the different inputs and clearly that's representing all the relevant data and this is often how

11:39

mathematicians define functions and in this kind of view it makes a bit more

11:45

sense because you could say well if that's what you mean by a function if we

11:51

just have to have all of the members of the source that appear once paired with some member of a destination set then a

12:02

function from the empty set is just a

12:09

sort of empty thing like this it doesn't have to have any entries in here because

12:17

there aren't any entries in here so a sort of empty object like this would be

12:24

describing a function say from here to here I mean obviously that would also be

12:31

an empty set in this kind of view the function from an empty sets to some

12:37

other set will be described by a empty set of of input elements output element

12:46

pairs so hopefully this convinces you that yes the empty set is in fact an

12:53

initial object of this category sets and when later we see connections between

13:01

what's going on in set theory and what's going on in logic this gets quite

13:07

interesting but I'll leave that for another time okay so I've been claiming

13:15

that this notion of duality is a very powerful one but so far all I've got out

13:23

of it is this idea of an initial objects is the sort of dual

13:29

notion of a terminal objects and you might say well I could have come up with

13:36

the definition of an initial object on my own it's it's not really such a great

13:42

jump for words but what we're going to do now is ask ourselves what's the dual

13:48

notion of a product and we've already seen that the product is a very powerful

13:55

concept and we're going to get a equally powerful kind of idea by thinking about

14:01

what the dual of this notion is and that's called the co products of a pair

14:08

of objects so now you know what a product is I can tell you a very

14:14

succinct definition of what a co product of a pair of objects is and essentially

14:22

the Co products of a pair of objects is just going to be their product in the

14:29

opposite category so say you have a pair of objects and you're interested in

14:34

determining the Co product of them well if you go to the opposite category by

14:41

reversing the directions of the arrows if you're able to find the product of these objects and that's going to be

14:47

something with an arrow into each which satisfies special properties then you

14:54

can take that and then just go back into the original category so you reverse

15:00

these arrows and now these three things

15:06

the arrows into the thing and the object constitute the code products of these

15:13

two objects so these are dual notions and so to really get a proper a sort of

15:20

full description of what the co product is we just take this description and

15:27

sort of jewel is it reversing the directions in our explanatory diagram and so on so here we have again the

15:36

definition of a product recall that it's an object a thumbs be with an arrow PI

15:44

one ins a and an arrow PI 2 into B that has a property that for any other object

15:51

with arrows into a and into B there's a unique intermediary arrow from that

15:57

object into our product that makes this diagram commute in the sense that if we

16:03

compose it with PI 1 we get our arrow F into a and if we compose it with PI 2 we

16:10

get our ro G into B oh and I should also say that I'm drawing this H arrow dotted

16:18

to represent that it's the unique arrow that makes this diagram commute ok so

16:25

now to get this idea of a co-product the co product of a pair of objects in a

16:34

category is just the product in the opposite category so what we're going to

16:42

do is just to reverse all the directions of the arrows in this picture

16:59

so we're going to denote this coke product as a plus B instead of a times B

17:43

so this is going to be our diagram which defines what the co product of a pair of

17:50

objects is so I'll simply write down the definition I've basically just redrawn

18:03

this diagram here but I've reversed the directions of the arrows and now I'm

18:09

calling these arrows here p1 and p2 whereas I called them PI 1 and PI 2 over

18:15

here and so here we have the definition of the co product so just like we had

18:24

the notion of a candidate's products being something that had arrows into a

18:30

and arrows into B we can similarly say that a candidate's count products is

18:38

something that has arrows into its from a and from B so then the co product of a

18:46

and B which we write as ay plus B has an arrow p1 into it from a and an arrow p2

18:55

into it from B and it has the property that for any other candidate Co products

19:02

like this on X there it clears with its arrows into it at from a and an arrow

19:10

into it G from B well for any candidate co-product there's going to be a unique

19:16

intermediary arrow H this time from the

19:22

co products into this candidate X which has the property that it makes this

19:29

diagram commute okay so it's such that

19:34

for any of the candidates X with its arrows from a to X and B to ax there's

19:43

going to be a unique H from our actual co-products into the candidates which

19:50

has the property that H after p1 is e is equal to F and H after p2 is

20:01

equal to g okay so hopefully you can

20:07

convince yourself that if something is the product of a pair of objects within

20:18

a category C then it's going to be the co product of those objects within the

20:27

category C op that we obtained by reversing all the directions of the

20:34

arrows in the category and conversely of course if something is the co product in

