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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeAug 15th 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItexBoth at [[string Lie 2-algebra]] and at [[nonabelian Lie algebra cohomology]] there is a (highlighted in both cases) gap in the displayed argument about homotopy pushouts. Effectively, the claim in both cases is that a pushout along a generating cofibration in the [[model structure on dg-algebras]] (for graded-commutative dg-algebras) is a homotopy pushout. I think one can see by hand that this is true, but I don't have a satisfactory proof yet. The statement would follow immediately if $dgAlg$ were a left proper model category -- for instance by the argument spelled out in detail at [[proper model category]], or even by simpler arguments . As emphasized in the article by Charles Rezk that is linked to at [[proper model category]], we have that the Quillen equivalent model structure of simplicial commutative algebras is indeed left proper (whereas the article focuses on the problem that for general non-commutative algebras this is not the case). Maybe using the explicit Quillen equivalence from Schwede-Schipley we can transfer the desired statement to dgAlg? There is an article by Mark Hovey on model structures on chain complexes where he says explicitly that he does not know if dgAlg is left proper. I don't actually need full left properness for the gap in the above two entries, just that pushout along generating cofibrations preserves weak equivalences would be sufficient. Probably it's even an easy exercise to work that out, but for the moment I'll just record that thought here. (Just came back from vacation and am in the process of getting back to speed.)

   Both at string Lie 2-algebra and at nonabelian Lie algebra cohomology there is a (highlighted in both cases) gap in the displayed argument about homotopy pushouts.

   Effectively, the claim in both cases is that a pushout along a generating cofibration in the model structure on dg-algebras (for graded-commutative dg-algebras) is a homotopy pushout.

   I think one can see by hand that this is true, but I don’t have a satisfactory proof yet.

   The statement would follow immediately if $dgAlg$ were a left proper model category – for instance by the argument spelled out in detail at proper model category, or even by simpler arguments .

   As emphasized in the article by Charles Rezk that is linked to at proper model category, we have that the Quillen equivalent model structure of simplicial commutative algebras is indeed left proper (whereas the article focuses on the problem that for general non-commutative algebras this is not the case). Maybe using the explicit Quillen equivalence from Schwede-Schipley we can transfer the desired statement to dgAlg?

   There is an article by Mark Hovey on model structures on chain complexes where he says explicitly that he does not know if dgAlg is left proper. I don’t actually need full left properness for the gap in the above two entries, just that pushout along generating cofibrations preserves weak equivalences would be sufficient. Probably it’s even an easy exercise to work that out, but for the moment I’ll just record that thought here.

   (Just came back from vacation and am in the process of getting back to speed.)

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeAug 15th 2010
   • (edited Aug 15th 2010)
   • PermaLink
   Author: Urs
   Format: MarkdownItexWait, I mixed up handedness in the above: for the argument in question I need non-negatively graded dg cochain algebras and their equivalent COsimplicial algebras. By <a href="http://arxiv.org/PS_cache/math/pdf/0306/0306289v3.pdf">Castiglioni-Cortinas</a> the dual-Dold-Kan functor $K : Ch^\bullet \to Ab^\Delta$ extends to a left Quillen functor $dgAlg \to Alg^\Delta$ which exhibits a Quillen equivalence. Using that, it should be sufficient to have that $Alg^\Delta$ is left proper (whereas we already know that $Alg^{\Delta^{op}}$ is, for commutative algebras).

   Wait, I mixed up handedness in the above: for the argument in question I need non-negatively graded dg cochain algebras and their equivalent COsimplicial algebras. By Castiglioni-Cortinas the dual-Dold-Kan functor $K : Ch^\bullet \to Ab^\Delta$ extends to a left Quillen functor $dgAlg \to Alg^\Delta$ which exhibits a Quillen equivalence. Using that, it should be sufficient to have that $Alg^\Delta$ is left proper (whereas we already know that $Alg^{\Delta^{op}}$ is, for commutative algebras).

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeAug 16th 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItexI think I got a decent solution, but have to go offline now. Will fill in the gaps of the two entries tomorrow. (Idea: embed $dgca^{op}$ via $(CAlg^\Delta)^{op}$ into simplicial presheaves on the site for the Cahiers topos, use that this is _right_ proper.)

   I think I got a decent solution, but have to go offline now. Will fill in the gaps of the two entries tomorrow.

   (Idea: embed $dgca^{op}$ via $(CAlg^\Delta)^{op}$ into simplicial presheaves on the site for the Cahiers topos, use that this is right proper.)

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeAug 16th 2010
   • (edited Aug 16th 2010)
   • PermaLink
   Author: Urs
   Format: MarkdownItexOkay, I filled in relevant discussion. First of all, I had remembered the issue at [[nonabelian Lie algebra cohomology]] incorrectly. In fact the gap there was no gap but a case of being dense: the left homotopy diagram in question is all fine already, since in $dgAlg^{op}$ all objects are cofibrant (and that's what is needed, not cofibrancy in $dgAlg$). I fixed the entry text accordingly. Then for the issue of homotopy fibers/extensions of $\infty$-Lie algebra cocycles, I added to [[∞-Lie algebra cohomology]] two new subsections with the relevant arguments. First in <a href="http://ncatlab.org/nlab/show/infinity-Lie+algebra+cohomology#Topos">(oo,1)-topos theoretic interpretation</a> I recall/summarize how we are to think of $\infty$-Lie algebras as sitting inside the $(\infty,1)$-topos of all synthetic $\infty$-Lie algebroids, and then in <a href="http://ncatlab.org/nlab/show/infinity-Lie+algebra+cohomology#Extensions">Extensions</a> I discuss how the "evident" pullbacks (such as appearing in the discussion at [[string Lie 2-algebra]]) indeed compute the homotopy fibers of the $\infty$-Lie algebra cocycles in the $(\infty,1)$-topos. There is plenty of room for polishing the discussion, and with a little luck I'll find the time to do so, but all the arguments are there.

   Okay, I filled in relevant discussion.

   First of all, I had remembered the issue at nonabelian Lie algebra cohomology incorrectly. In fact the gap there was no gap but a case of being dense: the left homotopy diagram in question is all fine already, since in $dgAlg^{op}$ all objects are cofibrant (and that’s what is needed, not cofibrancy in $dgAlg$). I fixed the entry text accordingly.

   Then for the issue of homotopy fibers/extensions of $\infty$-Lie algebra cocycles, I added to ∞-Lie algebra cohomology two new subsections with the relevant arguments.

   First in (oo,1)-topos theoretic interpretation I recall/summarize how we are to think of $\infty$-Lie algebras as sitting inside the $(\infty,1)$-topos of all synthetic $\infty$-Lie algebroids, and then in Extensions I discuss how the “evident” pullbacks (such as appearing in the discussion at string Lie 2-algebra) indeed compute the homotopy fibers of the $\infty$-Lie algebra cocycles in the $(\infty,1)$-topos.

   There is plenty of room for polishing the discussion, and with a little luck I’ll find the time to do so, but all the arguments are there.