1 Differentiable manifolds and smooth maps

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1 1 Differentiable manifolds and smooth maps Last updated: April 14, Examples and definitions Roughly, manifolds are sets where one can introduce coordinates. An n-dimensional manifold is a set whose points can be parameterized by arrays of n independent variables. Introducing such a parametrization is called a coordinate system or a chart. It may happen that that it is not possible to have a single coordinate system for the whole manifold and we need to consider instead overlapping patches each with its own system of coordinates. Therefore we are forced to consider changes of coordinates. (For a particular patch, a coordinate system is not unique either.) Later we shall see that there is topology involved as well, because each manifold automatically carries a structure of a topological space. Before giving precise definitions, let us discuss first the fundamental idea of coordinates. What are coordinates? Coordinates on familiar spaces. Example 1.1. A point x R n is by definition an array (x 1,..., x n ). The numbers x i are called the standard coordinates on R n. (The superscript is an index, not power.) On R 2 and R 3, the standard coordinates are traditionally denoted x, y, z. Example 1.2. A linear (more precisely, affine ) change of coordinates on R n : if we consider variables y i so that y 1 x 1 b 1... = A , y n x n b n where A is an invertible square matrix (det A 0) and b i some constants, then the standard coordinates x i can be expressed in terms of y j. Therefore there is a one-to-one correspondence x (y 1,..., y n ) and the variables y j may be regarded as new affine coordinates on R n. (This is the definition of affine coordinates on R n.) 1

2 Example 1.3. Polar coordinates on R 2 ; spherical coordinates on R 3, and on R n for n > 3 (which can be defined by induction). For example, for R 2, x = (x, y) (r, θ) where x = r cos θ, y = r sin θ. Note: coordinates (r, θ) serve not for the whole R 2, but only for a part of it (an open subset). Example 1.4. In general, for an open domain U R n, differentiable functions y 1,..., y n defined on U make a system of curvilinear coordinates on U if it is possible to express back the standard coordinates x i in terms of y j, again by differentiable functions. So there a one-to-one correspondence x (y 1,..., y n ) for x U differentiable in both directions. Then automatically the Jacobi matrix x 1 x... 1 y 1 y n x n x... n y 1 y n is invertible. (Polar and spherical coordinates are particular examples of curvilinear coordinates.) Remark 1.1. In this course we shall work with differentiable functions (of several variables). The standard notation for the class of functions possessing continuous derivatives of all orders k is C k. Though many statements remain valid for C k functions, for some finite differentiability order k, it is convenient to concentrate on functions of class C (those possessing derivatives of arbitrary order). They are called smooth. We shall use differentiable and smooth as synonyms, so for us all differentiable functions are supposed to have infinitely many derivatives. Affine coordinates on R n or curvilinear coordinates on open sets U R n have the property that one coordinate system works for the whole space, R n or U respectively. Consider now examples of coordinates on familiar objects where one coordinate system is not enough and one needs to introduce coordinate patches. Example 1.5. The circle S 1 : x 2 + y 2 = 1 in R 2. A coordinate on the circle: u R, where u = x, and conversely x = 2u, y = u2 1 (check!). Good 1 y u 2 +1 u 2 +1 for all points of the circle except N = (0, 1) (the north pole ). To have a coordinate system on S 1 covering the north pole, consider a different variable, denote it u R, such that u = x. It is defined at all points of the circle 1+y except for S = (0, 1) (the south pole ). In particular, u is defined for the 2

