Sigma limits in 2-categories and flat pseudofunctors

Autores
Descotte, María Emilia; Dubuc, Eduardo Julio; Szyld, Martín
Año de publicación
2018
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
In this paper we introduce sigma limits (which we write σ-limits), a concept that interpolates between lax and pseudolimits: for a fixed family Σ of arrows of a 2-category A, a σ-cone for a 2-functor A⟶FB is a lax cone such that the structural 2-cells corresponding to the arrows of Σ are invertible. The conical σ-limit of F is the universal σ-cone. Similarly we define σ-natural transformations and weighted σ-limits. We consider also the case of bilimits. We develop the theory of σ-limits and σ-bilimits, whose importance relies on the following key fact: any weighted σ-limit (or σ-bilimit) can be expressed as a conical one. From this we obtain, in particular, a canonical expression of an arbitrary Cat-valued 2-functor as a conical σ-bicolimit of representable 2-functors, for a suitable choice of Σ, which is equivalent to the well known bicoend formula. As an application, we establish the 2-dimensional theory of flat pseudofunctors. We define a Cat-valued pseudofunctor to be flat when its left bi-Kan extension along the Yoneda 2-functor preserves finite weighted bilimits. We introduce a notion of 2-filteredness of a 2-category with respect to a class Σ, which we call σ-filtered. Our main result is: A pseudofunctor A⟶Cat is flat if and only if it is a σ-filtered σ-bicolimit of representable 2-functors. In particular the reader will notice the relevance of this result for the development of a theory of 2-topoi.
Fil: Descotte, María Emilia. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentina
Fil: Dubuc, Eduardo Julio. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentina
Fil: Szyld, Martín. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentina
Materia
2-CATEGORIES
CONICAL LIMIT
FLAT
PSEUDOFUNCTOR
SIGMA FILTERED
SIGMA LIMITS
Nivel de accesibilidad
acceso abierto
Condiciones de uso
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
Repositorio
CONICET Digital (CONICET)
Institución
Consejo Nacional de Investigaciones Científicas y Técnicas
OAI Identificador
oai:ri.conicet.gov.ar:11336/159999
Acceder
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oai_identifier_str oai:ri.conicet.gov.ar:11336/159999
network_acronym_str CONICETDig
repository_id_str 3498
network_name_str CONICET Digital (CONICET)
spelling Sigma limits in 2-categories and flat pseudofunctorsDescotte, María EmiliaDubuc, Eduardo JulioSzyld, Martín2-CATEGORIESCONICAL LIMITFLATPSEUDOFUNCTORSIGMA FILTEREDSIGMA LIMITShttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1In this paper we introduce sigma limits (which we write σ-limits), a concept that interpolates between lax and pseudolimits: for a fixed family Σ of arrows of a 2-category A, a σ-cone for a 2-functor A⟶FB is a lax cone such that the structural 2-cells corresponding to the arrows of Σ are invertible. The conical σ-limit of F is the universal σ-cone. Similarly we define σ-natural transformations and weighted σ-limits. We consider also the case of bilimits. We develop the theory of σ-limits and σ-bilimits, whose importance relies on the following key fact: any weighted σ-limit (or σ-bilimit) can be expressed as a conical one. From this we obtain, in particular, a canonical expression of an arbitrary Cat-valued 2-functor as a conical σ-bicolimit of representable 2-functors, for a suitable choice of Σ, which is equivalent to the well known bicoend formula. As an application, we establish the 2-dimensional theory of flat pseudofunctors. We define a Cat-valued pseudofunctor to be flat when its left bi-Kan extension along the Yoneda 2-functor preserves finite weighted bilimits. We introduce a notion of 2-filteredness of a 2-category with respect to a class Σ, which we call σ-filtered. Our main result is: A pseudofunctor A⟶Cat is flat if and only if it is a σ-filtered σ-bicolimit of representable 2-functors. In particular the reader will notice the relevance of this result for the development of a theory of 2-topoi.Fil: Descotte, María Emilia. