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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2410.05921 |
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| _version_ | 1866914967948623872 |
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| author | Dias, João Dinis, Bruno Marques, Pedro Macias |
| author_facet | Dias, João Dinis, Bruno Marques, Pedro Macias |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_05921 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Bridging Meadows and Sheaves Dias, João Dinis, Bruno Marques, Pedro Macias Commutative Algebra Logic Rings and Algebras 16S60, 16U90, 06B15, 13L05 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. |
| title | Bridging Meadows and Sheaves |
| topic | Commutative Algebra Logic Rings and Algebras 16S60, 16U90, 06B15, 13L05 |
| url | https://arxiv.org/abs/2410.05921 |