3 north pole (where u = 0). Together the subsets where u or u are defined cover the whole S 1. On the intersection where both u and u are defined (it is S 1 \ {N, S}), the change of coordinates is u = 1 (check!). u Example 1.6. Similar stereographic coordinates can be defined for the unit 2-sphere S 2 in R 3 and, more generally, for the unit n-sphere S n R n+1. One needs to choose a point of the sphere S n as the center of projection and a plane (of dimension n) not containing this point as the plane of projection. Then the projection map send an arbitrary point P S n (except for the center) to the point P on the projection plane defined as the intersection with the straight line passing through P and the projection center. This is clearly a one-to-one correspondence P P. Say, if one takes the north pole N = (0,..., 0, 1) as the center and the coordinate plane of the first n coordinates as the projection plane, the corresponding projection will be given by formulas similar to the above. Exercise: check that they have the form u N = x 1 z and x = 2u N u N 2 +1, z = u N 2 1 u N 2 +1, where P = (x, z) Sn and P = (u N, 0), so that x and u are vectors in R n. Find the analogs for the projection from the south pole and calculate the change of coordinates. (Answer: u N = u S u S, if u 2 S R n corresponds to the projection from the south pole.) Example 1.7. Another way of introducing a coordinate on S 1 is to consider the polar angle θ. It is defined initially up to an integral multiple of 2π. To make it single-valued, we may restrict 0 < θ < 2π and thus we have to exclude the point (1, 0). To cover the whole circle, we may introduce θ so that π < θ < 3π and θ = θ for π < θ < 2π and θ = θ + 2π for 0 < θ < π. Example 1.8. Similarly, to obtain coordinates on S 2 R 3, one may use the angles θ, ϕ making part of the spherical coordinates on R 3. More generally, this can be done for S n R n+1 if one recalls the inductive construction of spherical coordinates on R n for any n. (Again, to be able to define such angular coordinates as single-valued functions, certain points have to be excluded from the sphere. To cover the whole S n, it will be necessary to consider several angular coordinate systems, each defined in a particular domain.) In each of the above examples, a familiar space (sphere) can be endowed with a collection of local coordinate systems. ( Local means that it covers only a part of the sphere.) The minimal number of coordinate patches here equals two; one can show that it is not possible to minimize it further, i.e., to cover the sphere by a single coordinate chart. 3

4 Consider more examples. Example 1.9. Recall the notion of a projective space. The real projective space RP n is defined as the set of all straight lines through the origin in R n+1. In other words, it is the set of all one-dimensional subspaces of R n+1 considered as a vector space. So a point of RP n is a line in R n+1. Fix a hyperplane (a plane of dimension n) H R n+1 not through the origin. For example, it is possible to take the hyperplane x n+1 = 1. Each line through the origin O intersects H at a unique point, except for the lines parallel to H, which do not intersect H. The hyperplane H can be identified with R n by dropping the last coordinate x n+1 = 1. Therefore the projective space RP n can be visualized as the ordinary ( affine ) n-dimensional space R n completed by adding extra points to it. 1 Notice that these extra points correspond to the straight lines through the origin in R n R n+1 considered as the coordinate hyperplane x n+1 = 0. Hence they themselves make RP n 1, and we have RP n = R n RP n 1 (so, by induction, RP n = R n R n 1... R 1 R 0 where R 0 is a single point). This construction introduces a coordinate system on the part RP n \RP n 1 of RP n. An inclusion RP n 1 RP n is equivalent to a choice of hyperplane H in R n+1. To cover by coordinates a different part of RP n, one has to choose a different H. It is not difficult to see that by taking as H the n + 1 coordinate hyperplanes x k = 1, where k = 1,..., n + 1, we obtain n + 1 coordinate systems covering together the whole RP n. Example The complex projective space CP n is defined similarly to RP n (with real numbers replaced by complex numbers). One can introduce coordinates into CP n in the same way as above Definition of a manifold. Recall that a set V R n is open is for each point x V there is an open ε-neighborhood entirely contained in V. (In a greater detail, there is ε > 0 such that V ε (x) V, where V ε (x) = {y R n x y < ε}. In other words, V ε (x) is an open ball of radius ε with center at x.) 1 Extra points that one adds to the affine space R n in order to obtain the projective space RP n are often referred to as points at infinity. Indeed, a line in R n+1 parallel to H may be visualized as the limiting position of lines intersecting H when the point of intersection goes farther and farther away from some point in H taken as an origin. 4

