I'm looking for conditions for a topological space $X$ to be a Spectrum. A topological space $X$ is a spectrum if it can be delooped infinitely (more accurately, «double-infinitely»).
The Loop Space of a space $X$ is the space of maps from $S^1$ into $X$ with the compact-open topology. In our case $X$ is a pointed space.
A space $(X,p)$ can be delooped if there is a space $(Y,p')$ so that $(Y,p')$ is the loop space of $(X,p)$.
A Spectrum is a space that can be infinitely-delooped in both directions, i.e., if we label $X$ as the $0$-th space, then there is both a space $Y$ labeled with a $1$, so that $Y$ is the loop space of $X$, and there is a space $Z$ labeled with a $-1$, so that $X$ is the loop space of $Z$. In a spectrum, this process goes on to both $+\infty$ and $-\infty$.
Does nayone know the necessary/sufficient conditions for a topological space to be a spectrum? Thanks.