 Having introduced categories and lots of examples, now we want to study relations between them. One simple relation we have introduced, namely subcategory. When is one category, subcategory of the other? It's just like inclusion maps, but then we want to have more maps. So now let us have a little more generalization of this concept. So that is functor. So C and Db, any two categories. There are two types of functors. One is called covariance, another one contravariant. This is classically, this has been named like this, there is no way to change it. Many people have objections for this, it doesn't matter, but this stands. So the people who object for these names, they want to say that this contravariant is the variant and other one is covariant. There is nothing like contra, it is a variant that is the foremost fundamental. And what you are saying is a covariant, like algebra and co-algebra, like finite or co-finite, whatever. Dimension and co-dimension, there are many co. So the contravariant is the actual variant, that is what they want to say. But what actually happened was the covariance was studied in the beginning and then they tried to attention to this other one. So they tried calling contravariant. This is just an unfortunate historical background. Okay, so let us see what is the difference between these. So let us look at just definition of covariant. It is a functor. A functor is C to D, but we will not write a just a simple arrow. A simple arrow is usually meant for a morphism, like a function. So that will not be indicated. So we have to have some other notation. So this is twisted arrow, okay, that will be the notation for the functor. F from C to D, we mean this is an association denoted by F itself again, okay. Several times we will have to use this one. It is an association from objects of C. It is a class, remember, objects of C is a class. Similarly, objects of D, that is another class. So here it is an association. If these were sets, then this would have been a function. So you cannot call it a function because the domain and co-domain are not necessarily sets, okay. So it is an association, which we will write it as suppose object is A, then this image will be written as F A under F, the associated thing, okay. So that much we do just like function theoretic notation. Each pair of objects, suppose you start with two of them, then you have what F A and F B, right. On the other hand, you have the set of all morphisms from A to B inside C. Then you have also the set of morphisms F A to F B inside D. So these morphisms to these morphisms, there is a function now. These are sets. This F again, we are writing the same F, the function, each morphism here is taken to a morphism there. So that is also written as F F, okay. Respectively MC of AD to MD. This is a difference here now for contravariance. The contravariance, what is happening is arrows are going the other way around. If MC AB to MD, F P F A instead of F A F B, okay. So that is also we are writing F F, but it is contravariance, okay. That is a property too, okay. Property one is same for both covariance and contravariance. Here contravariance is the other way around is what you have to remember. Now the third thing is what happens to the composite. The F of the composite is composites of the F's. Similarly, F of the composites is composites of the F and D, but you have to take it in the opposite direction because the other one may doesn't make sense. So this is contravariance, let's go variance. Finally, there is one more which is very important. F of identity itself is identity of the corresponding object in D, namely F A. This identity of A goes to identity of F A, okay. This must be true for wherever A, B, C, etc. are objects of C and all morphisms F and G wherever they are, namely they are inside the category C, okay. So basic thing is there is an assignment to each object an object in the other category and to each morphism here there is a morphism in the other category. The assignment must be respecting the compositions is this law, law A. The assignment must be respecting the identity is this law. This is just a group of morphisms remember. Identity goes to identity, the composition should go to composition. That is the group of morphisms. So that has been generalized here, okay. So such a thing is called a functor. If the arrows are reversed, it is a contravariant functor, it is still a functor. So co-variant, contravariant is what you have to say just to know whether the arrows are reversed or arrows are going in the same direction. That is what you have to do. Keep track of it, okay. The point is we are already familiar with plenty of examples and this definition has come much, much later now. We are just adjusting to the new definition, okay. This is actually an ideal thing. So let us examine what are all the known things for us which fit into this definition, okay. So I have already told you the difference between co-variant and contravariant is simply the fact that co-variant results in direction whereas contravariant is your system. However, in practice, it turns out that contravariant has more mathematical structure in it whereas co-variant is more geometrical and easy to understand. And obviously in the historically, study of contravariants was taken up by people a little later because this was easy to understand, co-variant. The second point is suppose F1 and F2, C1 to C2, C2 to C3 are both co-variant functors. Then there is an obvious way to define a composite of these functors. Namely, take an object in C1, you will associate directly an object in C3 by taking F2 of F1 of that object S. Likewise, if