Model structures on the category of small double categories

Thomas M. Fiore, Simona Paoli, Dorette Pronk

Research output: Contribution to journalArticlepeer-review

19 Citations (Scopus)


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 languageEnglish
Pages (from-to)1855-1959
Number of pages105
JournalAlgebraic and Geometric Topology
Issue number4
Publication statusPublished - 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 `
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.


  • 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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