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; ArgentinaFil: Dubuc, Eduardo Julio. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; ArgentinaFil: Szyld, Martín. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; ArgentinaAcademic Press Inc Elsevier Science2018-07info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfapplication/pdfapplication/pdfapplication/pdfhttp://hdl.handle.net/11336/159999Descotte, María Emilia; Dubuc, Eduardo Julio; Szyld, Martín; Sigma limits in 2-categories and flat pseudofunctors; Academic Press Inc Elsevier Science; Advances in Mathematics; 333; 7-2018; 266-3130001-8708CONICET DigitalCONICETenginfo:eu-repo/semantics/altIdentifier/url/https://www.sciencedirect.com/science/article/pii/S0001870818301968info:eu-repo/semantics/altIdentifier/doi/10.1016/j.aim.2018.05.021info:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-sa/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-10-22T11:12:49Zoai:ri.conicet.gov.ar:11336/159999instacron:CONICETInstitucionalhttp://ri.conicet.gov.ar/Organismo científico-tecnológicoNo correspondehttp://ri.conicet.gov.ar/oai/requestdasensio@conicet.gov.ar; lcarlino@conicet.gov.arArgentinaNo correspondeNo correspondeNo correspondeopendoar:34982025-10-22 11:12:50.223CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv Sigma limits in 2-categories and flat pseudofunctors
title Sigma limits in 2-categories and flat pseudofunctors
spellingShingle Sigma limits in 2-categories and flat pseudofunctors
Descotte, María Emilia
2-CATEGORIES
CONICAL LIMIT
FLAT
PSEUDOFUNCTOR
SIGMA FILTERED
SIGMA LIMITS
title_short Sigma limits in 2-categories and flat pseudofunctors
title_full Sigma limits in 2-categories and flat pseudofunctors
title_fullStr Sigma limits in 2-categories and flat pseudofunctors
title_full_unstemmed Sigma limits in 2-categories and flat pseudofunctors
title_sort Sigma limits in 2-categories and flat pseudofunctors
dc.creator.none.fl_str_mv Descotte, María Emilia
Dubuc, Eduardo Julio
Szyld, Martín
author Descotte, María Emilia
author_facet Descotte, María Emilia
Dubuc, Eduardo Julio
Szyld, Martín
author_role author
author2 Dubuc, Eduardo Julio
Szyld, Martín
author2_role author
author
dc.subject.none.fl_str_mv 2-CATEGORIES
CONICAL LIMIT
FLAT
PSEUDOFUNCTOR
SIGMA FILTERED
SIGMA LIMITS
topic 2-CATEGORIES
CONICAL LIMIT
FLAT
PSEUDOFUNCTOR
SIGMA FILTERED
SIGMA LIMITS
purl_subject.fl_str_mv https://purl.org/becyt/ford/1.1
https://purl.org/becyt/ford/1
dc.description.none.fl_txt_mv In this paper we introduce sigma limits (which we write σ-limits), a concept that interpolates between lax and pseudolimits: for a fixed family Σ of arrows of a 2-category A, a σ-cone for a 2-functor A⟶FB is a lax cone such that the structural 2-cells corresponding to the arrows of Σ are invertible. The conical σ-limit of F is the universal σ-cone. Similarly we define σ-natural transformations and weighted σ-limits. We consider also the case of bilimits. We develop the theory of σ-limits and σ-bilimits, whose importance relies on the following key fact: any weighted σ-limit (or σ-bilimit) can be expressed as a conical one. From this we obtain, in particular, a canonical expression of an arbitrary Cat-valued 2-functor as a conical σ-bicolimit of representable 2-functors, for a suitable choice of Σ, which is equivalent to the well known bicoend formula. As an application, we establish the 2-dimensional theory of flat pseudofunctors. We define a Cat-valued pseudofunctor to be flat when its left bi-Kan extension along the Yoneda 2-functor preserves finite weighted bilimits. We introduce a notion of 2-filteredness of a 2-category with respect to a class Σ, which we call σ-filtered. Our main result is: A pseudofunctor A⟶Cat is flat if and only if it is a σ-filtered σ-bicolimit of representable 2-functors. In particular the reader will notice the relevance of this result for the development of a theory of 2-topoi.
