categories: Re: bicategory of fractions and homotopy category

categories: Re: bicategory of fractions and homotopy category

[Michael Shulman]

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
>
>
>

categories: Re: bicategory of fractions and homotopy category

[Steve Lack]

Dear David,

I have written about this sort of thing in the paper

Homotopy-theoretic aspects of 2-monads, Journal of Homotopy and Related
Structures 2:229-260, 2007; also arXiv:math.CT/0607646.

Regards,

Steve Lack.

On 27/01/09 5:01 PM, "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
>
>