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 struc-tures 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 cofi-brant objects. As part of this program we give explicit descriptions and discuss properties of free double categories, quotient double cate-gories, colimits of double categories, and several nerves and categori