Study Guide for Midterm Exam
The midterm exam problems will be very similar to some of the problems listed below. The exam will consist of approximately five such problems.
Module Theory
Problem 1
Let
Problem 2
Let
Problem 3
Suppose
-
Show that for every
-bimodule there is an -bimodule morphism
defined on simple tensors by . -
Prove that if
is surjective, then so is . -
Show there are isomorphisms of abelian groups
and , and then explain why this proves tensor product does not in general preserve injections.
Problem 4
Let
Problem 5
Let
First prove the following lemma: If
Then suppose
Note: If necessary, you may use the fact that
Category Theory Problems
Problem 6
Let
- a natural bijection if for every
the set map is a bijection; - a natural isomorphism if there is a natural transformation
such that and .
Prove that
Problem 7
Suppose
- Prove that
has all (binary) products. - Prove that
has all equalizers.
- Show that for every pair of objects
in , the pullback satisfies the universal property of the product . - Show that the equalizer of a pair of arrows
may be constructed as the pullback of .
Problem 8
Suppose
- The map that assigns to each pair of parallel arrows
the equalizer object is the object function of a functor to . What is the arrow function of that functor? - The equalizer functor above is right adjoint to a certain diagonal (or constant) functor from
. Describe:- a) the other functor;
- b) the natural bijection of the adjunction; and
- c) the unit and counit of the adjunction.
Problem 9
To each category
-
Suppose
is a functor. Show that the maps and together define a functor . -
Consider a functor
. If we write for , show how the functor can be expressed directly in terms of the original category as maps (both denoted ) on objects and arrows in . How does interact with composition of arrows?Such maps are sometimes called contravariant functors from
to , in which case our usual functors are called covariant functors. -
For each fixed object
in a category , show how to define a functorThis functor is sometimes called the contravariant hom-functor (associated to the object
).
Problem 10
Suppose
Suppose
- Show there is a unique arrow
in such that .[1] - Show that the unique arrow
in part (a) is an isomorphism; i.e., there is an arrow in such that and .
Because of this result, we say that "representations of functors are unique up to unique isomorphism," and also "objects that satisfy a universal property are unique up to unique isomorphism."
For the first part, to show the existence first consider the natural isomorphism
For the second part of the problem, switch the roles of
This is if we're considering
as a contravariant functor from to . If we're considering as a functor from to , then this should be . ↩︎