A Formula for Codensity Monads and Density Comonads

A Formula for Codensity Monads and Density Comonads Appl Categor Struct https://doi.org/10.1007/s10485-018-9530-6 1 2,3 Jirí ˇ Adámek · Lurdes Sousa Received: 7 December 2017 / Accepted: 7 May 2018 © Springer Science+Business Media B.V., part of Springer Nature 2018 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 F F have a density comonad iff all natural transformations from X to 2 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. Keywords Codensity monad · Density comonad · Accessible functors Dedicated to Bob Lowen on his seventieth birthday Communicated by Walter Tholen. This work was partially supported by the Centre http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Categorical Structures Springer Journals

A Formula for Codensity Monads and Density Comonads

Loading next page...
 
/lp/springer_journal/a-formula-for-codensity-monads-and-density-comonads-hs0q5stYsm
Publisher
Springer Netherlands
Copyright
Copyright © 2018 by Springer Science+Business Media B.V., part of Springer Nature
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

Appl Categor Struct https://doi.org/10.1007/s10485-018-9530-6 1 2,3 Jirí ˇ Adámek · Lurdes Sousa Received: 7 December 2017 / Accepted: 7 May 2018 © Springer Science+Business Media B.V., part of Springer Nature 2018 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 F F have a density comonad iff all natural transformations from X to 2 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. Keywords Codensity monad · Density comonad · Accessible functors Dedicated to Bob Lowen on his seventieth birthday Communicated by Walter Tholen. This work was partially supported by the Centre

Journal

Applied Categorical StructuresSpringer Journals

Published: May 29, 2018

References

You’re reading a free preview. Subscribe to read the entire article.


DeepDyve is your
personal research library

It’s your single place to instantly
discover and read the research
that matters to you.

Enjoy affordable access to
over 18 million articles from more than
15,000 peer-reviewed journals.

All for just $49/month

Explore the DeepDyve Library

Search

Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly

Organize

Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place.

Access

Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals.

Your journals are on DeepDyve

Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more.

All the latest content is available, no embargo periods.

See the journals in your area

DeepDyve

Freelancer

DeepDyve

Pro

Price

FREE

$49/month
$360/year

Save searches from
Google Scholar,
PubMed

Create lists to
organize your research

Export lists, citations

Read DeepDyve articles

Abstract access only

Unlimited access to over
18 million full-text articles

Print

20 pages / month

PDF Discount

20% off