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34
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 47 (0 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
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 22 (6 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:
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 20 (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.
Strict model structures for pro–categories. Categorical decomposition techniques in algebraic topology
 Isle of Skye
, 2004
"... Abstract. We show that if C is a proper model category, then the procategory proC has a strict model structure in which the weak equivalences are the levelwise weak equivalences. This is related to a major result of [10]. The strict model structure is the starting point for many homotopy theories ..."
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Cited by 19 (4 self)
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Abstract. We show that if C is a proper model category, then the procategory proC has a strict model structure in which the weak equivalences are the levelwise weak equivalences. This is related to a major result of [10]. The strict model structure is the starting point for many homotopy theories of proobjects such as those described in [5], [17], and [19].
On ∞topoi
, 2003
"... 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, one has the singular cohomology H n sing(X, G), which is defined as the cohomology of a complex of Gvalu ..."
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Cited by 12 (0 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, one has the singular cohomology H n sing(X, G), which is defined as the cohomology of a complex of Gvalued singular cochains. Alternatively, one may regard H n (•, G) as a representable functor on the homotopy category of topological spaces, and thereby define H n rep(X, G) to be the set of homotopy classes of maps from X into an EilenbergMacLane space K(G, n). A third possibility is to use the sheaf cohomology H n sheaf (X, G) of X with coefficients in the constant sheaf G on X. If X is a sufficiently nice space (for example, a CW complex), then all three of these definitions agree. In general, however, all three give different answers. The singular cohomology of X is constructed using continuous maps from simplices ∆k into X. If there are not many maps into X (for example if every path in X is constant), then we cannot expect H n sing (X, G) to tell us very much about X. Similarly, the cohomology group H n rep(X, G) is defined using maps from X into a simplicial complex, which (ultimately) relies on the existence of continuous realvalued functions on X. If X does not admit many realvalued functions, we should not expect H n rep (X, G) to be a useful invariant. However, the sheaf cohomology of X seems to be a good invariant for arbitrary spaces: it has excellent formal properties in general and sometimes yields
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 10 (5 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 9 (2 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.
Computing homotopy types using crossed ncubes of groups
 in Adams Memorial Symposium on Algebraic Topology
, 1992
"... Dedicated to the memory of Frank Adams ..."
Etale realization on the A¹homotopy theory of schemes
 MATH
, 2001
"... We compare Friedlander’s definition of étale homotopy for simplicial schemes to another definition involving homotopy colimits of prosimplicial sets. This can be expressed as a notion of hypercover descent for étale homotopy. We use this result to construct a homotopy invariant functor from the ca ..."
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Cited by 7 (3 self)
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We compare Friedlander’s definition of étale homotopy for simplicial schemes to another definition involving homotopy colimits of prosimplicial sets. This can be expressed as a notion of hypercover descent for étale homotopy. We use this result to construct a homotopy invariant functor from the category of simplicial presheaves on the étale site of schemes over S to the category of prospaces. After completing away from the characteristics of the