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50
Universal homotopy theories
 Adv. Math
"... Abstract. Begin with a small category C. The goal of this short note is to point out that there is such a thing as a ‘universal model category built from C’. We describe applications of this to the study of homotopy colimits, the DwyerKan theory of framings, to sheaf theory, and to the homotopy the ..."
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Cited by 38 (3 self)
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Abstract. Begin with a small category C. The goal of this short note is to point out that there is such a thing as a ‘universal model category built from C’. We describe applications of this to the study of homotopy colimits, the DwyerKan theory of framings, to sheaf theory, and to the homotopy theory of schemes. Contents
Homotopical Algebraic Geometry I: Topos theory
, 2002
"... This is the first of a series of papers devoted to lay the foundations of Algebraic Geometry in homotopical and higher categorical contexts. In this first part we investigate a notion of higher topos. For this, we use Scategories (i.e. simplicially enriched categories) as models for certain kind of ..."
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Cited by 32 (20 self)
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This is the first of a series of papers devoted to lay the foundations of Algebraic Geometry in homotopical and higher categorical contexts. In this first part we investigate a notion of higher topos. For this, we use Scategories (i.e. simplicially enriched categories) as models for certain kind of ∞categories, and we develop the notions of Stopologies, Ssites and stacks over them. We prove in particular, that for an Scategory T endowed with an Stopology, there exists a model
HZalgebra spectra are differential graded algebras
 Amer. Jour. Math
, 2004
"... Abstract: We show that the homotopy theory of differential graded algebras coincides with the homotopy theory of HZalgebra spectra. Namely, we construct Quillen equivalences between the Quillen model categories of (unbounded) differential graded algebras and HZalgebra spectra. We also construct Qu ..."
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Cited by 32 (10 self)
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Abstract: We show that the homotopy theory of differential graded algebras coincides with the homotopy theory of HZalgebra spectra. Namely, we construct Quillen equivalences between the Quillen model categories of (unbounded) differential graded algebras and HZalgebra spectra. We also construct Quillen equivalences between the differential graded modules and module spectra over these algebras. We use these equivalences in turn to produce algebraic models for rational stable model categories. We show that bascially any rational stable model category is Quillen equivalent to modules over a differential graded Qalgebra (with many objects). 1.
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
Model structures and the Oka principle
 2004), 203–223. HOLOMORPHIC MAPPINGS 13
"... Stein manifolds, a prestack being a contravariant simplicial ..."
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Cited by 20 (6 self)
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Stein manifolds, a prestack being a contravariant simplicial
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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Cited by 19 (0 self)
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These notes are based on lectures given at the Workshop on Structured ring spectra and
Weak Factorization Systems and Topological Functors
 Appl. Categ. Struct
, 2000
"... Weak factorization systems, important in homotopy theory, are related to injective objects in comma{categories. Our main result is that full functors and topological functors form a weak factorization system in the category of small categories, and that this is not cobrantly generated. We also prese ..."
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Cited by 16 (6 self)
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Weak factorization systems, important in homotopy theory, are related to injective objects in comma{categories. Our main result is that full functors and topological functors form a weak factorization system in the category of small categories, and that this is not cobrantly generated. We also present a weak factorization system on the category of posets which is not cobrantly generated. No such weak factorization systems were known until recently. This answers an open problem posed by M. Hovey.
Homotopy limits and colimits and enriched homotopy theory
, 2006
"... Abstract. Homotopy limits and colimits are homotopical replacements for the usual limits and colimits of category theory, which can be approached either using classical explicit constructions or the modern abstract machinery of derived functors. Our first goal is to explain both and show their equiv ..."
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Cited by 15 (2 self)
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Abstract. Homotopy limits and colimits are homotopical replacements for the usual limits and colimits of category theory, which can be approached either using classical explicit constructions or the modern abstract machinery of derived functors. Our first goal is to explain both and show their equivalence. Our second goal is to generalize this result to enriched categories and homotopy weighted limits, showing that the classical explicit constructions still give the right answer in the abstract sense, thus partially bridging the gap between classical homotopy theory and modern abstract homotopy theory. To do this we introduce a notion of “enriched homotopical categories”, which are more general than enriched model categories, but are still a good place to do enriched homotopy theory. This demonstrates that the presence of enrichment often simplifies rather than complicates matters, and goes some way toward achieving a better understanding of “the role of homotopy in homotopy theory.” Contents
Homotopy theory of comodules over a Hopf algebroid
, 2003
"... Given a good homology theory E and a topological space X, E∗X is not just an E∗module but also a comodule over the Hopf algebroid (E∗, E∗E). We establish a framework for studying the homological algebra of comodules over a wellbehaved Hopf algebroid (A, Γ). That is, we construct ..."
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Cited by 13 (3 self)
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Given a good homology theory E and a topological space X, E∗X is not just an E∗module but also a comodule over the Hopf algebroid (E∗, E∗E). We establish a framework for studying the homological algebra of comodules over a wellbehaved Hopf algebroid (A, Γ). That is, we construct
LeftDetermined Model Categories and Universal Homotopy Theories
"... We say that a model category is leftdetermined if the weak equivalences are generated (in a sense specified below) by the cofibrations. While the model category of simplicial sets is not leftdetermined, we show that its nonoriented variant, the category of symmetric simplicial sets (in the sense ..."
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Cited by 13 (2 self)
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We say that a model category is leftdetermined if the weak equivalences are generated (in a sense specified below) by the cofibrations. While the model category of simplicial sets is not leftdetermined, we show that its nonoriented variant, the category of symmetric simplicial sets (in the sense of Lawvere and Grandis) carries a natural leftdetermined model category structure. This is used to give another and, as we believe simpler, proof of a recent result of D. Dugger about universal homotopy theories. 1