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68
Higher topos theory
, 2006
"... Let X be a topological space and G an abelian group. There are many different definitions for the cohomology group H n (X; G); we will single out three of them for discussion here. First of all, we have the singular cohomology groups H n sing (X; G), which are defined to be cohomology of a chain com ..."
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Cited by 183 (1 self)
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Let X be a topological space and G an abelian group. There are many different definitions for the cohomology group H n (X; G); we will single out three of them for discussion here. First of all, we have the singular cohomology groups H n sing (X; G), which are defined to be cohomology of a chain complex of Gvalued singular cochains on X. An alternative is to regard H n (•, G) as a representable functor on the homotopy category
Controlled algebra and the Novikov conjectures for K and Ltheory, Topology 34
, 1995
"... Abstract. The aim of this paper is to split the assembly map in K and Ltheory for a class of groups of finite cohomological dimension, containing the word hyperbolic groups In this paper we combine the methods of [5] with the continuously controlled algebra of [1] and the Ltheory of additive cate ..."
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Cited by 82 (13 self)
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Abstract. The aim of this paper is to split the assembly map in K and Ltheory for a class of groups of finite cohomological dimension, containing the word hyperbolic groups In this paper we combine the methods of [5] with the continuously controlled algebra of [1] and the Ltheory of additive categories with involution [23] to split assembly maps in Kand Ltheory. Specifically we prove the following theorems Let Γ be a group with finite classifying space BΓ. Assume EΓ admits a compactification
Homotopy Coherent Category Theory
, 1996
"... this paper we try to lay some of the foundations of such a theory of categories `up to homotopy' or more exactly `up to coherent homotopies'. The method we use is based on earlier work on: ..."
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Cited by 36 (7 self)
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this paper we try to lay some of the foundations of such a theory of categories `up to homotopy' or more exactly `up to coherent homotopies'. The method we use is based on earlier work on:
MODEL STRUCTURES ON PROCATEGORIES
 HOMOLOGY, HOMOTOPY AND APPLICATIONS, VOL. 9(1), 2007, PP.367–398
, 2007
"... We introduce a notion of a filtered model structure and use this notion to produce various model structures on procategories. We give several examples, including a homotopy theory for Gspaces, where G is a profinite group. The class of weak equivalences are an approximation to the class of underlyi ..."
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Cited by 32 (3 self)
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We introduce a notion of a filtered model structure and use this notion to produce various model structures on procategories. We give several examples, including a homotopy theory for Gspaces, where G is a profinite group. The class of weak equivalences are an approximation to the class of underlying weak equivalences.
A model structure on the category of prosimplicial sets
 Trans. Amer. Math. Soc
"... Abstract. We study the category proSS of prosimplicial sets, which arises in étale homotopy theory, shape theory, and profinite completion. We establish a model structure on proSS so that it is possible to do homotopy theory in this category. This model structure is closely related to the strict ..."
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Cited by 26 (5 self)
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Abstract. We study the category proSS of prosimplicial sets, which arises in étale homotopy theory, shape theory, and profinite completion. We establish a model structure on proSS so that it is possible to do homotopy theory in this category. This model structure is closely related to the strict structure of Edwards and Hastings. In order to understand the notion of homotopy groups for prospaces we use local systems on prospaces. We also give several alternative descriptions of weak equivalences, including a cohomological characterization. We outline dual constructions for indspaces.
Splitting with Continuous Control in Algebraic Ktheory
, 2002
"... Abstract. In this work, the continuously controlled assembly map in algebraic Ktheory, as developed by Carlsson and Pedersen, is proved to be a split injection for groups Γ that satisfy certain geometric conditions. The group Γ is allowed to have torsion, generalizing a result of Carlsson and Peder ..."
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Cited by 17 (6 self)
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Abstract. In this work, the continuously controlled assembly map in algebraic Ktheory, as developed by Carlsson and Pedersen, is proved to be a split injection for groups Γ that satisfy certain geometric conditions. The group Γ is allowed to have torsion, generalizing a result of Carlsson and Pedersen. Combining this with a result of John Moody, K0(kΓ) is proved to be isomorphic to the colimit of K0(kH) over the finite subgroups H of Γ, when Γ is a virtually polycyclic group and k is a field of characteristic zero. 1.
Homotopy theory of small diagrams over large categories
"... Abstract. Let D be a large category which is cocomplete. We construct a model structure (in the sense of Quillen) on the category of small functors from D to simplicial sets. As an application we construct homotopy localization functors on the category of simplicial sets which satisfy a stronger uni ..."
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Cited by 16 (4 self)
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Abstract. Let D be a large category which is cocomplete. We construct a model structure (in the sense of Quillen) on the category of small functors from D to simplicial sets. As an application we construct homotopy localization functors on the category of simplicial sets which satisfy a stronger universal property than the customary homotopy localization functors do. 1.
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 15 (6 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