HOMEWORK 3
ALGEBRA-I, MATH 5613, FALL 2009
Let Cbe a category. We say Ais a full subcategory of Cif it is a subcategory such
that for all objects Aand Bin Aone 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. ful l) if for all objects
A, B of C, the map f7→ F f from HomC(A, B ) to HomD(FA, F B) is injective (resp.
surjective). The same definition applies to a contravariant functor, except the map
is from HomC(A, B) to HomD(F B , F A). A functor is said to be fully faithful if it is
both full and faithful.
Exercise 2. Give two examples of covariant functors. For each example, decide if it
is full and/or faithful.
Exercise 3. Give two examples of contravariant functors. For each example, decide
if it is full and/or faithful.
Exercise 4. Let Vstand for the category whose objects are finite dimensional vector
spaces over a field F, and whose morphisms are F-linear maps. Show that double
dual functor from Vto itself, which assigns to every V∈Ob(V) its double dual V∗∗,
is naturally isomorphic to the identity functor on V. (Recall that a linear functional
on Vis a linear map `:V→F, and the dual V∗of Vconsists of all linear functionals
on V, i.e., V∗= HomF−linear(V , F ).)
Exercise 5. Let GRP be the category of all groups and group homomorphisms, and
let AB be the subcategory of all abelian groups. Let Gand Hbe abelian groups.
(1) Prove that the product of Gand Hin AB is the same as the product in
GRP.
(2) Explain why the coproduct of Gand Hin AB need not be same as the
coproduct in GRP. Give an example.
Exercise 6. In this exercise, for each i∈Nwe have a copy of integers Z.
(1) Show that the direct sum Li∈NZis countable.
(2) Show that the direct product Qi∈NZis uncountable.
1
