TITLE: What are iterated cobar constructions?
QUESTION [5 upvotes]: In Beck's paper "On H-spaces and Infinite Loop Spaces", he states that every algebra over the monad $\Omega^k$$\Sigma^k$ is a $k$-fold loop space. He proves the trivial case k = 0 when this is the identity monad, and says the remaining cases follow from "iterated cobar constructions". I'm hoping someone can elaborate on what exactly is meant by this statement.

REPLY [2 votes]: There is a map of mondas $D_k\longrightarrow \Omega^kS^k$ where $D_k$ is the little disks operad (see May's original lecture notes, for example), so any algebra over the target monad is one over the little disks, and hence a $k$-fold loop space.