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!
If not, I guess I have to do some work.
I hope it goes like this: you take the Moerdijk model structure
on bisimplicial sets and transfer it over to simplicial spaces via
the adjunction between these two categories. Is that right?
I'm far from expert at this stuff, but my friend Eugenia Cheng
pointed out theorem 3.10 in Clemens Berger's "A cellular nerve
for higher categories", which goes like this:
Let F, G be an adjoint pair such that the left adjoint F:C-->D preserves
finiteness. Assume that C is a closed model category finitely generated
by I and J, that D is cocomplete and finitely complete, and that G takes
any relative FJ-complex to a weak equivalence. Then (F,G) is a Quillen
pair for a closed model structure on D, which is finitely generated by FI
and FJ. The weak equivalences (resp. fibrations) of the transferred model
structure are the D-morphisms sent to weak equivalences (resp.
fibrations)
under G.
Is this the way I should try to get the desired model category
structure on simplicial spaces?
Best,
jb