Methods in Computer Science 2006,
Kananaskis Field Station (University of Calgary),
Abstract This is an introduction to the theory and applications of operads, with the emphasis on the theory.
In the first half I will give the basic definitions. Operads can be viewed in two ways: (i) as algebraic theories; (ii) as categorical structures in their own right. (Compare Lawvere theories.) I will explain the two viewpoints and how they can be useful in some diverse mathematical situations.
In the second half I will describe some generalizations of the notion of operad useful in higher category theory. Again, such generalized operads can be viewed in two ways: (i) as algebraic theories of a rather sophisticated kind (including, for instance, various theories of n-categories); (ii) as higher categorical structures in their own right, of independent interest.