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15
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 78 (1 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 27 (7 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:
Spaces of maps into classifying spaces for equivariant crossed complexes
 Indag. Math. (N.S
, 1997
"... Abstract. The results of a previous paper on the equivariant homotopy theory of crossed complexes are generalised from the case of a discrete group to general topological groups. The principal new ingredient necessary for this is an analysis of homotopy coherence theory for crossed complexes, using ..."
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Cited by 12 (7 self)
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Abstract. The results of a previous paper on the equivariant homotopy theory of crossed complexes are generalised from the case of a discrete group to general topological groups. The principal new ingredient necessary for this is an analysis of homotopy coherence theory for crossed complexes, using detailed results on the appropriate Eilenberg–Zilber theory, and of its relation to simplicial homotopy coherence. Again, our results give information not just on the homotopy classification of certain equivariant maps, but also on the weak equivariant homotopy type of the corresponding equivariant function spaces. Mathematics Subject Classifications (2001): 55P91, 55U10, 18G55. Key words: equivariant homotopy theory, classifying space, function space, crossed complex.
Quasicategories vs Segal spaces
 IN CATEGORIES IN ALGEBRA, GEOMETRY AND MATHEMATICAL
, 2006
"... We show that complete Segal spaces and Segal categories are Quillen equivalent to quasicategories. ..."
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We show that complete Segal spaces and Segal categories are Quillen equivalent to quasicategories.
Rigidification of quasicategories
"... We give a new construction for rigidifying a quasicategory into a simplicial category, and prove that it is weakly equivalent to the rigidification given by Lurie. Our construction comes from the use of necklaces, which are simplicial sets obtained by stringing simplices together. As an applicatio ..."
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Cited by 4 (1 self)
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We give a new construction for rigidifying a quasicategory into a simplicial category, and prove that it is weakly equivalent to the rigidification given by Lurie. Our construction comes from the use of necklaces, which are simplicial sets obtained by stringing simplices together. As an application of these methods, we use our model to reprove some basic facts from [L] about the rigidification process.
Higher homotopy operations and cohomology
"... Abstract. We explain how higher homotopy operations, defined topologically, may be identified under mild assumptions with (the last of) the DwyerKanSmith cohomological obstructions to rectifying homotopycommutative diagrams. ..."
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Abstract. We explain how higher homotopy operations, defined topologically, may be identified under mild assumptions with (the last of) the DwyerKanSmith cohomological obstructions to rectifying homotopycommutative diagrams.
A Riemann Hilbert correspondence for infinity local systems
"... We descibe a dgequivalence of dgcategories between Block’s PA, corresponding to the de Rham dga A of a compact manifold M and the dgcategory of ∞local systems on M. We understand this as a generalization of the RiemannHilbert correspondence to Zgraded connections (superconnections in some form ..."
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We descibe a dgequivalence of dgcategories between Block’s PA, corresponding to the de Rham dga A of a compact manifold M and the dgcategory of ∞local systems on M. We understand this as a generalization of the RiemannHilbert correspondence to Zgraded connections (superconnections in some formulations). An ∞local system is an (∞, 1) functor between the (∞, 1)categories π∞M and the linear simplicial nerve of the dgcategory of cochain complexes. This theory makes crucial use of Igusa’s notion of higher holonomy transport for Zgraded connections which is a derivative of Chen’s main idea of generalized holonomy. In the appendix we describe the linear simplicial nerve construction.
Khovanov homotopy type, Burnside category, and products
"... Abstract. In this paper, we give a new construction of a Khovanov homotopy type. We show that this construction gives a space stably homotopy equivalent to the Khovanov homotopy types constructed in [LS14a] and [HKK] and, as a corollary, that those two constructions give equivalent spaces. We show ..."
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Abstract. In this paper, we give a new construction of a Khovanov homotopy type. We show that this construction gives a space stably homotopy equivalent to the Khovanov homotopy types constructed in [LS14a] and [HKK] and, as a corollary, that those two constructions give equivalent spaces. We show that the construction behaves well with respect to disjoint unions, connected sums and mirrors, verifying several conjectures from [LS14a]. Finally, combining these results with computations from [LS14c] and the refined sinvariant from [LS14b] we obtain new results about the slice genera of certain knots.
Weak complicial sets, a simplicial weak ωcategory theory. Part II: nerves of complicial Graycategories. Available as arXiv:math/0604416
"... To Ross Street on the occasion of his 60 th birthday. Abstract. This paper develops the foundations of a simplicial theory of weak ωcategories, which builds upon the insights originally expounded by Ross Street in his 1987 paper on oriented simplices. The resulting theory of weak complicial sets pr ..."
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To Ross Street on the occasion of his 60 th birthday. Abstract. This paper develops the foundations of a simplicial theory of weak ωcategories, which builds upon the insights originally expounded by Ross Street in his 1987 paper on oriented simplices. The resulting theory of weak complicial sets provides a common generalisation of the theories of (strict) ωcategories, Kan complexes and Joyal’s quasicategories. We generalise a number of results due to the current author with regard to complicial sets and strict ωcategories to provide an armoury of well behaved technical devices, such as joins and Gray tensor products, which will be used to study these the weak ωcategory theory of these structures in a series of companion papers. In particular, we establish their basic homotopy theory by constructing a Quillen model structure on the category of stratified simplicial sets whose fibrant objects are the weak complicial sets. As a simple corollary of this work we provide an independent construction of Joyal’s model structure on simplicial sets for
speaks to graduates
, 1940
"... This accomplishment –no matter how grand or how modest – could not have been reached without the support of many individuals who formed an important part of my life –both mathematical and otherwise. The most immediate thanks go out to Jonathan Block, my advisor, who gently pushed me towards new math ..."
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This accomplishment –no matter how grand or how modest – could not have been reached without the support of many individuals who formed an important part of my life –both mathematical and otherwise. The most immediate thanks go out to Jonathan Block, my advisor, who gently pushed me towards new mathematical ventures and whose generosity and patience gave me some needed workingspace. He shared his insights and knowledge freely, and gave me a clear picture of an ideal career as a mathematician. I must also thank him for his warm hospitality, and introducing me to his family. I am also gracious for the presence of Tony Pantev, who was a kind of advisorfromafar during my tenure, and whose students were often my secondary mathematical mentors. Collectively that are responsible for a significant amount of learning on my part. I also