Appl Categor Struct https://doi.org/10.1007/s10485-018-9530-6 1 2,3 Jirí ˇ Adámek · Lurdes Sousa Received: 7 December 2017 / Accepted: 7 May 2018 © Springer Science+Business Media B.V., part of Springer Nature 2018 Abstract For a functor F whose codomain is a cocomplete, cowellpowered category K with a generator S we prove that a codensity monad exists iff for every object s in S all natural transformations from K(X, F −) to K(s, F −) form a set. Moreover, the codensity monad has an explicit description using the above natural transformations. Concrete examples are presented, e.g., the codensity monad of the power-set functor P assigns to every set X the set of all nonexpanding endofunctions of PX. Dually, a set-valued functor F is proved to F F have a density comonad iff all natural transformations from X to 2 form a set. Moreover, that comonad assigns to X the set of all those transformations. For preimages-preserving endofunctors F of Set we prove that F has a density comonad iff F is accessible. Keywords Codensity monad · Density comonad · Accessible functors Dedicated to Bob Lowen on his seventieth birthday Communicated by Walter Tholen. This work was partially supported by the Centre
Applied Categorical Structures – Springer Journals
Published: May 29, 2018
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