Download Resolution for Homework 1 - Topics in Geometry | MATH 6490 and more Assignments Geometry in PDF only on Docsity! TOPICS IN GEOMETRY: SHEAF THEORY MATH 6490, SPRING 2009 HOMEWORK 1 Let C be a category. A subcategory A of C is a category such that (1) Every object of A is an object of C, i.e., Ob(A) ⊂ Ob(C). (2) Every morphism in A is a morphism of C, i.e, HomA(A,B) ⊂ HomC(A,B). (3) The rule for composition of morphisms in A is the rule for composition in C. We say A is a full subcategory of C if it is a subcategory such that for all objects A and B in A one has HomA(A,B) = HomC(A,B). Exercise 1. Give an example of a category and a full subcategory. Give an example of a category and a subcategory which is not full. A covariant functor F : C → D is said to be faithful (resp. full) if for all objects A,B of C, the map f 7→ Ff from HomC(A,B) to HomD(FA,FB) is injective (resp. surjective). The same definition applies to a contravariant functor, except the map is from HomC(A,B) to HomD(FB,FA). A functor is said to be fully faithful if it is both full and faithful. Exercise 2. (1) Give two examples of covariant functors. For each example, decide if it is full and/or faithful. (2) Give two examples of contravariant functors. For each example, decide if it is full and/or faithful. Let C and D be two categories. Let F and G be two covariant functors from C to D. We say η : F → G is a natural transformation if for each A ∈ Ob(C) we are given ηA ∈ HomD(FA,GA) such that for all objects A,B of C and all f ∈ HomC(A,B) the following diagram commutes: FA Ff // ηA FB ηB GA Gf // GB If ηA is an isomorphism for all A then we say F and G are naturally isomorphic. Exercise 3. Let V stand for the category whose objects are finite-dimensional vector spaces over a field F , and whose morphisms are F -linear maps. Show that the identity functor on V is naturally isomorphic to the double-dual functor from V to itself. (The double-dual functor assigns to every V ∈ Ob(V) its double-dual V ∗∗; recall that a linear functional on V is a linear map ` : V → F , and the dual V ∗ of V consists of all linear functionals on V and is written as V ∗ = HomF (V, F ).)
