paper

Model Structures on the Category of Small Double Categories

arXiv:0711.0473 · doi:10.2140/agt.2008.8.1855

Abstract

In this paper we obtain several model structures on {\bf 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 {\bf Cat} and take as our weak equivalences various internal equivalences defined via Grothendieck topologies. Thirdly, {\bf 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 and discuss properties of free double categories, quotient double categories, colimits of double categories, several nerves, and horizontal categorification.

103 pages. Included Quillen adjunctions with Cat, improved characterization of flexible double categories, proved 2-cocompleteness of DblCat_v, proved horizontal nerve is 2-coskeletal, cut double categorification for a future article, removed identity squares from double derivation schemes, improved counterexample to Reedy transfer.

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