 So, this chapter is a quick introduction to the language of categories and functions. What we are going to do is just a limited introduction, interested reader may look into the book I have referred to at the end of this by three authors, one of them is Adam His book is freely downloadable and it is quite a readable book. The topic as I have told you language, so to master language of course it needs some time. So, I won't say that right now you will become master of whatever I am going to introduce here, though that itself is very minimal. But because I am going to use this language again and again, certainly by the end of this course I hope you will all know how to use this language. Doing categories and functions as a theory itself is not at all done here, you must be careful about that. I am just you know it is not like we are going to write poetry in this language, we are just trying to learn the market language of you know how to do day-to-day business that is all. All sciences are essentially study of patterns which is more so in mathematics. The fundamental concepts such as number and space combined to produce innumerable patterns that we come across in day-to-day life. All mathematical concepts involve a certain family of objects which fit into a pattern and then the study involves relations among those these objects. Let us consider some examples. In topology we study the so-called topological properties of topological spaces. The objects that we consider here that are topological spaces but the relations are among them is what are called as continuous functions. The fundamental property of continuous functions is that the composite of two continuous functions is a continuous function and identity map from the space to itself is also continuous. Similar statements can be done about groups. Suppose you are studying groups then what are the functions from one group to another group you are going to study? They will be homomorphisms. Once again composite of homomorphisms is again homomorphism and identity map from the group to the group itself is also homomorphism. We can go on listing such examples like vector spaces and linear maps, modules over a ring and then the module homomorphism which are called linear maps again. And so on. So what we shall do is give a strictly rigorous definition of categories and then reexamine some of these examples again in the light of this definition which extracts certain basic properties that we are trying to explain. Category is a language that sums up this aspect of mathematics in a technically precise way. Not surprisingly it can help you prove theorems also. This chapter is divided into let us say four sections briefly. First section you will see definition of a category and sufficiently many examples. In second and third we will study relations between them under the name of functors. In 4 and 5 we see some more examples of certain standard notions which can be expressed which have been expressed which have been studied without the categorical language. But when you put them in the categorical language how nice it becomes. So with this background let us begin with some brass stacks. A category C consists of like I can say it is a ordered quadruple the four things. But I do not want to use that kind of language. So it consists of an object C. So there are C objects you can say. What it is? It will come in the definition. The whole definition has to be understood before you understand anything any particular thing. So what are the properties? One by one I am going to list. To each ordered pair a, b of objects a, b. So a, b are members of this whatever this class. We do not say this is set that is the slight difference here. It is a collection or you can say class but not a set. But what makes sense is whether some other objects a and b belong to that or not. This belonging relation is well defined that is the meaning. Suppose you take two members of this class then assigned to that is another set. This time it is a set m, a, b is a set. And what is the property of this set? m, a, b, m, a prime, b prime they are disjoint unless a is equal to a prime and b equal to b prime. So this equality must make sense inside C. What is the meaning of two objects are equal? That is also understood. These are all axioms. These are all part of the definition. Each word, each comma here is a part of a definition. Elements of m, a, b are called morphisms. What do you mean by element? That this relation is epsilon here which are written belonging sign. a and b belong into this is equal to elements of elements of this. This sphere is elements of m, a, b. It is a set. So we understand what are the elements of a. These are called morphisms with domain a and co-domain b. A morphism is a domain a and a co-domain b. Now here is a practice. This is only a matter of what we do in practice. It is not a part of the definition. We do not need the part of the definition. The following practice, we follow the usual practice of writing f from a to b whenever f is inside m, a, b. This is imitating if a and b are sets, f is a function from with domain a and co-domain b. Then we write this symbol. So that we are following as a matter of practice. This does not mean that a and b are sets and f is function. What does mean? This f belongs to m, a, b of this category. So that is the meaning of this whenever you write like this. So definition is not yet over. So only one condition I have expressed this much. Now part 2 says if you take a triple of objects, then this m, a, b, m, b, c, m, a, c, these are all sets now. There is a binary operation from m, a, b cross m, b, c to m, a, c. Now these are all sets. So binary operation makes sense. So I can write it as f, g going to g composite f. G is here. So that will come on this side, the standard practice of f, g going to g composite f is the binary operation. These binary operations are for each triple there is one. So if you change that triples and so on, there will be many such binary operations. They are collectively associative. What does mean by collectively associative in the following sense namely f from a to b, g from b to c, h from c to d, then we have g composite f. We also have h composite g. These two things make sense. So what we want is g composite f and then followed by h behind then h of g composite f is same thing as h composite g first and then following the other side namely composite itself. So brackets can be interchanged like this as usual which is the associativity. The point is that they are not functions from same set to same set or something. So the compositions may not be defined unless the arrows match as usual just like in the case of sets. The third condition is that for each object a and c there exists a unique