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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.
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.
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
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.
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.
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
DIAGRAMS INDEXED BY GROTHENDIECK CONSTRUCTIONS AND STACKS ON STACKS
 HOMOLOGY, HOMOTOPY AND APPLICATIONS
"... Let I be a small indexing category, G: I op → Cat be a functor and BG ∈ Cat denote the Grothendieck construction on G. We define ..."
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Let I be a small indexing category, G: I op → Cat be a functor and BG ∈ Cat denote the Grothendieck construction on G. We define
HOMOTOPY TOPOI AND EQUIVARIANT ELLIPTIC COHOMOLOGY BY
"... 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. iv Acknowledgments I would like to thank the many people who have been both directly and indirectly involved with this project.
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.
Topological hypercovers and A¹realizations
, 2004
"... We show that if U ∗ is a hypercover of a topological space X then the natural map hocolim U∗→X is a weak equivalence. This fact is used to construct topological realization functors for the A1homotopy theory of schemes over real and complex fields. In an appendix, we also prove a theorem about co ..."
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We show that if U ∗ is a hypercover of a topological space X then the natural map hocolim U∗→X is a weak equivalence. This fact is used to construct topological realization functors for the A1homotopy theory of schemes over real and complex fields. In an appendix, we also prove a theorem about computing homotopy colimits of spaces that are not cofibrant.