5 Remark 1.2. There are many reasons why open sets in R n are important. For us the main motivation is differential calculus, where one studies how the function changes if its argument is given a small increment, i.e., a given initial value of the argument is replaced by adding a small vector (which can point in an arbitrary direction). Therefore its is necessary to be able to consider a function on a whole neighborhood of any given point. So domains of definitions of functions have to be open if we wish to apply to them differential calculus. Fix a natural number n. We shall now give a series of definitions leading to the notion of an n-dimensional smooth manifold. Let X be an abstract set. A chart (or n-dimensional chart) for X is an injective map 2 ϕ: V X where V R n is an open set in R n. Denote the image of a chart ϕ by U, so U = ϕ(v ) X and the inverse map ϕ 1 : U V R n makes sense. There is a one-to-one correspondence between points in U X and arrays (x 1,..., x n ) V R n given by the maps ϕ and ϕ 1 : X U x = ϕ(x 1,..., x n ) (x 1,..., x n ) V R n. We call the n real numbers x 1,..., x n, the coordinates of a point x U X. Formally, (x 1,..., x n ) = ϕ 1 (x). For this reason we also call a chart for X, a coordinate system on X. It is also called a local coordinate system on X to emphasize that ϕ 1 makes sense only for a subset U X. An atlas for X is a collection of charts (ϕ α : V α X) such that the images cover the whole of X: X = α U α where U α = ϕ α (V α ). All charts are supposed to have the same dimension n and if we need to stress this number we speak about an n-dimensional atlas. It is convenient to think that the R n s for different charts are different copies of the space R n and to denote them R n (α), so that V α R n (α). One should keep in mind a geographical atlas, pages of which correspond to different R n (α) (geographical maps of the Earth corresponding to mathematical charts ). Consider sets U α and U β such that U α U β. To the intersection U α U β correspond subsets ϕ 1 α (U α U β ) V α and ϕ 1 (U α U β ) V β. Any 2 Recall that a map ϕ being injective means that ϕ(a) = ϕ(b) implies a = b. β 5

6 point x U α U β has two coordinate descriptions: ϕ 1 α (x) = (x 1 α,..., x n α) and ϕ 1 β (x) = (x1 β,..., xn β ). Therefore there is an invertible map ϕ 1 α ϕ β : ϕ 1 β (U α U β ) ϕ 1 α (U α U β ), (x 1 β,..., x n β) (x 1 α,..., x n α), which we call the change of coordinates between charts ϕ a and ϕ β. Definition 1.1. An atlas A = (ϕ α : V α X) is differentiable or smooth if all the sets ϕ 1 α (U α U β ) are open and the changes of coordinates ϕ 1 α ϕ β are given by differentiable (smooth) functions. Recall that by our convention smooth or differentiable means C. Note that the first part of the condition (that the sets ϕ 1 α (U α U β ) R n α are open for all α) is necessary for the second part (that the changes of coordinates are smooth) to make sense. Coordinate changes are invertible smooth maps between open subsets of (different copies of) R n. Definition 1.2. We say that the set X is an n-dimensional differentiable (or smooth) manifold if X is endowed with a smooth n-dimensional atlas. The number n is called the dimension of X, n = dim X. Besides X, Y and Z, other traditional letters for denoting manifolds are M, N, P, and Q. The dimension of a manifold M is often indicated by a superscript, e.g., we write M = M n if dim M = n. It is important to understand that a manifold is not just a set, but rather a pair (M, A) consisting of a set M and a smooth atlas A on M. With a common abuse of language, we speak of a manifold M, where a certain atlas is implicitly understood. Remark 1.3. We shall normally omit the adjective and call differentiable manifolds simply manifolds because no other kinds of manifolds will be considered 3. It is worth at least to mention some options. If no smoothness condition is imposed, but the changes of coordinates are assumed to be just continuous maps (homeomorphisms) of open subsets of R n, such a version of manifolds is called topological manifolds. They properly belong to topology, not differential geometry. On the opposite extreme, one may require that 3 One can define differentiable manifolds of a particular class of smoothness C k, for a fixed given k (i.e., k continuous derivatives), but we shall not do that. 6

