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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
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
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
in monoidal categories
, 1995
"... We consider the theory of operads and their algebras in enriched category theory. We introduce the notion of simplicial A~cgraph and show that some important constructions of homotopy coherent category theory lead by a natural way to the use of such objects as the appropriate homotopy coherent coun ..."
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We consider the theory of operads and their algebras in enriched category theory. We introduce the notion of simplicial A~cgraph and show that some important constructions of homotopy coherent category theory lead by a natural way to the use of such objects as the appropriate homotopy coherent counterparts of the categories. @ 1998 Elsevier Science B.V. 1991 Math. Subj. Class.. " 18D20, 18D35, 18{330 Let A be a small simplicial category, and let F, G: A ~ K be two simplicial functors to a simplicial category K. Then we can consider, as the simplicial set of coherent natural transformations from F to G, the coherent end [8, 10, 12] (see Definition 6.2): Coh(F, G) =.fA K(F():) , G(2)).
Some Problems In NonAbelian Homotopical And Homological Algebra
, 1999
"... this paper is to convey some impression of the extent of an area of nonAbelian homotopical and homological algebra, by giving some of the problems, of varying degrees of diculty and of precision, which I have come across over the years and in which progress would be desirable. ..."
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this paper is to convey some impression of the extent of an area of nonAbelian homotopical and homological algebra, by giving some of the problems, of varying degrees of diculty and of precision, which I have come across over the years and in which progress would be desirable.
Scategories, Sgroupoids, Segal categories and quasicategories
, 2008
"... The notes were prepared for a series of talks that I gave in Hagen in late June and early July 2003, and, with some changes, in the University of La Laguña, the Canary Islands, in September, 2003. They assume the audience knows some abstract homotopy theory and as Heiner Kamps was in the audience in ..."
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The notes were prepared for a series of talks that I gave in Hagen in late June and early July 2003, and, with some changes, in the University of La Laguña, the Canary Islands, in September, 2003. They assume the audience knows some abstract homotopy theory and as Heiner Kamps was in the audience in Hagen, it is safe to assume that the notes assume a reasonable knowledge of our book, [26], or any equivalent text if one can be found! What do the notes set out to do? “Aims and Objectives! ” or should it be “Learning Outcomes”? • To revisit some oldish material on abstract homotopy and simplicially enriched categories, that seems to be being used in today’s resurgence of interest in the area and to try to view it in a new light, or perhaps from new directions; • To introduce Segal categories and various other tools used by the NiceToulouse group of abstract homotopy theorists and link them into some of the older ideas; • To introduce Joyal’s quasicategories, (previously called weak Kan complexes but I agree with André that his nomenclature is better so will adopt it) and show how that theory links in with some old ideas of Boardman and Vogt, Dwyer and Kan, and Cordier and myself; • To ask lots of questions of myself and of the reader. The notes include some material from the ‘Cubo ’ article, [35], which was itself based on notes for a course at the Corso estivo Categorie e Topologia in 1991, but the overlap has been kept as small as is feasible as the purpose and the audience of the two sets of notes are different and the abstract homotopy theory has ‘moved on’, in part, to try the new methods out on those same ‘old ’ problems and to attack new ones as well. As usual when you try to specify ‘learning outcomes ’ you end up asking who has done the learning, the audience? Perhaps. The lecturer, most certainly! 1
The
, 1998
"... Spaces of maps into classifying spaces for equivariant crossed complexes, II: ..."
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Spaces of maps into classifying spaces for equivariant crossed complexes, II:
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.