Right now I am interested in the "stabilization" endofunctor of the category of ∞-groupoids sending an object $X$ to $\text{colim } Ωⁿ Σⁿ X$. This colimit is related to $∃Y:X≅ΩY$. In Ω-spectra, this proposition is true.
I was trying to figure out why there are so many variations in categories of spectra and read here about five properties one might like of a category of spectra that were altogether impossible. I was wondering specifically what the case is for Ω-spectra along with its adjunction with CW-complexes. What properties does this lack and what are some selling points of these other longer definitions of spectra?