20:41

a category C then it's going to be the product in the opposite category so

20:46

these are dual notions to each other and they have roughly equal importance in

20:53

mathematics so we're going to now say we're going to now think about what the

21:00

co product would mean in this category set I mean it's not always the case that

21:09

you can determine the co product of a pair of objects that may not for example

21:15

be any arrows from the pair of objects you're interested in into a particular other object there may not be any

21:22

candidates but in many cases you can determine the code products there are

21:28

cases where you can determine the products of objects but you can't determine the co product of objects but

21:36

in this category set it's nice enough that not only can we determine the

21:42

product of any pair of objects we can also determine the Co products of any

21:48

pair of objects so let's discuss how we can do that ok so let's discuss what

21:56

this notion of code products means in this category sets so I just want to go

22:03

over something pretty basic in set theory that you probably all already

22:08

know but I don't want to assume any prior knowledge so I'm going to tell you anyway which is what the union of two

22:15

sets is so the union of a set a and the set B is usually written as a cup B and

22:25

this is for sets which just has all the elements which are either in a or in B

22:33

so is it true that the number of elements in the Union is equal to the

22:39

number of elements in a plus for number of elements in B well no not always

22:44

because for example in this case we have that 3 is an element of both of these

22:51

and so it's only going to appear once in the union okay

23:00

because sets have the property that they can only have one copy of any particular

23:06

elements so often we would write a sets

23:14

using curly brackets because it's easier for linear text of course so this would

23:23

also denote this set a and so now let me

23:28

tell you what the code product is and we can think we can write it as a Cartesian

23:39

product it with the singleton set just containing 0 Union B Cartesian product

23:51

it with the singleton set containing 1 ok so this is or we can think of this as

24:00

what the co product of a pair of objects

24:08

a and B is in this category set now just

24:13

like the product had the property that any two things satisfying the product

24:21

more thing to each other the co-product also has this property so it doesn't

24:27

exactly matter what we tagged our sets

24:32

with as long as we don't tag them with the same thing so what does this look like well essentially what would be

24:40

saying is that this set here any times the singleton set with zero in it looks

24:47

like 1 comma 0 2 comma 0 3 comma 0 and

25:01

if we Union this with this thing here 3

25:11

comma 1 4 comma 1 5 comma 1 well then we

25:20

do get something with 6 elements because there may be two copies of 3 here but

25:26

their tags differently this one's tagged with a 0 and this one's tagged over 1 so

25:33

this is the co-products or at least it can be thought of as representing the co

25:40

product of a and B so we're using a

25:52

little bit of sophistication here in a way because firstly we're sort of taking

26:03

it for granted that if we form these pairs here just by appending a 0 onto

26:13

the end of every element like this then we're getting a set which is isomorphic

26:18

so this is actually this idea itself is

26:26

actually quite deep because it too can be formalized in terms of cat

26:34

theory I think it's true in general that if you have a category where the

26:42

products of any pair of objects is defined and that category has a terminal

26:48

object remember the singleton set is a terminal object well I think it's always true in

26:54

such a case that when we take the categorical product of an object with

27:02

this terminal object we get something that's isomorphic to the original object so you might want to try and convince

27:09

yourself of that but that doesn't really matter in this context because I'm just

27:15

talking about the category set and we can see more clearly that it's true in

27:21

the category sets where we're doing the Cartesian product it's obviously true

27:27

that if we take the Cartesian products of a set with a singleton set we're just

27:34

going to get an isomorphic cept I mean this set basically looks the same as this one it's just that we've clipped

27:40

these zeros onto the end of all of our elements I mean actually defining the

27:52

union of two sets is a bit more tricky in terms of category theory really this

28:00

notion of the co-products this sort of

28:06

discriminated union is in a way a lot

28:12

more basic okay I mean think about it in computational terms if if you want to

28:18

take the union of two sets you have to go through quite a bit of effort to

28:23

establish whether they have any common elements and if so which where is to

28:31

take the to take this sort of co-products this sort of discriminated

28:38

union you simply have to get all the elements in a and tag them with

28:45

something get all the elements and be and tag them with something else and then just stick those two sets together

