I have a question about equivariant loop space that has been bothering me, and that I have not been able to find an answer to in the obvious places.

We know from the work of Segal that to give a loop space structure on $X$ is equivalent to producing a simplicial space $X_\bullet$ in which $X_0$ is weakly contractible, $X_1$ is weakly equivalent to $X$ and the map $X_n \to (X_1)^n$, corresponding to the order-preserving inclusion $[1] \to [n]$ taking $0$ to $0$, is a weak equivalence. (see Proposition 1.5 of the article CATEGORIES AND COHOMOLOGY THEORIES by G. Segal)

For the $n$-fold loop spaces case one may see the work of Peter Cobb. The approach is in the same spirit as G. Segal's investigating of the infinite loop spaces via special $\Gamma$-spaces.

Can we have a (similar) description for the equivariant loop space $\Omega^V X$ where $X$ is $G$-space and $V$ is a $G$-representation?

Thank you so much in advance. Any help will be appreciated.

  • 3
    $\begingroup$ whoa! Thanks for pointing out Cobb's work... it looks like he discovered Joyal's category Theta_n and proved results like Berger did, but way back in 1974! (or am I misreading?) I don't have an answer to your question, but my suspicion is that only (reduced) permutation representations would be amenable to a combinatorial description like Cobb's $\endgroup$ Aug 24, 2020 at 11:56
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    $\begingroup$ Thank you so much for your comment. Do you think that it follows from Shimakawa's work semanticscholar.org/paper/… $\endgroup$ Aug 24, 2020 at 13:34


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