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40
Stable étale realization and étale cobordism
 Adv. Math
"... We show that there is a stable homotopy theory of profinite spaces and use it for two main applications. On the one hand we construct an étale topological realization of the stable A 1homotopy theory of smooth schemes over a base field of arbitrary characteristic in analogy to the complex realizati ..."
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We show that there is a stable homotopy theory of profinite spaces and use it for two main applications. On the one hand we construct an étale topological realization of the stable A 1homotopy theory of smooth schemes over a base field of arbitrary characteristic in analogy to the complex realization functor for fields of characteristic zero. On the other hand we get a natural setting for étale cohomology theories. In particular, we define and discuss an étale topological cobordism theory for schemes. It is equipped with an AtiyahHirzebruch spectral sequence starting from étale cohomology. Finally, we construct maps from algebraic to étale cobordism and discuss algebraic cobordism with finite coefficients over an algebraically closed field after inverting a Bott element. 1
Context for models of concurrency
 in Preliminary Proceedings of the Workshop on Geometry and Topology in Concurrency and Distributed Computing GETCO 2004, vol NS042 of BRICS Notes
, 2004
"... Abstract. Many categories have been used to model concurrency. Using any of these, the challenge is to reduce a given model to a smaller representation which nevertheless preserves the relevant computerscientific information. That is, one wants to replace a given model with a simpler model with the ..."
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Abstract. Many categories have been used to model concurrency. Using any of these, the challenge is to reduce a given model to a smaller representation which nevertheless preserves the relevant computerscientific information. That is, one wants to replace a given model with a simpler model with the same directed homotopytype. Unfortunately, the obvious definition of directed homotopy equivalence is too coarse. This paper introduces the notion of context to refine this definition. 1.
A short note on models for equivariant homotopy theory
, 2006
"... These notes explore equivariant homotopy theory from the perspective of model categories in the case of a discrete group G. Section 2 reviews the situation for topological spaces, largely following [May]. In section 3, we discuss two approaches to equivariant homotopy theory in more general model ca ..."
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These notes explore equivariant homotopy theory from the perspective of model categories in the case of a discrete group G. Section 2 reviews the situation for topological spaces, largely following [May]. In section 3, we discuss two approaches to equivariant homotopy theory in more general model categories. Section 4 discusses
Morita theory in stable homotopy theory
, 2004
"... We discuss an analogue of Morita theory for ring spectra, a thickening of the category of rings inspired by stable homotopy theory. This follows work by Rickard and Keller on Morita theory for derived categories. We also discuss two results for derived equivalences of DGAs which show they differ fr ..."
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We discuss an analogue of Morita theory for ring spectra, a thickening of the category of rings inspired by stable homotopy theory. This follows work by Rickard and Keller on Morita theory for derived categories. We also discuss two results for derived equivalences of DGAs which show they differ from derived equivalences of rings.
Codescent theory II: Cofibrant approximations
, 2003
"... Abstract. We establish a general method to produce cofibrant approximations in the model category US(C, D) of Svalued Cindexed diagrams with Dweak equivalences and Dfibrations. We also present explicit examples of such approximations. Here, S is an arbitrary cofibrantly generated simplicial mode ..."
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Abstract. We establish a general method to produce cofibrant approximations in the model category US(C, D) of Svalued Cindexed diagrams with Dweak equivalences and Dfibrations. We also present explicit examples of such approximations. Here, S is an arbitrary cofibrantly generated simplicial model category and D ⊂ C are small categories. An application to the notion of homotopy colimit is presented. 1.
CODESCENT THEORY I: FOUNDATIONS
, 2003
"... C to S with a model structure, defining weak equivalences and fibrations objectwise but only on D. Our first concern is the effect of moving C, D and S. The main notion introduced here is the “Dcodescent ” property for objects in S C. Our longterm program aims at reformulating as codescent stateme ..."
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C to S with a model structure, defining weak equivalences and fibrations objectwise but only on D. Our first concern is the effect of moving C, D and S. The main notion introduced here is the “Dcodescent ” property for objects in S C. Our longterm program aims at reformulating as codescent statements the Conjectures of BaumConnes and FarrellJones, and at tackling them with new methods. Here, we set the grounds of a systematic theory of codescent, including pullbacks, pushforwards and various invariance properties. 1.
