Subject: Re: two postings From: "Vigleik Angeltveit" Date: Fri, 15 Sep 2006 08:58:07 -0500 In reply to Tom Goodwillie's question. The concept of a V-model category has been studied. A good place to start is with Mark Hovey's book on model categories. Vigleik > From: Tom Goodwillie > Date: Thu, 14 Sep 2006 21:54:29 -0400 > > Is there a literature on enriched model categories? > > A bit more precisely: If V is a symmetric monoidal category then one can > speak of a V-enriched category C. If V is also a model category, and if > its model structure is suitably compatible with its symmetric monoidal > structure, then one can imagine that there is then a serviceable notion > of V-enriched model category C. In the case when V is simplicial sets, > this should be the usual notion of simplicial model category. Has some > version of this been worked out? > > Thanks, > Tom Goodwillie _______________________________________________________________ Subject:Re: two postings From:Georg Biedermann Date:Fri, 15 Sep 2006 13:16:14 -0400 here is a short response to Tom Goodwillie's lates question: Phil Hirschhorn used to have a chapter about enriched model categories in his book, but very unfortunately it has vanished from the final version. Georg Biedermann ___________________________________________________________________ Subject: Re: two postings From: Mark Hovey Date: 15 Sep 2006 14:47:32 -0400 Tom, Michael Shulman, a student of Peter May's, has written a long paper that addresses this question. It is currently under revision, but if you write him you might be able to get a copy. His web page is http://www.math.uchicago.edu/~shulman/ Mark _________________________________________________________________________ Subject: Re: two postings From: "V. Schmitt" Date: Sat, 16 Sep 2006 10:19:01 +0100 Regarding enriched category theory, there is Max Kelly's book "basic concepts of enriched category theory". This is mainly about the theory of enriched categories based on a symmetric monoidal closed category. That i know a bit about. Regarding simplicial model category. I had a look and asked myself exactly the same question: Are they just simple simple categories enriched over V=SSet with tensors and cotensors defined as Kelly & Co did ? (and with extra axioms model categories + SM7). Actually I found many different definitions of simplicial model categories and never got clear answers from anyone (!?). So i really like to know! Anyway this is what i worked out. For what it is worth. For instance there are definitions of simplicial model cat C where the Hom is a bifunctor C^{op}xC -> SSet and the tensor is defined by means of *mere natural isomorphism* in SSet: C(k@a,b) ~ SSet(k, C(a,b)) (if i remember well!) where C(a,b) is in SSet. I am thinking for instance of the definition in chapter 2 of Goerss-Jardines'book - where they use many different homs C(-,-) defined for a simplicial model cat C. I doubt that this definition of simplicial (model) category corresponds to tensored and cotensored SSet-categories in the sense of Kelly & al. For instance, the definition of tensors for V-categories is as follows: a V-cat C is tensored iff and only the V-presheaf C -> V: b |--> [v,C(a,b)] is representable (and I should say *V-representable*), that one can write: C(v@a,b) iso_{in b} [v,C(a,b)] the latter iso being a *V-isomorphism* (i.e. a V-natural tranformation that is iso) and [-,-] above denotes the internal in V. The difference with the previous notion of tensors is the use of V-naturality. Now one can ask whether SSet-naturality amounts to the same thing as the mere naturality for SSet-presheaves A->SSet when A is an SSet-category. The answer is no. Actually these SSet-natural transformations between SSet-presheaves A->V=SSet are those called "maps" in Goerss_Jardines' book in chapter 9. Now if one considers the SSet-category [A,S] where A is a SSet-category and S is the SSet-category SSet enriched over itself then it happens that [A,S] is an SSet-category with all the SSet-indexed limits and colimits and thus it has in particular the cotensors and tensors as defined by Kelly and al. I hope that i did not write too many typos but i am in a rush this morning. By the way i really would like to know a clear answer on Tom Goodwillie's question: Are "simplicial model categories" exactly SSet-Complete & SSet-cocomplete SSet-categories with model categories axioms and SM7. Best regards, Vincent Schmitt.