# Conditions for a Topological space to be a Spectrum

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»).

Some definitions:

1. 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.

2. A space $(X,p)$ can be delooped if there is a space $(Y,p')$ so that $(Y,p')$ is the loop space of $(X,p)$.

3. 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.

• Up to homotopy, a space $X$ is a connective spectrum if and only if it is an algebra over some $E_\infty$ operad. Intuitively, this means that it has a multiplication that is associative and commutative up to all higher homotopies. See J.P. May, "The Geometry of Iterated Loop Spaces". – Justin Young Dec 15 '13 at 7:49
• Being spectrum is an extra structure, not just a condition (i.e. one should fix all deloopings) – Grigory M Dec 18 '13 at 6:09
• @Grigory M: Sorry, don't get your point; what do you mean by fixing all deloopings? – User Some Number Dec 26 '13 at 21:52
• @UserSomeNumber One space can have different structures of infinite loop space. So ($\Omega^\infty$-)spectrum is not just a space that can be infinitely delooped, it's a space together with some choice of such deloopings. – Grigory M Dec 26 '13 at 21:57
• @Grigory M: Thank you; just curious: is there any way of comparing different deloopings? For example, embeddings can be considered up to isotopy, maps up to homotopy, etc. Is there a way of comparing deloopings? – User Some Number Dec 26 '13 at 22:18