A question on recognition of equivariant loop spaces

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.

• 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 Aug 24, 2020 at 11:56
• Thank you so much for your comment. Do you think that it follows from Shimakawa's work semanticscholar.org/paper/… Aug 24, 2020 at 13:34