Hi, I've recently been interested in Stable Homotopy Theory and was reading this text to understand some basics: http://www.maths.ed.ac.uk/~aar/papers/carlmilg.pdf

Near the end of the text (p582) we learn that there is a functorial way of producing a connective spectrum out of a permutative category. Now, if we consider a (topological) group $G$, we can realize $BG$ as the geometric realization of the nerve of $G$ considered as a one object groupoid. Since there is only one object in $G$, call it $\ast$, we could just define $\ast\oplus\ast = \ast$ with the morphisms making this direct sum correspond to $G\times G$. This seems like it would satisfy the axioms of being a permutative category, and so it should give us some spectrum $E$. My question is what happens if we define a cohomology theory $h^n(X)=[X\wedge S^n\wedge \Sigma , E]$ where $\Sigma$ is the sphere spectrum? Does this give some sort of well known cohomology theory?

Edit: edited post to clarify notation