1.7. Initial and Terminal Objects 19 X → ∗ → Y it is called the trivial morphism, and it will be uniformly denoted ∗. In a pointed category, the sum of X and Y is sometimes denoted X ∨ Y and referred to as the wedge sum of X and Y . An important example of a pointed category is the category Sets∗ of pointed sets. An object of Sets∗ is a set X with a particular point x0 chosen and identified it is referred to as the basepoint of X. A morphism from (X, x0) to (Y, y0) is a function f : X → Y with the additional prop- erty that f(x0) = y0. In practice, we do not give individual names to the basepoints but just call them all ∗. Exercise 1.44. Verify that Sets∗ is a pointed category. What is a sum in Sets∗? What is a product? Problem 1.45. (a) Suppose C is a category with a terminal object τ, and let X, Y ∈ C, and suppose that a product P for X and Y exists in C. Show that P solves the problem expressed in the diagram Z f GFED ∃! t g P pr X pr Y X Y τ. (There is never any need to label a map into a terminal object!) (b) Formulate and prove the dual to part (a). Problem 1.46. Let C be pointed category in which products and coprod- ucts exist for all pairs of objects. (a) Give categorical definitions for the ‘axis’ maps inX : X → X × Y and inY : Y → X × Y . (b) In a pointed category, there is a particularly nice morphism w : X1 ∨ X2 → X1 × X2. Define it in terms of category theory, and check that the diagram X1 ∨ X2 f∨g w Y1 ∨ Y2 w X1 × X2 f×g Y2 × Y2 is commutative for any f : X1 → Y1 and g : X2 → Y2. Why is it necessary for the category to be pointed before you can define w?

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