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89
Triangulated categories of singularities and Dbranes in LandauGinzburg models
 Tr. Mat. Inst. Steklova, 246(Algebr. Geom. Metody, Svyazi i Prilozh.):240–262
, 2005
"... Dedicated to the blessed memory of Andrei Nikolaevich Tyurin – adviser and friend ..."
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Cited by 94 (4 self)
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Dedicated to the blessed memory of Andrei Nikolaevich Tyurin – adviser and friend
Colimit Theorems for Relative Homotopy Groups
, 2008
"... This is the second of two papers whose main purpose is to prove a generalisation to all dimensions of the SeifertVan Kampen theorem on the fundamental group of a union of spaces. The first paper [10] (whose results were announced in [8]) developed the necessary ‘algebra of cubes’. Categories G of ω ..."
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Cited by 77 (36 self)
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This is the second of two papers whose main purpose is to prove a generalisation to all dimensions of the SeifertVan Kampen theorem on the fundamental group of a union of spaces. The first paper [10] (whose results were announced in [8]) developed the necessary ‘algebra of cubes’. Categories G of ωgroupoids and C of crossed complexes were defined, and the principal result
Higher dimensional algebra III: ncategories and the algebra of opetopes
, 1997
"... We give a definition of weak ncategories based on the theory of operads. We work with operads having an arbitrary set S of types, or ‘Soperads’, and given such an operad O, we denote its set of operations by elt(O). Then for any Soperad O there is an elt(O)operad O + whose algebras are Soperads ..."
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Cited by 74 (6 self)
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We give a definition of weak ncategories based on the theory of operads. We work with operads having an arbitrary set S of types, or ‘Soperads’, and given such an operad O, we denote its set of operations by elt(O). Then for any Soperad O there is an elt(O)operad O + whose algebras are Soperads over O. Letting I be the initial operad with a oneelement set of types, and defining I 0+ = I, I (i+1)+ = (I i+) +, we call the operations of I (n−1)+ the ‘ndimensional opetopes’. Opetopes form a category, and presheaves on this category are called ‘opetopic sets’. A weak ncategory is defined as an opetopic set with certain properties, in a manner reminiscent of Street’s simplicial approach to weak ωcategories. In a similar manner, starting from an arbitrary operad O instead of I, we define ‘ncoherent Oalgebras’, which are n times categorified analogs of algebras of O. Examples include ‘monoidal ncategories’, ‘stable ncategories’, ‘virtual nfunctors ’ and ‘representable nprestacks’. We also describe how ncoherent Oalgebra objects may be defined in any (n + 1)coherent Oalgebra.
On differential graded categories
 INTERNATIONAL CONGRESS OF MATHEMATICIANS. VOL. II
, 2006
"... Differential graded categories enhance our understanding of triangulated categories appearing in algebra and geometry. In this survey, we review their foundations and report on recent work by Drinfeld, DuggerShipley,..., Toën and ToënVaquié. ..."
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Cited by 63 (3 self)
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Differential graded categories enhance our understanding of triangulated categories appearing in algebra and geometry. In this survey, we review their foundations and report on recent work by Drinfeld, DuggerShipley,..., Toën and ToënVaquié.
Combinatorial operad actions on cochains
, 2001
"... A classical Einfinity operad is formed by the bar construction of the symmetric groups. Such an operad has been introduced by M. Barratt and P. Eccles in the context of simplicial sets in order to have an analogue of the Milnor FKconstruction for infinite loop spaces. The purpose of this article i ..."
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Cited by 51 (18 self)
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A classical Einfinity operad is formed by the bar construction of the symmetric groups. Such an operad has been introduced by M. Barratt and P. Eccles in the context of simplicial sets in order to have an analogue of the Milnor FKconstruction for infinite loop spaces. The purpose of this article is to prove that the associative algebra structure on the normalized cochain complex of a simplicial set extends to the structure of an algebra over the BarrattEccles operad. We also prove that differential graded algebras over the BarrattEccles operad form a closed model category. Similar results hold for the normalized Hochschild cochain complex of an associative algebra. More precisely, the Hochschild cochain complex is acted on by a suboperad of the BarrattEccles operad which is equivalent to the classical little squares operad.
