Not signed in (Sign In)

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundle bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

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

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • 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

    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 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.

    diff, v32, current