A Formula for Codensity Monads and Density Comonads

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F68407700%3A21230%2F18%3A00324212" target="_blank" >RIV/68407700:21230/18:00324212 - isvavai.cz</a>
Result on the web
<a href="http://dx.doi.org/10.1007/s10485-018-9530-6" target="_blank" >http://dx.doi.org/10.1007/s10485-018-9530-6</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1007/s10485-018-9530-6" target="_blank" >10.1007/s10485-018-9530-6</a>

Result language
angličtina
Original language name
A Formula for Codensity Monads and Density Comonads
Original language description
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.
Czech name
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Czech description
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Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
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OECD FORD branch
10101 - Pure mathematics

Project
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Continuities
I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year
2018
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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