Results 1  10
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21
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 183 (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 36 (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:
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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Cited by 35 (0 self)
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We show that complete Segal spaces and Segal categories are Quillen equivalent to quasicategories.
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 16 (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.
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 5 (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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Cited by 5 (2 self)
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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.
HOMOTOPY COHERENT NERVE IN DEFORMATION THEORY
, 704
"... Abstract. The main object of the note is to fix an earlier error of the author, [H1], claiming that the (standard) simplicial nerve preserves fibrations of simplicially enriched categories. This erroneous claim was used by the author in his deformation theory constructions in [H1, H2, H3]. Fortunate ..."
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Abstract. The main object of the note is to fix an earlier error of the author, [H1], claiming that the (standard) simplicial nerve preserves fibrations of simplicially enriched categories. This erroneous claim was used by the author in his deformation theory constructions in [H1, H2, H3]. Fortunately, the problem disappears if one replaces the standard simplicial nerve with another one, called homotopy coherent nerve. In this note we recall the definition of homotopy coherent nerve and prove some its properties necessary to justify the papers [H1, H2, H3]. We present as well a generalization of the notion of fibered categories which is convenient once one works with enriched categories. 0.
The Higher RiemannHilbert Correspondence and Multiholomorphic Mappings
, 2011
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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.