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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.
This volume is an introductory textbook to K-theory, both algebraic
and topological, and to various current research topics within the
field, including Kasparov's bivariant K-theory, the Baum-Connes
conjecture, the comparison between algebraic and topological
K-theory of topological algebras, the K-theory of schemes, and the
theory of dg-categories.
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