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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeNov 11th 2009

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?

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeNov 11th 2009

I have a question at model structure on dg-algebras on the invariant characterization of commutative dg-algebras within all dg-algebras.

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeNov 11th 2009

I added an "Idea" section to model structure on dg-algebras

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeNov 11th 2009
• (edited Nov 11th 2009)

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)

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeNov 11th 2009

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.

• CommentRowNumber6.
• CommentAuthorzskoda
• CommentTimeNov 11th 2009
I would look into Hinich's 1995 or so paper, for a possible earlier reference.
• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeNov 12th 2009
• (edited Nov 12th 2009)

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 $(A,d) \hookrightarrow (A \otimes \wedge V, d')$ I suppose we do require that $d'$ restricted to $A$ acts like $d$ plus a term that contains elements in V ?

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeNov 29th 2010

I started at model structure on dg-algebras a new subsection on Unbounded dg-algebras, stating a basic existence result.

• CommentRowNumber9.
• CommentAuthorUrs
• CommentTimeDec 9th 2010

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).

• CommentRowNumber10.
• CommentAuthorUrs
• CommentTimeDec 9th 2010
• (edited Dec 9th 2010)

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.

• CommentRowNumber11.
• CommentAuthorUrs
• CommentTimeDec 10th 2010

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.

• CommentRowNumber12.
• CommentAuthorUrs
• CommentTimeJul 16th 2014
• (edited Jul 16th 2014)

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

I have added pointers to this here.

• CommentRowNumber13.
• CommentAuthorUrs
• CommentTimeFeb 21st 2017
• (edited Feb 21st 2017)

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.

• CommentRowNumber14.
• CommentAuthorUrs
• CommentTimeAug 19th 2020

trying to bring some order into the list of references, adding some subsections…

• CommentRowNumber15.
• CommentAuthorUrs
• CommentTimeAug 19th 2020

have equipped more of the Definitions/Propositions with pointers to page-and-verse in Bousfield-Gugenheim and in Gelfand-Manin.

• CommentRowNumber16.
• CommentAuthorUrs
• CommentTimeAug 21st 2020
• (edited Aug 21st 2020)

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?

• CommentRowNumber17.
• CommentAuthorUrs
• CommentTimeAug 23rd 2020
• CommentRowNumber18.
• CommentAuthorUrs
• CommentTimeSep 1st 2020

added statement (here) that quasi-isos are preserved by pushout along relative Sullivan algebras

• CommentRowNumber19.
• CommentAuthorUrs
• CommentTimeJul 7th 2021

I have added (here) statement and proof of the change-of-scalars Quillen adjunction

$\big( dgcAlg^{\geq 0}_k \big)_{proj} \underoverset {\underset{res_{\mathbb{Q}}}{\longrightarrow}} {\overset{ (-) \otimes_{\mathbb{Q}} \mathbb{R} }{\longleftarrow}} {\bot_{\mathrlap{Qu}}} \big( dgcAlg^{\geq 0}_{\mathbb{Q}} \big)_{proj}$
• CommentRowNumber20.
• CommentAuthorTodd_Trimble
• CommentTimeJul 11th 2021

Changed $\mathbb{R}$ to $k$.

• CommentRowNumber21.
• CommentAuthorUrs
• CommentTimeJul 11th 2021

Ah, right. Thanks.