Initially I wanted to call this question "Categorification of meromorphic functions?" but discovered so many questions about categorification that I became scared and decided to replace it with a more obscure title.

The question resulted from contemplating my own [answer](https://mathoverflow.net/a/442758/41291) to https://mathoverflow.net/q/440311/41291.

Let us work with some version of the notion of variety, say, complex varieties. Let $V$ be one such.

Given, say, a regular function on $V\setminus\{\text{a point}\}$, this function might have a pole of some order at that point, or an essential singularity. This is absolutely classical, one way to deal with it is to understand regular functions as morphisms to $\mathbb A^1$ and meromorphic functions as morphisms to $\mathbb P^1$.

**Slightly later:** as pointed out by [Alexandre Eremenko](https://mathoverflow.net/users/25510/alexandre-eremenko) in a comment, what I said above needs correction. Inverse image of infinity under a morphism to $\mathbb P^1$ only can reduce to a finite set of points when $V$ is 1-dimensional; for general $V$ it will have codimension 1. Whereas for morphisms to $\mathbb P^N$ with $N$ large considered later I am not sure what happens. Still let me leave the rest as is for the time being.

Suppose given a line bundle on $V\setminus\{\text{a point}\}$. What can happen at that point?

The analogy with functions might go like this. View "regular" line bundles as morphisms to $\mathbb P^N$ for large enough $N$, or maybe to $\mathbb P^\infty$ which is defined as an ind-scheme obtained from the union of all the $\mathbb P^N$ in a certain way.

Is there some $\mathbb X$ that is to $\mathbb P^\infty$ as $\mathbb P^1$ is to $\mathbb A^1$? That is, such that $\mathbb P^\infty$ maps to $\mathbb X$, representing a hypothetical map (functor?) from "regular" line bundles to "meromorphic" line bundles?