<Reference List> | |
Type: | Preprint |
National /International: | International |
Title: | A formula for codensity monads and density comonads |
Publication Date: | 2017-12-07 |
Authors: |
- Jirí Adámek
- Lurdes Sousa |
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. |
Institution: | DMUC 17-52 |
Online version: | http://www.mat.uc.pt...prints/eng_2017.html |
Download: | Not available |