you have a morphism from A to B, you will take F2 of F, then F1 of F and F2 of that from F1, F2, F2, F1 of A to F2, F1 of B, okay. So this is an obvious composition in there. Composition of something itself makes sense just like composition of homomorphisms makes sense. And if both of them are co-variant, then the composition will be co-variant from C1 to C3. Similarly, for contravariant functors, but if you compose two contravariant functors, it will become a co-variant functor. Now this is one of the reasons why co-variance has to be studied first before contravariance, okay. So co-variance is easy in that sense. But it is mandatory for you to study that before studying contravariance because if you compose two contravariant instincts, you get into co-variance, okay. Composite of co-contravariant functor will be co-variant. If you can also compose a co-variant and a contravariant, that will be contravariant. A contravariant and a co-variant, that will be also contravariant, okay. So this is like anti-homomorphism in group theory and anti-homomorphism functions in complex analysis, okay. So there are examples of these things all the while in mathematics, all right. So now suppose f is a functor from C to D. If two objects a, b are equivalent in C, remember what is an equivalent in C? There is a morphism from a to b and another from b to a which are inverses of each other. If there is such a such two objects which are equivalent, f a and f b will be automatically equivalent. What are the equivalents between them? Namely take little f from a to b, then f of that f of f will be an equivalent with f of g being its inverse. So this is one of the most effective way functors are exploited. Namely if you know f a and f b are in equivalent then a and b are in equivalent, okay. So the contra-positive of this statement will be used again and again in practice, okay, to derive lots of results. You have several such illustrations, okay. So already one of them we have discussed right in the beginning in part one also. Again I can repeat it here. Namely, okay, first let me do that and then come back here. Namely take x to be any topological space, okay. Then from topological space to a and s you assign the functor namely the set of path connected components. If you have a function from a to b, a and b are topological spaces, then it automatically induces a set theory function from path connected components to path connected components. So this association can be easily checked to be a functor. If you have identity map, the corresponding path connected components will go into the same path connected components. That will be identity map of a path connected components, okay, from x to x. The corresponding path components from x, x may be 20,000, there may be different number of them. Number is not another. So just the connected components. So that is a functor. From that you can extract just the cardinality because you are inside set. Two sets are equivalent, it just means they are in the cardinality. Suppose the cardinalities are different, okay. Then you know that the original spaces x and y or a and b are not homeomorphic to each other. In fact, they are not even homomorphic type of each other. From top you should take, they are not homeomorphic to each other. Because if there is homeomorphism, the number of path components will be the same. So this is the way, this is the very simplest way how a functor can be useful. This is used several times in ordinary topology. Like you showed that suppose you have a, the union of two x's, why it is not a one-dimensional manifold and so on. So many times it has been shown. Or classification of, that was one of the examples. Classification of alphabets as subsets of R2 and so on. Okay, so let me come back here for this important thing. One of the most important class of functors, okay, is the forgetful functor. What is the meaning of this forgetful functor? It is a very strange name here, okay. So in this one, a category is taken whose objects may also be considered as objects of a larger category. Okay, for instance, from abelian groups to the set of, to the category of sets. What you are doing, you are just forgetting the abelian group structure on it, only looking at the underlying set. Or you can take a vector space and forget the vector space structure and only look at the underlying set. Like this, you can have several such instances, okay. Like you can have an abelian group and forget that it is an abelian, then you will have, it has still a group, it is a group structure. You do not use abelian, you have just forgotten or you have been told and various things, right. So that functor is a forgetful functor. So such functors are called forgetful functor. It is a, it is a functor you can see. The whole idea is it is some category first of all, neutralized category. Morphisms are the same and compositions are the same and so on. Only certain objects are forgotten, certain structure, extra structures are forgotten. So that is the meaning of this forgetful functor, okay. So study of interlation between category and subcategory is a common feature in all mathematics and that is how the forgetful functors are important. The most widely studied forgetful functor in topology is the one from diff to top. You have manifold with a differentiable structure, but you can just consider it as the political space. So this is very, very important thing. First you say topological space. As a topological space is two of them are non-homomorphic. There is no chance that as diffeomorphism, as differential manifold, there will be diffeomorphism to each other, right. So first you