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Given a scheme $X$ and sum of divisors $D$, you can take the line bundle $$\mathcal{O}_X(D)\ =\ \{\text{functions }f\text{ with [conditions on zeroes/poles]}\}\ \subseteq\ j_*\mathcal{O}_\eta\ =\ \mathcal{M}$$ where $\mathcal{M}$ is the sheaf of rational functions on $X$. It satisfies $\mathcal{O}_X(D+D') =\mathcal{O}_X(D)\otimes\mathcal{O}_X(D')$. Moreover, under weak conditions these are the only coherent sheaves with an inverse under $\otimes$.

One categorical level up, you can consider sheaves of categories on $X$, with $\text{QCoh}_X$ being the analogue of $\mathcal{O}_X$.

My question is: what is the analogue "$\text{QCoh}_X(D)$" of $\mathcal{O}_X(D)$?

If at all possible, the definition should mimic the above, defining it as a subsheaf of categories of $j_*\text{QCoh}_\eta$ defined by similar [conditions], and with the same identity for $D+D'$. Ideally, under the same weak conditions as above, these are the only invertible sheaves of categories.

Given a scheme $X$ and sum of divisors $D$, you can take the line bundle $$\mathcal{O}_X(D)\ =\ \{\text{functions }f\text{ with [conditions on zeroes/poles]}\}\ \subseteq\ j_*\mathcal{O}_\eta\ =\ \mathcal{M}$$ where $\mathcal{M}$ is the sheaf of rational functions on $X$ and $j:\eta\to X$ is the generic point (or the union of generic points of the irreducible components). It satisfies $\mathcal{O}_X(D+D') =\mathcal{O}_X(D)\otimes\mathcal{O}_X(D')$. Moreover, under weak conditions these are the only coherent sheaves with an inverse under $\otimes$.

One categorical level up, you can consider sheaves of categories on $X$, with $\text{QCoh}_X$ being the analogue of $\mathcal{O}_X$.

My question is: what is the analogue "$\text{QCoh}_X(D)$" of $\mathcal{O}_X(D)$?

If at all possible, the definition should mimic the above, defining it as a subsheaf of categories of $j_*\text{QCoh}_\eta$ defined by similar [conditions], and with the same identity for $D+D'$. Ideally, under the same weak conditions as above, these are the only invertible sheaves of categories.

Given a scheme $X$ and divisor $D\subseteq X$, you can take the line bundle $$\mathcal{O}_X(D)\ =\ \{\text{functions }f\text{ with [conditions on zeroes/poles]}\}\ \subseteq\ j_*\mathcal{O}_\eta\ =\ \mathcal{M}$$ where $\mathcal{M}$ is the sheaf of rational functions on $X$. It satisfies $\mathcal{O}_X(D+D') =\mathcal{O}_X(D)\otimes\mathcal{O}_X(D')$. Moreover, under weak conditions these are the only coherent sheaves with an inverse under $\otimes$.

One categorical level up, you can consider sheaves of categories on $X$, with $\text{QCoh}_X$ being the analogue of $\mathcal{O}_X$.

My question is: what is the analogue "$\text{QCoh}_X(D)$" of $\mathcal{O}_X(D)$?

If at all possible, the definition should mimic the above, defining it as a subsheaf of categories of $j_*\text{QCoh}_\eta$ defined by similar [conditions], and with the same identity for $D+D'$. Ideally, under the same weak conditions as above, these are the only invertible sheaves of categories.

Given a scheme $X$ and sum of divisors $D$, you can take the line bundle $$\mathcal{O}_X(D)\ =\ \{\text{functions }f\text{ with [conditions on zeroes/poles]}\}\ \subseteq\ j_*\mathcal{O}_\eta\ =\ \mathcal{M}$$ where $\mathcal{M}$ is the sheaf of rational functions on $X$. It satisfies $\mathcal{O}_X(D+D') =\mathcal{O}_X(D)\otimes\mathcal{O}_X(D')$. Moreover, under weak conditions these are the only coherent sheaves with an inverse under $\otimes$.

One categorical level up, you can consider sheaves of categories on $X$, with $\text{QCoh}_X$ being the analogue of $\mathcal{O}_X$.

My question is: what is the analogue "$\text{QCoh}_X(D)$" of $\mathcal{O}_X(D)$?

If at all possible, the definition should mimic the above, defining it as a subsheaf of categories of $j_*\text{QCoh}_\eta$ defined by similar [conditions], and with the same identity for $D+D'$. Ideally, under the same weak conditions as above, these are the only invertible sheaves of categories.

Given a scheme $X$ and divisor $D\subseteq X$, you can take the line bundle $$\mathcal{O}_X(D)\ =\ \{\text{functions }f\text{ with [conditions on zeroes/poles]}\}\ \subseteq\ j_*\mathcal{O}_\eta\ =\ \mathcal{M}$$ where $\mathcal{M}$ is the sheaf of rational functions on $X$. It satisfies $\mathcal{O}_X(D+D') =\mathcal{O}_X(D)\otimes\mathcal{O}_X(D')$. Moreover, under weak conditions these are the only coherent sheaves with an inverse under $\otimes$.

One categorical level up, you can consider sheaves of categories on $X$, with $\text{QCoh}_X$ being the analogue of $\mathcal{O}_X$.

My question is: what is the analogue "$\text{QCoh}_X(D)$" of $\mathcal{O}_X(D)$?

Ideally, the definition will mimic the above, with $\text{QCoh}_X(D)$ being a subsheaf of categories of $j_*\text{QCoh}_\eta$ defined by similar [conditions], and having $\text{QCoh}_X(D)\otimes\text{QCoh}(D')=\text{QCoh}(D+D')$. Ideally, under the same weak conditions as above, these are the only invertible sheaves of categories.

Given a scheme $X$ and divisor $D\subseteq X$, you can take the line bundle $$\mathcal{O}_X(D)\ =\ \{\text{functions }f\text{ with [conditions on zeroes/poles]}\}\ \subseteq\ j_*\mathcal{O}_\eta\ =\ \mathcal{M}$$ where $\mathcal{M}$ is the sheaf of rational functions on $X$. It satisfies $\mathcal{O}_X(D+D') =\mathcal{O}_X(D)\otimes\mathcal{O}_X(D')$. Moreover, under weak conditions these are the only coherent sheaves with an inverse under $\otimes$.

One categorical level up, you can consider sheaves of categories on $X$, with $\text{QCoh}_X$ being the analogue of $\mathcal{O}_X$.

My question is: what is the analogue "$\text{QCoh}_X(D)$" of $\mathcal{O}_X(D)$?

If at all possible, the definition should mimic the above, defining it as a subsheaf of categories of $j_*\text{QCoh}_\eta$ defined by similar [conditions], and with the same identity for $D+D'$. Ideally, under the same weak conditions as above, these are the only invertible sheaves of categories.