Higher topos theory Lurie, Jacob; Lurie, Jacob
2009., 20090706, 2009, 2009-07-06, Letnik:
170
eBook
Higher category theory is generally regarded as technical and forbidding, but part of it is considerably more tractable: the theory of infinity-categories, higher categories in which all higher ...morphisms are assumed to be invertible. In Higher Topos Theory, Jacob Lurie presents the foundations of this theory, using the language of weak Kan complexes introduced by Boardman and Vogt, and shows how existing theorems in algebraic topology can be reformulated and generalized in the theory's new language. The result is a powerful theory with applications in many areas of mathematics.
The emergent mathematical philosophy of categorification is reshaping our view of modern mathematics by uncovering a hidden layer of structure in mathematics, revealing richer and more robust ...structures capable of describing more complex phenomena. Categorification is a powerful tool for relating various branches of mathematics and exploiting the commonalities between fields. It provides a language emphasizing essential features and allowing precise relationships between vastly different fields.This volume focuses on the role categorification plays in geometry, topology, and physics. These articles illustrate many important trends for the field including geometric representation theory, homotopical methods in link homology, interactions between higher representation theory and gauge theory, and double affine Hecke algebra approaches to link homology.The companion volume (Contemporary Mathematics, Volume 683) is devoted to categorification and higher representation theory.
The emergent mathematical philosophy of categorification is reshaping our view of modern mathematics by uncovering a hidden layer of structure in mathematics, revealing richer and more robust ...structures capable of describing more complex phenomena. Categorified representation theory, or higher representation theory, aims to understand a new level of structure present in representation theory. Rather than studying actions of algebras on vector spaces where algebra elements act by linear endomorphisms of the vector space, higher representation theory describes the structure present when algebras act on categories, with algebra elements acting by functors. The new level of structure in higher representation theory arises by studying the natural transformations between functors. This enhanced perspective brings into play a powerful new set of tools that deepens our understanding of traditional representation theory.This volume exhibits some of the current trends in higher representation theory and the diverse techniques that are being employed in this field with the aim of showcasing the many applications of higher representation theory.The companion volume (Contemporary Mathematics, Volume 684) is devoted to categorification in geometry, topology, and physics.
Vor den deutschen Zivilgerichten klagen zunehmend Spieler, die vor der weiteren Liberalisierung des Glücksspielrechts im Jahr 2021 an Glücksspielen im Internet teilgenommen haben. Sie fordern von ...Unternehmen, die solche Glücksspiele anbieten, die Rückzahlung geleisteter Spieleinsätze. Diese so genannten "Spielerklagen" waren bisher unterschiedlich erfolgreich. Die deutschen Gerichte sind sich zwar weitgehend einig, dass die Spielverträge wegen Verstoßes gegen die bisherigen Glücksspielstaatsverträge nach § 134 BGB unwirksam sind, obwohl die Glücksspielaufsicht bewusst das bisherige Glücksspielverbot im Internet nicht durchgesetzt hat. Vielmehr hat die staatliche Aufsicht Verstöße der Unternehmen geduldet, wenn sich die Unternehmen an bestimmte Vorgaben der Behörden halten, um zu verhindern, dass Spieler auf ein gänzlich unreguliertes Angebot ausweichen. Uneinigkeit besteht aber, ob etwaige bereicherungsrechtliche Ansprüche nach § 817 Satz 2 BGB gesperrt sind. Dagegen haben die Gerichte deliktische Schadensersatzansprüche der Spieler bisher nur vereinzelt bejaht. Auch die Verjährung etwaiger Ansprüche wirft Fragen auf. Zu diesen privatrechtlichen Aspekten der Spielerklagen, die bisher noch nicht umfassend diskutiert wurden, nehmen die Beiträge dieses Bandes Stellung.
This book presents the complete proof of the Bloch-Kato conjecture and several related conjectures of Beilinson and Lichtenbaum in algebraic geometry. Brought together here for the first time, these ...conjectures describe the structure of étale cohomology and its relation to motivic cohomology and Chow groups.
Although the proof relies on the work of several people, it is credited primarily to Vladimir Voevodsky. The authors draw on a multitude of published and unpublished sources to explain the large-scale structure of Voevodsky's proof and introduce the key figures behind its development. They proceed to describe the highly innovative geometric constructions of Markus Rost, including the construction of norm varieties, which play a crucial role in the proof. The book then addresses symmetric powers of motives and motivic cohomology operations.
Comprehensive and self-contained,The Norm Residue Theorem in Motivic Cohomologyunites various components of the proof that until now were scattered across many sources of varying accessibility, often with differing hypotheses, definitions, and language.
The corrosion behavior of the AA6016/SM490 couple in NaCl solutions was analyzed by experiments and simulations. In experiments, galvanic corrosion occurred between anodic AA6016 and cathodic SM490. ...In addition, as decreasing in NaCl concentration, the anodic dissolution of SM490 proceeded. From FEM simulations, it was found that as decreasing in NaCl concentration, the anodic current on AA6016 became insufficient to commensurate with the cathodic current on SM490; therefore, the anodic dissolution of SM490 proceeded simultaneously. The anodic dissolution rate of SM490 in 0.5 and 0.05 wt% NaCl was 13 and 18 times larger, respectively, than that in 5 wt% NaCl.
•The corrosion behavior of the AA6016/SM490 couple in NaCl solutions was analyzed by both experiments and simulations.•Besides galvanic corrosion (AA6016: anode, SM490: cathode), the self-corrosion of SM490 occurred in low NaCl solution.•The balance between galvanic and self-corrosion depending on the NaCl concentration was analyzed by FEM simulations.
This volume contains the proceedings of the CATS4 Conference on Higher Categorical Structures and their Interactions with Algebraic Geometry, Algebraic Topology and Algebra, held from July 2-7, 2012, ...at CIRM in Luminy, France.Over the past several years, the CATS conference series has brought together top level researchers from around the world interested in relative and higher category theory and its applications to classical mathematical domains.Included in this volume is a collection of articles covering the applications of categories and stacks to geometry, topology and algebra. Techniques such as localization, model categories, simplicial objects, sheaves of categories, mapping stacks, dg structures, hereditary categories, and derived stacks, are applied to give new insight on cluster algebra, Lagrangians, trace theories, loop spaces, structured surfaces, stability, ind-coherent complexes and 1-affineness showing up in geometric Langlands, branching out to many related topics along the way.
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