7 the changes of coordinates are given by analytic rather than C functions. (A function is analytic if it can be written as the sum of a power series.) Such a version of manifolds is called real-analytic manifolds. Many of our examples are in fact real-analytic. Also, one may consider open domains of C n instead of R n, introducing complex numbers as coordinates, and require that the changes of coordinates are given by holomorphic (i.e., complexanalytic) functions. In such a way complex manifolds are defined. Any complex manifold of complex dimension n is automatically a smooth (in fact, real-analytic) manifold of dimension 2n. Some of our examples will be complex manifolds. Example Consider S 1. Introduce two charts: ϕ N : R S 1 and ϕ S : R S 1, where ( 2uN ϕ N : u N N 1 ) u 2 N + 1, u 2 N + 1 and ϕ S : u S ( 2uS u 2 + 1, 1 ) u2 S u 2 S + 1 (see Example 1.5). They correspond to the stereographic projections from the north pole N = (0, 1) and the south pole S = (0, 1) respectively. We have V N = R, U N = S 1 \{N}, V S = R, U S = S 1 \{S}, and U N U S = S 1 \{N, S}. For the change of coordinates we obtain ϕ 1 N ϕ S : R \ {0} R \ {0}, u S u N = 1 u S (check!). Therefore it is smooth, and we conclude that S 1 with this atlas is a smooth manifold of dimension 1. Example In the same way we can obtain a smooth atlas consisting of two charts on any sphere S n R n+1. (Find the explicit formulas and check the smoothness!) This makes S n a smooth manifold of dimension n. Example A point of RP n can be identified with a non-zero vector v = R n+1 considered up to a non-zero scalar factor, v kv, k 0. The coordinates of v considered up to a factor are written as (x 1 :... : x n : x n+1 ) and traditionally called the homogeneous coordinates on RP n. (They are not 7

8 coordinates in the true sense, because are defined only up to a factor.) The construction described in Example 1.9 gives a chart ϕ: R n RP n, (y 1,..., y n ) (y 1 :... : y n : 1) with the image V = V (n+1) RP n specified by the inequality x n+1 0 in homogeneous coordinates. The inverse map ϕ 1 : V R n is ( x (x 1 :... : x n : x n+1 1 ) x,..., x n ). n+1 x n+1 Similarly we can define other charts ϕ (k), k = 1,..., n, corresponding to the choice of x k as a non-zero homogeneous coordinate (i.e., to the choice {x k = 1} as the hyperplane H in Example 1.9). Together with ϕ (n+1) they make an atlas for RP n consisting of n + 1 charts. It is smooth. (Write down the explicit formulas and check!) Hence, RP n with this atlas becomes an n-dimensional smooth manifold. Coordinates in any of these charts are traditionally called the inhomogeneous coordinates on RP n. Example Acting similarly for CP n, we obtain the n + 1 charts ϕ (k) : C n CP n again giving a smooth atlas. Hence CP n has the structure of a 2n-dimensional manifold. (Each complex coordinate gives two real coordinates.) Suppose on one set M, two smooth atlases are defined, so we have two smooth manifolds, (M, A 1 ) and (M, A 2 ).For example, for the circle S 1 (or the sphere S n ) we can consider the atlas consisting of the two stereographic charts as above or an atlas constructed using angular coordinates. Are these manifold structures the same? Definition 1.3. Two smooth atlases A 1 and A 2 on the same set M are equivalent if their union A 1 A 2 is also a smooth atlas. This is equivalent to saying that all changes of coordinates between charts from A 1 and charts from A 2 are smooth. Manifolds (M, A 1 ) and (M, A 2 ) (with the same underlying set M) are regarded the same if the atlases A 1 and A 2 are equivalent. We see that, strictly speaking, a manifold should be regarded as a pair consisting of a set and an equivalence class of a smooth atlas (rather than just an atlas). This is important for theoretical considerations. Practically, we always work with some particular fixed atlas. 8

