Equivalences of monoidal model categories

By S. Schwede, B. Shipley. 2002

256 citations recorded by the source corpus.

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, modules and algebras are Quillen equivalent to the associated model categories of connected differential graded rings, modules and algebras. We also show that our classification results from [Stable model categories are categories of modules, Topology, 42 (2003) 103-153] concerning stable model categories translate to any one of the known symmetric monoidal model categories of spectra.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/2611db1651e588bfe5b029442baa583dec2382e3

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

1. Bertrand Toën
2. S. Schwede
3. B. Keller
4. B. Shipley
5. Michael Shulman
6. David White
7. Fernando Muro
8. C. Berger

In Geometry and Topology

1. Bertrand Toën
2. S. Schwede
3. B. Keller
4. B. Shipley
5. M. Hill
6. David White
7. Michael Shulman
8. Fernando Muro

In Mathematical Physics

1. Bertrand Toën
2. S. Schwede
3. B. Keller
4. B. Shipley
5. Jacob Lurie
6. M. Hill
7. David White
8. Fernando Muro

Citations in the corpus

• Topological equivalences for differential graded algebras (2006, background): “If M and N are model categories, a Quillen pair L: M N: R will also be referred to as a Quillen map L: M! N. The terms ‘strong monoidal-’ and ‘weak monoidal Quillen pair’ will be used often—they are defined in [ SS3 , 3.6].”
• Homotopy Theory of Higher Categories: THE MODEL STRUCTURE (2010, background): “Our first axiom says that the unit object for the direct product is cofibrant, which is stronger than the unit axiom of [188].”
• K-THEORY AND DERIVED EQUIVALENCES (2002, background): “In particular, one gets a projective model structure onChA by lifting the one onChA across the equivalence; see [ SS3 , 6.1].”
• Moduli of objects in dg-categories (2005, background): “ial A-modules to quasi-isomorphisms. It even induces a fully faithful functor Ho(A−Mods) −→Ho(N(A) −Mod), whose essential image consists of all N(A)-dg-modules E such that Hi(E) = 0 for i > 0 (see [S-S3]). Any A ∈sk −CAlg possesses a natural augmentation A −→π0(A), and thus we can define a base ch”
• The Local Structure of Algebraic K-Theory (2012, background): “A more refined approach gives rise to (suitably monoidal) Quillen equivalences between HZ-modules, spectra of simplicial abelian groups and chain complexes, see Schwede and Shipley [254].”
• On the homotopy theory of enriched categories (2012, background): “gory: the objects of V−CatS are V-enriched categorieswith object-set S, and the morphisms of V−CatS are V-functors which are the identity on objects. The following proposition was shown in [4] (resp. [22, 16]) under the first (resp. second) hypothesis of the preceding proposition. Several other autho”
• Stable model categories are categories of modules (2003, methodology): “This is carried out in [58] and applied in [61] [19].”
• Derived algebraic geometry (2014, methodology): “The simplicial ring AU can be normalized to a commutative dg-algebra N(AU ), for which we can consider the category N(AU )-Mod of (unbounded) N(AU )-dg-modules (see [124] for more about the monoidal properties of the normalization functor).”
• On differential graded categories (2006, background): “Then the lemma shows that RG is an equivalence iff E is a quasi-equivalence.”
• Localization theorems in topological Hochschild homology and topological cyclic homology (2008, unknown): “Using the results of [28] extracted in Proposition 2.”
• HOMOTOPY LIMITS AND COLIMITS AND ENRICHED HOMOTOPY THEORY (2006, result): “For example, essentially the same result for the case V = M can be found in [ SS03a , 6.1].”
• On the nonexistence of elements of Kervaire invariant one (2009, methodology): “It is more or less tailor made to apply to the spectra K ′m. Proposition 5.22.”
• HZ -algebra spectra are differential graded algebras (2002, methodology): “ension to modules. In this section we construct Quillen equivalences between the categories of modules over differential graded algebras and HR-algebra spectra. Again, we use the criteria developed in [SS03b] to show that weak monoidal Quillen equivalences induce equivalences on these categories of m”
• Dwyer–Kan homotopy theory of enriched categories (2012, methodology): “S. Schwede and B. Shipley, ‘Equivalences of monoidal model categories’, Algebr.”