1.4. Natural Transformations 11 (b) For f : X → Y , write f∗ : morC(A, X) → morC(A, X) for the function f∗ : g → f ◦ g. Then the rules G(X) = morC(A, X) and G(f) = f∗ : G(X) → G(Y ) define a covariant functor from C to Sets. A functor that is constructed in either of these ways is called a repre- sented functor—that is, the functor F is represented by the object B, and the functor G is represented by the object A. Problem 1.22. Prove Proposition 1.21. Hint. Simply generalize your work from Exercise 1.20. Problem 1.23. Let f : A → B and g : X → Y in a category C. Show that f ∗ ◦ g∗ = g∗ ◦ f ∗ : morC(B, X) −→ morC(A, Y ). Problem 1.24. Let f : A → B be a morphism in the category C. (a) Suppose the induced map f∗ : morC(X, A) → morC(X, B) is a bijection for every X. Show that f is an equivalence. (b) Suppose the induced map f ∗ : morC(B, X) → morC(A, X) is a bijection for every X. Show that f is an equivalence. Hint. Try plugging in X = A and X = B. 1.4. Natural Transformations Category theory was invented by Saunders Mac Lane and Samuel Eilenberg in the early 1940s, largely motivated by the desire to be precise about what is meant by (or should be meant by) a ‘natural construction’. For many years before then, mathematicians had used the intuitive notion of a natural construction to mean that the construction is done in exactly the same way for all spaces, groups, or whatever. For example, for any vector space V , you can construct the dual vector space V ∗ = Hom(V, R) since this is done in the same way for every vector space, it is ‘naturally defined’ and so it will ‘of course’ (ha!) convert commutative diagrams to other commutative diagrams. This idea is formalized in the idea of a functor. Now how do we relate two different ‘natural’ constructions to one another? Let F : C → D and G : C → D be two covariant5 functors. A natural transformation Φ : F → G is a rule that associates to each X ∈ ob(C) a morphism ΦX : F (X) −→ G(X) 5 They could also both be contravariant. I’ll leave the formulation to you.

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