Exercises for the Course in Algebraic Geometry , Fall 2014

(In first case we say (F,G2) are an adjoint pair and in the second case we say that (G1, F ) are an adjoint pair.) (2) An object A in a category C is called initial (resp. final) if for every object B in C, HomC(A,B) (resp. HomC(B,A)) is a singleton. Prove that an initial (resp. final) object, if it exists, is unique up to unique isomorphism. Determine if such objects exist in the following categories: Set, Top, VSk, Ring (the categories of sets, topological spaces, vector spaces over a fixed field k and rings). (3) Let C be a category with objects X1, X2, Z and morphisms f : X1 → Z, g : X2 → Z. Prove that if X1 ×Z X2 exists it is unique up unique isomorphism (the product is taken relative to f , g). Show that X1 ×Z X2 always exists in Set, Top, VSk, Ring. (4) Define X1 ×X2 using the diagram