22 1. Categories and Functors Exercise 1.51. Are group objects domain-type or target-type gadgets? Suppose G is a group object in C. What conditions must you impose on a functor F : C D in order to conclude that F (G) D is also a group object? Let’s check that these things are correctly named. Exercise 1.52. (a) Check that, in the category of pointed sets, a group object is just an ordinary group. (b) Show that a group G G is a group object if and only if G is abelian. (c) Write GLn(R) to denote the set of all n × n invertible matrices. It is a subset of Rn2, so we can give it the subspace topology. Show that matrix multiplication makes GLn(R) into a group object in the category of pointed topological spaces. Exercise 1.53. You know that in the category of pointed sets and their maps, the inverse map ν for a group object G is uniquely determined by its multiplication μ. Prove that this is true for group objects in any category. The reason group objects are so important is that they provide morphism sets with group structures. Problem 1.54. Let G be a group object in a pointed category C. (a) Show that the composite map M in the diagram morC(X, G) × morC(X, G) M morC(X, G) morC(X, G × G) μ∗ makes morC(X, G) into a group object in the category of pointed sets (i.e., morC(X, G) is a group with multiplication M). (b) Draw a diagram that shows all the maps involved in the definition of the product of α, β morC(X, G) and how they fit together. (In other words, write down explicitly what α · β is.) (c) Let f : X Y be a morphism in C. Show that f : morC(Y, G) morC(X, G) is a group homomorphism. Hint. Use Problem 1.37 and part (b). Let’s think about what you have just proved. We have two functors F : C −→ Sets∗ and forget : G −→ Sets∗
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