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.