constr - module omega-monoids

Trans-categorical representations

Example (module $\omega$-algebras of $E_{\infty }$-rings)

We have

$nS{t}_A\simeq En{d}_{{(n+1)\mathcal{S}\mathcal{T}}_A}^{\otimes }({n\mathcal{S}\mathcal{T}}_A)\simeq {\omega }_n((n+1)S{t}_A))$ Hence these assemble into an $\omega$-algebra ${\mathfrak{mod}}^A$ with ${\mathfrak{mod}}_n^A\simeq nS{t}_A^{\otimes }$.

Verbally we’ll refer to this as the “universal $\omega$-algebra of $A$-modules”: it encodes $n$-categorical modules of the $E_{\infty }$-algebra $A$, for each $n$, and the de-categorification comparison maps.

Example (comodule $\omega$-algebras of $E_{\infty }$-corings) Dually, we have for an $E_{\infty }$-coalgebra $H$, an $\omega$-algebra ${\mathfrak{comod}}^H$,

Example (representations of affine commutative algebraic groups) Let $\mathcal{H}\in coAl{g}_{E_{\infty }}(Al{g}_{E_{\infty }}(Sp)))$ be a cocoommutative Hopf algebra. We’ll view it geometrically as an affine $E_{\infty }$-algebraic group in the world of non-connective spectral algebraic geometry, $\mathbb{G}:=Spec(\mathcal{H})$. We’ll then define

${\mathfrak{rep}}^{\mathbb{G}}:={\mathfrak{comod}}^H$ where $H$ is the image of $\mathcal{H}$ under $coAl{g}_{E_{\infty }}(Al{g}_{E_{\infty }}(Sp)))\to Al{g}_{E_{\infty }}(Sp)$

Warning An $E_1$-coalgebra $B$ does NOT give an $\omega$ algebra. So only $E_{\infty }$-commutative algebraic groups get a infinite tower of categorified representations. For example, this construction applies to Quillen-oriented spectral formal groups, objects appearing in TMF and TAF. It does NOT apply to a general descent groupoid, which is in general, very non-$E_2$.