#
On Finitary Functors

##
Jiri Adamek, Stefan Milius, Lurdes Sousa and Thorsten Wissmann

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.

Keywords:
Finitely presentable object, finitely generatd object,
(strictly) locally finitely presentable category, finitary functor,
finitely bounded functor

2010 MSC:
18C35, 18A30, 08C05

*Theory and Applications of Categories,*
Vol. 34, 2019,
No. 35, pp 1134-1164.

Published 2019-10-25.

http://www.tac.mta.ca/tac/volumes/34/35/34-35.pdf

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