9 Example Numerous natural ways of introducing a manifold structure on the sphere S n, such as, by using angular coordinates or by using the stereographic projection, all give equivalent atlases, as one can check. Therefore we may unambiguously speak of S n as of a smooth manifold. Example Consider two atlases on R each consisting of a single chart: for A 1 = (φ: R R), φ(x) = x (the identity map), and for A 2 = (ψ : R R), ψ(x) = x 3. Check that these atlases are NOT equivalent. (This is just for an illustration. Of course, when speaking about R as a manifold, we always use the natural structure given by the identity map.) 1.2 Smooth functions and smooth maps Consider a manifold M. In the sequel we assume for each manifold a particular smooth atlas is chosen, and when we speak of charts, we mean charts from that atlas. Definition 1.4. A function f : M R is called smooth (or differentiable, note remarks made above) if for all charts ϕ α : V α M the compositions f ϕ α : V α R are smooth functions on open sets V α R n. The set of all smooth functions on M is denoted C (M). The functions f α = f ϕ α : V α R are the coordinate representations of the function f. Such representations in any two different charts are related by the equality f α = f β ϕ βα where ϕ βα = ϕ 1 β ϕ α is the change of coordinates between the charts. Simply speaking, if we describe points of M by their coordinates (w.r.t. a particular chart), then smooth functions on M are expressed as smooth function of coordinates. The requirement that all changes of coordinates ϕ 1 α ϕ β are smooth can be seen as a compatibility condition: if a function on M is smooth in one coordinate system, it will be smooth also in any other coordinate system. (The composition of smooth functions defined on open domains of Euclidean spaces is smooth, as it follows from the chain rule.) The set of all smooth functions on a manifold M is denoted by C (M). Example Elements of C (S 1 ) can be locally represented by smooth functions f = f(ϕ) of the polar angle ϕ (0, 2π) or by smooth functions of the stereographic coordinate u = u N introduced above. 9

10 Example Similarly, functions on S 2 can be represented locally as functions of the two variables u 1, u 2 defined by a stereographic projection, or as functions of the angles θ, ϕ, (or of any other local coordinates that can be introduced on the sphere). In the same way as scalar-valued functions, we can define smooth maps between manifolds. Consider manifolds M1 n and M2 m. We shall denote charts on M1 n and M2 m by adding subscripts, such as ϕ 1α : V 1α M 1, etc. Let F : M 1 M 2 be a map. Consider the subset F 1 (U 2µ ) U 1α and assume that it is not empty. Then F maps it to U 2µ ; on the other hand, consider its preimage ϕ 1 1α (F 1 (U 2µ ) U 1α ) = (F ϕ 1α ) 1 (U 2µ ) V 1α. We have a map ϕ 1 2µ F ϕ 1α : (F ϕ 1α ) 1 (U 2µ ) V 2µ. Definition 1.5. A map F : M 1 M 2 is smooth if all sets (F ϕ 1α ) 1 (U 2µ ) are open and the above map ϕ 1 2µ F ϕ 1α is a smooth map (from an open set of R n to an open set in R m ), for all indices α, µ. The same as for functions f : M R, a map F : M 1 M 2 is smooth, simply speaking, if it is smooth when expressed in coordinates (using arbitrary coordinate systems on both manifolds). The set of all smooth maps from a manifold M 1 to a manifold M 2 is denoted by C (M 1, M 2 ). In particular, for smooth functions, which are maps to R, we have C (M) = C (M, R). Theorem 1.1. The composition of smooth maps of manifolds is smooth. The identity map, for any manifold, is smooth. Hence we have a category of smooth manifolds (see Appendix). Isomorphisms in this category are called diffeomorphisms: Definition 1.6. A map of manifolds F : M 1 M 2 is a diffeomorphism if it is smooth, invertible, and the inverse map F 1 : M 2 M 1 is also smooth. Manifolds M 1 and M 2 are called diffeomorphic if there is a diffeomorphism F : M 1 M 2. Notation: M 1 = M2. Example R + = R. Use the maps exp and log. Let us consider elementary properties of smooth functions. Theorem 1.2. Consider a manifold M n. All constants are smooth functions. The sum and product of smooth functions are smooth functions. 10

