Summary: Theory and Applications of Categories, Vol. 8, No. 13, 2001, pp. 377--390.
HOW LARGE ARE LEFT EXACT FUNCTORS?
J. AD ’
AMEK # , V. KOUBEK + AND V. TRNKOV ’
ABSTRACT. For a broad collection of categories K, including all presheaf categories,
the following statement is proved to be consistent: every left exact (i.e. finitelimits
preserving) functor from K to Set is small, that is, a small colimit of representables.
In contrast, for the (presheaf) category K = Alg(1, 1) of unary algebras we construct a
functor from Alg(1, 1) to Set which preserves finite products and is not small. We also
describe all left exact setvalued functors as directed unions of ``reduced representables'',
generalizing reduced products.
We study left exact (i.e. finitelimits preserving) setvalued functors on a category K,
and ask whether they all are small, i.e., small colimits of homfunctors. This depends of
the category K, of course, since even so wellbehaved categories as Grp, the category of
groups, have easy counterexamples: recall the wellknown example
F = #
Grp(A i , -) : Grp -# Set