Differential cohomology in a cohesive ∞topos
"... We formulate differential cohomology and ChernWeil theory the theory of connections on fiber bundles and of gauge fields abstractly in the context of a certain class of higher toposes that we call cohesive. Cocycles in this differential cohomology classify higher principal bundles equipped with c ..."
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We formulate differential cohomology and ChernWeil theory the theory of connections on fiber bundles and of gauge fields abstractly in the context of a certain class of higher toposes that we call cohesive. Cocycles in this differential cohomology classify higher principal bundles equipped with cohesive structure (topological, smooth, synthetic differential, supergeometric, etc.) and equipped with connections. We discuss various models of the axioms and wealth of applications revolving around fundamental notions and constructions in prequantum field theory and string theory. In particular we show that the cohesive and differential refinement of universal characteristic cocycles constitutes a higher ChernWeil homomorphism refined from secondary caracteristic classes to morphisms of higher moduli stacks of higher gauge fields, and at the same time constitutes extended geometric prequantization – in the sense of extended/multitiered quantum field theory – of higher dimensional ChernSimonstype field theories and WessZuminoWittentype field theories. This document, and accompanying material, is kept online at ncatlab.org/schreiber/show/differential+cohomology+in+a+cohesive+topos 1 We formulate differential cohomology and ChernWeil theory the theory of connections on fiber bundles
Hodge filtered complex bordism
, 2012
"... Abstract. We construct Hodge filtered cohomology groups for complex manifolds that combine the topological information of generalized cohomology theories with geometric data of Hodge filtered holomorphic forms. This theory provides a natural generalization of Deligne cohomology. For smooth complex a ..."
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Abstract. We construct Hodge filtered cohomology groups for complex manifolds that combine the topological information of generalized cohomology theories with geometric data of Hodge filtered holomorphic forms. This theory provides a natural generalization of Deligne cohomology. For smooth complex algebraic varieties, we show that the theory satisfies a projective bundle formula and A 1homotopy invariance. Moreover, we obtain transfer maps along projective morphisms. 1.
ORBIFOLDS AND STABLE HOMOTOPY GROUPS
, 2005
"... Abstract. Lie groupoids generalize transformation groups, and so provide a natural language for studying orbifolds [13] and other noncommutative geometries. In this paper, we investigate a connection between orbifolds and equivariant stable homotopy theory using such groupoids. A different sort of t ..."
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Abstract. Lie groupoids generalize transformation groups, and so provide a natural language for studying orbifolds [13] and other noncommutative geometries. In this paper, we investigate a connection between orbifolds and equivariant stable homotopy theory using such groupoids. A different sort of twisted sector, along with a classical theorem of tom Dieck [17], allows for a natural definition of stable orbifold homotopy groups, and motivates defining extended unstable orbifold homotopy groups generalizing previous definitions. 1.
HOMOTOPY TOPOI AND EQUIVARIANT ELLIPTIC COHOMOLOGY
, 2005
"... We use the language of homotopy topoi, as developed by Lurie [17], Rezk [21], Simpson [23], and TöenVezossi [24], in order to provide a common foundation for equivariant homotopy theory and derived algebraic geometry. In particular, we obtain the categories of Gspaces, for a topological group G, a ..."
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We use the language of homotopy topoi, as developed by Lurie [17], Rezk [21], Simpson [23], and TöenVezossi [24], in order to provide a common foundation for equivariant homotopy theory and derived algebraic geometry. In particular, we obtain the categories of Gspaces, for a topological group G, and Eschemes, for an E∞ring spectrum E, as full topological subcategories of the homotopy topoi associated to sheaves of spaces on certain small topological sites. This allows for a particularly elegant construction of the equivariant elliptic cohomology associated to an oriented elliptic curve A and a compact abelian Lie group G as an essential geometric morphism of homotopy topoi. It follows that our definition satisfies a conceptually simpler homotopytheoretic analogue of the GinzburgKapranovVasserot axioms [8], which allows us to calculate the cohomology of the equivariant Gspectra S V associated to representations V of G. iii To my parents.