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124
Algebraic geometry over model categories  A general approach to derived algebraic geometry
, 2001
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Model category structures on chain complexes of sheaves
 Trans. Amer. Math. Soc
"... of unbounded chain complexes, where the cofibrations are the injections. This folk theorem is apparently due to Joyal, and has been generalized recently ..."
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Cited by 27 (0 self)
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of unbounded chain complexes, where the cofibrations are the injections. This folk theorem is apparently due to Joyal, and has been generalized recently
Equivalences of monoidal model categories
 Algebr. Geom. Topol
, 2002
"... Abstract: 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 [ ..."
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Cited by 23 (8 self)
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Abstract: 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 [SS00]. As an application we extend the DoldKan 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 [SS] concerning stable model categories translate to any one of the known symmetric monoidal model categories of spectra. 1.
algebras and modules in general model categories, preprint arXiv:math.AT/0101102
, 2001
"... Abstract. In this paper we develop the theory of operads, algebras and modules in cofibrantly generated symmetric monoidal model categories. We give Jsemi model structures, which are a slightly weaker version of model structures, for operads and algebras and model structures for modules. We prove h ..."
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Abstract. In this paper we develop the theory of operads, algebras and modules in cofibrantly generated symmetric monoidal model categories. We give Jsemi model structures, which are a slightly weaker version of model structures, for operads and algebras and model structures for modules. We prove homotopy invariance properties for the categories of algebras and modules. In a second part we develop the theory of Smodules and algebras of [EKMM] and [KM], which allows a general homotopy theory for commutative algebras and pseudo unital symmetric monoidal categories of modules over them. Finally we prove a base change and projection formula.
ON DAVISJANUSZKIEWICZ HOMOTOPY TYPES II; COMPLETION AND GLOBALISATION
, 2009
"... For any finite simplicial complex K, Davis and Januszkiewicz have defined a family of homotopy equivalent CWcomplexes whose integral cohomology rings are isomorphic to the StanleyReisner algebra of K. Subsequently, Buchstaber and Panov gave an alternative construction, which they showed to be hom ..."
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Cited by 19 (6 self)
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For any finite simplicial complex K, Davis and Januszkiewicz have defined a family of homotopy equivalent CWcomplexes whose integral cohomology rings are isomorphic to the StanleyReisner algebra of K. Subsequently, Buchstaber and Panov gave an alternative construction, which they showed to be homotopy equivalent to the original examples. It is therefore natural to investigate the extent to which the homotopy type of a space X is determined by such a cohomology ring. Having analysed this problem rationally in Part I, we here consider it prime by prime, and utilise Lannes’ T functor and BousfieldKan type obstruction theory to study the pcompletion of X. We find the situation to be more subtle than for rationalisation, and confirm the uniqueness of the completion whenever K is a join of skeleta of simplices. We apply our results to the global problem by appealing to Sullivan’s arithmetic square, and deduce integral uniqueness whenever the StanleyReisner algebra is a complete intersection.
Morita theory in abelian, derived and stable model categories, Structured ring spectra
 London Math. Soc. Lecture Note Ser
, 2004
"... These notes are based on lectures given at the Workshop on Structured ring spectra and ..."
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These notes are based on lectures given at the Workshop on Structured ring spectra and
Resolution of coloured operads and rectification of homotopy algebras
 CONTEMPORARY MATHEMATICS
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Simplicial Structures on Model Categories and Functors
 Amer.J.Math.123
, 2001
"... We produce a highly structured way of associating a simplicial category to a model category which improves on work of Dwyer and Kan and answers a question of Hovey. We show that model categories satisfying a certain axiom are Quillen equivalent to simplicial model categories. A simplicial model cate ..."
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Cited by 15 (3 self)
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We produce a highly structured way of associating a simplicial category to a model category which improves on work of Dwyer and Kan and answers a question of Hovey. We show that model categories satisfying a certain axiom are Quillen equivalent to simplicial model categories. A simplicial model category provides higher order structure such as composable mapping spaces and homotopy colimits. We also show that certain homotopy invariant functors can be replaced by weakly equivalent simplicial, or "continuous," functors. This is used to show that if a simplicial model category structure exists on a model category then it is unique up to simplicial Quillen equivalence.