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Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeNov 11th 2010
• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeNov 17th 2010

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

• CommentRowNumber3.
• CommentAuthorDmitri Pavlov
• CommentTimeAug 10th 2022

## 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 $O$ 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.