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Functors and natural transformations A covariant functor F : C → D is a set map F : Ob(C) → Ob(D) together with a set map F : M or(A, B) → M or(F (A), F (B)) for each morphism f : A → B in C such that 1. F (idA) = idF (A). 2. F respects composition of morphisms, i.e. F (f ◦ g) = F (f ) ◦ F (g) if g : T → A and f : A → B are morphisms in C. Thus covariant functors preserve the direction of arrows: f : A → B goes to F (f ) : F (A) → F (B). 1 A contravariant functor H : C → D reverses the direction of arrows. Thus one has a set map H : Ob(C) → Ob(D) together with a set map F : M or(A, B) → M or(F (B), F (A)) for each morphism f : A → B in C such that 1. F (idA) = idF (A). 2. F respects composition of morphisms, i.e. F (f ◦ g) = F (g) ◦ F (f ) if g : T → A and f : A → B are morphisms in C, where F (g) : F (A) → F (T ) and F (f ) : F (B) → F (A). Example 1: Forgetful functors come from forgetting extra structure required for an object of D to give an object of C. For example, if C is the category of groups and D is the category of sets, we have a forgetful functor F : C → D defined by letting F (A) be the underlying set of a group A. Given a group homomorphism f : A → B we just let F (f ) be the underlying set map. Similarly, one has a forgetful functor from the category of vector spaces over a given field L and the category of abelian groups. 2 Example 2: The category T of pointed topological spaces consists of pairs (X, x) in which X is a topological space and x ∈ X. A morphism f : (X, x) → (Y, y) is a continuous map f : X → Y such that f (x) = y. The coprod` uct (X, x) (X 0, x0) is the so-called connected sum, which is the quotient of the disjoint union ` 0 X X by the equivalence relation generated by setting x∼x0. Is there a product in the category T ? 3 There is a covariant functor F : T → Groups defined by F (X, x) → π1(X, x) If f : (X, x) → (Y, y) is a morphism in T , then F (f ) : π1(X, x) → π1(Y, Y ) results from taking images under f of loops based at x. The Van-Kampen theorem says F respects coproducts: F ((X, x) a 0 0 (X , x )) = π1(X, x) T a π1(X 0, x0). Groups 4 Example 3: Representation functors Suppose C is a category and X is an object of C. One gets covariant and contravariant functors FA : C → Sets and F A : C → Sets defined by FA(B) = M or(A, B) and F A(B) = M or(B, A) If f : B → B 0 is a morphism in C, FA(f ) : M or(A, B) → M or(A, B 0 ) is just the composition h → f ◦ h. Similarly, F A(f ) : M or(B 0, A) → M or(B, A) is the composition h→h◦f If the morphisms M or(A, B) have additional structure, we can do better than using Sets as the range. For example, if C is the category of abelian groups, we can replace Sets by C. Problem: What is FZ : Groups → Groups? 5 Natural transformations, isomorphisms and equivalences Suppose F and J are covariant functors from C to D. A natural transformation (or morphism of functors ) from F to J is a map which assigns to each object A of C a morphism ηA ∈ M orD (F (A), J(A)) such that the following is true. For each pair of objects A, B of C and each f ∈ M orC (A, B), there is a commutative diagram in D F (A) ηA −→ F (f )↓ F (B) J(A) ↓J(f ) ηB −→ . (1) J(B) 6 If each ηA is an isomorphism, say η is an isomoprhism of functors. One says that C and D are isomorphic categories if there are functors F : C → D and H : C → D such that the compositions H ◦ F and F ◦ H are the identity functors. One says that C and D are equivalent categories if there are functors F : C → D and H : C → D such that the compositions H ◦ F and F ◦ H are naturally isomorphic to the identity functors. 7 Intuition for Isomorphisms of categories: Isomorphic categories C and D are the same up to relabelling specified by F : C → D and H : D → C. In order for H ◦ F = IdC and F ◦ H = IdD , the functors F and H have to induce a bijection on objects. Further, for objects A and B of C, the map F : M or(A, B) → M or(F (A), F (B)) is a bijection. 8 Intuition for Equivalences of Categories: Suppose F : C → D and H : D → A define an equivalence of categories. Then F (A) is a “model” of F (A) inside D, in the following sense. The composition H ◦F :C →C is naturally isomorphic to IdC via some η : H ◦ F → IdC . Thus η : H(F (A)) → IdC (A) = A is an isomorphism in C (but not necessarily the identity map). Example: One of the homework problems is to show that if G is a group, then the category of C of all finite G-sets is equivalent by not isomorphic to the category D of all finite Gsets contained in Z. 9 Applications of representation functors Various problems in group theory and geometry can be phrased by asking whether a covariant functor T : C → Sets is isomorphic to a the representation functor FA : C → Sets defined by FA(∗) = M orC (A, ∗) for some object A of C. Such an isomorphism comes about from compatible isomorphisms (= set bijections) ηB : T (B) → FA(B) = M orC (A, B) for all objects B of C. In this way, one can reconstruct T from the arrows in C from A to the other objects of C. 10 Example: Suppose T : Groups → Sets sends a group B to the set of elements of B of order 1 or 2. This is a functor, because a group homomorphism B → B 0 sends such elements to elements of order 1 or 2. Let’s check that T is represented by A = Z/2. We need a bijection ηB : T (B) → FZ/2(B) = HomGroups (Z/2, B) for all B. One can let η(b) be the homomorphism which sends the non-trivial element of Z/2 to b. 11 Example: Classifying spaces. A topological group G is a group with a topology such that the group law and the map giving inverses are continuous maps. For example, one can always take the discrete topology on G, in which every set is open. The real topology on GL(n, R) makes this a topological group. 12 A principal G-bundle is a continuous surjection of topological spaces f : E → B with these properties: 1. G acts continuously on E and trivially on B 2. f respects this action. So f (gα) = gf (α) = f (α) for G ∈ G and α ∈ E. 3. Each b ∈ B has a neighborhood Ub over which f is trivial, in the following sense. f −1(Ub) is isomorphic to G × Ub as a topological G-space. 13 For example suppose E is the unit two sphere S 2 in R3, and B = RP 2 is the real projective plane. So points of B are equivalence classes {x, −x} = [x] of x ∈ S 2 under the antipodal map. Define f : S 2 → RP 2 by x → [x] = {x, −x} Then f is a principal G = Z/2 bundle which is not trivial (i.e. not G-isomorphic to RP 2 × Z/2.) 14 We now fix G. We have a contravariant functor FG : Topological spaces = T → Sets defined by FG(B) = isomophism classes of topological G − bundles overB For B 0 → B, one constructs F (B) → F (B 0 ) by pulling back G-bundles on B to G-bundles on B 0. Exercise: Given a G-bundle E → B and a continuous map B 0 → B, construct the pullback G-bundle E 0 → B 0 having the appropriate universal property. 15 For W a topological space, let [B, W ] be the set of homotopy classes of maps f : B → W . Theorem For each topological group G, there is a space BG called the classifying space of G and an isomorphism of functors between FG and the functor B → [B, BG]. The homotopy class of the identity map BG → BG then gives a “universal” G-bundle EG → BG. For all topological G-bundles E → B, there is a continuous funtion t : B → BG unique up to homotopy for which E is G isomorphic to the pullback of EG via t. For example, if G = Z/2, then BG = RP ∞ is the union of over k the quotients RP k of the ksphere S k by the antipode map x → −x, where one puts these spheres in bigger and bigger real Euclidean spaces. 16