Bridging Meadows and Sheaves

TL;DR

This paper establishes a formal connection between sheaves of rings over topological spaces and algebraic structures called common meadows, providing constructions and correspondences that unify these mathematical concepts.

Contribution

It introduces a novel equivalence between presheaves of rings and pre-meadows, and constructs a method to produce common meadows from sheaves, linking sheafification with meadow formation.

Findings

Equivalence between presheaves of rings and pre-meadows.

Construction of common meadows from sheaves of rings.

Correspondence between sheafification and meadow formation.

Abstract

We bridge sheaves of rings over a topological space with common meadows (algebraic structures where the inverse for multiplication is a total operation). More specifically, we show that the subclass of pre-meadows with $\mathbf{a}$, coming from the lattice of open sets of a topological space $X$, and presheaves over $X$ are the same structure. Furthermore, we provide a construction that, given a sheaf of rings $\mathcal{F}$ on $X$ produces a common meadow as a disjoint union of elements of the form $\mathcal{F}(U)$ indexed over the open subsets of $X$. We also establish a correspondence between the process of going from a presheaf to a sheaf (called sheafification) and the process of going from a pre-meadow with $\mathbf{a}$ to a common meadow.