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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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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.
Inverting weak dihomotopy equivalence using homotopy continuous flow
 Theory Appl. Categ
"... Abstract. A flow is homotopy continuous if it is indefinitely divisible up to Shomotopy. The full subcategory of cofibrant homotopy continuous flows has nice features. Not only it is big enough to contain all dihomotopy types, but also a morphism between them is a weak dihomotopy equivalence if and ..."
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Cited by 5 (3 self)
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Abstract. A flow is homotopy continuous if it is indefinitely divisible up to Shomotopy. The full subcategory of cofibrant homotopy continuous flows has nice features. Not only it is big enough to contain all dihomotopy types, but also a morphism between them is a weak dihomotopy equivalence if and only if it is invertible up to dihomotopy. Thus, the category of cofibrant homotopy continuous flows provides an implementation of Whitehead’s theorem for the full dihomotopy relation, and not only for Shomotopy as in previous works of the author. This fact is not the consequence of the existence of a model structure on the category of flows because it is known that there does not exist any model structure on it whose weak equivalences are exactly the weak dihomotopy equivalences. This fact is an application of a general result for the localization of a model category with respect to a weak factorization system. Contents
A folk model structure on omegacat
, 2009
"... The primary aim of this work is an intrinsic homotopy theory of strict ωcategories. We establish a model structure on ωCat, the category of strict ωcategories. The constructions leading to the model structure in question are expressed entirely within the scope of ωCat, building on a set of generat ..."
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The primary aim of this work is an intrinsic homotopy theory of strict ωcategories. We establish a model structure on ωCat, the category of strict ωcategories. The constructions leading to the model structure in question are expressed entirely within the scope of ωCat, building on a set of generating cofibrations and a class of weak equivalences as basic items. All object are fibrant while free objects are cofibrant. We further exhibit model structures of this type on ncategories for arbitrary n ∈ N, as specialisations of the ωcategorical one along right adjoints. In particular, known cases for n = 1 and n = 2 nicely fit into the scheme.
ABSTRACT CELLULARIZATION AS A CELLULARIZATION WITH RESPECT TO A SET OF OBJECTS
, 2006
"... Abstract. Given a simplicial idempotent augmented endofunctor F on a simplicial combinatorial model category M, under the assumption of Vopěnka’s principle, we exhibit a set A of cofibrant objects in Msuch that F is equivalent to CWA, the cellularization with respect to A. ..."
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Abstract. Given a simplicial idempotent augmented endofunctor F on a simplicial combinatorial model category M, under the assumption of Vopěnka’s principle, we exhibit a set A of cofibrant objects in Msuch that F is equivalent to CWA, the cellularization with respect to A.
GENERALIZED BROWN REPRESETABILITY IN HOMOTOPY CATEGORIES
, 2005
"... Abstract. Brown representability approximates the homotopy ..."
GENERALIZED BROWN REPRESENTABILITY IN HOMOTOPY CATEGORIES
, 2008
"... Abstract. Brown representability approximates the homotopy ..."
LEFT DETERMINED MODEL STRUCTURES FOR LOCALLY PRESENTABLE CATEGORIES
, 901
"... Abstract. We extend a result of Cisinski on the construction of cofibrantly generated model structures from (Grothendieck) toposes to locally presentable categories and from monomorphism to more general cofibrations. As in the original case, under additional conditions, the resulting model structure ..."
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Abstract. We extend a result of Cisinski on the construction of cofibrantly generated model structures from (Grothendieck) toposes to locally presentable categories and from monomorphism to more general cofibrations. As in the original case, under additional conditions, the resulting model structures are ”left determined ” in the sense of Rosick´y and Tholen. 1.
Accessible categories and . . .
, 2007
"... Accessible categories have recently turned out to be useful in homotopy theory. This text is prepared as notes for a series of lectures at ..."
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Accessible categories have recently turned out to be useful in homotopy theory. This text is prepared as notes for a series of lectures at
Homotopy Theory of Higher Categories
"... portant la référence ANR09BLAN015102 (HODAG). This is draft material from a forthcoming book to be published by Cambridge University Press in the New Mathematical Monographs series. This publication is in copyright. c○Carlos T. Simpson 2010. v This is the first draft of a book about higher categ ..."
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portant la référence ANR09BLAN015102 (HODAG). This is draft material from a forthcoming book to be published by Cambridge University Press in the New Mathematical Monographs series. This publication is in copyright. c○Carlos T. Simpson 2010. v This is the first draft of a book about higher categories approached by iterating Segal’s method, as in Tamsamani’s definition of nnerve and Pelissier’s thesis. If M is a tractable left proper cartesian model