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14
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
A generalization of Quillen’s small object argument
 J. Pure Appl. Algebra
"... Abstract. We generalize the small object argument in order to allow for its application to proper classes of maps (as opposed to sets of maps in Quillen’s small object argument). The necessity of such a generalization arose with appearance of several important examples of model categories which were ..."
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Cited by 7 (3 self)
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Abstract. We generalize the small object argument in order to allow for its application to proper classes of maps (as opposed to sets of maps in Quillen’s small object argument). The necessity of such a generalization arose with appearance of several important examples of model categories which were proven to be noncofibrantly generated [2, 6, 8, 20]. Our current approach allows for construction of functorial factorizations and localizations in the equivariant model structures on diagrams of spaces [10] and diagrams of chain complexes. We also formulate a nonfunctorial version of the argument, which applies in two different model structures on the category of prospaces [11, 20]. The examples above suggest a natural extension of the framework of cofibrantly generated model categories. We introduce the concept of a classcofibrantly
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 the category of unbounded chain complexes of modules over a ring. In this paper we develop methods for ..."
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Cited by 6 (5 self)
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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 the category of unbounded chain complexes of modules over a ring. In this paper we develop methods for deciding when two stable model categories represent `the same homotopy theory'. We show that stable model categories with a single compact generator are equivalent to modules over a ring spectrum. More generally stable model categories with a set of generators are characterized as modules over a `ring spectrum with several objects', i.e., as spectrum valued diagram categories. We also prove a Morita theorem which shows how equivalences between module categories over ring spectra can be realized by smashing with a pair of bimodules. Finally, we characterize stable model categories which represent the derived category of a ring. This is a slight generalization of Rickard's work on derived equivalent rings. We also include a proof of the model category equivalence of modules over the EilenbergMac Lane spectrum HR and (unbounded) chain complexes of Rmodules for a ring R. 1.
Localization with respect to a class of maps I – Equivariant localization of diagrams of spaces
"... Abstract. Homotopical localizations with respect to a set of maps are known to exist in cofibrantly generated model categories (satisfying additional assumptions) [3, 12, 20, 29]. In this paper we expand the existing framework, so that it will apply to not necessarily cofibrantly generated model cat ..."
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Cited by 4 (4 self)
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Abstract. Homotopical localizations with respect to a set of maps are known to exist in cofibrantly generated model categories (satisfying additional assumptions) [3, 12, 20, 29]. In this paper we expand the existing framework, so that it will apply to not necessarily cofibrantly generated model categories and, more important, will allow for a localization with respect to a class of maps (satisfying some restrictive conditions). We illustrate our technique by applying it to the equivariant model category of diagrams of spaces [11]. This model category is not cofibrantly generated [7]. We give conditions on a class of maps which ensure the existence of the localization functor; these conditions are satisfied by any set of maps and by the class of maps which induces ordinary localizations on the generalized fixedpoints sets.
Obstruction theory in model categories
 ADV. MATH
, 2004
"... Many examples of obstruction theory can be formulated as the study of when a lift exists in a commutative square. Typically, one of the maps is a cofibration of some sort and the opposite map is a fibration, and there is a functorial obstruction class that determines whether a lift exists. Workingin ..."
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Cited by 2 (1 self)
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Many examples of obstruction theory can be formulated as the study of when a lift exists in a commutative square. Typically, one of the maps is a cofibration of some sort and the opposite map is a fibration, and there is a functorial obstruction class that determines whether a lift exists. Workingin an arbitrary pointed proper model category, we classify the cofibrations that have such an obstruction theory with respect to all fibrations. Up to weak equivalence, retract, and cobase change, they are the cofibrations with weakly contractible target. Equivalently, they are the retracts of principal cofibrations. Without properness, the same classification holds for cofibrations with cofibrant source. Our results dualize to give a classification of fibrations that have an obstruction theory.
SMASH PRODUCTS OF E(1)LOCAL SPECTRA AT AN ODD PRIME
, 2004
"... The two categories are not Quillen equivalent, and his proof uses systems of triangulated diagram categories rather than model categories. Our main result is that in the case n = 1 Franke’s functor maps the derived tensor product to the smash product. It can however not be an associative equivalence ..."
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Cited by 1 (0 self)
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The two categories are not Quillen equivalent, and his proof uses systems of triangulated diagram categories rather than model categories. Our main result is that in the case n = 1 Franke’s functor maps the derived tensor product to the smash product. It can however not be an associative equivalence of monoidal categories. The first part of our paper sets up a monoidal version of Franke’s systems of triangulated diagram categories and explores its properties. The second part applies these results to the specific construction of Franke’s functor in order to prove the above result. 1.
TORSION INVARIANTS FOR TRIANGULATED CATEGORIES
"... The most commonly known triangulated categories arise from chain complexes in an abelian category by passing to chain homotopy classes or inverting quasiisomorphisms. Such examples are called ‘algebraic ’ because they have underlying abelian (or at least additive) categories. Stable homotopy theory ..."
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The most commonly known triangulated categories arise from chain complexes in an abelian category by passing to chain homotopy classes or inverting quasiisomorphisms. Such examples are called ‘algebraic ’ because they have underlying abelian (or at least additive) categories. Stable homotopy theory produces examples of triangulated categories by quite different means, and in this context the underlying categories are usually very ‘nonadditive ’ before passing to homotopy classes of morphisms. We call such triangulated categories topological, compare Definition 3.1; this class includes the algebraic triangulated categories. The purpose of this paper is to explain some systematic differences between these two kinds of triangulated categories. There are certain properties – defined entirely in terms of the triangulated structure – which hold in all algebraic examples, but which can fail in general. These differences are all torsion phenomena, and rationally every topological triangulated category is algebraic (at least under mild size restrictions). Our main tool is a new numerical invariant, the norder of an object in a triangulated category, for n a natural number (see Definition 1.1). The norder is a nonnegative integer (or infinity), and an object Y has positive norder if and only if n · Y = 0; the norder can be thought of
AN EXAMPLE OF A NONCOFIBRANTLY GENERATED MODEL CATEGORY
, 2001
"... Abstract. We show that the model category of diagrams of spaces generated by a proper class of orbits is not cofibrantly generated. In particular the category of maps between spaces may be given a noncofibrantly generated model structure. ..."
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Abstract. We show that the model category of diagrams of spaces generated by a proper class of orbits is not cofibrantly generated. In particular the category of maps between spaces may be given a noncofibrantly generated model structure.