# New model Structure on $E_{\infty}$-algebras?

Let $\mathbf{sSet}$ be the category of simplicial sets. Is it possible to put a new model structure on $\mathrm{E}_{\infty}$-algebra (of simplicial sets) such that the weak equivalences and fibrations are those defined by Joyal for quasi-categories ($\infty$-categories)?

To be more clear, I'll explain the motivation. Lets start with some notations: $\mathbf{sSet}^{K}$ is the standard model structure on simplicial sets where the fibrant objects are Kan complexes and $E_{\infty}^{K}$ the standard $E_{\infty}$-operad on $\mathbf{sSet}^{K}$. Lets $\mathbf{M}$ be the category of $E_{\infty}^{K}$-algebras.

Lets $\mathbf{sSet}^{Q}$ be the category of simplicial sets with the Joyal model structure (quasi-categories).

First question: Suppose the $(\mathrm{C},\otimes)$ is symmetric monoidal enriched category over $\mathbf{sSet}$, is it true that the coherent nerve $N\mathrm{C}$ is an $E_{\infty}^{K}$-algebra?

second question: Can we put a model structure on $\mathbf{M}$ such that the weak equivalences (fibrations) are weak equivalences (fibrations) of $\mathbf{sSet}^{Q}$?

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Presumably, for this to be meaningful, one should require the $E_\infty$ operad to be Kan in each operadic degree. –  André Henriques Oct 6 '11 at 13:32
Thanks André, what I meant is just transferring the model structure from $\mathrm{sSet}$ with the Joyal model structure to the category of $E_{\infty}$-algebras (in $\mathrm{sSet}$) via the forgetful functor! maybe we need other assumptions as you motioned before! But maybe my question does not make sense for some obvious raisons that I don't understand yet! –  Gos Oct 6 '11 at 13:48
Maybe this comments is very naive but I want to be sure I understand the question. It seems to me that the projective model structure on $E$-algebras ($E$ an operad) in some symmetric monoidal model category $M$ exists if $M$ satisfies some reasonable conditions. Are you claiming that standard theorem don't work for simplicial sets with Joyal model structure ? –  Geoffroy Horel Oct 6 '11 at 14:07
I added some clarifications... –  Gos Oct 6 '11 at 15:02