6 1. Categories and Functors for any g ∈ morC(X, Y ). In diagram form: W f f X id X g X g Y. Here are some simple examples to think about. (a) The category G whose objects are groups and whose morphisms are group homomorphisms also the subcategory ab G whose objects are the abelian groups. (b) The category Top whose objects are topological spaces and whose mor- phisms are continuous functions. (c) The category whose objects are the numbers 1, 2, 3,... and such that there is a unique morphism n → m if n divides m and no morphisms n → m if n does not divide m. (d) The category whose objects are the real numbers and such that there is a unique morphism x → y if x ≤ y and no morphism if x y. There is a lot of shorthand that is often used when confusion is unlikely. For example, we usually write X ∈ C instead of X ∈ ob(C) and rather than f ∈ morC(X, Y ), we write f : X → Y . Exercise 1.3. (a) Give five examples of categories besides the ones already mentioned. (b) Find a way to interpret a group G as a category with a single object. (c) Let X be a topological space. Show how to make a category whose objects are the points of X and such that the set of morphisms from a to b is the set of all paths ω : [0,d] → X (where d ≥ 0) such that ω(0) = a and ω(d) = b. (d) Suppose D is a diagram in the sense of Section 1.1. Show that D is a category. Lots of basic mathematical ideas are ‘best’ expressed in the language of categories. For example: a morphism f : X → Y is an equivalence if there is a morphism g : Y → X such that g ◦ f = idX and f ◦ g = idY . It is the usual practice to write g = f −1 in this case. Exercise 1.4. Show that if such a g exists, it is unique. Problem 1.5. Let’s say X ∼ Y if there is an equivalence f : X → Y . (a) Show that ∼ is an equivalence relation.

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