11 Proof. Consider, for example, the sum of two functions f, g C (M). We need to check that f + g also belongs to C (M). By definition that means that for any chart ϕ: V M (where V R n ), the composition (f + g) ϕ belongs to C (V ). We have ((f + g) ϕ)(x) = (f + g)(ϕ(x)) = f(ϕ(x)) + g(ϕ(x)) = (f ϕ)(x)+(g ϕ)(x) = (f ϕ+g ϕ)(x) for all x V. Therefore (f + g) ϕ = f ϕ + g ϕ C (V ), since the sum of two smooth function of n variables is a smooth function. Recall that an algebra over any field is a vector space which is also a ring, such that the multiplication is a linear operation, i.e. (ka)b = k(ab) for all algebra elements a, b and field elements k. Theorem 1.2 means that C (M) is an algebra over the field of real numbers R. The algebra C (M) is associative and commutative, and there is a unit 1. Recall that a homomorphism of algebras is a linear map (of algebras regarded as vector spaces) preserving products: ψ(ab) = ψ(a)ψ(b). An isomorphism of algebras is an algebra homomorphism which is invertible. Remark 1.4. As we saw, elements of C (M n ), for any manifold of dimension n, have the local appearance as smooth functions of n real variables. Note, however, that the algebra C (M n ) for an n-dimensional manifold and the algebra C (R n ) are not, in general, isomorphic in spite of their elements having the same local form. Moreover (we do not prove that here), the algebras of functions are different (non-isomorphic) for different (non-diffeomorphic) manifolds, and the algebra C (M) completely defines a manifold M. We shall see now that smooth maps of manifolds induce homomorphisms of the corresponding algebras of smooth functions (in the opposite direction). We first analyze the case of arbitrary sets, because certain facts are not peculiar for manifolds. Let X and Y be sets, and F : X Y be an arbitrary map of sets. Denote by Fun(X) and Fun(Y ) the sets of all functions X R and Y R, respectively. Obviously, they are algebras. Consider g Fun(Y ). The operation g g F is a map Fun(Y ) Fun(X). Definition 1.7. The operation g g F is called the pullback w.r.t. F. Notation: F, so F (g) = g F. 11

12 The pullback F is a map of algebras Fun(Y ) Fun(X) in the opposite direction to the original map of sets F : X Y. Lemma 1.1. The pullback F : Fun(Y ) Fun(X) is an algebra homomorphism. Proof. We need to check that F is linear and sends products to products. For example, consider F (fg). We have ( F (fg) ) (x) = ((fg) F )(x) = (fg)(f (x) = f(f (x))g(f (x)) = ( (f F )(g F ) ) (x) = ( F f F g ) (x) for all x X. Hence F (fg) = F f F g. In the same way we check the linearity. Lemma 1.2. For arbitrary two maps F : X Y and G: Y Z, (G F ) = F G. (1) Proof. Indeed, for an arbitrary h Fun(Z), we have (G F ) (h) = h (G F ) = (h G) F = G (h) F = F (G (h)) = (F G )(h). Note the compositions of F and G in the LHS and of F and G in the RHS of formula (1) are in the opposite orders. Returning to manifolds, we obtain the following theorem. Theorem 1.3. Suppose F : M 1 M 2 is a smooth map of manifolds. Then the pull-back of functions F is an algebra homomorphism C (M 2 ) C (M 1 ), in the reverse order. For two smooth maps, F : M 1 M 2 and G: M 2 M 3, the pull-back under the composition is the composition of pull-backs in the opposite order, (G F ) = F G. Proof. We first have to check that the pull-back of a smooth function on M 2 under a smooth map F : M 1 M 2 is a smooth function on M 1. Indeed, by the definition, for f C (M 2 ), we have F f = f F, which as smooth as the composition of smooth maps. The rest follows from the two lemmas above. Example Consider a map F : R S 1 that sends t R to F (t) = (cos t, sin t) S 1 R 2. It is smooth. Because it is onto, the pullback is injective. (Check!) It follows that C (S 1 ) can be identified with its image in C (R), which is the subalgebra consisting of all 2π-periodic functions. 12

