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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeOct 7th 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItexsplit off [[model structure on simplicial T-algebras]] from [[(infinity,1)-algebraic theory]]

   split off model structure on simplicial T-algebras from (infinity,1)-algebraic theory

2. • CommentRowNumber2.
   • CommentAuthoradeelkh
   • CommentTimeNov 7th 2014
   • (edited Nov 7th 2014)
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   Author: adeelkh
   Format: MarkdownItexAdded a reference to section 3.1 of the paper * [[Stefan Schwede]], _Spectra in model categories and applications to the algebraic cotangent complex_, Journal of Pure and Applied Algebra 120 (1997), pp. 77-104, [pdf](http://www.math.uni-bonn.de/people/schwede/modelspec.pdf). which discusses properness of the model structure in question. (Thanks to [this MO question](http://mathoverflow.net/questions/186463/proper-model-category).)

   Added a reference to section 3.1 of the paper

   • Stefan Schwede, Spectra in model categories and applications to the algebraic cotangent complex, Journal of Pure and Applied Algebra 120 (1997), pp. 77-104, pdf.

   which discusses properness of the model structure in question. (Thanks to this MO question.)

3. • CommentRowNumber3.
   • CommentAuthoradeelkh
   • CommentTimeMar 15th 2015
   • (edited Mar 15th 2015)
   • PermaLink
   Author: adeelkh
   Format: MarkdownItexAt [[model structure on simplicial algebras]], it says that it is possible to describe the $\infty$-category of simplicial commutative rings as the $\infty$-category of $(\infty,1)$-algebras over the algebraic theory of commutative rings. I don't know the first thing about algebraic theories, so I am wondering: is it also possible to recover the symmetric monoidal structure on the $\infty$-category of simplicial commutative rings via this description? Is it also possible to talk about simplicial modules and simplicial algebras in this language? I am wondering because this would seem to get around the problem of not being able to talk about _strictly_ commutative monoids in the $\infty$-category of simplicial abelian groups.

   At model structure on simplicial algebras, it says that it is possible to describe the $\infty$-category of simplicial commutative rings as the $\infty$-category of $(\infty,1)$-algebras over the algebraic theory of commutative rings. I don’t know the first thing about algebraic theories, so I am wondering: is it also possible to recover the symmetric monoidal structure on the $\infty$-category of simplicial commutative rings via this description? Is it also possible to talk about simplicial modules and simplicial algebras in this language?

   I am wondering because this would seem to get around the problem of not being able to talk about strictly commutative monoids in the $\infty$-category of simplicial abelian groups.

4. • CommentRowNumber4.
   • CommentAuthorDmitri Pavlov
   • CommentTimeMar 15th 2015
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   Author: Dmitri Pavlov
   Format: MarkdownItex>is it also possible to recover the symmetric monoidal structure on the ∞-category of simplicial commutative rings via this description Isn't the monoidal structure just the (derived) coproduct? In any case, any category of ∞-algebras over an E_∞-commutative algebraic theory has a symmetric monoidal structure, in the same way as in the 1-categorical case (see [[commutative algebraic theory]]). >Is it also possible to talk about simplicial modules and simplicial algebras in this language? Yes, simplicial modules and algebras are ∞-algebras for the corresponding algebraic theories. An underlying result in all these cases is that a homotopy coherent algebraic structure can always be rectified to a strict one. For simplicial sets or abelian groups this always true as long as no symmetries are involved (which is the case for modules and associative rings). In fact, for algebraic structures that are specified as ∞-algebras over an algebraic theory this is always true, this is a theorem due to Badzioch, I believe. >strictly commutative monoids in the ∞-category of simplicial abelian groups Strictly commutative monoids are (Quillen equivalent to) ∞-algebras over the algebraic theory of commutative monoids whereas E_∞ commutative monoids are ∞-algebras over the [[(2,1)-algebraic theory of E-infinity algebras]].

   is it also possible to recover the symmetric monoidal structure on the ∞-category of simplicial commutative rings via this description

   Isn’t the monoidal structure just the (derived) coproduct?

   In any case, any category of ∞-algebras over an E_∞-commutative algebraic theory has a symmetric monoidal structure, in the same way as in the 1-categorical case (see commutative algebraic theory).

   Is it also possible to talk about simplicial modules and simplicial algebras in this language?

   Yes, simplicial modules and algebras are ∞-algebras for the corresponding algebraic theories. An underlying result in all these cases is that a homotopy coherent algebraic structure can always be rectified to a strict one. For simplicial sets or abelian groups this always true as long as no symmetries are involved (which is the case for modules and associative rings).

   In fact, for algebraic structures that are specified as ∞-algebras over an algebraic theory this is always true, this is a theorem due to Badzioch, I believe.

   strictly commutative monoids in the ∞-category of simplicial abelian groups

   Strictly commutative monoids are (Quillen equivalent to) ∞-algebras over the algebraic theory of commutative monoids whereas E_∞ commutative monoids are ∞-algebras over the (2,1)-algebraic theory of E-infinity algebras.

5. • CommentRowNumber5.
   • CommentAuthoradeelkh
   • CommentTimeMar 15th 2015
   • (edited Mar 15th 2015)
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   Author: adeelkh
   Format: MarkdownItexThanks! Is there any reference that discusses algebras and modules in this language? I just want to be sure that I get the kind of general results one finds in Higher Algebra in this framework as well. For example, are there abstract nonsense results of the type that tell me that $Alg(A) \to Mod(A)$ creates limits and sifted colimits (for $A$ a simplicial commutative ring)? Edit: also, do I get the free-forgetful adjunctions for free this way?

