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    • 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 A_\infty-algebras, A 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 VV is a symmetric monoidal model category equipped with a subcategory V V^\flat of flat objects. Given a flat admissible Σ-cofibrant VV-operad OO, the canonical comparison functor

    Alg O(V) c[W O 1]Alg O(V[W 1])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 VV^\flat\subset V is a subcategory of flat objects (Haugseng \cite{Haugseng}, Definition 4.1) if it contains all cofibrant objects of VV, is closed under monoidal products, and tensoring a weak equivalence with an object produce a weak equivalence (in V V^\flat).

    Here a VV-operad is flat if it is enriched in the subcategory V VV^\flat\subset V.

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

    An operad OO is Σ-cofibrant if the unit map 1O(1)1\to O(1) is a cofibration and the object O(n)O(n) is cofibrant in the projective model structure on Σ n\Sigma_n-objects in VV.

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

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