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 ’
A +
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.
1. Introduction
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 = #
i#Ord
Grp(A i , -) : Grp -# Set

Source: Adámek, Jiri - Institut für Theoretische Informatik, Fachbereich Mathematik und Informatik, Technische Universität Braunschweig

Collections: Computer Technologies and Information Sciences