idea, this is notation. Inside m a, m a a are morphisms from a to a. There is one unique element there which has the property that it is a two sided identity. For all f from a to b or g from c to a, you have f composite identity. This makes sense equal to f identity composite g equal to g. So this identity of a is a two sided identity. It is the meaning of this one. Morphism a to b is called an equivalence. Now I am defining equivalence in the category c or you may say invertible. This word is also defined. If there exists a morphism g, morphism means what? An element of m b a. So I am going to write it as in this notation such that g composite f, f composite g is identity of b, g composite f is identity of a. Then g is said to be the inverse of f. In fact, you can say this is n inverse but now it is elementary algebra that you have to verify that this inverse as well as the identities are unique. Here identity is, you can, is already axiomatized as unique. Uniqueness actually follows. Similarly, the inverse will be also unique if it exists. There may not be an inverse. So if there is an equivalence or invertible map from a to b, then we say a and b are equivalent objects in the category. That is the end of the definition. So we have defined what is a category, what is a morphism in it, what are objects in it, what is an invertible map and what are equivalences. Now I want to make a few elementary remarks. Often when more than one category is involved in the discussion, we write instead of just m x y, we write m c x y, morphisms x y instead of that home m c x y that c is indicated. Here also morphism c x y should be to denote the set of all morphisms from x to y, where x and y are considered as objects in a particular category. Suppose I am having two different categories, then if I just say morphism x y, you do not know whether it is in category c or category d. So that is why in that case we write. Otherwise when the discussion is going on, a particular thing we understand and then we reduce the notation as usual. This is a practice, but this kind of thing should not be done when you are dealing with a computer or one thing. Computer won't understand the card unless you define writing till here, till we redefine it is this one. That is what you have to define like that and so on. By the way, the computer scientists use the category theory very much. So it follows easily that identity morphism is unique in MMA for all A. What we have seen is there exists one, but now you can prove and you should try your hands that it is unique. You cannot have two different identities. Note that MAB is a set may or may not be empty. If MAA if you take, it is not empty because identity is there, but otherwise MAB may be empty, may not be empty. It is not necessary that it is not empty. Whereas object C need not be sets. So we will see some examples later on. Neither is the collection of objects. Of course, this is definitely most often it is not a fact. You need not bother about these purely logical problems at this stage. That is the key word here because if you get into that, you will never learn how category theory. Category theory you should keep learning. Whether you call it a class or a set and so on, you should bother about only when you are talking about foundational mathematics like logic. The logicians are more interested in this, this aspect of that. Till then you will not have any problems. You should not bother about that one. Okay, right now. Otherwise you will never learn the subject. It also follows that if f is invertible, it is inverse is unique. Therefore I can use the notation f inverse. Any g such that f composite g and g composite f are identity of the corresponding domain or co-domain that g will have to be the same. Therefore you can write g as f inverse. The equivalence of any two objects defines an equivalence relation namely reflexive, transitive and symmetric. So you can verify this and what one would like to do, one of the central problems in category theory is when you have defined all this category, whether given two objects they are anywhere near being equivalent. So if you know all equivalence classes in this in the particular category, you will pretend to know that I know this category completely. Okay, so that is pattern studying is precisely what science is all about. So one more remark is the condition of disjointness. Remember I tell m, a, b, m, a prime, b prime are different and disjoint unless a is equal to a prime and b is equal to b prime. There is no overlap among us then that is very important here. Right now you may not understand the importance of this because in standard mathematics that we keep studying in calculus and so on right from the days of Euler, Gauss and so on, we have to keep that practice hasn't been given up. Okay, maybe for good we haven't given up namely for example I will give you, we write sine function. The sine function could be when you are doing just the real, real value sine function, it could be just function from r2 minus 1 plus 1 okay. But you can treat it as function from r to r also still you write sine. Then there is a complex value to sine function defined all over the complex number c but that also you write sine okay. In all the these three for example are denoted by sine. You take a function then you restrict it to a subset use a sine. You take a function a to b and treat b inside c okay the function still written as a to c the same function okay. This practice is a denote all writing all these three as sine sine okay which is a sin according to categorical language this is not allowed. If you didn't domain r to code domain minus 1 plus 1 you change any one of them the function there you take will not be nothing to do with the function in the other side okay. They are disjoint elements. So this is what we have to insist. So however as I told you cutting down clumsy notations for all these things may obscure the way we do mathematics okay. But for example this is not at all done in computer science computers okay. On the other hand one of the basic objects of language of category theory is to provide us rigor without being two verbose to achieve this one write in the beginning it is very very verbose like insisting on having MAB and AA prime B prime disjoint all the time and so in the beginning you very very verbose then you will see the reward coming you know you just tell one single statement it will mean 50,000 theorems that is the kind of kind of you know achievement which category theory has has done okay all right. Let C be a category what is the meaning of a sub category that is what I am going to define now okay. A sub category we mean a category D satisfying the following condition so so D is going to be a category by itself