18 1. Categories and Functors Exercise 1.42. (a) Give number-theoretical interpretations of products and sums in the category of positive integers 1, 2, 3,... with morphisms corresponding to divisibility. (b) Repeat (a) with the category of real numbers with morphisms corre- sponding to inequalities x ≤ y. (c) Is there a category structure on the set N so that the categorical product is the same as the numerical product? Larger Sums and Products. The sum of a set {Xi | i ∈ I} of objects in the category C is an object I Xi which comes equipped with morphisms ini : Xi → I Xi satisfying the universal property that every collection {fi : Xi → Y | i ∈ I} gives rise to a unique morphism f : I Xi → Y such that f ◦ ini = fi for each i ∈ I. Dually, the product of the collection {Xi | i ∈ I} is an object I Xi having maps pr i : I Xi → Xi such that for every collection of maps {gi : W → Xi | i ∈ I}, there is a unique map g : W → I Xi such that pr i ◦ g = gi for each i ∈ I. If J is some set, then we can define the J-fold sum and product of X with itself, and (if they exist) there will be a diagonal map ΔJ : X → j∈J X and a fold map ∇J : j∈J X → X. 1.7. Initial and Terminal Objects An object τ ∈ C is called a terminal object if the set morC(X, τ) has exactly one element, no matter what X ∈ C we plug in.7 Dually, an object ι ∈ C is called an initial object if the set morC(ι, Y ) has exactly one element, no matter what Y ∈ C we plug in. Exercise 1.43. Find initial and terminal objects in the following contexts. (a) The category of sets and functions. (b) The category of topological spaces and continuous functions. (c) The category of groups and homomorphisms. A pointed category is a category C in which there is an object, gener- ally8 denoted ∗, which is simultaneously initial and terminal. If X, Y ∈ C, where C is a pointed category, then there is a unique morphism of the form 7 So a terminal object is a target-type concept. 8 Though in algebraic contexts, it is often 0 or {1} or something even more substantial.

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