Hi David, This is something I've thought about as well. If your model category is a Cat-model category, then by Hovey's general results on enriched model categories, its homotopy category is automatically enriched over Ho(Cat), the category of categories and natural-isomorphism-classes of functors. A Ho(Cat)-enriched category is like a "bicategory without coherence," and the question is about lifting that structure to a coherent bicategory. However, in this case I believe you can actually always obtain a strict 2-category equivalent to the bicategory of fractions by just looking at the full sub-2-category of your model 2-category spanned by the fibrant and cofibrant objects. Since any Ho(Cat)-category that is equivalent (as a Ho(Cat)-category) to a bicategory must itself underlie a bicategory, you can use this to get a "homotopy bicategory" without needing the calculus of fractions (which model category theory is basically designed to avoid). There is lots of good stuff about Cat-model categories in Steve Lack's paper "Homotopy-theoretic aspects of 2-monads": http://arxiv.org/abs/math.CT/0607646. Best, Mike On Tue, Jan 27, 2009 at 12:01 AM, David Roberts <droberts@maths.adelaide.edu.au> wrote:

Hi all,

has anyone come across this situation? I have a 2-category where the underlying category has a model structure, and the class of equivalences (from the 2-cat structure) is contained in the weak equivalences. The class of weak equivalences admits a bicategory of fractions, and so one can consider that bicategory as the homotopy 'category' in some sense.

Cheers,

David Roberts