Simplicial Methods for Operads and Algebraic Geometry

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.