1.3. Functors 7 (b) Interpret ‘equivalence’ in each of the categories that have been discussed in the text so far, including the ones you found in Exercise 1.3. Problem 1.6. Let C be a category and let f : X Y be a map which has a left inverse g : Y X, and suppose g also has a left inverse. Show that f and g are two-sided inverses of each other. Another simple—but extremely useful—idea is that of a retract. If A, X C, then A is a retract of X if there is a commutative diagram A i id A X r A. If f : A B and g : X Y , then f is a retract of g if there is a commutative diagram A i f idA X g r A f B j idB Y s B. Exercise 1.7. Whenever we use the term ‘retract’, we should be referring to the definition above, where an object A was a retract of another object X in some category. By setting up an appropriate category, show that our definition of f being a retract of g can be thought of as an instance of that general categorical definition. What is does it mean, in terms of the category C, for two objects to be equivalent in your new category? Hint. Obviously, f and g must be among the objects in your category! Exercise 1.8. Find examples of retracts in algebra, topology, and other contexts. Problem 1.9. Let f : A B and g : X Y be morphisms in a category C. Assume that f is a retract of g. (a) Show that if g is an equivalence, then f is also an equivalence. (b) Show by example that f can be an equivalence even if g is not an equiv- alence. 1.3. Functors As you have no doubt experienced, it seldom happens that any serious math- ematical study is performed entirely inside a single category. For example, when Galois set out to study fields, he was forced to also work in the cat- egory of groups it was the relationship between these two categories that
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