Model structures on the category of small double categories

Thomas M. Fiore; Simona Paoli; Dorette Pronk

doi:10.2140/agt.2008.8.1855

Model structures on the category of small double categories

Thomas M. Fiore, Simona Paoli, Dorette Pronk

Mathematical Science

Research output: Contribution to journal › Article › peer-review

20 Citations (Scopus)

Abstract

In this paper we obtain several model structures on DblCat, the category of small double categories. Our model structures have three sources. We first transfer across a categorificationnerve adjunction. Secondly, we view double categories as internal categories in Cat and take as our weak equivalences various internal equivalences defined via Grothendieck topologies. Thirdly, DblCat inherits a model structure as a category of algebras over a 2 -monad. Some of these model structures coincide and the different points of view give us further results about cofibrant replacements and cofibrant objects. As part of this program we give explicit descriptions for and discuss properties of free double categories, quotient double categories, colimits of double categories, horizontal nerve and horizontal categorification.

Original language	English
Pages (from-to)	1855-1959
Number of pages	105
Journal	Algebraic and Geometric Topology
Volume	8
Issue number	4
DOIs	https://doi.org/10.2140/agt.2008.8.1855
Publication status	Published - 21 Oct 2008

Bibliographical note

Acknowledgements Thomas M Fiore was supported by National Science Foundation
Grant DMS 0501208 at the University of Chicago. At the Universitat Autonoma de `
Barcelona he was supported by Grant SB2006-0085 of the Spanish Ministerio de
Educacion y Ciencia under the Programa Nacional de ayudas para la movilidad de ´
profesores de universidad e investigadores espaznoles y extranjeros. Simona Paoli
Algebraic & Geometric Topology, Volume 8 (2008)
Model Structures on DblCat 1861
was supported by an Australian Research Council Postdoctoral Fellowship (project
number DP0558598) and by a Macquarie University New Staff Grant Scheme. Dorette
Pronk was supported by an NSERC Discovery Grant, and she also thanks Macquarie
University and the University of Chicago for their hospitality and financial support, as
well as Calvin College and Utrecht University for their hospitality during her sabbatical
visits. All three authors gratefully acknowledge the financial support and hospitality
of the Fields Institute during the Thematic Program on Geometric Applications of
Homotopy Theory in 2007, at which a significant portion of this work was completed.
Thomas M Fiore and Simona Paoli also thank the Centre de Recerca Matematica in `
Bellaterra (Barcelona) for its hospitality during the CRM Research Program on Higher
Categories and Homotopy Theory in 2007–2008. Additional material in this article
was completed during that time. Simona Paoli thanks the Universitat Autonoma de `
Barcelona and the Centre de Recerca Matematica for the financial support during her `
visits.
The authors thank Steve Lack for suggesting the comparison of the model structure
induced by the categorically surjective topology and the algebra model structure.
They also thank Michael Shulman for the simplified proof of Corollary 2.14. The
authors express their gratitude to Peter May, Robert Pare and Robert Dawson for some ´
discussion of this work. Additionally, they thank the anonymous referee for many
helpful suggestions.

Keywords

2-category
2-monad
Categorification
Colimit
Double category
Fundamental category
Fundamental double category
Horizontal categorification
Internal category
Model structure
Transfer of model structure

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10.2140/agt.2008.8.1855

Cite this

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abstract = "In this paper we obtain several model structures on DblCat, the category of small double categories. Our model structures have three sources. We first transfer across a categorificationnerve adjunction. Secondly, we view double categories as internal categories in Cat and take as our weak equivalences various internal equivalences defined via Grothendieck topologies. Thirdly, DblCat inherits a model structure as a category of algebras over a 2 -monad. Some of these model structures coincide and the different points of view give us further results about cofibrant replacements and cofibrant objects. As part of this program we give explicit descriptions for and discuss properties of free double categories, quotient double categories, colimits of double categories, horizontal nerve and horizontal categorification.",

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author = "Fiore, {Thomas M.} and Simona Paoli and Dorette Pronk",

note = "Acknowledgements Thomas M Fiore was supported by National Science Foundation Grant DMS 0501208 at the University of Chicago. At the Universitat Autonoma de ` Barcelona he was supported by Grant SB2006-0085 of the Spanish Ministerio de Educacion y Ciencia under the Programa Nacional de ayudas para la movilidad de ´ profesores de universidad e investigadores espaznoles y extranjeros. Simona Paoli Algebraic & Geometric Topology, Volume 8 (2008) Model Structures on DblCat 1861 was supported by an Australian Research Council Postdoctoral Fellowship (project number DP0558598) and by a Macquarie University New Staff Grant Scheme. Dorette Pronk was supported by an NSERC Discovery Grant, and she also thanks Macquarie University and the University of Chicago for their hospitality and financial support, as well as Calvin College and Utrecht University for their hospitality during her sabbatical visits. All three authors gratefully acknowledge the financial support and hospitality of the Fields Institute during the Thematic Program on Geometric Applications of Homotopy Theory in 2007, at which a significant portion of this work was completed. Thomas M Fiore and Simona Paoli also thank the Centre de Recerca Matematica in ` Bellaterra (Barcelona) for its hospitality during the CRM Research Program on Higher Categories and Homotopy Theory in 2007–2008. Additional material in this article was completed during that time. Simona Paoli thanks the Universitat Autonoma de ` Barcelona and the Centre de Recerca Matematica for the financial support during her ` visits. The authors thank Steve Lack for suggesting the comparison of the model structure induced by the categorically surjective topology and the algebra model structure. They also thank Michael Shulman for the simplified proof of Corollary 2.14. The authors express their gratitude to Peter May, Robert Pare and Robert Dawson for some ´ discussion of this work. Additionally, they thank the anonymous referee for many helpful suggestions.",

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N2 - In this paper we obtain several model structures on DblCat, the category of small double categories. Our model structures have three sources. We first transfer across a categorificationnerve adjunction. Secondly, we view double categories as internal categories in Cat and take as our weak equivalences various internal equivalences defined via Grothendieck topologies. Thirdly, DblCat inherits a model structure as a category of algebras over a 2 -monad. Some of these model structures coincide and the different points of view give us further results about cofibrant replacements and cofibrant objects. As part of this program we give explicit descriptions for and discuss properties of free double categories, quotient double categories, colimits of double categories, horizontal nerve and horizontal categorification.

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KW - Colimit

KW - Double category

KW - Fundamental category

KW - Fundamental double category

KW - Horizontal categorification

KW - Internal category

KW - Model structure

KW - Transfer of model structure

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DO - 10.2140/agt.2008.8.1855

M3 - Article

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SN - 1472-2747

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SP - 1855

EP - 1959

JO - Algebraic and Geometric Topology

JF - Algebraic and Geometric Topology

IS - 4

ER -

Model structures on the category of small double categories

Abstract

Bibliographical note

Keywords

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