A 4-fold categorical equivalence

TL;DR

This paper establishes a 4-fold categorical equivalence linking quiver grassmannians, smooth projective curves, Riemann surfaces, and fields of transcendence degree 1, with implications for elliptic curves.

Contribution

It constructs a novel 4-fold categorical equivalence connecting geometric and algebraic structures, extending Reineke's work to include elliptic curves.

Findings

Category of elliptic curves is isomorphic to a category of quiver grassmannians

Provides an analytic group structure to certain quiver grassmannians

Establishes a 4-fold categorical equivalence among key mathematical objects

Abstract

In this note, we will illuminate some immediate consequences of work done by Reineke that may prove to be useful in the study of elliptic curves. In particular, we will construct an isomorphism between the category of smooth projective curves with a category of quiver grassmannians. We will use this to provide a 4-fold categorical equivalence between a category of quiver grassmannians, smooth projective curves, compact Riemann surfaces and fields of transcendence degree 1 over $\mathbb{C}$. We finish with noting that the category of elliptic curves is isomorphic to a category of quiver grassmannians, whence providing an analytic group structure to a class of quiver grassmannians.