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Pareigis, Bodo and Sweedler, M. (1970): On Generators and Congenerators. In: Manuscripta Mathematica, Vol. 2: pp. 49-66
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Abstract

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 ISBELL has given an example (unpublished) which shows t h a t the existence of a cogenerator i s a necessary part of the hypothesis of the special adjoint functor theorem.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.