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Justin Young

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 Name Justin Young Member for 2 years Seen 4 hours ago Website Location Age
 May25 awarded ● Enthusiast May6 comment Seeking errata for Berger-Moerdijk Axiomatic Homotopy Theory for OperadsBy studying free algebras, one can prove using direct point set arguments, and knowledge of how to build pushouts in $\mathcal P$ algebras from pushouts in pointed spaces, to prove that the pushout in $\mathcal P$ algebras of the inclusion $\mathbb P(S^n_+) \to \mathbb P(D^{n+1}_+)$ along a map $\mathbb P(S^n_+) \to X$ gives a closed inclusion $X \to Y$. This is enough to get the small object argument working in pointed spaces, so one can do a transfer. Feb24 comment Non-commutativity of certain Hopf spaces$R$ is an arbitrary ring, but $R=\mathbb R$ suffices as does $R= \mathbb Z$. And yes, the James construction works if you already know about it (which you do). Feb23 comment Non-commutativity of certain Hopf spacesEdit: in the above I should specify that tensor algebras are not commutative provided the generators are more than one dimensional, or in odd degree. Remember, to be commutative, odd degree elements must satisfy $2x^2 = 0$. Feb23 comment Non-commutativity of certain Hopf spacesIn general, if $\tilde H_*(X, R)$ is free then $H_*(\Omega \Sigma X, R)$ is a tensor algebra on $\tilde H_*(X,R)$, and tensor algebras are not commutative. You can prove this using your Serre SS. With $X = S^1$ and $X = S^1 \vee S^1$ you get your result. Hi Prasit! Feb23 comment Non-$\Sigma$ $E_n$ algebras?Thank you for responding. I do not have any examples in mind other than the obvious free algebras. This question was put to me and I thought it was surprising and interesting, and I had never thought about it, either. I thought that there should be a simple answer, but I was unable to come up with one quickly myself or to find anything in the literature about it. Feb22 comment Non-$\Sigma$ $E_n$ algebras?My ideal answer would be something analogous to the characterization of symmetric $C_n$ algebras as $n$-fold loop spaces. Perhaps a nice description of the free algebras. It is clear you have a loop space for all $n$, and then at $C_\infty$ you get a loop space again, but does anything interesting happen in the middle? Feb22 asked Non-$\Sigma$ $E_n$ algebras?