In recent notes of complex geometry by Clausen–Scholze, they gave a theory of analytification of finite type $\mathbb C$-schemes. It seems to me that there is a non-commutative analogue which works for any DG-category over $\mathbb C$ (we take large categories, not just perfect objects there). For sake of simplicity, we ignore the size issues in the sequel.

$\DeclareMathOperator\Cond{Cond}\DeclareMathOperator\Liq{Liq}\DeclareMathOperator\an{an}$Let $\mathcal C$ be a DG-category over $\mathbb C$ (or more precisely, a presentable stable $\mathbb C$-linear $\infty$-category). Then $\Cond(\mathcal C)$ is tensored with $D(\Cond(\mathbb C))$, and via base change along $D(\Cond(\mathbb C))\to D(\Liq(\mathbb C))$, we get a $\Liq(\mathbb C)$-linear category $\mathcal C^{\Liq}$.

Recall that the analytification of $A:=\mathbb C[T]$ is obtained by killing the boundary idempotent algebra $A_\infty:=A(\{|T|\gg O(1)\})$ in $D(\Liq(\mathbb C[T]))$. The idempotent algebra $A_\infty$ gives rise to a split Verdier sequence (we adopt the definition in Calmès–Dotto–Harpez–Hebestreit–Land–Moi–Nardin–Nikolaus–Steimle, *Hermitian $K$-theory for stable $\infty$-categories II: cobordism categories and additivity*, including the splitness below)
$$
D(A_\infty)\longrightarrow D(\Liq(\mathbb C[T]))\longrightarrow C(\mathbb C[T],\mathbb C[T]).
$$
The analytification of every finitely generated $\mathbb C$-algebra $B$ can be understood as killing $A_\infty$ in $D(\Liq(B))$ for every map $A\to B$ of $\mathbb C$-algebras. This also works for $\mathcal C^{\Liq}$. We form the full subcategory $\mathcal C^{\an}\subseteq\mathcal C^{\Liq}$ spanned by objects $M\in\mathcal C^{\Liq}$ such that, for every $\Liq(\mathbb C)$-linear functor $F\colon D(\Liq(A))\to\mathcal C^{\Liq}$ with right adjoint $G\colon\mathcal C^{\Liq}\to D(\Liq(A))$, we have $G(M)\in C(\mathbb C[T],\mathbb C[T])$. Under finiteness conditions on $\mathcal C$ (a possible option is *of finite type* in Toën–Vaquié, *Moduli of objects in DG-categories*), it seems that this could be tested on finitely many $F$, and thus $\mathcal C^{\Liq}\to\mathcal C^{\an}$ is a split Verdier quotient.

Here are some mutually related questions:

- Can we compare this construction with any classical notion of analytification in noncommutative algebraic geometry?
- Can we apply any variant of Rosenberg's spectrum to this situation? This might be closely related to Soibelman,
*On non-commutative analytic spaces over non-archimedean fields*. Rosenberg's spectrum usually only applies to non-derived settings, although he has also worked out some version which might be applicable to derived settings.