3
$\begingroup$

By a (finite) 2-group $X$, I mean a finite group $G$, a finite abelian group $A$, an action of $G$ on $\operatorname{Aut}(A)$, as well as a 3-cocycle $\alpha\in H^3(BG, A)$. They are also equivalent to crossed modules and represent the "fusion rules" of fusion 2-categories that are "pointed" in a strong enough sense.

I think of a group $n$-cocycle $\omega\in H^n(BG,\mathbb{R}/\mathbb{Z})$ as a map that associates a $\mathbb{R}/\mathbb{Z}$-value to each cellular 1-cocycle on the $n$-simplex, such that the sum over $n$-simplices equals $0$ for every 1-cocycle in the $n+1$-simplex. A 1-cocycle on the $n$-simplex is specified by the $G$-elements on $n$ of the edges (usually taken the $01$, $12$, $\ldots$, $(n-1)n$ edges).

Similarily, I believe it is correct to think of a "2-group 4-cocycle" $\omega\in H^4(BX, \mathbb{R}/\mathbb{Z})$ as a map that associates a $\mathbb{R}/\mathbb{Z}$-value to every configuration of (1) a cellular 1-cocycle $g$ on the 4-simplex with values in $G$, and (2) a cellular 2-cochain $a$ with values in $A$ such that $d_g a=\alpha(g)$, where $d_g$ is the coboundary operator twisted by $g$ via the $G$-action. Such a configuration is specified by 4 $G$-values and 6 $A$-values. The sum over all 4-simplices inside a 5-simplex has to be $0$ for each configuration as described above on the 5-simplex.

Question: In this context, what are some explicit examples of (non-trivial) 2-group 4-cocycles for some "small" finite 2-groups. For example, what's the cohomology of the 2-group with $G=A=\mathbb{Z}_2$ and $\omega(a,b,c)=abc$? I'd be most happy about direct formulas depending on the $G$ and $A$-values on the 4-simplex, but also expressions involving higher cup products etc.

I'm aware that any element in $H^4(BG,\mathbb{R}/\mathbb{Z})$ gives an example. If the action of $G$ on $A$ is trivial, then the Pontryagin square gives a full classification of $H^4(B^2A,\mathbb{R}/\mathbb{Z})$ which yields further examples. Is there any way to "twist" the last example if the action is non-trivial? In general, I'm looking for examples that depend on both $g$ and $a$.

(Btw, the reason why I'm interested in this is that in physics, these describe discrete "2-group gauge theories" in 3+1 dimensions.)

Improve this question
$\endgroup$
2
  • $\begingroup$ (Sorry for being a bit sloppy and letting $H(.)$ denote both the cohomology group and the cocycles.) $\endgroup$
    – Andi Bauer
    Commented Dec 7, 2023 at 23:28
  • 4
    $\begingroup$ As a topologist, the way I'd try and compute this cohomology is via the spectral sequence of the fibration $K(A,2)\to BX\to BG$. At least when $A=\mathbb{Z}/2$ this should be tractable, as the cohomology of $K(\mathbb{Z}/2,2)$ is known. The fundamental class transgresses to the given class in $H^3(BG,A)$. $\endgroup$
    – Mark Grant
    Commented Dec 8, 2023 at 7:47

0

Reset to default

You must log in to answer this question.