Point-free Construction of Real Exponentiation

Ming-fai Ng, S. Vickers
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引用次数: 4

Abstract

We define a point-free construction of real exponentiation and logarithms, i.e.\ we construct the maps $\exp\colon (0, \infty)\times \mathbb{R} \rightarrow \!(0,\infty),\, (x, \zeta) \mapsto x^\zeta$ and $\log\colon (1,\infty)\times (0, \infty) \rightarrow\mathbb{R},\, (b, y) \mapsto \log_b(y)$, and we develop familiar algebraic rules for them. The point-free approach is constructive, and defines the points of a space as models of a geometric theory, rather than as elements of a set - in particular, this allows geometric constructions to be applied to points living in toposes other than Set. Our geometric development includes new lifting and gluing techniques in point-free topology, which highlight how properties of $\mathbb{Q}$ determine properties of real exponentiation. This work is motivated by our broader research programme of developing a version of adelic geometry via topos theory. In particular, we wish to construct the classifying topos of places of $\mathbb{Q}$, which will provide a geometric perspective into the subtle relationship between $\mathbb{R}$ and $\mathbb{Q}_p$, a question of longstanding number-theoretic interest.
实幂的无点构造
我们定义了实幂和对数的无点构造,即。我们构造了映射$\exp\colon (0, \infty)\times \mathbb{R}\rightarrow \!(0,\infty),\, (x, \zeta) \mapsto x^\zeta$和$\log\colon(1,\infty)\times (0, \infty) \rightarrow\mathbb{R},\, (b, y) \mapsto\log_b(y)$,并为它们制定了熟悉的代数规则。无点方法是建设性的,并将空间中的点定义为几何理论的模型,而不是作为集合的元素-特别是,这允许几何结构应用于生活在集合以外的拓扑中的点。我们的几何发展包括新的提升和粘接技术在无点拓扑中,这突出了$\mathbb{Q}$的性质如何决定实幂的性质。这项工作的动机是我们更广泛的研究计划,即通过拓扑理论发展对adelic几何的厌恶。特别是,我们希望构建$\mathbb{Q}$的位置分类拓扑,这将为$\mathbb{R}$和$\mathbb{Q}_p$之间的微妙关系提供几何视角,这是一个长期存在的数论兴趣问题。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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