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28
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
Complete modules and torsion modules
 Amer. J. Math
"... Abstract. Suppose that R is a ring and that A is a chain complex over R. Inside the derived category of differential graded Rmodules there are naturally defined subcategories of Atorsion objects and of Acomplete objects. Under a finiteness condition on A, we develop a Morita theory for these subc ..."
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Cited by 34 (5 self)
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Abstract. Suppose that R is a ring and that A is a chain complex over R. Inside the derived category of differential graded Rmodules there are naturally defined subcategories of Atorsion objects and of Acomplete objects. Under a finiteness condition on A, we develop a Morita theory for these subcategories, find conceptual interpretations for some associated algebraic functors, and, in appropriate commutative situations, identify the associated functors as local homology or local cohomology. Some of the results are suprising even in the case R = Z and A = Z/p. 1.
A Cellular Nerve for Higher Categories
, 2002
"... ... categories. The associated cellular nerve of an ocategory extends the wellknown simplicial nerve of a small category. Cellular sets (like simplicial sets) carry a closed model structure in Quillen’s sense with weak equivalences induced by a geometric realisation functor. More generally, there ..."
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Cited by 21 (2 self)
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... categories. The associated cellular nerve of an ocategory extends the wellknown simplicial nerve of a small category. Cellular sets (like simplicial sets) carry a closed model structure in Quillen’s sense with weak equivalences induced by a geometric realisation functor. More generally, there exists a dense subcategory YA of the category of Aalgebras for each ooperad A in Batanin’s sense. Whenever A is contractible, the resulting homotopy category of Aalgebras (i.e. weak ocategories) is
Homotopy fixed point methods for Lie groups and finite loop spaces
"... this paper we prove the following theorem. ..."
Implications of largecardinal principles in homotopical localization
 Adv. Math
"... The existence of arbitrary cohomological localizations on the homotopy category of spaces has remained unproved since Bousfield settled the same problem for homology theories in the decade of 1970. This is related with another open question, namely whether or not every homotopy idempotent functor on ..."
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Cited by 19 (3 self)
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The existence of arbitrary cohomological localizations on the homotopy category of spaces has remained unproved since Bousfield settled the same problem for homology theories in the decade of 1970. This is related with another open question, namely whether or not every homotopy idempotent functor on spaces is an flocalization for some map f. We prove that both questions have an affirmative answer assuming the validity of a suitable largecardinal axiom from set theory (Vopěnka’s principle). We also show that it is impossible to prove that all homotopy idempotent functors are flocalizations using the ordinary ZFC axioms of set theory (Zermelo–Fraenkel axioms with the axiom of choice), since a counterexample can be displayed under the assumption that all cardinals are nonmeasurable, which is consistent with ZFC.
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
Product Splittings For pCompact Groups
"... this paper is to prove a theorem similar to 1.1 for pcompact ..."
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Cited by 14 (3 self)
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this paper is to prove a theorem similar to 1.1 for pcompact
Convergence Of The Homology Spectral Sequence Of A Cosimplicial Space
 AMER. J. MATH
, 1996
"... In this paper we study the convergence properties of the homology spectral sequence of a cosimplicial space. The EilenbergMoore spectral sequence is an example of this spectral sequence applied to the cobar construction of a fibre square. Hence this spectral sequence is also called the generalized ..."
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Cited by 13 (0 self)
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In this paper we study the convergence properties of the homology spectral sequence of a cosimplicial space. The EilenbergMoore spectral sequence is an example of this spectral sequence applied to the cobar construction of a fibre square. Hence this spectral sequence is also called the generalized EilenbergMoore spectral sequence. This work builds on results due to W. Dwyer and A. K. Bousfield. W. Dwyer considered the convergence properties of the EilenbergMoore spectral sequence for a fibration in [D1]. Then A. K. Bousfield used these results as a basis for finding convergence conditions for the generalized EilenbergMoore spectral sequence [B2]. We continue in this direction. In section 3 we consider new convergence conditions for the EilenbergMoore spectral sequence of a fibre square. Using these results and adding a finite type assumption we obtain a new proconvergence result, Theorem 5.3. Combining this result with Corollary 1.2 from [S] gives conditions
Stable Homotopy of Algebraic Theories
 Topology
, 2001
"... The simplicial objects in an algebraic category admit an abstract homotopy theory via a Quillen model category structure. We show that the associated stable homotopy theory is completely determined by a ring spectrum functorially associated with the algebraic theory. For several familiar algebraic t ..."
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Cited by 11 (1 self)
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The simplicial objects in an algebraic category admit an abstract homotopy theory via a Quillen model category structure. We show that the associated stable homotopy theory is completely determined by a ring spectrum functorially associated with the algebraic theory. For several familiar algebraic theories we can identify the parameterizing ring spectrum; for other theories we obtain new examples of ring spectra. For the theory of commutative algebras we obtain a ring spectrum which is related to AndreH}Quillen homology via certain spectral sequences. We show that the (co)homology of an algebraic theory is isomorphic to the topological Hochschild (co)homology of the parameterizing ring spectrum. # 2000 Elsevier Science Ltd. All rights reserved. MSC: 55U35; 18C10 Keywords: Algebraic theories; Ring spectra; AndreH}Quillen homology; #spaces The original motivation for this paper came from the attempt to generalize a rational result about the homotopy theory of commutative rings. For...
The orthogonal subcategory problem in homotopy theory
 Proceedings of the Arolla conference on Algebraic Topology 2004
"... Abstract. It is known that the existence of localization with respect to an arbitrary (possibly proper) class of maps in the category of simplicial sets is implied by a largecardinal axiom called Vopěnka’s principle. In this article we extend the validity of this result to any left proper, combinat ..."
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Cited by 7 (1 self)
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Abstract. It is known that the existence of localization with respect to an arbitrary (possibly proper) class of maps in the category of simplicial sets is implied by a largecardinal axiom called Vopěnka’s principle. In this article we extend the validity of this result to any left proper, combinatorial, simplicial model category M and show that, under additional assumptions on M, every homotopy idempotent functor is in fact a localization with respect to some set of maps. These results are valid for the homotopy category of spectra, among other applications.