   Thanks! Is there any reference that discusses algebras and modules in this language? I just want to be sure that I get the kind of general results one finds in Higher Algebra in this framework as well. For example, are there abstract nonsense results of the type that tell me that $Alg(A) \to Mod(A)$ creates limits and sifted colimits (for $A$ a simplicial commutative ring)?

   Edit: also, do I get the free-forgetful adjunctions for free this way?

6. • CommentRowNumber6.
   • CommentAuthorDmitri Pavlov
   • CommentTimeMar 15th 2015
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItex>Alg(A)→Mod(A)Alg(A) \to Mod(A) creates limits and sifted colimits (for AA a simplicial commutative ring)? In this case one can simply use that the forgetful functors from Alg(A) and Mod(A) to spaces create limits and sifted colimits. >do I get the free-forgetful adjunctions for free this way? Yes, every algebraic ∞-theory induced a free-forgetful adjunction, see, for example, the paper by Cranch “Algebraic theories and (∞,1)-categories”, Proposition 3.15.

   Alg(A)→Mod(A)Alg(A) \to Mod(A) creates limits and sifted colimits (for AA a simplicial commutative ring)?

   In this case one can simply use that the forgetful functors from Alg(A) and Mod(A) to spaces create limits and sifted colimits.

   do I get the free-forgetful adjunctions for free this way?

   Yes, every algebraic ∞-theory induced a free-forgetful adjunction, see, for example, the paper by Cranch “Algebraic theories and (∞,1)-categories”, Proposition 3.15.

7. • CommentRowNumber7.
   • CommentAuthoradeelkh
   • CommentTimeMar 16th 2015
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   Author: adeelkh
   Format: MarkdownItexPerfect, thanks! This seems to be just what I was looking for.

   Perfect, thanks! This seems to be just what I was looking for.

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeFeb 22nd 2017
   • (edited Feb 22nd 2017)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have added pointers to work by Reedy, Schwänzl-Vogt and Schwede and then added a section _[For algebras over simplicial Lawvere theories](https://ncatlab.org/nlab/show/model+structure+on+simplicial+algebras#ForAlgebrasOverSimplicialLawvereTheories)_, so far stating Reedy's result, that the projective model structure on the simplicial $T$-algebras in this generality exists and is simplicial.

   I have added pointers to work by Reedy, Schwänzl-Vogt and Schwede and then added a section For algebras over simplicial Lawvere theories, so far stating Reedy’s result, that the projective model structure on the simplicial $T$-algebras in this generality exists and is simplicial.

9. • CommentRowNumber9.
   • CommentAuthorDmitri Pavlov
   • CommentTimeFeb 22nd 2017
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   Author: Dmitri Pavlov
   Format: MarkdownItexThe reference to Bergner's paper states: > The fact that the model structure on simplicial T T-algebras serves to model ∞ \infty-algebras is in >Julie Bergner, Rigidification of algebras over multi-sorted theories , Algebraic and Geometric Topoogy 7, 2007. > > The Quillen equivalence to the model structure on homotopy T T-algebras is in >Bernard Badzioch, Algebraic theories in homotopy theory, Annals of Mathematics, 155 (2002), 895-913 (JSTOR) It appears to me that Bergner's paper is nothing else than Badzioch's paper extended to multisorted theories. There is nothing about ∞-categories or ∞-algebras in Bergner's paper.

   The reference to Bergner’s paper states:

   The fact that the model structure on simplicial T T-algebras serves to model ∞ \infty-algebras is in Julie Bergner, Rigidification of algebras over multi-sorted theories , Algebraic and Geometric Topoogy 7, 2007.

   The Quillen equivalence to the model structure on homotopy T T-algebras is in Bernard Badzioch, Algebraic theories in homotopy theory, Annals of Mathematics, 155 (2002), 895-913 (JSTOR)

   It appears to me that Bergner’s paper is nothing else than Badzioch’s paper extended to multisorted theories. There is nothing about ∞-categories or ∞-algebras in Bergner’s paper.

10. • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeFeb 22nd 2017
    • (edited Feb 22nd 2017)
    • PermaLink
    Author: Urs
    Format: MarkdownItexThanks for the alert! I forget what happened there, as this is a line from years ago. But I have fixed it now, both in the references and below prop. 2.2. (The whole entry would deserve to be expanded and polished more. But I won't do it at the moment.)

    Thanks for the alert! I forget what happened there, as this is a line from years ago. But I have fixed it now, both in the references and below prop. 2.2.

    (The whole entry would deserve to be expanded and polished more. But I won’t do it at the moment.)

11. • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeMar 1st 2017
    • (edited Mar 1st 2017)
    • PermaLink
    Author: Urs
    Format: MarkdownItexI have added statement of Quillen's recognition theorem [here](https://ncatlab.org/nlab/show/model+structure+on+simplicial+algebras#QuillenCriterion) and statement of the example of simplicial complete rational Hopf algebras ([here](https://ncatlab.org/nlab/show/model+structure+on+simplicial+algebras#SimplicialCommutativeHopfAlgebras))

    I have added statement of Quillen’s recognition theorem here and statement of the example of simplicial complete rational Hopf algebras (here)