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10 1. Categories and Functors Exercise 1.17. Show that there is a universal example for contravariant functors out of a category C. That is, show that there is a category Cop and a contravariant functor C → Cop so that every other contravariant functor C → D has a unique factorization C → Cop → D, where the functor Cop → D is covariant. The category Cop is known as the opposite category of C. Some au- thors choose not to use contravariant functors at all and instead use covariant functors Cop → D. Let’s look at some simple functors. Exercise 1.18. Consider the categories ab G of abelian groups (and homo- morphisms) and Sets of sets (and functions). Since an abelian group is a set together with extra structure, we can define F : ab G → Sets by G −→ G, but completely forgetting the group structure . Complete the definition of F on morphisms, and show that F is a functor. Any functor of this kind, in which the target category is a dumbed-down version of the domain, and the functor consists of just getting dumber, is called a forgetful functor. Exercise 1.19. Try to make a formal definition of ‘forgetful functor’. Exercise 1.20. Let V denote the category of all vector spaces (over the real numbers, say) and all linear transformations. Thus morV(V, W ) = HomR(V, W ) = {T : V → W | T is R-linear}. (a) Define F : V → V by the rules F (V ) = HomR(R,V ) and F (f) : g → f ◦ g. Show that F is a covariant functor. (b) Define G : V → V by the rules G(V ) = HomR(V, R) and G(f) : g → g ◦ f. Show that G is a contravariant functor. The functors described in Exercise 1.20 are specific examples of what are, for us, the two most important general kinds of functors. Proposition 1.21. Let C be a category, and let A, B ∈ ob(C). (a) For f : X → Y , write f ∗ : morC(Y, B) → morC(X, B) for the function f ∗ : g → g ◦ f. Then the rules F (X) = morC(X, B) and F (f) = f ∗ : F (Y ) → F (X) define a contravariant functor from C to Sets.