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Model structures on the category of small double categories

Thomas M Fiore, Simona Paoli and Dorette Pronk

Algebraic & Geometric Topology 8 (2008) 1855–1959

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 categorification-nerve 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.

categorification, colimit, double category, fundamental category, fundamental double category, horizontal categorification, internal category, model structure, transfer of model structure, $2$–category, $2$–monad
Mathematical Subject Classification 2000
Primary: 18D05, 18G55
Secondary: 55P99, 55U10
Received: 26 October 2007
Revised: 27 August 2008
Accepted: 27 August 2008
Published: 21 October 2008
Thomas M Fiore
Department of Mathematics
University of Chicago
5734 South University
Chicago, IL 60637
and Departament de Matemàtiques
Universitat Autònoma de Barcelona
08193 Bellaterra (Barcelona)
Simona Paoli
Department of Mathematics
Macquarie University
NSW 2109
Dorette Pronk
Chase Building
Department of Mathematics and Statistics
Dalhousie University
Halifax, Nova Scotia
Canada B3H 3J5