 So, we shall now continue to give some more examples, familiar examples of categories. We studied topological category, the homotopy category and the simplicial category and CW category. So, one more most important category for topologist is the smooth category denoted by DF by me, but other people may denoted slightly differently, notation is not all that all that uniformized. So, what are the objects of this category? They are smooth manifolds and morphisms are smooth functions. So, this category is called the smooth category and denoted by DF, DIFF, DF. So, some of you who have not studied any, you know, difference to power of your soul, for them this may not be quite familiar object. In any case we are not going to do anything in this category right now in this course. But however I would like to introduce a closely related category to this one which is actually somewhat larger than this category in some sense and that you will be familiar with. So, that is denoted, let us denoted by small DF, DIFF is actually larger than DF in some sense. So, what is this? The objects of this DIFF are subspace is of some Euclidean space, there is no other condition, subspace is of some Euclidean space, they are all allowed here. And what are the maps from one object to another object, they are all smooth functions. Now, this word smooth functions on orbital subsets you may not know what is it, so here it is namely I am saying to recall but if you do not know this is the first time you should know. A function from X to Rm where X is subspace of some Rm is said to be smooth if it exists if there exists an open subset U inside Rm, see X is inside Rm, so there must be an open subset inside Rm and a smooth function G from U to Rm such that X is subset of U and U is open and G is a function on U, G restricted to X must be F. So, why I am giving this, you all know what is the differentiability, second differentiability, first differentiability or all that smooth functions, three infinity functions or open subsets but you may not know what is the meaning of that on an arbitrary subset and this is the meaning, with this the DIFF category is very much familiar to you in doing calculus courses, you do in the calculus course. So, this is one of the important category, the category of open sets in a single topological space, this is a wonderful category but this is again not going to be used in this course but this will be useful when you do commodity theories, shift theory and so on. What is it? UX which is notation for this category that the objects of this UX are nothing but open subsets of X, if you say X tau then tau is the set of objects here. So, object sets are open subsets, each object is an open subset of X, what is a morphism from U to V? Whenever U is contained inside V, take the inclusion map, if it is not contained in take this as a empty set. So, UV is empty if U is not contained in it. So, this is the object, very simple object. Of course, M U U, U is contained inside U and what is the inclusion map? There is only one inclusion map, the identity map and that will serve as the identity, two-sided identity. So, this category is called UX, it is quite useful in the study of shift theory. Now, I come to examples from algebra. The first example is category of groups, GR, all groups are taken as objects. Morphisms between one group to another group is homomorphism and composition is just like set theory composition. Does that convince you that it is a category? Associative law is there, two-sided identity is there and that is what it makes a category. What is an equivalence here? Isomorphism of groups. What are isomorphism? What are equivalence objects? Isomorphism classes of groups. And there is a very nice subcategory here, namely you only take abelian groups. The rest of the things is same, namely even any two abelian groups, what is morphism A to B? All homomorphisms. There is no restriction there, there are no abelian homomorphisms, every homomorphism is fine. And identity is the two-sided identity as usual, identity homomorphism, the identity of the group. So, that is a category which is a subcategory of category of all groups GR and it is a full subcategory. Because whenever you have abelian group, homomorphism from A to B is the same as homomorphism A to B as whether they are abelian or not, is a full subcategory. Similarly, another one important one which is very familiar to you is, take a field, fix that field, then take a vector space over it, all vector space over that, that is the object. What are morphisms? Vector space linear maps, linear maps from one vector space to another vector space over K. The K field has to be fixed to make it a category. Then if you take the composition of linear maps, it will be again linear map, right. Identity is a two-sided identity, all that is fine. Similar, exactly similar, instead of K is a field, you take a K to be a commutative ring, okay. And the objects now are what are called as morphisms instead of vector spaces. So, this is another important category which we will be using all the time. The vector spaces as well as categories, we will be using in this course also, okay. So, homomorphisms are what? A and B are modules. What are homomorphisms? The so called R linear maps. The R is fixed here. You cannot change one module over R, another module over R prime where R and R prime are different rates. That is a entirely different concept that needs more algebra to be handled, okay. But there is no category as such. Next comes a cooked up category. It is an algebraic category, okay. However, it depends upon a topological space, okay. Later on we will conceptualize this also. But right now it is just an example of a category. What is this category? Fix a topological space. Then, what are the objects? Objects are elements of this set. So, object C, okay, is nothing but x itself. Elements of x are objects. What are maps from, so m of x, y, what is it? When x and y are elements of x, what is mx, y? They will be path is from x to y. A path is a continuous function defined on the closed interval 0, 1, okay, say omega. Omega 0 must be x, omega 1 must be y. All such path is will be taken as collection of m, x, y. So, this does not look like functions, you see. But it works out