 So, today will be the last meeting for categories and punctives, today we shall demonstrate how certain classical constructions in mathematics can be put in categorical language beneficially. Like any category, an object in this category is called an initial object. If it admits exactly one morphism into every other object, say A is an object, then MAB must be a singleton set for every B inside C. So, it is called initial object. Exactly due to this, if A admits exactly one morphism from every member B into itself, then we will call A a terminal object. So, it is same thing as saying M B A is a singleton for every B inside C. If A happens to be both initial object and a terminal object, it is possible of course, then this is called a zero object. So, this terminology is again copied from like abelian groups or any other abelian kind of things or even groups theory also like rings, fields and so on. So, we will see why these these terminologies help us and how they help us. In the category of all sets, the empty set is an initial object, but it is not a terminal object. Also, every singleton set in ANS is a terminal object, but not an initial object. Categories such as the groups, abelian groups, vector space over a field, R modules, etc. They all have zero objects. The zero space or whatever the zero group will have exactly one homomorphism from it to any other group. Similarly, there will be exactly one homomorphism from any group into the trivial group, zero group. So, zero element which is if we multiply it, if you write it will be the identity element, the singleton identity group is both an initial object and a terminal object. So, this is called a zero object. There is a simple way to turn initial object into a terminal object by considering the opposite category. In category C, if A is a initial object, category C of it will become a terminal object and vice versa. So, you see, ENS, ENS had an empty set as an initial object, but not a terminal object. But if you take ENS of it will become a terminal object, but not a initial object. So, it is easy to produce categories which are, which may or may not have any initial object or may have many of them also and so on. In any category, it is easy to see that any two initial objects are equivalent in the sense of that category. So, this is simple observation which helps to understand many part of mathematics very easily. Likewise, a terminal object will be also unique if it exists. Unique is a sense of equivalent class. Equivalent, within that category, they will be equivalent or you can say isomorphic or whatever. Now, a zero object has some more structure inside it. For every A, B inside C, there is a special morphism 0, A to B. The morphism itself will be denoted by 0, A to B, which is the composite of unique morphism A to 0 to B. Now, this 0 I am using to denote the zero object which is obviously a unique element. So, I can have a symbol for it. To take any A and B in that category, A to 0, there is a unique morphism and 0 to B, there is a unique morphism. Take the composite of this, that will be denoted by the zero morphism from A to B. So, 0, A, B. So, nobody told you this one, but it is there as now consequence of these axioms that such an element exists. These morphisms have the additional property that for any morphism F from D to A, G from B to E in C, we have if you pre-compose F with 0 A, B, it will be 0, 0 D, B. Similarly, if you post-compose G with 0 A, B, it will be 0 A, E. So, G is from B to E, whatever. So, they are all different zero morphisms, but composing with zero morphism any morphism, either this side or that side, right side or left side of a position 0. So, this conforms with our experience with zero morphisms of abelian groups, rings and so on, or vector spaces and so on. So, we will come back to initial object and treadmill object, but right now I will give you a construction now, because that is what my aim was in this lecture. So, pullbacks. We had introduced this one in part one in a very special circumstance, namely when you have a covering spaces or G covering spaces and so on and you have a map from any space into the bottom space, the base space. Then the covering could be pulled back onto the space X. Now, we would like to do the same kind of thing in any category, at least we do not know the existence and so on, but we will know that there is a concept like this. So, take any category, then fix a morphism from X to B. And a morphism from E to B. So, P is inside M E B and F is inside M X B. So, B is here the base for both the maps, B is the codome, that is the picture. Once you have fixed this, you will define another category itself now. In that category, which I am denoting by CFP. So, it depends upon the category C, where you are working. Then it depends upon F and P, which you have fixed, the two morphisms which you have fixed. So, CFP is a notation. So, this is going to be category, I have defined how. So, whose objects are commutative diagrams like this. Namely, F and P are fixed. Z is any other object, but then you have to take morphisms from Z to E as well as Z to X, such that this square is commutative. P composite alpha is equal to F composite beta. Take such things, they will be the objects of this category. Next I have defined what are the morphisms, verifying that the compositions and all that will be automatic because they depend upon the property of C being a category. So, what is a morphism from one such object to another object? So, I have to take two such squares. So, here Z1 alpha 1, Z1 beta 1, F and P are the same. Similarly, Z2 alpha 2 here, Z2 beta 2 there. So, this is one commutative square. That is another, this is the first one, that is another second one. A morphism will consist of now, a morphism from Z2 to Z1, one more extra area, all these things are given. So, a morphism from this square to that square consists of one morphism Z2 to Z1, gamma such that the entire thing is commutative. Namely, gamma followed by beta 1 is beta 