14 1. Categories and Functors Equivalence of Categories. The obvious notion of equivalence of cate- gories, where we ask for functors F : C → D and G : D → C whose compos- ites are the identity functors, has proven to be more rigid than is necessary, and too rigid for many applications. Instead, a functor F : C → D is an equivalence of categories if there is a functor G : D → C and two natural isomorphisms idC −→ G ◦ F and F ◦ G −→ idD. Exercise 1.30. Show that an equivalence of categories need be neither injective nor surjective on objects. 1.5. Duality In studying categories, you should keep your eyes open for instances of du- ality. The dual of a category-theoretical expression is the result of reversing all the arrows, changing each reference to a domain to refer to the target (and vice versa), and reversing the order of composition. Exercise 1.31. (a) The notation f : X → Y is shorthand for the sentence: ‘f is a morphism with domain X and target Y .’ What is the dual of this statement? (b) Find instances of duality in the previous sections. For example, consider lifting problem: you are given maps f : A → Y and p : X → Y , and you would like to find a map λ : A → X such that p ◦ λ = f. This problem is neatly expressed in the diagram X p A f λ Y (when expressing problems in this way, the map you hope to find is usually dotted or dashed). The dual problem is expressed by the diagram V B U. g q This is known as an extension problem, because you hope to extend the map g to the ‘larger’ thing V .6 6 Norman Steenrod, one of the architects of modern algebraic topology, used the extension and lifting problems to frame the entire subject. It can be argued that a great deal of mathematics is about lifting and extension problems. Exercise. Show how the problem: ‘decide whether f : X → Y is a homeomorphism’ can be written in terms of extension and/or lifting problems.

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