Equivalences of monoidal model categories

  title={Equivalences of monoidal model categories},
  author={Stefan Schwede and Brooke E. Shipley},
  journal={arXiv: Algebraic Topology},
We construct Quillen equivalences between the model categories of monoids (rings), modules and algebras over two Quillen equivalent model categories under certain conditions. This is a continuation of our earlier work where we established model categories of monoids, modules and algebras [Algebras and modules in monoidal model categories, Proc. London Math. Soc. 80 (2000), 491-511]. As an application we extend the Dold-Kan equivalence to show that the model categories of simplicial rings… 
Model structures on commutative monoids in general model categories
Coalgebras in the Dwyer-Kan localization of a model category
We show that weak monoidal Quillen equivalences induce equivalences of symmetric monoidal $\infty$-categories with respect to the Dwyer-Kan localization of the symmetric monoidal model categories.
Stable model categories are categories of modules
Classification of stable model categories
A stable model category is a setting for homotopy theory where the suspension functor is invertible. The prototypical examples are the category of spectra in the sense of stable homotopy theory and
Koszul duality for categories with a fixed object set
We define a notion of Koszul dual of a monoid object in a monoidal biclosed model category. Our construction generalizes the classic Yoneda algebra Ext A ( k , k ). We apply this general construction
We define the notion of an additive model category and prove that any stable, additive, combinatorial model category M has a model enrichment over Sp (sAb) (symmetric spectra based on simplicial
Admissible replacements for simplicial monoidal model categories.
Using Dugger's construction of universal model categories, we produce replacements for simplicial and combinatorial symmetric monoidal model categories with better operadic properties. Namely, these
We collect in one place a variety of known and folklore results in enriched model category theory and add a few new twists. The central theme is a general procedure for constructing a Quillen
Two results from Morita theory of stable model categories
We prove two results from Morita theory of stable model categories. Both can be regarded as topological versions of recent algebraic theorems. One is on recollements of triangulated categories, which
Enrichments of additive model categories
We prove that any stable, additive, combinatorial model category M has a canonical model enrichment over Sp(sAb) (symmetric spectra based on simplicial abelian groups). So to any object X ∈ M one can


Algebras and Modules in Monoidal Model Categories
In recent years the theory of structured ring spectra (formerly known as A∞‐ and E∞‐ring spectra) has been simplified by the discovery of categories of spectra with strictly associative and
Monoidal model categories
A monoidal model category is a model category with a compatible closed monoidal structure. Such things abound in nature; simplicial sets and chain complexes of abelian groups are examples. Given a
Operads, Algebras and Modules in General Model Categories
In this paper we develop the theory of operads, algebras and modules in cofibrantly generated symmetric monoidal model categories. We give J-semi model strucures, which are a slightly weaker version
S-modules and symmetric spectra
We study a symmetric monoidal adjoint functor pair between the category of S-modules and the category of symmetric spectra. The functors induce equivalences between the respective homotopy categories
Model Categories of Diagram Spectra
Working in the category T of based spaces, we give the basic theory of diagram spaces and diagram spectra. These are functorsD→T for a suitable small topological categoryD. WhenD is symmetric
An Algebraic Model for Rational S1‐Equivariant Stable Homotopy Theory
Greenlees dened an abelian categoryA whose derived category is equivalent to the rational S 1 -equivariant stable homotopy category whose objects represent rational S 1 - equivariant cohomology
HZ -algebra spectra are differential graded algebras
We show that the homotopy theory of differential graded algebras coincides with the homotopy theory of HZ-algebra spectra. Namely, we construct Quillen equivalences between the Quillen model
Monoidal Uniqueness of Stable Homotopy Theory
We show that the monoidal product on the stable homotopy category of spectra is essentially unique. This strengthens work of this author with Schwede on the uniqueness of models of the stable