2, gamma followed by alpha 1 is alpha 2. Then rest of them will be automatic. So, such a thing will be a morphism. If you have another Z3 and you have defined like this, then obviously, the compositor will be the compositor of gamma 1 and gamma 2 from previous case. Because we are using the category C, all that is obvious what we are going to do. So, this itself becomes a category. So, CFP is a category of diagrams of the squares. Now, a terminal object in this category will be called the pullback of P under F. So, a terminal object means it is some object like this, which will admit exactly one morphism from every other object. So, if Z1 satisfies that property, for every other Z2, there is only one more morphism like this, exactly one, not utmost one, remember that. So, then Z1 will be called the total space of this pullback. The pullback map itself will be this beta 1. The notation will be F star of P, fully P under F, F star of P. So, that is notation F star of P. So, the total space will be denoted by F star of P. The map to X will be denoted by projection map F star of P, whatever. By this, we mean the codomain of the morphism F star of P is X and whereas its domain is denoted by F star of P. The uniqueness of the pullback, you see this being a terminal object, the uniqueness of the pullback is obvious. Okay. So, in part one, we gave a constructive proof of the fact that in the category top, namely topological category, pullbacks exist. The same construction you can try in various other small categories, but they may fail to give you the full satisfactory answer that they may not be terminal objects in that sense. That part we have to verify, but quite often it may work. Okay. So, you see now because of this terminology, I have reduced the work of defining pullbacks, what I did for just the covering spaces. Okay. And prototype of that, I have defined it for umpteen number of cases, all categories, all morphisms. You construct a CFP in this way and look for the terminal object in that category. So, one single definition takes forever. So, this is the beauty of category for language. Okay. So, for example, recall that a category is called small category, if its objects, each object is a set. Like ENS is a small category. Top is a small category. Many of them rings, groups, they are all small categories because the object sets can be thought of as sets. They are subsets of subsets of the no, the underlying set is a subset of, can be thought of as a set. Okay. So, or you can say that they are in some sub-categories of edness. But that is not necessary. Small category just the object sets are sets, that is enough. Okay. Let A be any set. Okay. And consider category C A. Now, I am going to define another category, just another example. Whose objects are pairs A, G. You understand? C is a category. I pick up A is any set and then I am taking A, G is a pair where G is an object in C. And A is a subset of G. For example, this category C could be topological category. Then G is an object means what? G is a topological space. But I can talk about subset of that, just a subset. A is A must be subset of that. Or this category may be a groups. Then every object is a group there. But then I can talk about a subset of that group. So, this A must be subset of that. Okay. So, take such pairs A, G, where G is an object in C and A is a subset of G. Now, these are the objects of this C A I am going to define. What are the morphisms? Morphisms are again commutative diagrams. See A to G1, I have this picture namely inclusion map. A to G2 also I have. And G1 and G2 are what? They are objects in this category. So, this F must be a morphism in this category such that these diagrams come out. Now, this thing makes sense because this is a set theoretic, it is a small category. So, you have to pass on to the subsets A to F composition as an ENS you have to do. Here also, this is some set inclusion. So, composition here is nothing but what? Composition in the small category, in the ENS. So, this diagram should come out and F must be a morphism in the given category. If you are taking topological category, this must be a continuous function. If you are taking groups, it must be a what? Homomorphism. If this is a category of vector spaces, this must be linear map and so on. So, this must be a morphism in that category. So, such diagrams will be called morphisms of this category. Objects are like inclusion maps like A to G1, the pairs of A, G is a super set of A or A is a subset of G. So, this is a category. In this category, take an initial object. If it exists, it is unique. If it may not exist, I do not know. If it exists, it is unique. That unique object, the initial object will be called free object in C with A as a basis. This is a definition. What is the meaning of free object in any category? Now, let us look at the examples that I am telling about namely groups or abelian groups or vector spaces or R models and so on. What happens to this? This should be nothing but free group, free abelian group, vector space or a free module over R with A as a basis. So, all these four different things we have studied very thoroughly at different places. All of them can be studied in one single go by this or express, right. Do not say it can be studied or express the ideas all about all of them in one single go. They are all free objects in a particular category, whatever category you are looking at. So, if you have proved a theorem for free objects, it will be true for all of them once in Galway. Construction could be also similar existence of constructions. I mean constructions for existence theorems but it may not work. So, you may have to keep modifying it. They are similar but you know always existence part you will have to use very special properties of the particular category that you are looking at, okay, that cannot be generalized. So, pullbacks etc are new but these free objects were somewhat old. I will ask you one simple question. If you take CDS topological spaces, what