EXERCISES IN CATEGORY THEORY 2
1. Products
Let % be a category with products.
(1) Suppose that X admits a terminal object 1. Show that there are isomorphisms Ax 1 = A and lxi~A
(2) Find an isomorphism 4xB~i]~A
(3) Define the product A x B x C of three objects A, B and C using a universal property and show that it is unique up to isomorphism.
(4) Given objects A, B and C find an isomorphism (A x B) x C = A x (B x C). Show that these are isomorphic (one way is to show that both have the universal product oi Ax B xC.)
(5) Given / : A\ —>• A2 and g : B\ —>• B2 find a map / x g : A\ x B\ —>• A2 x B2- In the category of sets this is the map sending the ordered pair (a,b) to (fa,gb).
(6) Show that in any category with products and coproducts there exists a canonical map
(A x B) + (A x C)     A x (B + C)
(To construct this, use maps of the form / x g as constructed in the previous question.)
(7) Show that, in the category of sets, the above map is an isomorphism.
Date: September 24, 2014.
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