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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
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.
Dualizability in low dimensional higher category theory. Notre Dame Graduate Summer School on Topology and Field Theories. University of Notre Dame, Notre Dame
, 2013
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Multidimensional Interleavings and Applications to Topological Inference
, 2012
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Ordinal subdivision and special pasting in quasicategories
"... Quasicategories are simplicial sets with properties generalising those of the nerve of a category. They model weak∞categories. Using a combinatorially defined ordinal subdivision, we examine composition rules for certain special pasting diagrams in quasicategories. The subdivision is of combinatori ..."
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Quasicategories are simplicial sets with properties generalising those of the nerve of a category. They model weak∞categories. Using a combinatorially defined ordinal subdivision, we examine composition rules for certain special pasting diagrams in quasicategories. The subdivision is of combinatorial interest in its own right and is linked with various combinatorial constructions. 1 Introduction. The most usual method of subdivision for a simplicial complex used in elementary algebraic and geometric topology is the barycentric subdivision. There is however another very well structured subdivision construction encountered from time to time. The basic geometric construction involves chopping up a
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.
Models for (∞, n)Categories and the Cobordism Hypothesis
 Mathematical Foundations of Quantum Field Theory and Perturbative String Theory, Proceedings of Symposia in Pure Mathematics, volume 83 AMS (2011), arXiv:1011.0110
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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
Two Models for the Homotopy Theory of Cocomplete Homotopy Theories
 Ph.D. Thesis, Rheinische FriedrichWilhelmsUniversität Bonn, 2014, http://hss.ulb.unibonn.de/2014/3692/3692.htm
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