28:53

and so this is this could be thought of as what the code product looks like

29:00

in the category sets although it doesn't

29:06

really matter what you tagged things with I mean essentially code products is

29:14

this kind of thing which is just defined up to isomorphism any anything

29:19

isomorphic to this type of form can also be considered to be a co-product to

29:25

those sets okay so I want to illustrate how this notion of code products works

29:32

in the category of sets so what we're going to do is have a look at how we're

29:38

going have a look at the co-product of this a and this set B so here's the sort

29:47

of categorical picture of what's going on now the first thing we need for this

29:55

thing here to be a code product is we need it to have a we need there to be a

30:01

arrow from a into it and an arrow from B into it

30:18

so I'm calling those P 1 and P 2 and

30:23

remember when we were doing products we were calling the arrows projections well

30:33

it would probably be good to call these arrows injections because essentially

30:39

they're just going to inject elements from these sets into the place where

30:44

they are floating around tagged within a plus B so one is going to go to 1 0 to s

30:55

going to go to 2 0 under P 1 and he is

31:02

going to go to u 1 under P 2 so these

31:08

are the special kind of injections which

31:13

come with a plus B and now I have to try and demonstrate to you that this is a

31:23

co-product okay so this co-products comes with these injections here and we

31:30

have to show that for any other candidates co-products say x

31:45

say for example we have something of elements are an S well let's say that

31:57

this is a candidate's co-products so then there must be some arrow some

32:04

functions from a to axon from B to X so let's just make one up

32:28

so we've just made up a function f from a into X and G from B into X so now in

32:53

order for this to really be a co-product we must have that for any of a candidate

32:59

like this there's a unique arrow so I'll draw a dotted because it's unique it's

33:06

unique with respect to the property that it makes this diagram commute okay so we

33:12

want that if we do this injection and then we apply H we get if we do this

33:18

injection p1 and then we apply H those give us F as a composition and similarly

33:25

if we do the p2 projection and then apply h we get the same thing as if we

33:31

just applied G so how can we make such an injection from a plus B into X well I

33:41

propose we try H of X comma Y is equal

33:54

to you f of X if y equals zero and G of X if y

34:09

equals one so remember the elements in

34:14

this set a plus B here are attacked so the ones that came from a attacked with

34:21

a zero and the ones that came from be a tag to a one so this kind of what I'm

34:28

proposing for this intermediary arrow it will have a look at 1 comma zero it'll

34:34

say okay your last entries are zero I know you came from a so I'm going to apply F to you

34:51

I'm going to apply after your first entry similarly this came from a so I'll

34:57

apply after you to the first entry and

35:05

this one ends with a one so we're going to apply G to its first entry

35:25

so now hopefully you can see that that H makes this diagram commute in the sense

35:33

that's F equals H after p1 and G equals

35:44

H after p2 and that's what we want but

35:52

recall that in order for this to be a co-product we also need that this is the only such

36:01

H that makes this commute so is that clear well suppose we had in

36:08

elements X Y in here and let's suppose

36:15

without loss of generality that Y was 0 okay so we'll suppose that it came from

36:20

this set a well then we better apply F to its first entry if we sent this

36:29

element X comma 0 to something different than f of X well then it wouldn't match

36:37

up because it's it's getting sent to so

36:42

X gets sent to X comma 0 on the p1 and then it gets sent wherever we send me to

36:50

H better be f of X so we really have to

36:56

specify our intermediary our intermediate kind of function H here

37:02

like this nothing else will do so indeed there is just one intermediary arrow

37:10

given any other candidate co-products which makes this diagram commutes so

37:16

indeed this is going to be a categorical

37:22

co-products okay I just want to make

37:28

sure the terminology is clear again so I suppose in set theory terms this should

37:35

be called discriminated union so I've just tried to show you that a discriminated union with the injections

37:44

associated corresponds to a co-product okay and just like in the case with

37:53

products any two Co products they're going to be isomorphic to each other so

38:02

that kind of shows you the converse because if something is a co-product

38:15

well then it must be isomorphic to this kind of discriminated union okay so now

38:22

we know about this duality and these relationships with Co products there's

38:30

even more reason for us to be interested in what are the properties of products

38:35

because we now see that whatever we learn about products we can sort of generalize to Co products as well in

38:44

some sense now what I want to do next is to connect together a few of the things

38:54

that we've been learning about so basically I want to relate the idea of

39:00

products with the idea of functors I want to show that in some sense the

39:07

categorical product can be viewed as a functor and to do this we're going to have a

39:15

look firstly at some fairly simple example in sets and then we're going to build up in

39:22

generality until we get a view of a product a general categorical product as

39:30

a func talk and this is going to be very

39:36

useful because for example we already have a notion of what it means to

39:41

multiply two objects together in a category but we're also going to get a notion of what it means in some sense to

