Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Username
• Password
• Remember me

• Sign in using OpenID
• Remember me

• Forgot your password?
• Apply for membership

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nforum nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf sheaves simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

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.