Fil: Descotte, María Emilia. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentina
Fil: Dubuc, Eduardo Julio. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentina
Fil: Szyld, Martín. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentina
description In this paper we introduce sigma limits (which we write σ-limits), a concept that interpolates between lax and pseudolimits: for a fixed family Σ of arrows of a 2-category A, a σ-cone for a 2-functor A⟶FB is a lax cone such that the structural 2-cells corresponding to the arrows of Σ are invertible. The conical σ-limit of F is the universal σ-cone. Similarly we define σ-natural transformations and weighted σ-limits. We consider also the case of bilimits. We develop the theory of σ-limits and σ-bilimits, whose importance relies on the following key fact: any weighted σ-limit (or σ-bilimit) can be expressed as a conical one. From this we obtain, in particular, a canonical expression of an arbitrary Cat-valued 2-functor as a conical σ-bicolimit of representable 2-functors, for a suitable choice of Σ, which is equivalent to the well known bicoend formula. As an application, we establish the 2-dimensional theory of flat pseudofunctors. We define a Cat-valued pseudofunctor to be flat when its left bi-Kan extension along the Yoneda 2-functor preserves finite weighted bilimits. We introduce a notion of 2-filteredness of a 2-category with respect to a class Σ, which we call σ-filtered. Our main result is: A pseudofunctor A⟶Cat is flat if and only if it is a σ-filtered σ-bicolimit of representable 2-functors. In particular the reader will notice the relevance of this result for the development of a theory of 2-topoi.
publishDate 2018
dc.date.none.fl_str_mv 2018-07
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
http://purl.org/coar/resource_type/c_6501
info:ar-repo/semantics/articulo
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv http://hdl.handle.net/11336/159999
Descotte, María Emilia; Dubuc, Eduardo Julio; Szyld, Martín; Sigma limits in 2-categories and flat pseudofunctors; Academic Press Inc Elsevier Science; Advances in Mathematics; 333; 7-2018; 266-313
0001-8708
CONICET Digital
CONICET
url http://hdl.handle.net/11336/159999
identifier_str_mv Descotte, María Emilia; Dubuc, Eduardo Julio; Szyld, Martín; Sigma limits in 2-categories and flat pseudofunctors; Academic Press Inc Elsevier Science; Advances in Mathematics; 333; 7-2018; 266-313
0001-8708
CONICET Digital
CONICET
dc.language.none.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv info:eu-repo/semantics/altIdentifier/url/https://www.sciencedirect.com/science/article/pii/S0001870818301968
info:eu-repo/semantics/altIdentifier/doi/10.1016/j.aim.2018.05.021
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
eu_rights_str_mv openAccess
rights_invalid_str_mv https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
dc.format.none.fl_str_mv application/pdf
application/pdf
application/pdf
application/pdf
dc.publisher.none.fl_str_mv Academic Press Inc Elsevier Science
publisher.none.fl_str_mv Academic Press Inc Elsevier Science
dc.source.none.fl_str_mv reponame:CONICET Digital (CONICET)
instname:Consejo Nacional de Investigaciones Científicas y Técnicas
reponame_str CONICET Digital (CONICET)
collection CONICET Digital (CONICET)
instname_str Consejo Nacional de Investigaciones Científicas y Técnicas
repository.name.fl_str_mv CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicas
repository.mail.fl_str_mv dasensio@conicet.gov.ar; lcarlino@conicet.gov.ar
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score 12.982451

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