Pareigis, Bodo; Sweedler, M.
On Generators and Congenerators.
In: Manuscripta Mathematica, Vol. 2: pp. 49-66
The question of the existence of generators and cogenerators
i n a category is of i n t e r e s t i n view of the special
adjoint functor theorem. ISBELL has given an example (unpublished)
which shows t h a t the existence of a cogenerator
i s a necessary part of the hypothesis of the special
adjoint functor theorem. This example also shows t h a t the
category of groups has no cogenerator. (Clearly the f r e e
group on one element i s a generator i n the category of
groups.) It is well known t h a t there e x i s t generators and
cogenerators i n the categories of commutative groups,
Comrnutative Lie algebras (over a f i e l d ) and commutative
r e s t r i c t e d Lie algebras, because a l l of these categories
are module categories. By ISBELL1s r e s u l t when one drops
the condition of cornmutativity for the category of commut
a t i v e groups there i s no longer a cogenerator. We have
Proved similar r e s u l t s for the categories of commutative
Lie algebras and commutative r e s t r i c t e d Lie algebras. The
r e s u l t s are summarized i n the l i s t below where we have
included some r e l a t e d categories.