Subject: Re: 3 postings Date: 19 Jan 2004 11:04:18 -0500 From: Mark Hovey Reply-To: mhovey@wesleyan.edu To: dmd1@lehigh.edu In reply to Philippe Gaucher's message: > Could I have please a reference for the general fact (if this is true > !) that the homotopy colimit is a colimit in Ho(C) where C is a model > category ? If for any given small category I, there exists a model > structure on C^I such that the constant diagram functor is a right > Quillen functor, it's OK (for example if C is cofibrantly > generated). Same question for the homotopy limit. Homotopy categories don't have colimits and limits, as a rule. They have honest coproducts and products, but only have weak colimits and weak limits. The homotopy colimit is left adjoint to a constant diagram functor, but it is still not a colimit because the domain of the homotopy colimit functor is NOT a diagram category. The domain is Ho(C^I) not (Ho(C))^I. OK, so what you must have meant is a reference for the fact that hocolim is left adjoint to the constant diagram functor. I agree with you that it is hard to find this in the literature. It must follow from the stuff that Hirschhorn does in the last chapter, but a better write-up can be found, I bet, in the new book "Homotopy limit functors on model categories and homotopical categories" by Dwyer, Hirschhorn, Kan, and J. Smith, which can be found on Phil's web page: http://www-math.mit.edu/~psh/ Mark Hovey