@UNPUBLISHED{publication1, title = "A formula for codensity monads and density comonads", author = "{AD{\'{A}}MEK, Jir{\ยด{i}}} and {SOUSA, Lurdes}", year = "2017-12-07", 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 all natural transformations from K(X,F\−) to K(s,F\−) form a set (given objects s \∈ S and X arbitrary). 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 have a density comonad iff all natural transformations from XF to 2F form a set (for every set X). Moreover, that comonad assigns to X the set Nat(XF,2F). For preimages-preserving endofunctors of Set we prove that the existence of a density comonad is equivalent to the accessibility of F.

", url = "http://www.mat.uc.pt/preprints/eng\_2017.html", note = "" }