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

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    $\begingroup$ It would seem $G$ would need to be abelian in order to give a permutative category in the way you describe. How are you going to define $\oplus$ on morphisms? Since it must preserve compositions, it looks like a homomorphism $G\times G\to G$. As to your actual question, I don't know. $\endgroup$ – Mark Grant Aug 22 '12 at 19:59
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    $\begingroup$ Mark is right. Your category is connected, so its classifying space would be the zeroth space of the spectrum you would get if it were permutative. Then the fundamental group G would of course be commutative. Ignoring the notational problem with your definition of $h^n(X)$ ($\Sigma S^n$ is just $S^{n+1}$ and the difference between spaces and spectra is fudged) the spectrum you get when $G$ is abelian is just the suspension of the Eilenberg-Maclane spectrum $HG$, so it represents ordinary cohomology with coefficients in $G$. But you should learn the basics first, ask questions later. $\endgroup$ – Peter May Aug 22 '12 at 20:45
  • $\begingroup$ Thanks for the pointer; as for the notation I was treating $X$ as a regular space and smashing it with the suspension spectrum of $S^n$ and then the bracket notation would denote homotopy classes of spectra. I believe this is the notation used in the paper I posted but it may idiosyncratic. In any case, I'll think about how it is the suspension of $HG$. $\endgroup$ – Geoffrey Aug 22 '12 at 20:57

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