Given a commutative ring $k$ there is a bicategory with

- algebras over $k$ as objects,
- bimodules as morphisms,
- bimodule homomorphisms as 2-morphisms.

This is a *monoidal* bicategory, since we can take the tensor product of algebras, and everything else gets along nicely with that.

Given any monoidal bicategory we can take its **core**: that is, the sub-monoidal bicategory where we only keep *invertible* objects (invertible up to equivalence), *invertible* morphisms (invertible up to 2-isomorphism), and *invertible* 2-morphisms.

The core is a monoidal bicategory where everything is invertible in a suitably weakened sense so it's called a **3-group**.

The particular 3-group we get from a commutative ring $k$ could be called its **Brauer 3-group** and denoted $\mathbf{Br}(k)$. It's discussed on the $n$Lab: there it's called the Picard 3-group of $k$ but denoted as $\mathbf{Br}(k)$.

Like any 3-group, $\mathbf{Br}(k)$ has homotopy groups which I will call $\pi_1, \pi_2, \pi_3$ (though there are choices of where we start numbering). These are well-known things:

$\pi_1$ is the Brauer group of $k$.

$\pi_2$ is the Picard group of $k$.

$\pi_3$ is the group of units of $k$.

My question is whether people have studied, or computed, the Postnikov invariants involving these things. The simplest is the map

$$ a : \pi_1^3 \to \pi_2$$

coming from the associator in the monoidal category of $k$-algebras (with isomorphism classes of bimodules as morphisms). Since the associator obeys the pentagon identity this is a 3-cocycle on $\pi_1$ with values in its module $\pi_2$, so it gives an element of $ H^3(\pi_1, \pi_2)$.

Is this element trivial? If not, what is it?

But in fact $\mathbf{Br}(k)$ is not just a 3-group but also a symmetric monoidal bicategory. So, it's what I call a **symmetric 3-group**, though some others call it a **Picard 2-category**. These have a number of other Postnikov invariants:

- Nick Gurski, Niles Johnson, Angélica M. Osorno and Marc Stephan, Stable Postnikov data of Picard 2-categories.

Has anyone figured out any of these for $\mathbf{Br}(k)$?