Math 726
Fall 2002
Homework 2
Due
Monday, September 23, 2002. Do not forget to mark your “favorite” problem.
- Consider the category
from Example 1.8 (Let 
be a monoid, 
(a single
object) and 
with composition given by multiplication in 
). Find all equivalences in 
- Let RINGS be the
category of commutative rings with
Let RNGS
be the category of commutative rings. Show that RNGS has a
universal and couniversal object, but RINGS has only a universal
object.
- A pointed set is
a pair
with 
a set and 
A morphism of
pointed sets (
) is a triple 
where 
is a function
such that 
Show that
pointed sets with these morphisms form a category.
- Let
be the category
of all finite-dimensional vector spaces over a field,
(of characteristic not equal to 2 or 3) and let the
morphisms be vector space isomorphisms. We denote 
by 
a)
Show
that 
is naturally
isomorphic to 
b)
If
is a vector space
isomorphism, then so is the dual map 
(and hence so is its
inverse).
c)
Show
that 
is a covariant
functor (where 
and 
).
d)
For
each 
choose a basis 
and let 
be the dual basis
(that is 
where 
) of 
Then the map 
given by 
induces an
isomorphism 
e)
Show
that 
is not natural (that is, the assignment 
is not a natural
isomorphism from the identity functor to 
5. Let 
be a category and 
a family of objects in 
Show that any two
coproducts of the family 
(if they exist) are
equivalent.