try to understand it as a topological space, forgetting about its differential structure. So whenever you are in trouble, you will maybe use the more structure. Namely that this being a differential structure, you will use. Like a function may be C infinity, but you are going to differentiate it only once. So you forget about C infinity, but you are just using C1-S and so on. So all the time inherent in our approach to mathematics, we are using forgetful functor without even aware of it. Giving a name to it and pinpointing that this is what is happening, makes the concepts much more clear and much more powerful, that's all. It has its advantage, okay. Long, long back this I learned as a student of chess. I used to play chess very reasonably well and without knowing any of these theories. But when I met some theorists and I beat him, he was very much impressed and then he was using certain terms. I said, oh, is that what I am doing? So he explained certain things. Oh, this is called this, this is called this. After that my own game became even better than what it was. So that made me read some books and read some theory also, okay. So that is the story, how it happens that when you know whatever weapon you are using, you know look it is two-sided weapon, you are using two-sided weapon here, you can use the other side also. Oh, yeah, I am unaware I was using but now I know what I am doing. So then it become more powerful. So that is what it is, okay. So in this way not only forgetful functor, many of these categorical things will help you to make you a better mathematician if not teach you new mathematics as such, okay. So let us go ahead. Consider the category top, I already told you this example. As a, you know, cardinality of the underlying set and cardinality of the connector components, all these things are some kind of functors, okay. There are several of them. Now let C be any category, I am going to construct another category out of this, namely C-hat, you know, the category whose single object is C itself, okay. The singleton C is the objects of, object of the C-hat, okay. There is only one of them. Now I have to define morphisms. There is only one object. So I have to define MCC. MCC also you just take the identity element there, namely identity functor now. C-hat to C-hat, it should be the identity functor. So this is a category, okay, because the rest of them composition, etc. are obviously defined, okay. And what is the object set singleton? What is the element of this thing, element of this one that is not a set possibly, okay. Because we know category for example ENL, set of all sets is not a set. Now that becomes one single object. So objects themselves may not be on sets, okay. So this is an example for that. Once you have this, you can make many more examples. So just to illustrate that objects of a category may not be sets, okay. Now, I mean I could have given many other examples, more complicated examples. I waited for this one so that now we know what is the meaning of functor. Once you have an object, you have to define the morphisms. A morphism from one category to another category, we know what is it. What is it? It is called a functor. Functors play that role. Let C be any category and A be an object, one single object in C, okay. Now I am going to define two functors here. One is covariant, another is contravariant. These are not for fun. These examples are very, very important. But the way I divide is similar to, you know, it is cooking off something, okay. It looks like. So A is fixed. You vary the objects, the second coordinate, second slot here on objects of C. For each, say, B inside objects of C, you have home A, B, that is a set. So A to home A, B, you can assign. Or you can do the other way around, namely A to home B, A. There are two of them, right. So this will be an assignment from the set of, from the category C to the category of sets, E and S. Because you know their home is always a set, okay. So this is the definition. Home A operating upon B is home A, B. So this, this dash I have kept as a slot. So B occupies that slot. Here B comes on the left side, home B, A, okay. Now having defined the association on the objects, I have defined the association of, of morphisms. So take any morphism from B to C, home A is a first slot. Now morphism you put here, F, from A, B to A, C is nothing but composition first. F star of home A, B are given as follows. What is it? F star of G is F composite G. F is on the left. F upper star is on the right. So you will see that the, that is affected by this twisting of the arrows A to B. This is B to A. The compositions you know are automatically associative. From which it follows that all these things are actually functors. And one is covariant functor and another is contrivariant functor, okay. So these are called representable functors. Out of a single category, I have cooked up functors. In fact so many functor, for each A there are two functors, covariant and contrivariant. The whole idea is if you know these two functors for all A, whole lot of them, then you know the category C perfectly. Or in other words, these two functors will bring out various things happening inside C. You may not know everything because there are everything you have to understand. So maybe a few of them if you understand, then the rest of them will automatically follow and so on. So representable functors are key to understand a given category, okay. So given, so this is where I will stop. So these, now the next time I have to explain you the canonicalness of G by Karnel and V to V double star, okay. So this is where I will stop today. Thank you.