but what is its relation with C is what we are going to see each objects of D is an object in C also for each pair of objects AB inside D the set of morphisms from A to B is a subset of set of all morphisms C morphisms from A to B that is MD AB is a subset of MC AB the third thing is the binary operation inside D is the same as the binary operation inside C quite similar to you can see that what is the meaning of subgroup and so on or what is the meaning of subspace and so on all that is included in a sub category okay further if equality holds in this one namely once A and B are inside D MD of AB is equal to MC of AB if this hope happens for all pairs of A and B whenever A and B are inside D then we call D as a full sub category. So we will have examples of this no problem right now we will examples of categories as you might have guessed the foremost and easiest category which one calls you said mother of many many categories mother of the idea of category and that is you know by ENS ENS is short one for ensembles which is a French word it just means sets take the collection of all sets that is the object of this category object set okay what are the morphisms morphisms are the usual set functions. So verification of the axioms is totally easy because the axioms have been modeled on what is happening in the sets okay so we will see that many other other examples that we are going to discuss they are all in some sense sub categories of this category that is why I call this as mother category. So the next category I will give you a very simple one consider a category you know single object how to denote this object put it as just a star there is only this object it has no structure nothing but it is an object okay then I have defined a morphism so where do I have to define morphisms from star to star so m star star right I have defined take that also as a single element you have to take a single element without because the axiom says that m a a has to be non-empty and what element I have to put identity element identity morphism what is the meaning of this identity it must be two sides identity blah blah blah so the composition is also well defined here namely identity composite identity has to be identity there is no other choice okay so this is the in some sense the smallest category maybe one can make a category out of empty objects also I am not going to discuss but that is also allowed okay empty objects other than that this is a nice category sometimes the category here is to write this as 1 the empty object as 0 okay I am not going to do all that but I am aware of such things okay the opposite category the opposite category means there is a category already see I am going to construct a category it is called C op the opposite category what is it objects of opposite category are the same as the objects of the original category C but whenever a and b are members of this category C C op of m m of c of the word or the morphism from a they will be the morphisms from b to a okay and the composition law is the same except you write it in the opposite direction so you can easily verify that identity left identity becomes right identity right individual left identity so identity will be there two sides identity no problem okay associativity also no problem so these are all easy to verify not only that if you take C op op to take opposite of this opposite category then what you get is C category so this looks like totally useless thing but some of the very deep results are used by just doing this kind of thing in category namely what are called a duality principles we do not have time for discussing that one but just important because of the importance of this in general category I have just introduced this this very simple idea of constructing it in the of category which is opposite category similarly this singleton category is also it is very important in that sense we may not have much use of them the category of topological spaces is one that is what we are interested in we are all the time in what are the the objects all topological spaces that collection by the way it is not a set okay collection of all topological spaces is not a set but it is a class what are morphisms continuous function okay what are the equivalence is homeomorphisms what are equivalent classes of objects they are homeomorphism types okay so this is the first example we started with even before the definition so now we can see that it is a good example the next category now you will see the power of categories how one can just change a little bit and express lots of ideas consider a category whose objects are topological spaces but morphisms from x to y x and y are topological spaces are not maps continuous maps but the homotopy class of a continuous the homotopy class of a continuous map is treated as a morphism from x to y okay this category is called homotopy category so the elementary observations that if f is homotopy to g and g h is homotopy to say h prime then f composite h is homotopy to g composite h prime and so on we have verified that and those things are necessary to verify that this category it makes sense the composition is the same f composite g the classes is same thing as you take f composite g the class of that okay what is a equivalence here the homotopy equivalence what are the equivalence classes of objects homotopy equivalent spaces so suddenly it has totally different meaning okay and this is the category that algebraic topology is all the time interested okay so this can happen suppose you change the topological category slightly there are various ways of changing it like you can take simple shell complexes CW complex I am going to discuss them okay or just horse door spaces objects you have changed just the only horse door space you have taken but continuous functions you remain continuous functions or now only you can take homotopy classes okay and so on so so there is a lot of scope in this language okay so now these two categories are of importance simple shell category we have introduced in part one now we did not have this terminology then now we have to just verify that terminology CW category we introduced just a couple of days back but what is this objects are topological spaces which have a structure extra structure namely a CW structure a topological space itself is a set with an extra structure now we have one more extra structure namely the CW structure and what are the maps cellular maps component of cellular maps is cellular identity map is always cellular this is what I have to verify and identity we have such a 200 identity so that is a CW category similarly the simple shell category in which morphisms are simple shell maps okay so we will discuss more examples next time thank you