is the free object there? Think about it. So, direct limit and inverse limit is the next object that I want to discuss. So, go back to this example wherein we started with a partially ordered set and then converted into a category. So, that conversion was equivalent. You can recover it to get the partially ordered set. Namely, the category associated to a partially ordered set at objects are points of this set, members of this set. So, axis is set, okay. And morphisms are precisely one single element if filled only if axis less than or equal to y. If x and y are incomparable, MX y will be empty. So, that was the partially ordered set and that was the category that associated it, okay, right. So, we will use that now. We have seen how to view a poset as a smaller category. A directed set, okay, J comma instead of x comma something I have J comma less than or equal to is a poset such that for every two elements i and j, there is a k such that i is less than 2k and j is less than 2k. You know, i and j may not be comparable. If they are comparable, then I do not need another k there. One of them is bigger than that. But if they are not comparable, this is an extra assumption on the directed set. It is not a or it is not always true in a partially ordered set. Given any two elements, there is an element which is bigger than both of them. So, that is the meaning of it. There is a k such that i is less than 2k, j is less than 2k. Such a thing is called directed set. It is a partially ordered set with extra condition, okay. It is not difficult to reformulate this in terms of the category, corresponding category, okay. All that it means is given i and j inside j, there is a k such that M i k and M j k are non-empty. That is all. So, i less than 2k is if you are doing M i k is non-empty. j j k is non-empty, okay. You can talk about the same thing in slightly different language category. Now, a directed system I am going to define just as a covariant functor from this category. Now, this is your think of this as a category to any other category. If you have covariant functor, okay, taking values inside c, then this directed system is in the category c. So, a directed system is nothing but a covariant functor from a directed set. What are its constituents? For each element i in j in j, you would have an object here f j, right. And for each i less than or equal to j, there will be a morphism from f i, f j, f i to f j, right. And then because it is directed set, f i, f j are given, there will be some f k for which f i to f k and f j to f k also there will be a morphism. So, that is the picture of a directed set, directed system, okay. Associated to a directed system, we can define a category now, okay. See, like I started with with a and contain is a g and then a subset a, a set a and a category, then I define this category of pairs and so on, okay. So, c a I define. Similarly, fixing a map f and p, I define c, f p and so on. So, this game you have to keep playing. So, whatever you concept you have, first fix this category and look at the data to begin with and then out of that you have to construct the correct category. That category you look for initial or terminal objects according to your test. So, now what I am going to do? So, I am going to define another category which depends upon this system. The system if I, okay. So, I am just writing c f because it is taking value in c, just not to come to, neither c or nara, it depends upon c f. So, as follows. So, what is the objects first of all? The objects are the pairs here, first object here is a, a itself which is the category, in the category, then all these alpha j which are morphisms for each j and j. Where a is an object in c and alpha j's are all morphisms from f j to a. Remember f j's are given by this c f, okay. So, you have to have these morphisms, okay, such that alpha i is alpha j composed of f i j. Whenever f i j exists, f i j may not exist all the time, f i j is to exist only if i is less than or equal to j, remember that, okay. So, we can represent it by this kind of diagram here. For each pair i j wherein j is bigger than or equal to i, or i is less than or equal to j, you have a diagram. Diagram means what? This composite this, alpha j composite f i j is alpha i. So, this is your category now, this is your objects, okay. So, what are morphisms again? From one triangle like this to another triangle, you must have morphisms. So, morphisms are expressed like this, okay. Suppose you have a here and another on b here, for a you have all these f i's, f j is going into a and beta is coming into b. So, there must be a morphism in the category tau, let us say a to b is tau which fits all these diagrams for all j's and all betas alpha j's beta j's. So, one single tau should commute. So, that is a morphism. By the direct limit of the directed system f, with a direct limit now I am talking about, the directed system f, we mean an initial object in this category Cf, okay. Note that in general, the category Cf may be empty. Why you see, there is a lot of things you have to have, the category must have morphisms like this. For each i, this f i m of f i a is empty then there will not be any such thing, even if one of them is empty, there is nothing like that. So, there is lots of conditions here, okay. If this category itself may be empty, then there is no question of having an initial object, okay. Even if it is non-empty, there may not be initial object. If it exists, then only you say it is a directed, direct limit. Once it exists, it is unique, okay. So, a direct limit a, you mean an initial object in this category. We note that in general, the category Cf may be empty. If c has terminal objects, suppose I take a terminal object a, then automatically all these morphisms will be there and they will be communicative also because terminal object has this property. There is a unique map here. The composite has to be there, so their composite will be automatically this one, okay. So, it is non-empty