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Let $G$ be a group and $A$ a group with a $G$-action. Then in general, $H^0(G;A)=A^G$ is a group, and $H^1(G;A)$ is simply a pointed set. If $A$ is an abelian group, then $H^i(G;A)$ exists and is an abelian group for all $i \ge 0$.

This resembles the fact that if $X$ is a pointed space, then $\pi_1(X)$ is always a group, while $\pi_0(X)$ is just a pointed set. But if $X$ has an abelian group structure (more generally, a double loop space structure), then all $\pi_i(X)$ are abelian groups.

We can extend the analogy further to the non-pointed situation if we consider as given not a group $A$ with an action of $G$, but an extension $0 \to A \to E \to G \to G$. Then there is $H^2(G;A)$, which is simply a truth value, depending on whether the sequence splits or not. This is analogous to the fact that for a non-pointed space $X$, $\pi_{-1}(X)$ is a truth value, depending on whether $X$ is empty or not.

The relation can be made more explicit if we let $X$ be the space (groupoid) of sections of $E \to G$. Then $H^i(G;A)$ is naturally $\pi_{1-i}(X)$, at least for $0 \le i \le 2$.

Is there a generalization to all natural numbers $i$ in the case that $A$ is abelian? This would seem to require producing a non-connective spectrum of sections, but I'm not sure how to do this. My best attempt is to deloop and think about $0 \to BA \to BE \to BG \to 0$. However, how do I deal with an infinite delooping of $BA$ in this sequence when $BG$ is not a loop space (because $G$ might be nonabelian)?

Also, is there a group cohomology analogue of the case where $X$ is a single loop space? I.e., where $H^0(G;A)$ is an abelian group, $H^1(G;A)$ is a group, $H^2(G;A)$ is a pointed set, and $H^3(G;A)$ (if you set things up correctly) is a truth value?

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    $\begingroup$ Isn't the observation in your second paragraph actually a theorem? Specifically, for a space $X$ and a sheaf $A$ of abelian groups on $X$, $H^n (X, A) \cong \pi_{m - n} \mathbf{R} \Gamma (X, K(A, m))$, where $K(A, m)$ is the simplicial sheaf version of the usual thing. Group cohomology comes about when you have $X = B G$. The existence of $H^1 (G, A)$ for non-abelian $A$ comes about because we can define $K (A, 1) = B A$ for non-abelian $A$ as well, etc. $\endgroup$
    – Zhen Lin
    Commented Jul 12, 2020 at 22:27
  • $\begingroup$ Do you mean in my fourth paragraph? $\endgroup$
    – David Corwin
    Commented Jul 12, 2020 at 22:39
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    $\begingroup$ I mean second. I'm not familiar with $H^2$ for group extensions. $\endgroup$
    – Zhen Lin
    Commented Jul 12, 2020 at 22:44
  • $\begingroup$ Can you give a reference for the definition $H^i(G,A)$ when for a group extension? Or do you just mean normal group cohomology and the extension is just an extension with abelian first term? $\endgroup$
    – Denis Nardin
    Commented Jul 15, 2020 at 6:21
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    $\begingroup$ In paragraph 5, I think you may be looking for the notion of a parametrized spectrum over $BG$. You can think of such a thing as a functor $BG \to Spectra$, or equivalently as an infinite loop object in $Spaces_{/BG}$. More generally, for a base space $X$, if $X$ has an $E_n$-monoidal structure (e.g. an $n$-fold loopspace) you can contemplate $E_n$-maps $X \to Spectra$. This sort of situation (in the case where the functor is valued in invertible spectra) is studied in Antolin-Camarena--Barthel arXiv:1411.7988. $\endgroup$
    – Aaron Mazel-Gee
    Commented Jul 15, 2020 at 17:52

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