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arXiv · 2410.05921

Bridging Meadows and Sheaves

João Dias·Bruno Dinis·Pedro Macias Marques
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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.

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João Dias, Bruno Dinis, Pedro Macias Marques. 2024-10-08. Bridging Meadows and Sheaves. https://arxiv.org/abs/2410.05921

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