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I expanded proper model category a bit.
In particular I added statement and (simple) proof that in a left proper model category pushouts along cofibrations out of cofibrants are homotopy pushouts. This is at Proper model category -- properties
On page 9 here Clark Barwick supposedly proves the stronger statement that pushouts along all cofibrations in a left proper model category are homotopy pushouts, but for the time being I am failing to follow his proof.
(??)
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<p>I wrote:</p>
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but for the time being I am failing to follow his proof.
</blockquote>
<p>Oh, I get it. I was being stupid. Will add the statement and proof now.</p>
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okay, here now -- in full beauty -- the proof that pushouts along cofibrations in left proper model categories are homotopy pushouts.
Added more examples and counter-examples to proper model category.
Stated Charles Rezk's theorem about passing to proper Quillen equivalent models for simplicial algebras over simplicial theories. Also stated Thomas Nikolaus' theorem about Quillen equivalent models of fibrant objects (which are in particular right proper).
Made the statement that “all objects (co)fibrant” implies (left)right properness more explicit in Properties and added a citation.
I added to proper model category the result discussed at this MO question that to prove right properness we are free to assume the base object of the pullback to be fibrant.
Which of the standard dg-algebra categories are proper, such as dg-(co)algebras or dg-Lie algebras?
[edit: I see here on MO discussion that the projective model structure on unbounded dgc-algebras is proper.]
BTW, there is a bug in this page: the link “General” from the table of contents does not work. I think the problem is that there is also a heading hidden in the “Context” menu called “general”, and so that’s the one that gets linked to. Is there anything that we can do about this? E.g. could the software detect duplicate header names and disambiguate them somehow, say with numbers?
Thanks for raising this. I will try to remember to look into it when I get the chance. It will not be for at least a few days.
changed this slightly, from
to
Added by Urs Schreiber in Revision 8 on March 25, 2010.
If the standard Dwyer-Kan-Bergner model structure is used, then few simplicial categories are cofibrant.
I am not sure if one can enlarge the class of cofibrations to all (local) monomorphisms (say), which would make all objects cofibrant, but it seems difficult: acyclic cofibrations must be closed under cobase changes, and the latter are rather complicated in simplicial categories.
Thanks for the alert. Checking, I don’t find the paragraph which this is about. I guess you guys fixed it? But I also don’t see it in that revision 8 of “proper model category”.
(BTW, you can link to specific points in an entry by adding an anchor.)
No, it is still here: https://ncatlab.org/nlab/show/proper+model+category#left_proper_model_categories
Thanks!
So I have replaced the line with
Checking whether the error was induced from an error at model structure on sSet-categories… But that speaks of right-properness (here)
Added:
The following criterion shows that the notion of left or right properness only depends on the underlying relative category of a model category, i.e., does not depend on fibrations or cofibrations. This is clear once we observe that the notion of a Quillen equivalence in the statement below can be replaced by the notion of a Dwyer–Kan equivalence of underlying relative categories.
\begin{theorem} (Rezk \cite{Rezk02}, Proposition 2.7 (arXiv), Proposition 2.5 (journal).) A model category $M$ is left proper if and only if for every weak equivalence $f\colon X\to Y$ the induced Quillen adjunction
$X/M\leftrightarrows Y/M$is a Quillen equivalence. A model category $M$ is right proper if and only if for every weak equivalence $f\colon X\to Y$ the induced Quillen adjunction
$M/X\leftrightarrows M/Y$is a Quillen equivalence. \end{theorem}
added pointer to:
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