1.6. Products and Sums 17 It is tempting to prove these by chasing elements around, which is fine if your objects are sets. But you should prove that these diagrams commute only using the category-theoretical definitions of the maps involved. Exercise 1.38. Explain how to view the diagonal map as a natural trans- formation. Let’s look at some specific examples of products. Exercise 1.39. (a) Show that the product of two sets X and Y is simply the ordinary cartesian product X × Y = {(x, y) | x ∈ X, y ∈ Y }. (b) What is the product of two abelian groups G and H? Let’s look at the dual concept. Problem 1.40. (a) Formulate the (dual) definition of a coproduct, which is denoted X Y . Coproducts are also known as (categorical) sums. (b) Prove that if X, Y ∈ ob(C), then morC(X Y, B) ∼ morC(X, B) × morC(Y, B). Write down the isomorphism explicitly. Thus we can (and will) describe maps F : X Y → B with the notation (f, g), where f : X → B and g : Y → B. Hint. This is formally dual to Exercise 1.34, so you should be able to prove this by simply inverting all the arrows in your previous proof. (c) Write down the definition of f g : A B → X Y . (d) Explain how to view as a functor. (e) The dual of the diagonal map is called the folding map, and we will denote it by the symbol ∇. Define it explicitly in category-theoretical language, and explain how to view it as a natural transformation. The coproduct, being dual to the product, is a domain-type construction. Exercise 1.41. (a) Show that the sum of two sets X and Y is simply the disjoint union of X and Y . Conclude that X Y and X ×Y are not generally equivalent. (b) What is the folding map in the case X = {a, b, c}? (c) What is the sum of abelian groups G and H? Construct a nice map w : G H → G × H what can you say about it?

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