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Simplicial Methods for Operads and Algebraic Geometry (Advanced Courses in Mathematics - CRM Barcelona)

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en Limba Engleză Carte Paperback – 02 Dec 2010
An introduction to two topics in homotopy theory: Dendroidal Sets and Derived Algebraic Geometry.
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Specificații

ISBN-13: 9783034800518
ISBN-10: 3034800517
Pagini: 196
Dimensiuni: 168 x 240 x 10 mm
Greutate: 0.30 kg
Ediția: 2010
Editura: Springer
Colecția Birkhäuser
Seria Advanced Courses in Mathematics - CRM Barcelona

Locul publicării: Basel, Switzerland

Public țintă

Graduate

Cuprins

Lectures on Dendroidal Sets.- Operads.- Trees as operads.- Dendroidal sets.- Tensor product of dendroidal sets.- A Reedy model structure on dendroidal spaces.- Boardman–Vogt resolution and homotopy coherent nerve.- Inner Kan complexes and normal dendroidal sets.- Model structures on dendroidal sets.- Simplicial Presheaves and Derived Algebraic Geometry.- Motivation and objectives.- Simplicial presheaves as stacks.- Algebraic stacks.- Simplicial commutative algebras.- Derived stacks and derived algebraic stacks.- Examples of derived algebraic stacks.

Textul de pe ultima copertă

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 Toën 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 Toën’s lectures, some  background in algebraic geometry is also necessary.

Caracteristici

The material is not available elsewhere, except in the original research articles
The material is presented by the best experts, in fact the inventors themselves
The exposition has maintained the directness of the original lectures