well for morphisms. Now I have to define what should be the operation, binary operation. Binary operation, I take concatenation of maps of path is. Whenever it, how is defined? If x to y, if take and another path is from y to z, then concatenation is defined, composition of path is. Not as morphism, this composition is concatenation. You have to follow x to y, the path and then trace y to z that another path, okay. The only problem is here now, associativity as well as identity. You see, we tried just taking morphisms, then it does not work. Therefore, we have to modify it. Namely, we know how to modify it anyway. Namely, instead of taking all path is, what you do is take only homotopy classes as a one single object. Namely, path homotopy classes. Path homotopy is what? A homotopy which keeps the endpoint fixed. Okay, take that as a one single member. Then you know that concatenation is associative as well as there are two-sided identity, namely the constant functions, the constant maps are two-sided. So, at x, you take the constant map at x, okay, that will be two-sided identity for path is which end there as well as path is which start from there. So, that will complete the definition of this category. This category is denoted by this curly px. It depends upon x, this p corresponding to whatever path is, okay. And this is called the fundamental groupoid, this category. Why it is called groupoid? Because every morphism is invertible. You know that if you take a path from x to y, there is a tracing the path from the other way around from y to x, okay. So, omega tau of t equal to omega of 1 minus t, that will define a homotopy inverse for the path omega. Therefore, every morphism is invertible. So, such things are called groupoids in more general set up, okay. So, concatenation of a path, path homotopy path is or path homotopy becomes a category. The class of constant path at x forms a two-sided identity, okay. And every omega has an inverse, a unique inverse up to homotopy everything, okay. More generally, any category in which every morphism is invertible is called a groupoid. That is why we call this group. In this particular case, look at mxx. They will be what? Loops at x, the class, homotopy class of loops at x and that forms a group namely pi 1 of xx. This is the notation. This is called the fundamental group of x at x, which we have studied thoroughly in part 1, okay. So, more generally, you have fundamental groupoid. Mxy is empty if x and y are in different path components. You should observe this also. Which is non-empty means they are in the same path component. So, there is no assumption on x, x may be path connected or may not be path connected. But groupoid is well defined, okay. Now, I come to some other kind of example from slightly different area. This is from common adorix. This is another very important example. We start with a partially ordered set, okay. Associated to that, you can define a category and associated to that category of some certain category, you can define a partially ordered set. This way, partially ordered sets can be converted into category and you can apply the theory of categories to derive theorems in partially ordered sets or concerning partially ordered sets and so on. So, how do we do this? Let us define a category associated to this x less than or equal to, that is the partial order. What are the elements of this new category? They are just elements of x. For any two elements x and y, take m x y, we have defined the morphism. The m x y will have one single element if and only if x is less than or equal to y. Otherwise, it is empty. This is similar to what we had, the u x, the topology, given a topological space x, we had the category u x. It is similar to this one you can see, okay. So, it is actually, this is more general than that one. That can be obtained as a special case of this one. So, m x y to be single attorney if x is less than or equal to y. Otherwise, it is empty. The binary operations are defined in an obvious way because what is it? You have a transitivity if x is less than or equal to y and y is less than or equal to z, then you know x is less than or equal to z. That is the transitivity of the partial order, okay. So, this element, there is one single element, x is less than or equal to y and y less than or equal to one single element, composite it to get the x is equal to z. So, that is the composition. So, finally, x is less than or equal to x, okay. So, m x x has one element there, which is a two sided identity. If you composite with x less than or equal to y, you get again x less than or equal to y. So, this is a category, okay, the category of posets. Posets means what? The partial order. So, what I want to say, these steps can be reversed. Starting with a category whose family of objects is a set, that is a messed up. The start to suppose the family of objects is a set. In general, it is not that. For each pair of objects, A, B, the set of morphisms is either empty or a single attorney. If it is empty, do not do anything. If it is a single turn, define x is less than or equal to y. If we do not leave, the m x y is non-empty. Then you get a partial order. So, now let us look at another example from algebra. Recall that by a semi-group, we mean a set with an associative binary operation with a two sided identity, like the set of non-negative integers with addition. That is simple example, okay. So, such thing is called semi-group. In other words, inverse may not exist. In particular, every group is also a semi-group binary. A semi-group may not be a semi-group because inverse does not exist. But inverse, if it exists, it is allowed. I do not say that. So, all groups are semi-groups also. Semi-groups are larger than. The category of semi-groups is larger than groups. That is all. So, semi-groups are very important in function theory and so on, okay. So, in any category C, for any object A in C, MAA is a semi-group in two different ways. What is this? Either taking operation f star g, I want to define the binary operation to be the fg which is defined in the category or gf. We will define it as gf also, okay. Because it is MAA, both of them will give you a semi-group structure on MAA, okay. So, in a, I am now doing nothing. I start with