New model Structure on $E_{\infty}$-algebras? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T09:36:42Z http://mathoverflow.net/feeds/question/77350 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/77350/new-model-structure-on-e-infty-algebras New model Structure on $E_{\infty}$-algebras? Ilias 2011-10-06T10:44:20Z 2011-10-06T15:01:39Z <p>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)?</p> <p>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.</p> <p>Lets $\mathbf{sSet}^{Q}$ be the category of simplicial sets with the Joyal model structure (quasi-categories).</p> <p>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?</p> <p>second question: Can we put a model structure on $\mathbf{M}$ such that the weak equivalences (fibrations) are weak equivalences (fibrations) of $\mathbf{sSet}^{Q}$?</p>