Abstract
For a functor F whose codomain is a cocomplete, cowellpowered category \(\mathcal {K}\) with a generator S we prove that a codensity monad exists iff for every object s in S all natural transformations from \(\mathcal {K}(X, F-)\) to \(\mathcal {K}(s, F-)\) 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 \(\mathcal {P}\) assigns to every set X the set of all nonexpanding endofunctions of \(\mathcal {P}X\). Dually, a set-valued functor F is proved to have a density comonad iff all natural transformations from \(X^F\) to \(2^F\) form a set. Moreover, that comonad assigns to X the set of all those transformations. For preimages-preserving endofunctors F of \({\mathsf {Set}}\) we prove that F has a density comonad iff F is accessible.
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Communicated by Walter Tholen.
Dedicated to Bob Lowen on his seventieth birthday
This work was partially supported by the Centre for Mathematics of the University of Coimbra—UID/MAT/00324/2013, funded by the Portuguese Government through FCT/MEC and co-funded by the European Regional Development Fund through the Partnership Agreement PT2020.
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Adámek, J., Sousa, L. A Formula for Codensity Monads and Density Comonads. Appl Categor Struct 26, 855–872 (2018). https://doi.org/10.1007/s10485-018-9530-6
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DOI: https://doi.org/10.1007/s10485-018-9530-6