Operads and Cosimplicial Objects: an Introduction

This paper is an introduction to the series of papers [26, 28, 29, 30], in which we develop a combinatorial theory of certain important operads and their actions. The operads we consider are A∞ operads, E∞ operads, the little n-cubes operad and the framed little disks operad. Sections 2, 6 and 9, which can be read independently, are an introduction to the theory of operads. The reader is also referred to the very interesting papers of Batanin ([3, 4]), which treat similar questions from a categorical point of view. Here is an outline of the paper. In Section 2 we motivate the concept of non-symmetric operad. We give the definition of A∞ space and state the characterization (up to weak equivalence) of loop spaces: a space is weakly equivalent to a loop space if and only if it is a grouplike A∞ space. In Section 3 we introduce the total space construction Tot for cosimplicial spaces and the related conormalization construction for cosimplicial abelian groups. In Section 4 we address the question: when is Tot of a cosimplicial space an A∞ space? We obtain a useful sufficient condition for this to happen. In Section 5 we reformulate the main result of Section 4 in a way which is convenient for generalization. In Section 6 we motivate the concept of operad. We give the definition of E∞ space and state the characterization (up to weak equivalence) of infinite loop spaces: a space is weakly equivalent to an infinite loop space if and only if it is a grouplike E∞ space. Section 7 contains motivation for the main result of Section 8. In Section 8 we give a sufficient condition for Tot of a cosimplicial space to be an E∞ space. In Section 9 we introduce the little n-cubes operad Cn. Operads weakly equivalent to Cn are called En operads. We give the characterization (up to weak equivalence) of n-fold loop