Subject: Re: two questions From: Charles Rezk Date: Mon, 16 Apr 2007 10:08:07 -0500 Some answers to John's questions. 1) Geometric realization is *not* a Quillen equivalence between simplicial spaces with the Reedy model structure and Top. The right adjoint to geometric realization sends a space X to a the simplicial space Y with Y_n = map( n-simplex, X); in particular, this is a simplicial space in which every face and degeneracy map is a weak equivalence. There are clearly simplicial spaces which are not Reedy weak equivalent (i.e., levelwise weak equivalent) to ones of this form, so its not a Quillen equivalence. 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. >> >> 1') The same question, but with Top replaced by the category of >> simplicial spaces: >> >> With the Reedy model category structure on bisimplicial sets, is >> the "geometric realization" functor from bisimplicial sets to simplicial >> sets a Quillen equivalence? No. See above. >> >> 2) If two model categories are Quillen equivalent, are their >> categories of simplicial objects also Quillen equivalent, with the >> Reedy model category structure? (If not, someone should be fired.) Yes this is true. >> Finally, what I'd really like are some references that settle these >> questions! Goerss and Jardine's book discusses the Reedy model structure >> and a general concept of geometric realization, but I don't see these >> questions being addressed. There is a chapter on Reedy model structures in Hirschhorn's book on localization; question 2 is answered in Prop. 16.11.2 in the preliminary electronic version on my computer. Hovey's book Model Categories also deals with these notions, esp. in the chapter on "Framings". -- Charles ________________________________________________________________________ Subject: Re: two questions From: "Clark Barwick" Date: Mon, 16 Apr 2007 17:32:42 +0200 This is in response to the questions of John Baez on 15 April. The references I know for information about the Reedy model structure are Hovey's book, section 5.2, Hirschhorn's book, chapter 15, and the original paper of Reedy, which has been TeXed and is available from Hirschhorn's website < http://www-math.mit.edu/~psh/#Reedy>. There are also some terse remarks about them in section 22 of the Dwyer-Hirschhorn-Kan Smith book. I think virtually all of what if below is already in one of these references. Left Quillen functors f: M --> N induce left Quillen functors sf: sM --> sN on the Reedy model categories of simplicial objects, and if f is a Quillen equivalence, then sf is too. This follows easily from the "implicit description" of the Reedy model structure, ( 22.3 of DHKS). As far as I understand, the "geometric realization" functor from simplicial spaces to spaces is nothing more than the homotopy colimit over Delta^{op}. Let S be the category of simplicial sets with the usual model structure. The functor d^*: sS --> S (induced by the diagonal functor d: Delta --> Delta x Delta) provides a model for hocolim, essentially because it is left adjoint to the functor X |--> Map(Delta[-], X). It is left Quillen, but it is not a Quillen equivalence. (Observe that the functor colim: sS --> S is NOT left Quillen, for if it were, it would preserve weak equivalences!) Essentially by taking a left Bousfleld localization of the Reedy model category sS with respect to certain components of the unit of the adjunction (d^*, Map(Delta[-], -)), one obtains a model structure in which the cofibrations are the Reedy cofibrations, the weak equivalences are precisely those morphisms F --> G such that hocolim F --> hocolim G is a weak equivalence, and the fibrations are what they have to be. With this "hocolim" model structure, the functor d^*: sS --> S is now a Quillen equivalence. A homotopically equivalent way of doing all this is to use the _projective_ model structure on sS (with weak equivalences and fibrations defined objectwise), with the functor colim: sS --> S, which left Quillen, but, again, not a Quillen equivalence. By taking the Bousfield localization of this with respect to certain components of the unit of the adjunction (colim, const), one again obtains a hocolim model structure in which the cofibrations are the projective cofibrations, and the weak equivalences are again those morphisms F --> G such that hocolim F --> hocolim G is a weak equivalence. Now the functor colim is a Quillen equivalence. The advantage of this method is that now one may replace S with any left proper combinatorial model category, and the same arguments will work. I believe this was first observed by D. Dugger. Hope some of that was helpful. Best, Clark Barwick Matematisk Institutt Universitetet i Oslo Boks 1053 Blindern NO 0316 Oslo Norway Tel: +47 22 85 59 12 Fax: +47 22 85 43 49 