Geodesic
How large are left exact functors?
Theory and applications of categories, Tome 8 (2001), pp. 377-390

Voir la notice de l'article provenant de la source Theory and Applications of Categories website

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.

Classification : 18A35, 18C99, 04A10.
Keywords: left exact functor, small functor, regular ultrafilter. 
@article{TAC_2001_8_a12,
     author = {J. Adamek and V. Koubek and V. Trnkova},
     title = {How large are left exact functors?},
     journal = {Theory and applications of categories},
     pages = {377--390},
     publisher = {mathdoc},
     volume = {8},
     year = {2001},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TAC_2001_8_a12/}
}
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J. Adamek; V. Koubek; V. Trnkova. How large are left exact functors?. Theory and applications of categories, Tome 8 (2001), pp. 377-390. http://geodesic.mathdoc.fr/item/TAC_2001_8_a12/