13 Appendix. The notion of a category A category is an algebraic structure (generalizing groups and semigroups) consisting of the following data: a set whose elements are called objects and a set whose elements are called arrows or morphisms, so that for each arrow there are two uniquely defined objects called its source and target (it is said that an arrow goes from its source to its target); there is a binary operation called the product or composition of arrows, defined for any two arrows a 1 and a 2 if the target of a 2 coincides with the source of a 1 ; then the source of the composition a 1 a 2 is equal to the source of a 2 and the target of a 1 a 2, to that of a 1. Two properties are satisfied: composition is associative and for each object X there is a morphism 1 X, called the identity for X, such that a 1 X = a and 1 X a = a for all arrows a going from, and to X respectively. (Categorical constructions are best understood by drawing diagrams where arrows are represented by actual arrows joining letters denoting the corresponding sources and targets.) One obtains a group if there is only one object and all arrows are assumed to be invertible, i.e., for each a there is a 1 such that a a 1 = 1 and a 1 a = 1, where 1 stands for the identity corresponding to the single object. The arrows are the group elements in this case. If no invertibility of arrows is assumed, a category with a single object is what is known as a monoid (or a semigroup with identity ). In the same way as groups appear as transformation groups, i.e., families of certain invertible transformations of a given object (a set, a vector space, etc.), semigroups or monoids appear as families of not necessarily invertible transformations of a fixed object. Categories can appear as families of transformations between different objects. For example, for a fixed vector space V, all invertible linear operators A: V V make a group (called the general linear group of V ); all linear operators A: V V, not necessarily invertible, make a monoid, where the identity is the identity operator 1 V : V V ; finally, the collection of arbitrary linear transformations A: V W between all vector spaces make a category. The objects for this category are the vector spaces and the arrows, the linear maps. The source and the target of a map have their natural meanings. The composition is the composition of maps in the natural sense. This category is called the category of vector spaces. Similarly appear such examples of categories as the category of sets (objects are sets, morphisms are arbitrary maps of sets), the category of topological spaces (objects are topological spaces, morphisms are continuous maps), the category of groups (objects are groups, morphisms are group homomorphisms), the category of associative rings (objects are associative rings, morphisms are ring homomorphisms), etc. From the viewpoint of category theory, in all these examples objects 13

14 are treated as whole entities without internal structure (which is necessary only for specifying the set of arrows); it is morphisms that play a role. One should also note that, the same as abstract groups not arising as transformation groups, there are abstract categories where arrows are not defined as maps between sets. An example is the pair category of a given set S where the objects are defined as the elements of S and the arrows are defined as the pairs (X, Y ) S S, with the composition law (X, Y ) (Y, Z) = (X, Z). Category theory plays a very important role in modern mathematics, primarily as a unifying language for many algebraic and geometric constructions. A simplest abstract notion from category theory is that of an isomorphism. A morphism a in a given category is called an isomorphism if there is a morphism b going in the opposite direction such that a b and b a are the identities for the respective objects. It follows that in this case b is defined uniquely (check from the axioms of a category!). It is called the inverse of a and denoted a 1. Two objects X and Y are called isomorphic if there is at least one isomorphism going from X to Y. One can see that this defines an equivalence relation on the set of objects (check!), also called isomorphism. Examples: in the category of groups, the isomorphisms are the isomorphisms of groups; in the category of sets, the isomorphisms are the bijections; in the category of topological spaces, the isomorphisms are the homeomorphisms. In the category of smooth manifolds, the isomorphisms are the diffeomorphisms. 14