Neuware -This book is an introduction to two higher-categorical topics in algebraic topology and algebraic geometry relying on simplicial methods. It is based on lectures - livered at the Centre de Recerca Matem ati ca in February 2008, as part of a special year on Homotopy Theory and Higher Categories. Ieke Moerdijk�s lectures constitute an introduction to the theory ofdendroidal sets, an extension of the theory of simplicial sets designed as a foundation for the homotopy theory of operads. The theory has many features analogous to the theory of simplicial sets, but it also reveals many new phenomena, thanks to the presence of automorphisms of trees. Dendroidal sets admit a closed symmetric monoidal structure related to the Boardman{Vogt tensor product. The lecture notes develop the theory very carefully, starting from scratch with the combinatorics of trees, and culminating with a model structure on the category of dendroidal sets for which the brant objects are the inner Kan dendroidal sets. The important concepts are illustrated with detailed examples.Springer Basel AG in Springer Science + Business Media, Heidelberger Platz 3, 14197 Berlin 196 pp. Englisch.

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Detalles bibliogr�ficos

T�tulo: Simplicial Methods for Operads and Algebraic Geometry
Editor: Springer Basel, Birkh�user Basel Dez 2010
Fecha de publicaci�n: 2010
Idioma: Ingl�s
ISBN 10: 3034800517
ISBN 13: 9783034800518
Encuadernaci�n: Taschenbuch
Estado: Neu

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Sinopsis

This book is an introduction to two new topics in homotopy theory: Dendroidal Sets (by Ieke Moerdijk) and Derived Algebraic Geometry (by Bertrand To�n). The category of dendroidal sets is an extension of that of simplicial sets, based on rooted trees instead of linear orders, suitable as a model category for higher topological structures. Derived algebraic geometry deals with functors from simplicial commutative rings to simplicial sets subject to a homotopical descent condition. The material in the book is an enhanced version of lecture notes from courses given within a special year on Homotopy Theory and Higher Categories at the CRM in Barcelona.

De la contraportada

This book is an introduction to two higher-categorical topics in algebraic topology and algebraic geometry relying on simplicial methods.

Moerdijk s lectures offer a detailed introduction to dendroidal sets, which were introduced by himself and Weiss as a foundation for the homotopy theory of operads. The theory of dendroidal sets is based on trees instead of linear orders and has many features analogous to the theory of simplicial sets, but it also reveals new phenomena. For example, dendroidal sets admit a closed symmetric monoidal structure related to the Boardman Vogt tensor product of operads. The lecture notes start with the combinatorics of trees and culminate with a suitable model structure on the category of dendroidal sets. Important concepts are illustrated with pictures and examples.

The lecture series by To�n presents derived algebraic geometry. While classical algebraic geometry studies functors from the category of commutative rings to the category of sets, derived algebraic geometry is concerned with functors from simplicial commutative rings (to allow derived tensor products) to simplicial sets (to allow derived quotients). The central objects are derived (higher) stacks, which are functors satisfying a certain up-to-homotopy descent condition. These lectures provide a concise and focused introduction to this vast subject, glossing over many of the technicalities that make the subject s research literature so overwhelming.

Both sets of lectures assume a working knowledge of model categories in the sense of Quillen. For To�n s lectures, some background in algebraic geometry is also necessary.

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