1.8. Group and Cogroup Objects 23 which can be arranged as: G forget C F Sets∗. You have proved that the dotted arrow can be filled in, and thereby proved the following theorem. Theorem 1.55. If G is a group object in a pointed category C, then the con- travariant functor F (X) = morC(X, G) factors through the forgetful func- tor from the category G of groups and homomorphisms to the category of pointed sets Sets∗. We usually use the same symbol, F , for the dotted functor C → G. It is sometimes said in this situation that F ‘takes its values in the category G’. This phrasing, though not entirely accurate, makes sense because G can be considered to be a subcategory of Sets∗ (via the forgetful functor). Exercise 1.56. Let G be a group object in a category C. Work out the product pr1 · pr2 ∈ morC(G × G, G). You should be able to express it as a specific map you already know. Now let’s dualize. Problem 1.57. Write down the definition of a cogroup object. What is a cogroup object in the category of groups and homomorphisms? What about abelian groups and homomorphisms? What is a cogroup object in the category of pointed sets? What if you replace ‘cogroup’ with ‘comonoid’? By dualizing our discussion of group objects, we can immediately derive the following result. Theorem 1.58. If C is a cogroup object in a pointed category C, then the covariant functor G(Y ) = morC(C, Y ) takes its values in the category of groups and homomorphisms. Problem 1.59. Prove Theorem 1.58. Suppose C is a cogroup object and G is a group object. Then the set mor(X, Y ) has two ways to multiply. More precisely, morC(C, G) is a group because C is a cogroup object—we’ll write α♠β for this product and morC(C, G) is a group because G is a group object—we’ll write α♥β for this product. Problem 1.60. Show that, with the setup above, the products ♠ and ♥ are the same. That is, show that for any f, g ∈ morC(C, G), f♠g = f♥g.

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