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Title: A Formula for Codensity Monads and Density Comonads
Authors: Adámek, Jiří
Sousa, Lurdes
Keywords: Codensity monad; Density comonad; Accessible functors
Issue Date: May-2018
Publisher: Springer
Project: UID/MAT/00324/2013 
Serial title, monograph or event: Applied Categorical Structures
Volume: 26
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 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 Set we prove that F has a density comonad iff F is accessible.
DOI: 10.1007/s10485-018-9530-6
Rights: openAccess
Appears in Collections:I&D CMUC - Artigos em Revistas Internacionais

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