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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.
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