at least. Terminal objects then, of course, this category is non-empty. However, there is still no guarantee that the direct limit will exist. Of course, an initial object exists, namely direct system. We have seen that this unit, okay. Then we can have some notation, direct limit of fj. Thus, the direct limit of a direct system in C is an object a in C together with the collection of morphisms alpha j from fj to a because it is after all it is an object in this new category that I have defined. Satisfying compatibility condition, this condition, okay, and then it must have these things, conditions are there. This must be also true, okay. Such that for every other compatible family beta j, there is a unique morphism from a to b, making the up diagram b. So, if it is an initial object, for every b which has satisfied this property, there must be a single tau. That is an initial object. So, this is the way, okay, a directed limit is defined every time whether it is a direct system of groups, direct system of topological spaces, direct system of rings, direct system of algebras and so on. So, all of them can be talked about in one single language. So, again, in the category sets itself, okay, every directed system has a direct limit. That may be due to the fact that there are lots of terminal objects there. Every single term is a terminal object. It may have some such, you know, impact on that, that why the direct system exists. I am not saying that is a proof or anything, okay. So, how do you do that? Take b to be the quotient of the disjoint union of Fj's. Now, Fj's are just sets. Take the disjoint union, then identify x with Fij of x whenever x is in Fy. Fy, there is a map from Fy to Fj. So, Fij of x should be identified with x itself. So, do this for all possible Fij's. Whenever x belongs to some Fj and there is amorphism further, you identify that map, identify this element with the image of that and so on, okay. That quotient set will satisfy the requirement. This being a quotient of all these automatically from each Fj there will be a map, okay, the quotient map. So, you will get a morphism whatever requirement is there. The same kind of construction, when I am same, it is not the same because you have to worry about the group operations and module operations, upper space operations and so on. But similar construction works in many other categories. These are all what are called as abelian categories or algebraic categories which are abelian, abelian categories. There are various terms, such terms which generalize many, many algebra, topology, analysis, computer science and all that together. So, that is the whole idea of of category theory. Perhaps we are coming to a close now, we will start talking about other things now, okay. So, let me just give you one more example. In top also, for example, this works, okay. What I have to do? Take disjoint union just like we did in the sets. Do the identification also. But what is the topology? On the disjoint union, take the disjoint union topology and on the quotient set, you take the quotient topology. So, that topology where you take direct system is a very special case namely it can be thought of as the co-induced topology, okay. So, many of these topological things that you have studied could have been worked in with category theory. On the other hand, topological ideas themselves will come back and generalize all this category theory, what are called as tapas. So, there are so many other things to do in the category theory. Very closely goes, in the beginning it looks like algebra but later on it becomes topology. In modern algebraic geometry, all these things are a must, we have to study all these things. So, for directed system, there is one small technical thing which helps, it has nothing to do with the category theory. So, it is better to know that since I am talking about directed system, I will tell you that namely suppose you have a sub-posset of a poset i, okay. So, this is a partially ordered set and this is some partially ordered set. We say this j is final in i, okay, it is called a final family usually in the terms but as a poset is suppose sub-posset is final in i, whenever for every i in i, you will have a j which will supersede that namely j is bigger than i. For example, if you take natural numbers for i and take only the odd integers here, odd natural number, this will be a sub-posset but it is final because given any natural number, there is an odd number which is bigger than that, it is like that, okay. It is like this is this concept is similar to subsequence, the concept of subsequence, okay. It is completely generalized like this, okay. You do not need to go to category theory, this is done even in the classically, okay. So, what happens is suppose you have a directed system f i and then you consider subsystem f j. If one of the direct limit exists, then both of them will exist and they will be equal, okay. So, this is, I said it is similar to subsequence. Now, for subsequence, we know subsequence limit may exist, sequence limit may not exist in general, right. So, there is a problem there, okay. But here, if for final family, if this exists, then this will also exist, okay. So, directed systems are already like Cauchy sequences, not just general sequences, that is why this works, okay. You can verify this one, this is just a remark. Finally, I said only directed system, I did not talk about inverse system. There is no need because all these things, all that you have to do is take C of namely reverse the arrows. Instead of covariant functor, take the contrivariant functor and do the same thing, what you get is the inverse system. And a terminal object will be a direct, is the inverse limit, okay. So, that is easy exercise for you to think about, okay. So, after that, I have a number of exercise here. You can keep trying them. Some of them we will discuss in assignments and so on. So, our next topic will be homology theory, that will be for another five weeks. Thank you.