Let $X,Y$ be topological spaces, both of which admit an action by a group $G$. What is the definition of the space denoted by $ X \times_G Y $?
In particular, let $C_n(k)$ denote the space of ordered $k$-tuples of $n$-dimensional cubes in $I^n$ with disjoint interiors. This space admits an action by the symmetric group $\Sigma_k$ by permutation of the cubes. Similarly given a space $X$, the $k$-fold product $X^k$ admits an action by $\Sigma_k$ by permuting the terms. What is the definition of the space $C_n(k) \times_{\Sigma_k} X^k$?