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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeNov 11th 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItexam working on [[model structure on algebras over an operad]].

   am working on model structure on algebras over an operad.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeNov 17th 2010
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   Author: Urs
   Format: MarkdownItexin reply to an MO-qustion I added a <a href="http://ncatlab.org/nlab/show/model+structure+on+algebras+over+an+operad#InfCatOfMods">section</a> on the constructon of the simplicial category of $A_\infty$-algebras, $A_\infty$-bimodules, etc.

   in reply to an MO-qustion I added a section on the constructon of the simplicial category of -algebras, -bimodules, etc.

3. • CommentRowNumber3.
   • CommentAuthorDmitri Pavlov
   • CommentTimeAug 10th 2022
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   Author: Dmitri Pavlov
   Format: MarkdownItexAdded: ## Equivalence of model-categorical algebras and quasicategorical algebras The results of Lurie \cite{Lurie.HA}, Pavlov–Scholbach \cite{PavlovScholbach18}, and Haugseng \cite{Haugseng} establish an equivalence of [[quasicategories]] between the underlying quasicategory of the model category of algebras over an operad and the quasicategory of quasicategorical algebras over the underlying quasicategorical operad, provided some mild conditions are met. \begin{theorem} (Theorem 7.11 in Pavlov–Scholbach \cite{PavlovScholbach18}, Theorem 4.10 in Haugseng \cite{Haugseng}.) Suppose $V$ is a [[symmetric monoidal model category]] equipped with a subcategory $V^\flat$ of flat objects. Given a flat admissible Σ-cofibrant $V$-operad $O$, the canonical comparison functor $$Alg_O(V)^c[W_O^{-1}]\to Alg_O(V[W^{-1}])$$ is an equivalence of [[quasicategories]]. \end{theorem} Here a full subcategory $V^\flat\subset V$ is a __subcategory of flat objects__ (Haugseng \cite{Haugseng}, Definition 4.1) if it contains all cofibrant objects of $V$, is closed under monoidal products, and tensoring a weak equivalence with an object produce a weak equivalence (in $V^\flat$). Here a $V$-operad is __flat__ if it is enriched in the subcategory $V^\flat\subset V$. An operad is __admissible__ if the category of algebras admits a [[transferred model structure]]. An operad $O$ is __Σ-cofibrant__ if the unit map $1\to O(1)$ is a cofibration and the object $O(n)$ is cofibrant in the projective model structure on $\Sigma_n$-objects in $V$. By Remark 4.9 in Haugseng \cite{Haugseng}, a Σ-cofibrant operad is flat whenever the objects of unary endomorphisms $O(x,x)$ are flat. <a href="https://ncatlab.org/nlab/revision/diff/model+structure+on+algebras+over+an+operad/32">diff</a>, <a href="https://ncatlab.org/nlab/revision/model+structure+on+algebras+over+an+operad/32">v32</a>, <a href="https://ncatlab.org/nlab/show/model+structure+on+algebras+over+an+operad">current</a>

   Added:

   Equivalence of model-categorical algebras and quasicategorical algebras

   The results of Lurie \cite{Lurie.HA}, Pavlov–Scholbach \cite{PavlovScholbach18}, and Haugseng \cite{Haugseng} establish an equivalence of quasicategories between the underlying quasicategory of the model category of algebras over an operad and the quasicategory of quasicategorical algebras over the underlying quasicategorical operad, provided some mild conditions are met.

   \begin{theorem} (Theorem 7.11 in Pavlov–Scholbach \cite{PavlovScholbach18}, Theorem 4.10 in Haugseng \cite{Haugseng}.) Suppose is a symmetric monoidal model category equipped with a subcategory of flat objects. Given a flat admissible Σ-cofibrant -operad , the canonical comparison functor

   is an equivalence of quasicategories. \end{theorem}

   Here a full subcategory is a subcategory of flat objects (Haugseng \cite{Haugseng}, Definition 4.1) if it contains all cofibrant objects of , is closed under monoidal products, and tensoring a weak equivalence with an object produce a weak equivalence (in ).

   Here a -operad is flat if it is enriched in the subcategory .

   An operad is admissible if the category of algebras admits a transferred model structure.

   An operad is Σ-cofibrant if the unit map is a cofibration and the object is cofibrant in the projective model structure on -objects in .

   By Remark 4.9 in Haugseng \cite{Haugseng}, a Σ-cofibrant operad is flat whenever the objects of unary endomorphisms are flat.

   diff, v32, current