defn - moduli-completion of a category

Definition The moduli-completion of an $\infty$-category in $C$ is given by

${\hat{C}}_{moduli}:=P(C{)}_{\ast }^{\%}$ (here the RHS is moduli completion of pointed presheaves, see defn - (formal) moduli-objects in a pointed category)

Project/conjecture

Let $\mathcal{O}$ be a non-unital $\infty$-operad, $V^{\otimes }$ a presentably symmetric monoidal stable $\infty$-category. Let $D\mathcal{O}$ denote the Koszul dual operad. There’s an equivalence

$${\widehat{Al{g}_{\mathcal{O}}(V)}}_{moduli}\simeq Al{g}_{D\mathcal{O}}(V)$$

That is, algebras over $\mathcal{O}$ moduli-complete to algebras over $D\mathcal{O}$.

Remark: even for $\mathcal{O}\in \{Comm,E_k\}$, the LHS above generalizes the category of formal moduli constructed by Lurie, Gatisgory-Rozenblyum in several ways

• It makes sense for $V\ne Mo{d}_{\mathbb{C}}$. Of particular interest are examples
  • $V=Mo{d}_R(Sp)$ for $R$ an $E_{\infty }$ ring spectrum. For example, $R\in \{\mathbb{S},T(n),K(n),TMF,H\mathbb{Z}/p^n\}$
  • It might contain “curved deformations/formal moduli equipped with (non-flat connections/twisted D-modules)“.

Also, we expect ${\widehat{Al{g}_{\mathcal{O}}(V)}}_{moduli}$ to admit various full subcategories constructed using a generalization of the notion in defn - nilpotent maps of commutative ring spectra