User talk:Alodyne/Model category

Alodyne: I got your message on my talk page and I will take a look as soon as I find some time. Unfortunately, that will not be until at least a week from now, as I am going on vacation and will be away from my computer. As for standards, I am in favor in developing these here (see also simplicial set, which I have heavily rewritten, for notation). My main reference for this stuff is Goerss/Jardine "Simplicial Homotopy Theory".

This neatly sidesteps the set-theoretic issues outlines above, for it will be apparent that the morphisms form a set in the category we construct.

How "apparent" this is is up for debate, I think. In fact, the difficulty with this approach is that the hom "things" we get may not be sets in the original universe. There is an explicit construction which gets around this, but it is too long to include here. See GJ pg. 74. The upshot is that there is a way, but it is not entirely trivial.

Probably what this article should include are descriptions of the model category versions of classical results in homotopy theory. A lot of this depends on the particular model category, though. Thomason's work on a model structure for the category of small categories may be worth mentioning just for "wow" effect. Most of the stronger results at least assume that such a category is "simplicially enriched" in that the hom sets are simplicial sets, and/or that the category is "cofibrantly generated", and I don't know whether results imposing such restrictions should be included here. The "small object argument" of Quillen should be mentioned and perhaps explained a bit, given its importance. That's all I have for now. Talk to you in a week. - Gauge 07:05, 10 August 2005 (UTC)

In the definition, some authors only require the category to have finite limits and colimits, instead of small limits and colimits. Path and cylinder objects should be mentioned for what they do, but it may be worthwhile to make them into separate articles. The article here looks good so far. I would move it into place and let people start adding to it. - Gauge 23:16, 25 August 2005 (UTC)

Hi; maybe you've lost enthusiasm for model categories by now, but what you've written already is much better than the current page. Since I hope you return to it, someday, let me criticize it: I think you should deemphasize the set-theoretic motivation. That "there are many cases in which constructions can only be made before passage to homotopy, even if one is primarily concerned with the homotopy categories" is a great way to motivate model categories. The set-theory issue deserves to be discussed, but I think of it as a reason that model categories are hard (i.e. that it is hard to prove that something is a model category) rather than a reason that one should think about model categories. Also, is it bad form to edit your page? If not I'd be happy to tinker with it. Changbao 01:12, 28 August 2006 (UTC)