[ALGTOP-L] Fwd: Re: Steve's question, Retraction of my last email
Steven R. Costenoble
Steven.R.Costenoble at Hofstra.edu
Mon Oct 5 17:43:20 EDT 2009
Thanks, Don. I believe that gives me what I was looking for. The key
is being able to construct a cylinder for B that works, and that's
what I was missing.
I'd also like to thank Mark Hovey, who sketched out a similar argument
in private email.
I'll end up using this as a step in something else I'm working on, but
I was also curious for the following reason: Shwanzl & Vogt showed
that you have to restrict to either strict cofibrations or strict
fibrations to get a model category on spaces using homotopy
equivalences as your weak equivalences, and similarly for simplicial
categories and other similar enriched categories. (Peter alluded to
this and I learned about it first from the treatment in his book with
Sigurdsson on parametrized homotopy theory. He'll probably know what
project I'm working on, again.) The question I asked and the argument
Don gives show that we're not requiring more than must be true in any
model category, which is nice.
--Steve
On Oct 5, 2009, at 12:21 PM, Donald Stanley wrote:
>
>
>>>> Donald Stanley 10/5/2009 8:35 AM >>>
> Below I've attached the email I send to Steve yesterday.
> It gives an answer to Steve's question if the model category is
> proper or
> the space A is cofibrant. I (mistakenly) thought the fact that
> B \coprod_{A} AxI -> BxI is a cofibration was not so hard so I
> didn't prove that part.
> Here's a proof.
>
> I use the notation CylA \cup B instead of B \coprod_{A} AxI
>
> Consider the diagram
>
> A v A --f-> Cyl A
> | |
> v v
> A --> A v B --g-> Cyl A \cup B
> | | |
> j k m
> v v v
> B --> B v B --h-> B\cup Cyl A\cup B
>
>
> All squares are pushouts. Since f is a cofibration so are g and h.
> Since
> j is a cofibration so are k and m.
>
> The maps CylA --> A --> B and BvB -fold-> B agree on AvA and hence
> induce a map p:B\cup Cyl A\cup B--> B.
>
> Factor p into an cofibraion i:B\cup Cyl A\cup B---> Cyl B followed
> by an acyclic fibration.
> Then Cyl B is a valid cylinder object for B. (Since ih is a
> cofibration)
> Also im Cyl A \cup B -> Cyl B is a composition of cofibrations and
> thus a cofibration.
> It is the cofibration B \coprod_{A} AxI -> BxI (in Phil's notation)
>
> Don Stanley
>
>
>
>>>> Philip Hirschhorn <psh at math.mit.edu> 10/4/2009 11:01 PM >>>
> My apologies: My last email was a hallucination. That last assertion
> about how the second of the two maps must be a trivial cofibration
> used what I thought was a very clever and roundabout proof, that I
> even thought I'd used somewhere in my book. I couldn't find it
> anywhere in my book, and when I tried writing it down I found that it
> doesn't actually work unless you know that map to be a cofibration
> (which, of course, is the only hard part in what's asked for here).
>
> Please ignore my previous email. In the future, I'll try not to post
> things I haven't written down and thought about a bit more than I did
> here.
>
> Sorry about that.
>
> Phil
>
>
> From: Donald Stanley
> To: Steven.R.Costenoble at hofstra.edu
> Date: 10/4/2009 10:23 AM
> Subject: Re: [ALGTOP-L] Strong fibrations in model categories
>
> Hi Steve,
>
> I guess this is true if A is cofibrant or the model category is
> proper.
> Here is a sketch of the proof.
>
> Proper (actually this is left or right proper I forget which) says
> that if
>
> C -f->D
> | |
> g s
> v v
> H-t-> K
>
> is a push out, f is a cofibration and g is a weak equivalence then s
> is a weak equivalence.
>
> Then you can take the pushout
>
> A --> Cyl A
> | |
> v v
> B--->Cyl A \cup B
>
> and the lower horizontal map will be a weak equivalence
> this will imply that the map Cyl A\cup B into Cyl B will be a weak
> equivalence and hence
> an acyclic cofibration. I left out a few (standard?) details on that
> point.
