In this paper I continue to look at the connections between 2-category theory and homotopy theory, using the notion of Quillen model category, which is designed to provide sufficient structure on a category to allow one to "do homotopy theory" inside that category.
This paper concerns the case of Quillen model categories enriched in Cat: the enrichment in Cat gives a 2-category structure, and there is a model structure on the underlyingordinary category which is required to be compatible with the enrichment. The compatibility conditions themselves are expressed using the model structure on Cat.
It turns out that every 2-category with finite limits and colimits has a canonical model structure of this type, which we call the trivial model structure. Such trivial structures can then be lifted along adjunctions to obtain non-trivial model structures.
We study 2-monads and their algebras using these Cat-enriched Quillen model categories, emphasizing the parallels between the homotopical and 2-categorical points of view. In particular, there is a model structure on the 2-category of algebras for a 2-monad T, and a model structure on a 2-category of 2-monads on a fixed 2-category K.
Download it from the arXiv at math.CT/0607646 (these links are from the mirror in Australia).