clarkbar@gmail.com clarkbar@math.uio.no http://folk.uio.no/clarkbar ________________________________________________________________ Subject: Response to John Baez From: Philip Hirschhorn Date: Mon, 16 Apr 2007 20:50:05 -0400 (EDT) This is in response to John Baez's questions: >> Subject: simplicial spaces >> From: John Baez >> Date: Sun, 15 Apr 2007 12:53:35 -0700 >> >> Hi - >> >> I have a question about simplicial spaces: >> >> 1) With the Reedy model category structure on simplicial spaces, is >> the "geometric realization" functor from simplicial spaces to Top >> a Quillen equivalence? Alas, no. It is a left Quillen functor, though: This is Theorem 18.6.7 (on page 396) of my book "Model Categories and Their Localizations". To see that it's not a Quillen equivalence, see below. >> Some subsidiary questions: >> >> 1') The same question, but with Top replaced by the category of >> simplicial spaces: >> >> With the Reedy model category structure on bisimplicial sets, is >> the "geometric realization" functor from bisimplicial sets to simplicial >> sets a Quillen equivalence? Same answer. That Theorem 18.6.7 shows that for any simplicial model category M, the geometric realization functor from simplicial objects over M to M preserves cofibrations and trivial cofibrations. >> 2) If two model categories are Quillen equivalent, are their >> categories of simplicial objects also Quillen equivalent, with the >> Reedy model category structure? (If not, someone should be fired.) This one is yes, though: That's Proposition 15.4.1 (on page 294) of my book. -------------------------------------------------------------------- As for why those Quillen functors aren't Quillen equivalences: Let U be the functor from spaces to simplicial spaces that's the right adjoint of realization; if X is a space, UX in degree n is the space of maps from \Delta[n] to X (where \Delta[n] is a topological space if we're in Question 1 and a simplicial set if we're in Question 1'). We'll construct a cofibrant simplicial space B, a fibrant space X, and a weak equivalence B -> UX whose adjoint is not a Reedy weak equivalence. Let X be a fibrant version of a circle. (If we're talking actual topological spaces, let it be a plain old circle, since everything's fibrant.) For the simplicial space B, we start with a simplicial set A whose realization is the circle: Let it have a single vertex, a single nondegenerate 1-simplex, and everything else degenerate. We then let B be the simplicial space that in every simplicial degree is discrete, with B_n equal to A_n. That is, if we view B as a horizontal simplicial object of vertical spaces, and if we're talking simplicial sets, then each B_n is a constant vertical simplicial set with B_{n,k}=A_n. The realization of B is a circle; since X is fibrant, we can choose a weak equivalence f:|B| -> X. The adjoint of f is a map g:B->UX. In each simplicial degree, B is discrete, whereas UX in each simplicial degree is the space of maps from \Delta[n] to a circle and is thus weakly equivalent to a circle. Thus, g isn't a weak equivalence in any simplicial degree. -------------------------------------------------------------------- I hope that helps. In particular, I hope it's right. When I sat down to type this, I realized that what I had in my head was a bit off. I *think* the above isn't confused, but... Phil __________________________________________________________________ Subject: Re: simplicial spaces From: Tore August Kro Date: Tue, 17 Apr 2007 10:13:57 +0200 The answer to 1) is no! To see this let S be the right adjoint to geometric realization. Given a space Y, notice that SY in simplicial degree n is the topological space of maps from the topological n-simplex into Y, thus (SY)_n is homotopic to Y. Now consider a simplicial set X as a simplicial space. Observe that X is Reedy cofibrant, |X| is fibrant, and the identity map id:|X| -> |X| is a weak equivalence. However, if X has a non-degenerate simplex of positive dimension, then the counit of the adjunction, X -> S|X|, is not a pointwise weak equivalence (=Reedy weak equivalence). This gives a counterexample showing that geometric realization is not part of a Quillen equivelence between the Reedy model structure on simplicial spaces and Top. Similarly, the answer to 1') is also no! 2) Yes, see Hirschhorn, Model Categories and Their Localizations, Proposition 15.4.1. If you want a model structure on simplicial spaces (or bisimplicial sets) which is Quillen equivalent to the usual model structure on Top (or simplicial sets), you should have a look at the realization model structure considered by Rezk, Schwede and Shipley in their article Simplicial structures on model categories and functors, Amer.J.Math.123(2001), no.3, 551-575. Best, Tore A Kro