A formula for codensity monads and density comonads (Preprint)

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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
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