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 $ \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.