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Group completion and the answer to your questions (for $k\geq 2$) are probaby best understood homologically. An ancient definition is that a map $X\rightarrow Y$ of homotopy commutative $H$-spaces …

$C_n(X)$ is not a monoid in any natural way; I never said it was. And the target $\Omega^n\Sigma^n X$ of $\alpha_n$ has $n$ different loop products. The question is not meaningful as posed. Neverthe …

There is a very careful analysis of this question in Lemma 3.4.2, page 57, of More Concise Algebraic Topology, by Kate Ponto and myself. Assuming that $E$ and $B$ are connected, a fibration $E\longrig …

This is an edited extract from a book in preparation (Bruner, Catanzaro, May) tentatively titled Characteristic Classes and is therefore overlong for an answer. This is similar to Denis Nardin's answe …

This is a longish comment on James Schwass's answer, not an answer to the original questions. Have to be a little careful here. We are deliberately informal (p.97), but we are working in a catego …

Charles, thanks for asking. This is not an answer, but it is too long for a comment. Like you, I encourage others to pursue the question and closely related ones I'll raise here. The paper of mine yo …

Something very close to this and quite possibly relevant occurs in the theory of $E_{\infty}$ ring spaces. There is a notion of an action of an operad $Q$ on an operad $P$. When $Q$ acts on $P$, the …

I don't have a real answer but possibly relevant observations and speculations. I imagine $BG$ can be pretty horrible if $G$ is not well-pointed, unless of course one uses the fat realization, when I …

That guy that keeps getting mentioned here never claimed it in general because he does not believe it in general. Think about $G=U(n)$ and about equivariant $K$-theory. This is very close to the At …

In view of the references to my Memoir, Classifying spaces and fibrations, in other answers, I guess I should answer too. The requested answer is implicit but not quite explicit there. Fix a groupli …