Model Structures on the Category of Small Double Categories

TL;DR

This paper develops multiple model structures on the category of small double categories, using various approaches including categorification, internal categories, and 2-monads, providing new insights into their properties and constructions.

Contribution

It introduces several new model structures on small double categories derived from different theoretical frameworks, and explores their properties and relationships.

Findings

Multiple model structures on DblCat are established.

Descriptions and properties of free and quotient double categories are provided.

Connections between different model structures and their implications are discussed.

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.