A Formula for Codensity Monads and Density Comonads
Autor(a) principal: | |
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Data de Publicação: | 2018 |
Outros Autores: | |
Tipo de documento: | Artigo |
Idioma: | eng |
Título da fonte: | Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
Texto Completo: | http://hdl.handle.net/10316/89487 https://doi.org/10.1007/s10485-018-9530-6 |
Resumo: | 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. |
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A Formula for Codensity Monads and Density ComonadsCodensity monad; Density comonad; Accessible functorsFor 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.Springer2018-05info:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/articlehttp://hdl.handle.net/10316/89487http://hdl.handle.net/10316/89487https://doi.org/10.1007/s10485-018-9530-6enghttps://link.springer.com/article/10.1007/s10485-018-9530-6Adámek, JiříSousa, Lurdesinfo:eu-repo/semantics/openAccessreponame:Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)instname:Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informaçãoinstacron:RCAAP2022-05-25T01:36:27Zoai:estudogeral.uc.pt:10316/89487Portal AgregadorONGhttps://www.rcaap.pt/oai/openaireopendoar:71602024-03-19T21:09:47.039477Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) - Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informaçãofalse |
dc.title.none.fl_str_mv |
A Formula for Codensity Monads and Density Comonads |
title |
A Formula for Codensity Monads and Density Comonads |
spellingShingle |
A Formula for Codensity Monads and Density Comonads Adámek, Jiří Codensity monad; Density comonad; Accessible functors |
title_short |
A Formula for Codensity Monads and Density Comonads |
title_full |
A Formula for Codensity Monads and Density Comonads |
title_fullStr |
A Formula for Codensity Monads and Density Comonads |
title_full_unstemmed |
A Formula for Codensity Monads and Density Comonads |
title_sort |
A Formula for Codensity Monads and Density Comonads |
author |
Adámek, Jiří |
author_facet |
Adámek, Jiří Sousa, Lurdes |
author_role |
author |
author2 |
Sousa, Lurdes |
author2_role |
author |
dc.contributor.author.fl_str_mv |
Adámek, Jiří Sousa, Lurdes |
dc.subject.por.fl_str_mv |
Codensity monad; Density comonad; Accessible functors |
topic |
Codensity monad; Density comonad; Accessible functors |
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. |
publishDate |
2018 |
dc.date.none.fl_str_mv |
2018-05 |
dc.type.status.fl_str_mv |
info:eu-repo/semantics/publishedVersion |
dc.type.driver.fl_str_mv |
info:eu-repo/semantics/article |
format |
article |
status_str |
publishedVersion |
dc.identifier.uri.fl_str_mv |
http://hdl.handle.net/10316/89487 http://hdl.handle.net/10316/89487 https://doi.org/10.1007/s10485-018-9530-6 |
url |
http://hdl.handle.net/10316/89487 https://doi.org/10.1007/s10485-018-9530-6 |
dc.language.iso.fl_str_mv |
eng |
language |
eng |
dc.relation.none.fl_str_mv |
https://link.springer.com/article/10.1007/s10485-018-9530-6 |
dc.rights.driver.fl_str_mv |
info:eu-repo/semantics/openAccess |
eu_rights_str_mv |
openAccess |
dc.publisher.none.fl_str_mv |
Springer |
publisher.none.fl_str_mv |
Springer |
dc.source.none.fl_str_mv |
reponame:Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) instname:Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação instacron:RCAAP |
instname_str |
Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação |
instacron_str |
RCAAP |
institution |
RCAAP |
reponame_str |
Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
collection |
Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
repository.name.fl_str_mv |
Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) - Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação |
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