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39
Limits in ncategories
, 1997
"... One of the main notions in category theory is the notion of limit. Similarly, one of the most commonly used techniques in homotopy theory is the notion of “homotopy limit” commonly called “holim ” for short. The purpose of the this paper is to begin to develop ..."
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One of the main notions in category theory is the notion of limit. Similarly, one of the most commonly used techniques in homotopy theory is the notion of “homotopy limit” commonly called “holim ” for short. The purpose of the this paper is to begin to develop
Flexible sheaves
, 1996
"... This is an unfinished explanation of the notion of “flexible sheaf”, that is a homotopical notion of sheaf of topological spaces (homotopy types) over a site. See “Homotopy over the complex numbers and generalized de Rham cohomology ” (submitted to proceedings of the Taniguchi conference on vector b ..."
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This is an unfinished explanation of the notion of “flexible sheaf”, that is a homotopical notion of sheaf of topological spaces (homotopy types) over a site. See “Homotopy over the complex numbers and generalized de Rham cohomology ” (submitted to proceedings of the Taniguchi conference on vector bundles, preprint of Toulouse 3, and also available at my homepage 2) for a more detailed introduction, and also for a further development of the ideas presented here. The present paper was finished in December 1993 while I was visiting MIT. Since writing this, I have realized that the theory sketched here is essentially equivalent to JardineIllusie’s theory of presheaves of topological spaces (although they talk about presheaves of simplicial sets which is the same thing). This is the point of view adopted in “Homotopy over the complex numbers and generalized de Rham cohomology”. 1 Added in August 1996: I am finally sending this to “Duke eprints ” because it now seems that there may be several useful features of the treatment given here. First of all, the direct construction of the homotopysheafification (by doing a certain operation n+2 times) seems to be useful, for example I need
Twisted differential nonabelian cohomology Twisted (n−1)brane nbundles and their ChernSimons (n+1)bundles with characteristic (n + 2)classes
, 2008
"... We introduce nonabelian differential cohomology classifying ∞bundles with smooth connection and their higher gerbes of sections, generalizing [138]. We construct classes of examples of these from lifts, twisted lifts and obstructions to lifts through shifted central extensions of groups by the shif ..."
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We introduce nonabelian differential cohomology classifying ∞bundles with smooth connection and their higher gerbes of sections, generalizing [138]. We construct classes of examples of these from lifts, twisted lifts and obstructions to lifts through shifted central extensions of groups by the shifted abelian ngroup B n−1 U(1). Notable examples are String 2bundles [9] and Fivebrane 6bundles [133]. The obstructions to lifting ordinary principal bundles to these, hence in particular the obstructions to lifting Spinstructures to Stringstructures [13] and further to Fivebranestructures [133, 52], are abelian ChernSimons 3 and 7bundles with characteristic class the first and second fractional Pontryagin class, whose abelian cocycles have been constructed explicitly by Brylinski and McLaughlin [35, 36]. We realize their construction as an abelian component of obstruction theory in nonabelian cohomology by ∞Lieintegrating the L∞algebraic data in [132]. As a result, even if the lift fails, we obtain twisted String 2 and twisted Fivebrane 6bundles classified in twisted nonabelian (differential) cohomology and generalizing the twisted bundles appearing in twisted Ktheory. We explain the GreenSchwarz mechanism in heterotic string theory in terms of twisted String 2bundles and its magnetic dual version – according to [133] – in terms of twisted Fivebrane 6bundles. We close by transgressing differential cocycles to mapping
Filtrations On Higher Algebraic KTheory
 In Algebraic Ktheory
, 1983
"... this paper is to compare the analogous ltrations for the higher ..."
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this paper is to compare the analogous ltrations for the higher
Presheaves of Chain Complexes
, 2003
"... This paper was written to express a personal attitude about derived categories of presheaves and sheaves of chain complexes, much of which has existed for some time but has not previously appeared in the literature ..."
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This paper was written to express a personal attitude about derived categories of presheaves and sheaves of chain complexes, much of which has existed for some time but has not previously appeared in the literature
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
Secondary KodairaSpencer classes and nonabelian Dolbeault cohomology
, 1998
"... One of the nicest things about variations of Hodge structure is the “infinitesimal variation of Hodge structure ” point of view [25]. A variation of Hodge structure (V = V p,q, ∇) over a base S gives rise at any point s ∈ S to the KodairaSpencer map κs: T(S)s → Hom(V p,q s, V p−1,q+1 s). In the geo ..."
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One of the nicest things about variations of Hodge structure is the “infinitesimal variation of Hodge structure ” point of view [25]. A variation of Hodge structure (V = V p,q, ∇) over a base S gives rise at any point s ∈ S to the KodairaSpencer map κs: T(S)s → Hom(V p,q s, V p−1,q+1 s). In the geometric situation of a family X → S we have V p,q s = Hq (Xs, Ω p
THE PORDER OF TOPOLOGICAL TRIANGULATED CATEGORIES
"... p annihilates objects of the form Y/p. In this paper we show that the porder of a topological ..."
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p annihilates objects of the form Y/p. In this paper we show that the porder of a topological