the category and look at and one of the object and look at MAA. It is already a semi-group. But there is no unique way. There are two different ways of looking at it, okay. So, conversely, suppose you are given a semi-group M, okay. There are essentially two different ways of thinking this as MC of AA where C is a category. What is that category? The category has only A as object and MCAA is the semi-group. But unfortunately, you do not know whether you are going to take the left multiplication as a operation or right multiplication, okay. So, there is that much of ambiguity. So, actually, both of them can be taken. So, there are two different ways of getting it. Sometimes may group like in a case of groupoid, Mxx became a fundamental group because everything was invertible. A semi-group is called monoid. These are some words you may not pay much attention to them. If it is cancellative, okay, a group piece of course a monoid automatically. If inverse exists, it is automatically cancellative. Look at the semi-group of integers that you have taken, positive integers. It is cancellative, okay. You have already realized that if 5 plus 10 is equal to something equal to say 5 plus some other number, then that other number must be equal to 5 without knowing the traction. So, this is built in the beginning of a child knows, okay. This plus this, this plus something else, that something else must be this. So, this is easier to understand by cutting down. That cutting down later on becomes the traction, okay. So, cancellation law can be understood without being actually invertible. Invertibility will become cancellative will be forced after that. So, the example 5, 13, 14, 15, you state the fact that there are many interesting categories in which morphisms are not necessarily functions. The first category, this example 5 was our homotopy category. A homotopy class, you cannot call it a function, okay. So, what is example 13? Let us see what was example 13? The fundamental group point. Here morphisms are again equations, classes of pathies. They are not even function, equivalent classes of functions. So, they are also not functions, equations, classes of pathies, okay. In the 14, this is just a binary relation x less than or equal to y, that will be morphisms. You can think of this as morphism, okay. So, this kind of abstractness slowly have to sink inside this one. All these are concrete examples inside the category theory. There are what are called as abstract categories. You can imagine how abstract they are. So, these things are these come come into very concrete examples. There are abstract examples also, which we are not going to deal with, okay. So, since I have already introduced what is the meaning of a subcategory, let us have a quick look at some more examples of subcategories now. We have already indicated one such example, namely the AB. The category of abelian groups is a subcategory of category of groups. Not only that is subcategory, it is actually a full subcategory, right. Similarly, if you take diffeomorphism, dif class, what are the objects? They are manifolds. So, before being manifold, they are actually topological spaces. And a smooth function is automatically a continuous function. Therefore, dif can be thought of as a subcategory of topological category, set of all topological spaces and continuous maps. But this is not a full subcategory because to take all continuous map, they may not be smooth. Therefore, it is strictly smaller, right. Therefore, it is not a full subcategory. So, we have examples as a counter examples also. It is here understanding what going on has to be, right. Now, I introduce little df and then I said that this is in some sense larger than capital Df. Why? Because there is a theorem which says take any smooth manifold, it can be embedded inside some large Rn. When you say embedded, it will become homomorphic, difomorphic to a subset of Rn. Rn is very large depending upon the manifold. So, this is theorem of Whitney, okay. It tells you that every smooth manifold is a sub-object in some Rn. Therefore, in that sense, you can think of, you know, this capital Df objects in this capital Df are objects in small df also. And what are morphisms? They are smooth functions. So, this way you can think of this as a full subcategory of that category. But inside dif, there are more elements. Namely, you can take union of x axis and y axis. That is not a manifold. But you can define what is the meaning of smooth functions on it, okay. Similarly, if you take a square, that is not a smooth object, it is not a smooth manifold because it has corner, okay. So, you can take triangle. It is not a smooth manifold but it is a smooth object. You do not know what is the meaning of smooth functions on it, okay. So, there are many more objects in subspaces inside Rn which are not manifolds, okay. Manifolds is a small object, set of objects there inside that. So, in that sense dif, capital Df is a subcategory of small diff, okay. Now, this is an open kind of exercise for you while studying these two morphisms attaching. Go back and just keep seeing what are the examples of what category is an example can be thought of as a subcategory of the category of sets. You will see many of them are, you will see some of them are not, okay. And that is the, that is why I told this ENS is a mother category for many of them. This is what you have to then you know which one are, which one are not, okay. All structural categories, many of them we have introduced they are like that. Namely, you start with a set X then you put extra structure on it. Namely, put a topology, namely put a binary operation which makes it a group. May be put a double two operations which make them into a vector space. So, these are called extra structures. They are all to begin with a set. Even if you put extra structure, there will still be a set. So, objects are sets. What are morphisms? Again, they are functions. So, all structural categories are subcategories of the ENS, okay. So, to understand these comments, you know, understand takes a little more time. You have to read whatever has been done so far, which is very little again and again, a couple of times. Be sure of what is going on, okay. Thank you. So, that's all for today.