Model structures on the category of small double categories

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 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.