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.

  • 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. $\endgroup$ – user62675 Feb 16 '15 at 17:40
  • $\begingroup$ Thanks! I will look that over. It does indeed seem relevant. $\endgroup$ – Matthew Sartwell Feb 16 '15 at 18:45
  • 4
    $\begingroup$ A modern treatment of this is also in section 5.2.3 of math.harvard.edu/~lurie/papers/higheralgebra.pdf $\endgroup$ – Marc Hoyois Feb 16 '15 at 20:27
  • 1
    $\begingroup$ An older reference is May's Geometry of Iterated Loop Spaces. $\endgroup$ – Todd Trimble Feb 17 '15 at 23:54
  • $\begingroup$ @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 Sartwell Feb 18 '15 at 16:21

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.

| cite | improve this answer | |

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.