39:49

multiply two arrows together in a category because when we're dealing

39:54

with funk tools we have to know how they map objects and how they map arrows okay

40:02

then so let's have a look in this category of sets that's how we can view

40:09

product as a functor okay so we're going to start in this category of sets and

40:17

we're going to do a functor from this category of sets back into itself right

40:22

so I've just drawn two copies of it because that's easier to visualize and

40:29

basically what we have all these sets and we have different functions between

40:34

them and we want to sort of lift all those goings on into a new setting where

40:40

all of those sets that we're playing around with have all been Cartesian

40:46

product 'add with this w this is another set so just to put this into a bit more

40:54

practical terms let's say this you had a

40:59

list of people's names and next next to

41:06

their name you had let's say there's a GPS coordinates of their home okay and

41:12

then you made some function which takes in a GPS coordinate system on time and

41:19

outputs say the distance that they are from from your location okay so maybe

41:29

then you'd want to apply that function to everybody's information so you'd want

41:38

to you'd start off by taking in this GPS

41:45

coordinates and mave and then you'd want

41:52

to do some function of the GPS coordinates let's say that calculates

41:59

how far they are away or something and you want to keep the name right so

42:07

if you wanted to do that kind of thing you'd essentially want to sort of lift

42:12

this function so that it was being applied to it wouldn't just be mad at

42:18

being a to be you'd want to map a times a set of words for set of names well

42:26

including the set of names to be times W

42:32

and here this is the Cartesian products okay so we're just talking about pairs

42:38

so how can we think of this in context of in the context of category theory

42:44

well what we can do is we can make a functor which goes from set to set and

43:00

it's going to send an object a set to the Cartesian product of that set with

43:09

this set of with acceptable you so each one of these sets gets sent to earth

43:18

sets times W now remember this is not a function on these sets it's not this

43:27

functor is not sending it's not telling us how every element in here is getting

43:33

certain four elements in a year it's just something which takes a set

43:38

and sort of points it towards this other

43:44

say here a times W and so that's how

43:50

this thoughts are operates on objects but to define a function we have to say

43:56

how it operates on objects and also how it operates on arrows so we have an

44:03

arrow like this publishing its function from A to B and when we apply this long

44:11

talk it's going to change into a function from a times W two three times a week and in this case

44:19

it's defined like this so it takes in an inputs a member of this set that lay

44:28

comic W and it just applies this

44:34

function to the first member of the panel and leaves the second member

44:39

unaltered so it takes a comma W to f of a comma W okay so how do we know that

44:48

this way of sending category set to category sets is actually a functor well

44:55

it has to satisfy the two mores for for concern and that is basically the the

45:03

image of the composition of a couple of factors ought to be the composition of

45:10

their images and also the functor should send the identity arrow to the identity

45:16

error so I've tried to demonstrate this

45:23

here and basically the idea is just that we look at how this because these are

45:32

functions in this case write their functions over here and their functions over here so to prove that two functions

45:39

do the same thing you just look at how

45:45

they work on a sort of general elements and if they map that general element in

45:52

the same way then they're the same function okay so for example to prove

45:58

this one we can just look at how these functions work when we feed in a

46:04

elements a comic W and we see that we get the same results in each case and

46:12

similarly you can show that this law holds by thinking about how these things

46:19

work on the general elements I'm not going to talk through this you can pause

46:25

the video and have look at this okay so we're going to need

46:30

this idea of the product of two categories C and D so I wanted to remind

46:37

you about the definition of this so this product here is going to be a category

46:44

and it's going to have objects a comma a bar whenever a is an object of C and a

46:52

bar is an object of D and this category is also going to have arrows and the

47:00

arrows are written F comma f bar now that's going to be an arrow from a comma

47:06

a bar to B comma B bar whenever F is an

47:12

arrow from a to b in c and f bar is an arrow from a bar to B bar in D so now we

47:20

know what the objects of this category are they're essentially just the Cartesian product of the arrows and C

47:28

and the arrows in D and we also know what the arrows of its product are the

47:34

final thing we need to know is how to compose these arrows together and the

47:41

basic rule is that the composition of a pair is the power of the compositions

47:47

okay so if we want to compose G comma G

47:53

bar with F comma f bar then the result is G after F comma G bar after F bar so

48:05

to work out this composition we just take the first components in the pair and compose them in C and take the

48:12

second components in the path and compose them in D and then just pair those results together okay so what

