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.
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1$\begingroup$ I think p. 21 of maths.ed.ac.uk/~aar/surgery/uicc/drachman.pdf might be relevant. It's a very brief section, but is related to the question. $\endgroup$– user62675Feb 16, 2015 at 17:40
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$\begingroup$ Thanks! I will look that over. It does indeed seem relevant. $\endgroup$– Matthew SartwellFeb 16, 2015 at 18:45
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4$\begingroup$ A modern treatment of this is also in section 5.2.3 of math.harvard.edu/~lurie/papers/higheralgebra.pdf $\endgroup$– Marc HoyoisFeb 16, 2015 at 20:27
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1$\begingroup$ An older reference is May's Geometry of Iterated Loop Spaces. $\endgroup$– Todd Trimble ♦Feb 17, 2015 at 23:54
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$\begingroup$ @ToddTrimble Part of my question is whether or not the methods May employed were the iterated cobar constructions that Beck had in mind. I would be happy to see what argument people think he had in mind for the $k = 1$ and $k = 2$ cases. $\endgroup$– Matthew SartwellFeb 18, 2015 at 16:21
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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.
