24 1. Categories and Functors Hint. Write down the compositions which define f♠g and f♥g in a single commutative diagram. Use Problem 1.46. In view of Problem 1.60, we will never again use suits to denote products and will be content to write f · g, or simply fg for these products. 1.9. Homomorphisms As you know from your study of algebra, when you are studying groups, you inevitably find yourself studying homomorphisms. Our goal in this section is to establish definitions for homomorphisms of group and cogroup objects and to prove some simple but important facts about them. Let G and H be group objects in a pointed category C. A map f : G → H is a homomorphism if the diagram G × G μG f×f G f H × H μH H commutes.10 Exercise 1.61. Show that when C is the category of pointed sets, a homo- morphism of group objects is just the same as a homomorphism of groups. Exercise 1.62. Show that if f : G → H is a homomorphism of group objects in C, then f preserves inverses, in the sense that the diagram G f νG H νH G f H is commutative. If G and H are group objects and f : G → H is some map, then we automatically get an induced map f∗ : morC(X, G) −→ morC(X, H) From what we know already, the sets morC(X, G) and morC(X, H) are groups but what can we say about the map f∗? In general, there is nothing we can say, but if f is a homomorphism of group objects, then f∗ is a group homomorphism. 10 This is actually a perfectly good definition for a monoid homomorphism.

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