48:20

we've done so far is to show that the

48:26

Cartesian products can be viewed as a functor

48:32

in the sense that if you have some fixed set W then you can say well if I get

48:40

the sets multiply them by those I can think of those as the objects just func

48:46

to sense those sets too and the arrows

48:53

and sets the functions also have images under that functor which are going to be

49:00

functions from sets like a times W to B times to me so in that case we were

49:07

holding this set W fixed now we'll get

49:12

some more general view points if one allows that second set to vary as well

49:18

because then you've got a type of something like I sort of flunked it in two directions if you like but this can

49:27

all be formalized using this idea of the products of categories so set is a

49:36

category it has these objects as sets and these arrows as functions and we can

49:44

consider the product of the category set with itself so this is this notion of

49:51

products of categories okay so the result is going to be called set

49:57

time to set and if you recall from the definition of a product of categories

50:04

the objects in this thing set time set they're going to be pairs an object from

50:13

this category and an object from risk after me categories are the same so the

50:20

objects are going to be pairs of sets and the arrows are going to be pairs of

50:27

arrows and arrow from this category in an arrow from this category so we'd have

50:33

a an arrow F comma F bar which takes us from the object a comma a bar to the

50:40

object P comma D bar okay so this

50:46

category is not a

50:52

it's not a load of sets it's a load of these objects which is sort of

50:59

abstractly representing pairs of sets but remember the way we compose arrows

51:07

in this product of categories we'd have

51:15

that G comma G bar after F comma f bar is G after F comma G bar after F bar

51:24

okay so that's the critical way we compose arrows in this product of

51:31

categories now to get this bigger view

51:37

of this product operation as a func tour in two directions we're gonna consider

51:45

it to be a functor from set time set to set okay so what it's going to do is

51:55

it's going to translate an object in set

52:00

time set like a comma a bar and it's going to send that to the Cartesian

52:06

product of those sets so a comma a bar is going to get sent to a times a bar

52:16

similarly with the B's and what happens to this arrow here

52:23

well this arrow is going to get changed into a function which goes from a times

52:30

a bar to B times B bar and we call that function f times f bar now the way this

52:40

function works is that when its operating on some elements a comma a bar

52:51

we're just going to operate using these two functions f and f bar on the two

52:57

pairs on the two elements of the pair so we'll just do have on the lay

53:05

and we'll do a half bar on the little a bar so the rewards going to be F of a

53:11

and F bar of a bar paired together and that's going to be an element and it's

53:17

set here okay so here's the sort of

53:23

algebraic description of what's going on in this picture up here we're saying f

53:30

is a full table it goes from the product of the category

53:36

sets with the category set to the category set it's segments at a object

53:43

which is a pair a comma a bar over here to the Cartesian products of a and a bar

53:52

and how does this work on an arrow of this category well if we have an arrow F

53:59

comma f bar from a comma a bar to B

54:05

comma B bar that's an arrow over here its image under this from tour is going

54:13

to be F times F bar which is defined like this so it's a great exercise to

54:23

convince yourself that this is a functor okay it's just a few algebraic

54:32

manipulations and you can show that this satisfies these two laws other functions

54:39

that the composition of images is the image of the compositions and that's it

54:46

Maps identity arrows to identity arrows

54:51

okay so now I'm going to explain how to do what we just did for set but for

55:00

general categories okay so what did you

55:05

stood for set was to say that there's a sort of functor which goes in two

55:12

directions if you like which

55:18

basically works with this notion of products and I'm going to do the same

55:25

thing for a general category C for the

55:31

categorical products now we're going to suppose that we're lucky enough that this category C is such that for any

55:41

pair of objects we can find a categorical product recall that the

55:48

categorical products are pair of objects a and B is another object which we've

55:56

got a times B together with some arrows one called pi one which goes to a or

56:03

called piety which goes to B we often call those arrows projections and it has

56:10

the property that's for any other object which has arrow Y which has arrows into

56:22

a and into P there exists a unique arrow

56:28

which has the property that's this diagram commutes so going along H and

56:35

there going on PI 1 their composition is equal to P and going on H and they're

56:42

going on pi to their composition is equal to Q ok so we're going to suppose

56:48

that there is a products like that for any pair of objects and in that case we

56:56

have this kind of functor which is very similar to the kind of function we looked at we've set I mean remember that

57:03

one we've set it went from the product of the category sets with itself back in

57:11

to set this one is a functor from the product of C with the product of seen

57:17

this is a product of categories it goes from that to the category C so here's a

57:24

picture of it I haven't drawn the identity arrows and we have some objects and arrows and

