#
How large are left exact functors?

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J. Adamek, V. Koubek and V. Trnkova

For a broad collection of categories $\cal K$, including
all presheaf categories, the following statement is proved
to be consistent: every left exact (i.e. finite-limits
preserving) functor from $\cal K$ to $\Set$ is small, that is,
a small colimit of representables. In contrast, for the
(presheaf) category ${\cal K}=\Alg(1,1)$ 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
set-valued functors as directed unions of ``reduced
representables'', generalizing reduced products.

Keywords: left exact functor, small functor, regular ultrafilter.

2000 MSC: 18A35, 18C99, 04A10.

*Theory and Applications of Categories*, Vol. 8, 2001, No. 13, pp 377-390.

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