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 to Modern treatment of Dirac monopoles and related topics.

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 $\mathbb P^1$.

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?

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 to Modern treatment of Dirac monopoles and related topics.

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 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?

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 to Modern treatment of Dirac monopoles and related topics.

Let us work with some version of algebraic varieties. Let $X$ be one such.

Given, say, a regular function on $X\setminus\{\text{a point}\}$, 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 $\mathbb P^1$.

Suppose given a line bundle on $X\setminus\{\text{a point}\}$. 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 $\mathbb E^\infty$ that is to $\mathbb P^\infty$ as $\mathbb P^1$ is to $\mathbb A^1$? That is, such that $\mathbb P^\infty$ maps to $\mathbb E^\infty$, representing a hypothetical map (functor?) from "regular" line bundles to "meromorphic" line bundles?

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 to Modern treatment of Dirac monopoles and related topics.

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 $\mathbb P^1$.

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?