Given a full subcategory
[Fscr ] of a category [Ascr ], the existence
of left [Fscr ]-approximations (or
[Fscr ]-preenvelopes) completing diagrams in a unique way is
equivalent to the fact that [Fscr ] is reflective in
[Ascr ], in the classical terminology of category
theory.
In the first part of the paper we establish, for a rather general
[Ascr ], the relationship between reflectivity and covariant
finiteness of [Fscr ] in [Ascr ], and
generalize Freyd's adjoint functor theorem (for inclusion functors) to not
necessarily complete categories. Also, we study the good behaviour of reflections
with respect to direct limits. Most results in this part are dualizable, thus
providing corresponding versions for coreflective subcategories.
In the
second half of the paper we give several examples of reflective subcategories of
abelian and module categories, mainly of subcategories of the form Copres
(M) and Add (M). The second case covers the
study of all covariantly finite, generalized Krull-Schmidt subcategories of
{\rm Mod}_{R}, and has some connections with the
“pure-semisimple conjecture”.
1991 Mathematics Subject Classification 18A40, 16D90, 16E70.