丰富范畴与热带数学
Enriched categories and tropical mathematics
摘要：
本文综述了Quantale上的丰富范畴论与热带数学的联系。Quantale或完全幂等半环，以及系数矩阵，是这两个领域的基本对象。我们首先解释标准的范畴理论结构的矩阵，即组成，右扩展，右提升和Isbell船体。一路上，我们回顾已知的重新制定（由于埃利奥特和威勒顿）的热带多面体，有向紧跨度和勒让德-Fenchel变换通过这些建设，说明他们的无处不在的热带数学和相关领域。然后，我们考虑完全半模的Quantale $\mathcal{Q}$，热带模拟向量空间的领域，并提到Stubbe的结果确定他们的骨架和完整的$\mathcal{Q}$类别。为了弥合丰富的范畴论和热带数学之间的差距，我们假设在这两个领域都没有知识。
英文摘要：
This is a survey paper on the connection of enriched category theory over a quantale and tropical mathematics. Quantales or complete idempotent semirings, as well as matrices with coefficients in them, are fundamental objects in both fields. We first explain standard category-theoretic constructions on matrices, namely composition, right extension, right lifting and the Isbell hull. Along the way, we review known reformulations (due to Elliott and Willerton) of tropical polytopes, directed tight spans and the Legendre--Fenchel transform by means of these constructions, illustrating their ubiquity in tropical mathematics and related fields. We then consider complete semimodules over a quantale $\mathcal{Q}$, a tropical analogue of vector spaces over a field, and mention Stubbe's result identifying them with skeletal and complete $\mathcal{Q}$-categories. With the aim to bridge a gap between enriched category theory and tropical mathematics, we assume no knowledge in either field.
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