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https://hdl.handle.net/10316/89487
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. | URI: | https://hdl.handle.net/10316/89487 | DOI: | 10.1007/s10485-018-9530-6 | Rights: | openAccess |
Appears in Collections: | I&D CMUC - Artigos em Revistas Internacionais |
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