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Spaces and classical (single-object) operads are the easiest example. …

Here is something that I think is reasonably difficult to get around. As you observe, the unit is the initial object of commutative monoids, and so your request includes that the unit is cofibrant. Su …

(Similar remarks apply to operads.) …

Both papers choose a normalization that is different from yours: they define $$ [a,b] = (-1)^{na+1} s(a \otimes b). $$ This is definition 5.7 in Cohen's paper that you mentioned. The reason for this …

Unfortunately there is no such filtration. At first, this looks very similar (but not as strong as) asking for a map of $E_\infty$ spaces $\coprod BU(n) \to BU$ which would become a splitting map $ku …

I know this question is a little old but I just came across it. Roughly, as you say, from this data you get an object $v$ of $O^\otimes$ and an algebra $A$ of $Alg_{/O}(C)$. However, you also get an …

May gave a proof in the case $n = \infty$ in this followup paper to "Geometry of iterated loop spaces," which relies on knowing that a certain map (from a free algebra on $X$ to the free infinite loop …

One reason that operads are used is in obstruction theory. … However, it is difficult to attack such problems outside specific circumstances without using operads. (Perhaps someone else knows of methods that don't implicitly use operads; I don't.) …

You can do that. If an operad O acts on a space X, then the structure maps O(n) x Xn -> X induce homology operations H*O(n) ⊗ H*(X)⊗n -> H*(X). In particular, any path component in O(2) produces …