I am wondering if the following construction has appeared in the literature:
Let $X$ be a pointed space. Define the singular configuration space of $X$, $F^*(X,k)$ ,to be $\{(x_1,\dots,x_k)| x_i=x_j \implies i=j \: \mathrm{or} \: x_1=x_2=\dots=x_k=*\}$. If the basepoint of $X$ is disjoint, then this coincides with the classical configuration space $F(X,k)_+$.
Now consider the sequential spectrum $\{\Sigma^n X\}_n$ . We may take categorical products to achieve the spectrum $\{ (\Sigma^n X )^k\}_n$. The gluing maps $\Sigma(\Sigma^n X )^k \rightarrow (\Sigma^{n+1} X)^k$ are given on each factor by the suspension of the projection map. We now define configurations in a suspension spectrum, $F^*(\Sigma^\infty X,k)$, as $\{ F^*(\Sigma^n X ,k)\}_n$ with gluing maps given by the restriction of the product gluing maps.
A simple observation is that $H_i(F^*(\Sigma^\infty X,k))$ is the stabilized homology of singular configurations, i.e. $\operatorname{colim}\limits_n H_{i+n}(F^*(\Sigma^n X,k))$. Is it known if $F^*(\Sigma^\infty X,k)$ is equivalent to a suspension spectrum when $k \geq 2$?
