Brauer groups for omega-monoids

Background: Brauer Spectra

Let $R$ be an $E_{\infty }$ ring spectrum, we have the following important invariants

$g{l}_1(R):=Au{t}_{Mo{d}_R}(R)\subset En{d}_{Mo{d}_R}(R)\simeq R$ $Pic(R):=Au{t}_{Ca{t}_R}^L(Mo{d}_R)\subset En{d}_{Ca{t}_R}(Mo{d}_R)\simeq Mo{d}_R$ $Br(R):=Au{t}_{2Ca{t}_R}^L(Ca{t}_R)\subset En{d}_{2Ca{t}_R}(Ca{t}_R)\simeq Ca{t}_R$ (Invertible exact endofunctors of $Mo{d}_R$, which under Eilenberg-Watts, identifies with invertible modules.)

In our language, these are clearly extracted from the $\omega$-spectrum ${\mathfrak{mod}}_R$, (as in constr - module omega-monoids, with ${\mathfrak{mod}}_R^n\simeq nCat$).

General construction The functors $(-{)}^{\simeq }:nCat\to 0Cat$ that compute the internal groupoint of an $n$-category are monoidal with respect to the Cartesian monoidal structure (because it is a right adjoint, it preserves limits). Hence, these induce functors $CAlg(nCat)\to CAlg(S)$. Under this, diagram, we have

\begin{CD} CAlg(nCat) @>{\omega}>> CAlg((n-1)Cat)\\ @V(-)^\simeq VV @V (-)^\simeq VV \\ CAlg(S) @>{\Omega}>> CAlg(S) \end{CD}$$ Taking the limit as $n \to \infty$, we get a functor $$(-)^\simeq:\omega Alg \to CAlg(Sp)$$ **Example** $\mathfrak{mod}_R^{\simeq}$ is an $E_\infty$ ring spectrum whose $n$-th space is $nCat_R^\simeq$. The delooping data $$\Omega^{\infty - n}\mathfrak{mod}_R^\simeq \simeq \Omega^{\infty - (n+1)}\mathfrak{mod}_R^\simeq$$ is provided by equivalences $$nCat_R^\simeq \simeq \omega ((n+1)Cat_R)^\simeq \simeq \Omega ((n+1)Cat_R^\simeq)$$ **Example: shifts (finish...this should reproduce the Brauer spectra)** $(\omega\mathfrak{mod}_R)^{\simeq}$ is an $E_\infty$ ring spectrum whose $n$-th space is $(n-1)Cat_R^\simeq$. The delooping data $$\Omega^{\infty - n}\mathfrak{mod}_R^\simeq \simeq \Omega^{\infty - (n+1)}\mathfrak{mod}_R^\simeq$$ is provided by equivalences $$nCat_R^\simeq \simeq \omega ((n+1)Cat_R)^\simeq \simeq \Omega ((n+1)Cat_R^\simeq)$$