Mathematisches Institut WS 2001/02 der Universität München Set 7 Prof. Dr. B. Pareigis

Problem set for Advanced Algebra

(25) Let (A×B, p_{A}, p_{B}) be the product of A and B in C. Then there is a natural isomorphism

Mor(-, A × B) ≅ Mor_{C}(-, A) × Mor_{C}(-, B).

(26) Let C be a category with finite products. Show that there is a bifunctor - × - : C × C ⟶ C such that (- × -)(A, B) is the object of a product of A and B. We denote elements in the image of this functor by A × B := (- × -)(A, B) and similarly f × g.

(27) Let F : C ⟶ D be an equivalence with respect to G : D ⟶ C, φ : GF ≅ Id_{C}, and ψ : FG ≅ Id_{D}. Show that G : D ⟶ C is an equivalence. Show that G is uniquely determined by F up to a natural isomorphism.

(28) (a) Given V ∈ 𝕂- Mod. For A ∈ 𝕂- Alg define

F (A) := {f : V ⟶ A|f 𝕂-linear, ∀v, w ∈ V : f(v) · f(w) = 0}.

Show that this defines a functor F : 𝕂- Alg ⟶ Set.

(b) Show that F has the algebra D(V) as constructed in Exercise 2.1 (3) as a representing object.

Due date: Tuesday, 4.12.2001, 16:15 in Lecture Hall 138