Derived categories of coherent sheaves and triangulated categories of singularities
, 2005
"... ..."
Twisted Ktheory of differentiable stacks
 ANN. SCI. ÉCOLE NORM. SUP
, 2004
"... In this paper, we develop twisted Ktheory for stacks, where the twisted class is given by an S 1gerbe over the stack. General properties, including the Mayer–Vietoris property, Bott periodicity, and the product structure K i α ⊗K j β → Ki+j α+β are derived. Our approach provides a uniform framew ..."
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Cited by 44 (12 self)
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In this paper, we develop twisted Ktheory for stacks, where the twisted class is given by an S 1gerbe over the stack. General properties, including the Mayer–Vietoris property, Bott periodicity, and the product structure K i α ⊗K j β → Ki+j α+β are derived. Our approach provides a uniform framework for studying various twisted Ktheories including the usual twisted Ktheory of topological spaces, twisted equivariant Ktheory, and the twisted Ktheory of orbifolds. We also present a Fredholm picture, and discuss the conditions under which twisted Kgroups can be expressed by socalled “twisted vector bundles”. Our approach is to work on presentations of stacks, namely groupoids, and relies heavily on the machinery of Ktheory (KKtheory) of C ∗algebras.
TOPOLOGICAL DEFORMATION OF HIGHER DIMENSIONAL AUTOMATA
 HOMOLOGY, HOMOTOPY AND APPLICATIONS, VOL.5(2), 2003, PP.39–82
, 2003
"... A local pospace is a gluing of topological spaces which are equipped with a closed partial ordering representing the time flow. They are used as a formalization of higher dimensional automata (see for instance [6]) which model concurrent systems in computer science. It is known [11] that there are ..."
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Cited by 40 (18 self)
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A local pospace is a gluing of topological spaces which are equipped with a closed partial ordering representing the time flow. They are used as a formalization of higher dimensional automata (see for instance [6]) which model concurrent systems in computer science. It is known [11] that there are two distinct notions of deformation of higher dimensional automata, “spatial” and “temporal”, leaving invariant computer scientific properties like presence or absence of deadlocks. Unfortunately, the formalization of these notions is still unknown in the general case of local pospaces. We introduce here a particular kind of local pospace, the “globular CWcomplexes”, for which we formalize these notions of deformations and which are sufficient to formalize
Homotopical Algebraic Geometry I: Topos theory
, 2002
"... This is the first of a series of papers devoted to lay the foundations of Algebraic Geometry in homotopical and higher categorical contexts. In this first part we investigate a notion of higher topos. For this, we use Scategories (i.e. simplicially enriched categories) as models for certain kind of ..."
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Cited by 29 (18 self)
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This is the first of a series of papers devoted to lay the foundations of Algebraic Geometry in homotopical and higher categorical contexts. In this first part we investigate a notion of higher topos. For this, we use Scategories (i.e. simplicially enriched categories) as models for certain kind of ∞categories, and we develop the notions of Stopologies, Ssites and stacks over them. We prove in particular, that for an Scategory T endowed with an Stopology, there exists a model
Geometry and Concurrency: A User's Guide
, 2000
"... Introduction "Geometry and Concurrency" is not yet a wellestablished domain of research, but is rather made of a collection of seemingly related techniques, algorithms and formalizations, coming from different application areas, accumulated over a long period of time. There is currently a certain ..."
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Cited by 28 (7 self)
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Introduction "Geometry and Concurrency" is not yet a wellestablished domain of research, but is rather made of a collection of seemingly related techniques, algorithms and formalizations, coming from different application areas, accumulated over a long period of time. There is currently a certain amount of effort made for unifying these (in particular see the article (Gunawardena, 1994)), following the workshop "New Connections between Computer Science and Mathematics" held at the Newton Institute in Cambridge, England in November 1995 (and sponsored by HP/BRIMS). More recently, the first workshop on the very same subject has been held in Aalborg, Denmark (see http://www.math.auc.dk/~raussen/admin/workshop/workshop.html where the articles of this issue, among others, have been first sketched. But what is "Geometry and Concurrency" composed of then? It is an area of research made of techniques which use geometrical reasoning for describing and solving problems