> If A is cofibrant in any model category then the top map will be an
> acyclic cofibration so the bottom map
> will also be an acyclic cofibration and in particular a weak
> equivalence.
>
> Now your lifting problem is just a lifting in the diagram
>
> CylA \cup B -> X
> | |
> V V
> CylB --------->Y
>
> This has a lift from the definitions of model category.
>
> Best regards,
> Don Stanley
>
>>>> "Steven R. Costenoble" <Steven.R.Costenoble at hofstra.edu> 10/04/09
>>>> 5:59 AM >>>
> All,
>
> I'm probably missing an embarrassingly obvious proof or
> counterexample, but the following question is currently standing in my
> way: It's a standard and easy result to show in any model category
> that fibrations have the homotopy lifting property with respect to
> cofibrant objects. Is it true that they must also have the relative
> homotopy lifting property? By that I mean the following. Suppose that
> p: X -> Y is a fibration and i: A -> B is a cofibration. Assume also
> that A and B are cofibrant and X and Y are fibrant if necessary.
> Suppose given maps alpha: A -> X and beta: B -> Y, a lifting gamma:
> B -
>> X of beta, a homotopy H of alpha and an extension of pH to a
> homotopy K of beta. Does there necessarily exist a lifting of K to X
> starting at gamma and extending H? (Here, homotopies are general model
> category homotopies.)
>
> This is essentially asking if fibrations are always strong fibrations,
> in a sense generalizing that used by Schwanzl & Vogt in their "Strong
> cofibrations and fibrations in enriched categories." Their results
> show that this will be true in model categories of a certain kind,
> including categories of topological spaces with homotopy equivalences
> as the weak equivalences.
>
> Thanks for any proofs, counterexamples, or cites.
>
> --Steve
>
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>
>
> On Sun, 4 Oct 2009, Philip Hirschhorn wrote:
>
>> This is a continuation of Peter's answer to Steve: For something more
>> general, we first need to say what we mean by extending the homotopy
>> from A to B.
>>
>> The problem is that a random left homotopy can be defined using any
>> old cylinder object on the domain, so what does it mean to "extend"
>> the homotopy? I'll define that to mean that, in addition to having
>> the cofibration A -> B, we also have natural cylinder objects, which
>> I'll call AxI and BxI (not that either of them is constructed as a
>> product with an object named I) (That is, we have a map AxI -> BxI
>> that commutes with the structure of a cylinder object). We then have
>> a homotopy
>>
>> H: AxI -> X
>>
>> and an "extension" K: BxI -> Y of the composition pH. As Peter said,
>> we need to show that the map
>>
>> B \coprod_{A} AxI -> BxI
>>
>> is a trivial cofibration. The composition of that map with the
>> inclusion B -> B \coprod_{A} AxI is the inclusion B -> BxI. Since B
>> is cofibrant, the inclusion B -> BxI is a trivial cofibration. The
>> map B -> B \coprod_{A} AxI is also a trivial cofibration, because it
>> is the pushout of the map A -> AxI (and A is cofibrant). Thus, the
>> composition of the two maps is a trivial cofibration and the first of
>> the two maps is a trivial cofibration; this implies that the second
>> of
>> the two maps is a trivial cofibration.
>>
>> Phil
>>
>>
>> On Sun, 4 Oct 2009, Peter May wrote:
>>
>>> Steve is asking for a model theoretic version of the
>>> standard topological result that the inclusion of the
>>> mapping cylinder Mi of a cofibration i: A\to B in the
>>> cylinder BxI is an acyclic cofibration. One general
>>> situation where this holds is in a V-model category M,
>>> where V is a monoidal model category with a unit object
>>> U with a good cylinder object I. Then the analogue of the
>>> inclusion Mi \to BxI is the pushout product associated to
>>> the cofibrations
>>>
>>> i\colon A\to B and i_0\colon U\to I,
>>>
>>> the latter of which is of course acyclic. I imagine Steve
>>> is asking for something more general.
>>>
>>> Peter
>>>
>>>
>>>
>>> _______________________________________________
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>>> ALGTOP-L at lists.lehigh.edu
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>>>
>>
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