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28
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
Homotopical Algebraic Geometry I: Topos theory
, 2002
"... This is the first of a series of papers devoted to lay the foundations of Algebraic Geometry in homotopical and higher categorical contexts. In this first part we investigate a notion of higher topos. For this, we use Scategories (i.e. simplicially enriched categories) as models for certain kind of ..."
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Cited by 32 (20 self)
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This is the first of a series of papers devoted to lay the foundations of Algebraic Geometry in homotopical and higher categorical contexts. In this first part we investigate a notion of higher topos. For this, we use Scategories (i.e. simplicially enriched categories) as models for certain kind of ∞categories, and we develop the notions of Stopologies, Ssites and stacks over them. We prove in particular, that for an Scategory T endowed with an Stopology, there exists a model
Algebraic geometry over model categories  A general approach to derived algebraic geometry
, 2001
"... ..."
Higher and derived stacks: a global overview
, 2005
"... These are expended notes of my talk at the summer institute in algebraic geometry (Seattle, JulyAugust 2005), whose main purpose is to present a global overview on the theory of higher and derived stacks. This text is far from being exhaustive but is intended to cover a rather large part of the sub ..."
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Cited by 21 (6 self)
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These are expended notes of my talk at the summer institute in algebraic geometry (Seattle, JulyAugust 2005), whose main purpose is to present a global overview on the theory of higher and derived stacks. This text is far from being exhaustive but is intended to cover a rather large part of the subject, starting from the motivations and the foundational material, passing through some examples and basic notions, and ending with some more recent developments and open questions.
Quasismooth Derived Manifolds
"... products; for example the zeroset of a smooth function on a manifold is not necessarily a manifold, and the nontransverse intersection of submanifolds is ..."
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Cited by 14 (0 self)
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products; for example the zeroset of a smooth function on a manifold is not necessarily a manifold, and the nontransverse intersection of submanifolds is
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
Worytkiewicz: A model category for local pospaces
 Homology, Homotopy and Applications
, 506
"... Abstract. Locally partialordered spaces (local pospaces) have been used to model concurrent systems. We provide equivalences for these spaces by ..."
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Cited by 7 (2 self)
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Abstract. Locally partialordered spaces (local pospaces) have been used to model concurrent systems. We provide equivalences for these spaces by
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
Parametrized spaces model locally constant homotopy sheaves
 Topology Appl
, 2008
"... Abstract. We prove that the homotopy theory of parametrized spaces embeds fully and faithfully in the homotopy theory of simplicial presheaves, and that its essential image consists of the locally homotopically constant objects. This gives a homotopytheoretic version of the classical identification ..."
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Abstract. We prove that the homotopy theory of parametrized spaces embeds fully and faithfully in the homotopy theory of simplicial presheaves, and that its essential image consists of the locally homotopically constant objects. This gives a homotopytheoretic version of the classical identification of covering spaces with locally constant sheaves. We also prove a new version of the classical result that spaces parametrized over X are equivalent to spaces with an action of ΩX. This gives a homotopytheoretic version of the correspondence between covering spaces and π1sets. We then use these two equivalences to study base change functors for parametrized spaces. Contents
Derived Smooth Manifolds
"... ... and it is closed under taking arbitrary intersections in a manifold. A derived manifold is a space together with a sheaf of local C∞rings that is obtained by patching together homotopy zerosets of smooth functions on Euclidean spaces. We show that derived manifolds come equipped with a stable ..."
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Cited by 3 (1 self)
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... and it is closed under taking arbitrary intersections in a manifold. A derived manifold is a space together with a sheaf of local C∞rings that is obtained by patching together homotopy zerosets of smooth functions on Euclidean spaces. We show that derived manifolds come equipped with a stable normal bundle and can be imbedded into Euclidean space. We define a cohomology theory called derived cobordism, and use a PontrjaginThom argument to show that the derived cobordism theory is isomorphic to the classical cobordism theory. This allows us to define fundamental classes in cobordism for all derived manifolds. In particular, the intersection A ∩ B of submanifolds A, B ⊂ X exists on the categorical level in our theory, and a cup product formula [A] ⌣ [B] = [A ∩ B] holds, even if the submanifolds are not transverse. One can thus consider the theory of derived manifolds as a categorification of intersection theory.