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20 1. Categories and Functors (c) State (and prove?) the dual statements. Exercise 1.47. In what sense do the maps w constitute a natural transfor- mation? Problem 1.48. Show that for any f : X → Y in a pointed category C and ∗ : W → X, then f ◦ ∗ = ∗ : W → Y . Also show that if ∗ : Y → Z, then ∗ ◦ f = ∗ : X → Z. Conclude that the functors morC( ? , Y ) and morC(A, ? ) take their values in the category of pointed sets and pointed maps. Exercise 1.49. (a) Show that the trivial group {1} is simultaneously initial and terminal in the category G of groups and homomorphisms. Show that the vector space 0 is simultaneously initial and terminal in the category of vector spaces (over R, if you like) and linear transformations. (b) Show that in the category ab G of abelian groups and homomorphisms, the map w : G ∨ G → G × H is an isomorphism for any G and H. Also show that the analogous statement is true in the category of vector spaces and linear transformations. Matrix Representation of Morphisms. Since sums are domain-type constructions and products are target-type constructions, the maps from a sum to a target should be fairly easy to understand. Problem 1.50. (a) Show that, in any category C, there is a natural bijection between the morphism set morC(X1 X2,Y1 × Y2) and the set M of all matrices f11 f12 f21 f22 with fij ∈ morC(Xj,Yi). (b) Now suppose that C is a pointed category in which the canonical map w : X ∨Y → X ×Y is an isomorphism for each pair of objects X, Y ∈ C. Show that composition morC(Y1 Y2,Z1 × Z2) × morC(X1 X2,Y1 × Y2) ∼ morC(X1 X2,Z1 × Z2) morC(Y1 × Y2,Z1 × Z2) × morC(X1 X2,Y1 × Y2) corresponds to matrix multiplication in M.9 (c) Show that linear transformations R2 → R2 are in one-to-one correspon- dence with 2 × 2 matrices with real entries. 9 Exercise. What exactly do I mean by ‘matrix multiplication’?