Subject: Re: question about model categories From: Philip Hirschhorn Date: Thu, 3 May 2007 13:53:21 -0400 (EDT) A response to John Baez: >> Subject: model category of simplicial spaces >> From: John Baez >> Date: Wed, 2 May 2007 17:06:53 -0700 >> >> Thanks to everyone for their replies to my question a while back. >> I'd like to ask a followup question. >> >> Charles Rezk wrote: >> > >>>> > > There is a model category structure on simplicial spaces which is >>>> > > Quillen equivalent to Top. I don't know a proper reference for >>>> > > this. However, if you are willing to replace "space" with >>>> > > "simplicial set", then the "Moerdijk model structure" on >>>> > > bisimplicial sets discussed in Goerss-Jardine does exactly this. > >> >> I'm unwilling to replace "space" with "simplicial set". My >> intended audience will enjoy my results more if they're stated >> in terms of spaces. >> So, I really want to use a model category structure on simplicial >> spaces that's Quillen equivalent to Top. If anyone knows a reference, >> I'd like to hear about it! Take a look at either Rezk, Schwede, and Shipley: "Simplicial Structures on Model Categories and Functors", American Journal 123 (2001), 551-575 or Dugger: "Replacing Model Categories with Simplicial Ones", Transactions of the AMS 353 (2001), 5003-5027. Given a model category M satisfying some extra hypotheses (satisfied by the model category of topological spaces), both of those papers put a model category structure on the category of simplicial objects over M that's Quillen equivalent to M. Their purpose is to apply it to model categories M that aren't simplicial model categories, since the structures they construct on the category of simplicial objects is simplicial, but their theorems apply to the category of topological spaces. Best regards, Phil ____________________________________________________________________ Subject: Re: question about model categories From: Clemens.BERGER@unice.fr Date: Fri, 04 May 2007 11:01:24 +0200 just a few comments concerning the discussion about model structures on simplicial spaces: (1) There is a quite complete discussion of one possible model structure on simplicial spaces with the ``right'' weak equivalences in literature, namely (as already pointed out by Tore August) the paper by Rezk, Schwede and Shipley on model structures on simplicial functors. In this paper, the three authors give a general criterion, when the Reedy model structure on the simplicial objects sM of a model category M admits a left Bousfield localization with respect to hocolim-equivalences, and this is precisely what has to be done in order to get a Quillen equivalence sM <=> M. This criterion is called the ``realization axiom'' in loc. cit. If M=Top, the realization axiom holds by an old paper of V. Puppe. Therefore, there is a model structure on sTop with Reedy cofibrations as cofibrations, and hocolim-equivalences as weak equivalences, and with this model structure one has a Quillen equivalence sTop <=> Top. The nice feature of the paper by Rezk, Schwede and Shipley is that they characterize completely the fibrations of this localized Reedy model structure. These are called equifibered Reedy fibrations, i.e. Reedy fibrations X->Y with the supplementary property that any injective simplicial operator [m]->[n] induces a homotopy pullback square in Top: X_n->Y_n | | X_m->Y_m Also, the hocolim-equivalences are somewhat easier to define as those maps of simplicial spaces X->Y which become isomorphisms in Ho(Top) under the left derived functor of geometric realization sTop->Top. (2) Exactly the same construction works, if Top is replaced by sSets (simplicial sets), i.e. there is a model structure on bisimplicial sets with Reedy cofibrations as cofibrations, hocolim-equivalences (=realization weak equivalences=diagonal weak equivalences) as weak equivalences and equifibered Reedy fibrations as fibrations. There are at least four Quillen equivalent model structures on bisimplicial sets. The previous localized Reedy structure, Moerdijk's diagonal structure, Cegarra-Remedios' Wbar-structure, and Dugger's localised projective structure. All four model structures share the same class of weak equivalences. (3) It is of course some kind of folklore that topological spaces and simplicial sets are interchangeable with respect to homotopy theory. Nevertheless, it seems to me that John Baez' insistence of having geometric model structures on simplicial spaces is justified, and this topic is much less studied than the analogous one for bisimplicial sets. I would just mention one related problem I thought upon with Rainer Vogt without getting a satisfactory answer: One important feature of Segal's delooping machine is the fact for a 0-reduced Reedy fibrant simplicial space X, the associated path fibration OmegaX->PX->X is an equifibered Reedy fibration if and only if X is group like (i.e. pi_0(X) is a group). There is a generalization of this for n-fold loop spaces, where 0-reduced simplicial spaces are replaced by (n-1)-reduced simplicial spaces. This theory has been worked out by Bousfield in the bisimplicial case, and it has been stated (without proof) by Hopkins in the case of simplicial spaces. However, it is not anymore true, that one gets equifibered Reedy fibrations OmegaX->PX->X, but some kind of weaker fibrations. It would be good to have an explicit description of what kind of fibrations one gets in order to get rid of spectral sequence arguments which arise in Bousfield's approach. With best wishes, Clemens Berger.