A Formula for Codensity Monads and Density Comonads For a functor F whose codomain is a cocomplete, cowellpowered category $$\mathcal {K}$$ K with a generator S we prove that a codensity monad exists iff for every object s in S all natural transformations from $$\mathcal {K}(X, F-)$$ K ( X , F - ) to $$\mathcal {K}(s, F-)$$ 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 $$\mathcal {P}$$ P assigns to every set X the set of all nonexpanding endofunctions of $$\mathcal {P}X$$ P X . Dually, a set-valued functor F is proved to have a density comonad iff all natural transformations from $$X^F$$ X F to $$2^F$$ 2 F form a set. Moreover, that comonad assigns to X the set of all those transformations. For preimages-preserving endofunctors F of $${\mathsf {Set}}$$ Set we prove that F has a density comonad iff F is accessible. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Categorical Structures Springer Journals

Applied Categorical Structures, Volume 26 (5) – May 29, 2018
18 pages

Publisher
Springer Journals
Subject
Mathematics; Mathematical Logic and Foundations; Theory of Computation; Convex and Discrete Geometry; Geometry
ISSN
0927-2852
eISSN
1572-9095
D.O.I.
10.1007/s10485-018-9530-6
Publisher site
See Article on Publisher Site

### Abstract

For a functor F whose codomain is a cocomplete, cowellpowered category $$\mathcal {K}$$ K with a generator S we prove that a codensity monad exists iff for every object s in S all natural transformations from $$\mathcal {K}(X, F-)$$ K ( X , F - ) to $$\mathcal {K}(s, F-)$$ 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 $$\mathcal {P}$$ P assigns to every set X the set of all nonexpanding endofunctions of $$\mathcal {P}X$$ P X . Dually, a set-valued functor F is proved to have a density comonad iff all natural transformations from $$X^F$$ X F to $$2^F$$ 2 F form a set. Moreover, that comonad assigns to X the set of all those transformations. For preimages-preserving endofunctors F of $${\mathsf {Set}}$$ Set we prove that F has a density comonad iff F is accessible.

### Journal

Applied Categorical StructuresSpringer Journals

Published: May 29, 2018

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