1.10. Abelian Groups and Cogroups 25 Theorem 1.63. If f : G → H is a homomorphism of group objects in the pointed category C, then the induced map f∗ : morC(X, G) −→ morC(X, H) is a homomorphism of groups for every X ∈ C. Problem 1.64. Prove Theorem 1.63. Is the converse true? As usual, it is up to you to formulate the duals. Problem 1.65. Define homomorphisms of cogroup objects, and prove that they induce group homomorphisms on mapping sets. 1.10. Abelian Groups and Cogroups Since abelian groups are especially easy to work with, we establish the notion of commutative groups and cogroups. Abelian Objects. In any category, we can define a twist map for co- products T : X Y → Y X, which ‘switches the terms’. Problem 1.66. Write down a categorical description of T . Also define the twist map T : X × Y → Y × X for products. Show that both twist maps are equivalences. A cogroup object C in a pointed category C is cocommutative, or simply commutative, if the diagram C φ ①①①①①①①①① ❋❋❋ ❋❋❋φ ❋❋❋ C ∨ C T C ∨ C is commutative. Problem 1.67. Show that C is a cocommutative cogroup if and only if morC(C, Y ) is an abelian group for every Y . Problem 1.68. Dualize this discussion: define a commutative group object, and verify that morC(X, G) is abelian if and only if G is such an object. Products of Groups. As you know, the set-theoretical product of two groups can be made into a group using coordinatewise multiplication. The same can be done with group objects. Let G and H be group objects in the pointed category C, and denote their multiplications by μG and μH. Then

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