57:34

see another copy of see we take the product of these two categories we get

57:42

this category here which is C times C it has objects which are pairs of objects

57:50

from C and it has arrows which are pairs of arrows from C and we're going to

57:57

define this font which is basically the

58:02

products the categorical product so we're going to associate every object

58:08

over here an object of this category which is a but we're going to associate

58:16

that with the categorical product of the

58:25

well a categorical products of the two things in that pair okay so we can say

58:34

okay we've got a and a bar and we can

58:44

just do this kind of construction here

58:59

so we can just take a and a bar and then say okay well I can find an object which

59:09

I'll call a times a bar which has this property that you know has these

59:16

projections into a and into a bar and it has its property that for any other

59:22

objects Y which also has arrows into a and a bar we can find this unique into

59:30

me during an arrow that makes this diagram commute okay so we'll do that

59:37

for every object to in this category C times C so there's one for a comma a bar

59:45

there's one for B comma B bar and the other thing we have to do since we're

59:50

trying to make a folk guitar we've said how to send objects to objects but we

59:57

also have to send arrows to arrows so I'm saying that this pair of arrows F

1:00:03

comma F bar gets into this thing here F times f bar and I haven't told you what

1:00:10

that is I'm going to go through the definition now it's a very lovely definition I think so the way it works

1:00:17

is

1:00:25

we have a times 8 bar over here and we

1:00:31

know it has these projections PI 1 and

1:00:38

PI 2 which go into a and a black we also

1:00:52

have this thing here which has a similar type of nature well what else do we know

1:01:04

well we know that we have this arrow F which goes from A to B I mean we're

1:01:14

trying to we're saying we're starting

1:01:19

with F and f bar and we want to somehow construct this thing that we can think

1:01:25

of as the product of those arrows so we have our F and we also have our f bar

1:01:35

which goes from arrow a bar to arrow B

1:01:41

bar now here is the clever bits if we

1:01:46

compose this arrow and this arrow we get an arrow from a times a bar

1:02:00

so this arrow here is just going to be f

1:02:05

bar after high two we also can do the

1:02:13

same thing over here

1:02:23

when we get F after pi-1 so now what we

1:02:32

can do is look at things from the perspective of B times B bar and think

1:02:38

of this as a candidate for being a problems and we'll think well hey you're

1:02:45

a candidate you have these arrows going in to my feet in my B Baja so I can do

1:02:51

what you can do better if you that means there's going to be a unique intermediary arrow from this candidate

1:03:00

to this which is going to be like this

1:03:08

which has the property that's going on here and here as a composition gives us

1:03:15

this going on here and here as a composition gives us this and that's

1:03:20

what we call F times f bar it's this

1:03:29

unique arrow so that's the definition of F times F by RC intermediary arrow from

1:03:39

a times a bar to B times B bar and it's unique so we have a proper definition of

1:03:46

it and how do we compose these type of arrows well I think it's it's it's very

1:03:56

natural so you can see we have another thing G times G bar going to C times C

1:04:12

bar well we just extend this picture so

1:04:17

these go okay and go on G so ya see

1:04:26

we go on G bar get C bar there's going

1:04:33

to be a product of these two remember we're assuming that there is a product

1:04:38

for every pair of objects we have pi-1

1:04:54

squiggle squiggle and v to squiggle squiggle and again by the same type of

1:05:02

logic well hey let's draw it in

1:05:16

if we go along here and here

1:05:21

yeah like this if we go along here and here we get an hour like this

1:05:31

so there's G times G bar and we simply

1:05:38

compose these arrows that's what we have to do that's going to be the composing

1:05:44

these arrows is straight forward okay

1:05:49

then so this really brings together a lot of the things we've been talking

1:05:55

about today we have the idea of categorical products of objects so we

1:06:01

know that mammal which is very useful for us now we also have this idea of the

1:06:07

categorical product of arrows and they're both involved in this as a folk

1:06:13

tour and so I think it's a great exercise to try to bring these things

1:06:22

together to show that this is actually a functor so we need the composition we

1:06:30

need that the image of the composition is equal to the composition of the

1:06:36

images arrows and you can show that by looking at this picture here and then the other

1:06:45

thing you want to show is that this funk

1:06:52

tall sends identity arrows over here to identity arrows over here ok so now we

1:06:59

have this kind of deeper understanding of what the product is in terms of funk

1:07:06

tools we can of course exploit this

1:07:11

notion of duality we can basically do lies a lot of the things we've learned

1:07:18

and applied them to code products so code products can also be viewed as font

1:07:24

or

English (auto-generated)