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Fixed the comments in the reference list at model structure on dg-algebras: Gelfand-Manin just discuss the commutative case. The noncommutative case seems to be due to the Jardine reference. Or does anyone know an earlier one?
I have a question at model structure on dg-algebras on the invariant characterization of commutative dg-algebras within all dg-algebras.
I added an "Idea" section to model structure on dg-algebras
I added a theorem from the book by Igor Kriz and Peter May about how commutative dg-algebras already exhaust, up to weak equivalence, all homotopy-cmmutative dg-algebras to model structure on dg-algebras.
(this might better fit into some other entry eventually, though)
Added the statement that the forgetful functor from commutative dg-algebras to all dg-algebras is the right adjoint part of a Quillen adjunction with respect to the given modelstructures.
created a section on cofibrations in CdgAlg and hence on Sullivan algebras.
Created entry for Dennis Sullivan in that context. Always nice to create an entry and see that the "timeline"-entry was already requesting it.
I suppose I'll have the following sorted out in a minute, but maybe somebody is quicker with helping me:
in the definition of a relative Suillvan algebra I suppose we do require that restricted to acts like plus a term that contains elements in V ?
I started at model structure on dg-algebras a new subsection on Unbounded dg-algebras, stating a basic existence result.
added a section Simplicial hom-objects to model structure on dg-algebras on the derived hom-spaces for unbounded commutative dg-algebras (over a field of char 0).
okay, I think I finally have assembled the full proof that the derived copowering of undbounded commutative dg-algebras (over field of char 0) over degreewise finite simplicial sets is given by the polynomial-differential-forms-on-simplices-functors. I have written this out now at Derived powering over sSet.
This provides the missing detail for the discussion that $\mathcal{O}(S^1) \simeq k \oplus k[-1]$ over at Hochschild cohomology. This is supposed to be all very obvious, but it took me a bit to assemble all the details properly, anyway.
I now moved other discussion of the model structure on commutative unbounded dg-algebras from the Hochschild entry over to model structure on dg-algebras. This mainly constituting the other new section Derived co-powering over sSet.
I added statement and proof that $([n] \mapsto Hom(A, B \otimes \Omega^\bullet_{poly}(\Delta[n])))$ is a Kan complex when $A$ is cofibrant.
Also added an earlier reference by Ginot et al where it is discussed that their “derived copowering” of cdgAlg over sSet indeed lands in commutative dg-algebras. I’ll try to add more technical details on how this works later on.
there was an old question of how (to which extent) the homotopy theory of commutative dg-algebras is homotopy-faithful inside that of all dg-algebras.
Recently there appeared some discussion of this issue in
Ilias Amrani, Comparing commutative and associative unbounded differential graded algebras over Q from homotopical point of view (arXiv:1401.7285)
Ilias Amrani, Rational homotopy theory of function spaces and Hochschild cohomology (arXiv:1406.6269)
I have added pointers to this here.
I have added to the section on the Bousfield-Gugenheim model structure a subsection on its simplicial hom-complexes which almost but not quite, make for a simplicial model category structure.
Added full publication data to:
Around Example 3.7, these authors make (somewhat implicitly) the observation that the Bousfield-Gugenheim model structure on connective rational dgc-algebras (which B&G and later Gelfand&Manin establish by laborious checks) is simply that right transferred from the projective model structure on chain complexes – which makes the proof that relative Sullivan models are cofibrations a triviality.
So this is all very nice, and highlighted as such in Hess’s recview. But neither of these authors states this as a theorem that could be properly cited as such, instead they leave it at side remarks. Is there any author who has published this in more citeable form?
added paragraph on the Quillen adjunction between simplicial sets and connective dgc-algebras
I have added (here) statement and proof of the change-of-scalars Quillen adjunction
$\big( dgcAlg^{\geq 0}_k \big)_{proj} \underoverset {\underset{res_{\mathbb{Q}}}{\longrightarrow}} {\overset{ (-) \otimes_{\mathbb{Q}} \mathbb{R} }{\longleftarrow}} {\bot_{\mathrlap{Qu}}} \big( dgcAlg^{\geq 0}_{\mathbb{Q}} \big)_{proj}$Changed $\mathbb{R}$ to $k$.
Ah, right. Thanks.
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