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