Geodesic
On Finitary Functors
Theory and applications of categories, Tome 34 (2019), pp. 1134-1164

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

A simple criterion for a functor to be finitary is presented: we call F finitely bounded if for all objects X every finitely generated subobject of FX factorizes through the F-image of a finitely generated subobject of X. This is equivalent to F being finitary for all functors between `reasonable' locally finitely presentable categories, provided that F preserves monomorphisms. We also discuss the question when that last assumption can be dropped. The answer is affirmative for functors between categories such as Set, K-Vec (vector spaces), boolean algebras, and actions of any finite group either on Set or on K-Vec for fields K of characteristic 0.

All this generalizes to locally $\lambda$-presentable categories, $\lambda$-accessible functors and $\lambda$-presentable algebras. As an application we obtain an easy proof that the Hausdorff functor on the category of complete metric spaces is $\aleph_1$-accessible.

Publié le :
Classification : 18C35, 18A30, 08C05
Keywords: Finitely presentable object, finitely generatd object, (strictly) locally finitely presentable category, finitary functor, finitely bounded functor 
@article{TAC_2019_34_a34,
     author = {Jiri Adamek and Stefan Milius and Lurdes Sousa and Thorsten Wissmann},
     title = {On {Finitary} {Functors}},
     journal = {Theory and applications of categories},
     pages = {1134--1164},
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     year = {2019},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TAC_2019_34_a34/}
}
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Jiri Adamek; Stefan Milius; Lurdes Sousa; Thorsten Wissmann. On Finitary Functors. Theory and applications of categories, Tome 34 (2019), pp. 1134-1164. http://geodesic.mathdoc.fr/item/TAC_2019_34_a34/