EAPSI:Studying the Interplay between Localization and Categorical Algebra via Algebraic Topology

EAPSI：通过代数拓扑研究定位与分类代数之间的相互作用

• 批准号: 1414942
• 负责人: David White
• 金额: $ 0.51万
• 依托单位: White David
• 依托单位国家: 美国
• 项目类别: Fellowship Award
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-06-01 至 2015-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1414942&HistoricalAwards=false
• 关键词: EAPSI; Studying; Interplay; between; Localization

Localization is a fundamental tool in mathematics which allows one to zoom in on the pertinent information in a problem. In the context of category theory?a mathematical theory which studies objects and their relationships?localization is a way to view two different objects as equivalent. As humans we do this all the time. For example, if two different driving routes take the same amount of time we might view them as equivalent. It is often advantageous to study objects which have additional algebraic structure, e.g. objects with a multiplication. The research in this proposal studies the interplay between localization and algebraic structure using tools from homotopy theory, which views two objects as equivalent if one can be deformed into the other. Homotopy Theory is a subfield of algebraic topology, a fundamental area of mathematics which blends algebra and topology and uses tools from each to study the other. The applications of this work will strengthen the developing bond between homotopy theory and category theory, will allow tools from each area to be applied in the other, and will lead to a better understanding of both fields. The generality of this approach allows for applications in many different fields of mathematics, physics, and computer science.The setting of monoidal model categories guarantees the efficacy of the localization procedure described above. In this setting there is a further localization known as Bousfield localization, and algebraic structure is encoded via vehicles for universal algebra known as operads. The Principal Investigator?s recent work allows operad-algebras to be studied after Bousfield localization has been applied to the underlying category. Recent work of Michael Batanin?with whom this research will be jointly conducted at Macquarie University?allows for Bousfield localization to be applied to the category of operad-algebras. The project will attempt to link up these two approaches, to relate their outputs, to pool the two proof strategies, and to reduce the conditions required so that this work will apply to a wider class of model categories. This NSF EAPSI award is funded in collaboration with the Australian Academy of Science.

局部化是数学中的一个基本工具，它允许人们放大问题中的相关信息。在范畴理论的背景下？研究物体及其相互关系的数学理论？本地化是将两个不同的对象视为等同的一种方式。作为人类，我们一直在这样做。例如，如果两条不同的驾驶路线花费相同的时间，我们可能会认为它们是等效的。研究具有额外代数结构的对象通常是有利的，例如具有乘法的对象。该提案中的研究使用同伦理论的工具研究了局部化和代数结构之间的相互作用，同伦理论认为两个对象是等价的，如果一个可以变形为另一个。同伦理论是代数拓扑学的一个分支，是数学的一个基本领域，它融合了代数和拓扑学，并使用每一个工具来研究另一个。这项工作的应用将加强同伦理论和范畴理论之间的发展纽带，将允许从每个领域的工具被应用于其他领域，并将导致更好地理解这两个领域。这种方法的通用性允许在数学、物理和计算机科学的许多不同领域中应用。monoidal模型类别的设置保证了上述定位过程的有效性。在这种情况下，还有一种局部化称为Bousfield局部化，而代数结构是通过称为操作数的泛代数的载体来编码的。首席研究员？最近的工作允许运算代数的研究后，Bousfield本地化已被应用到基本范畴。Michael Batanin的作品这项研究将与谁在麦考瑞大学联合进行？允许Bousfield本地化被应用到运算代数的范畴。该项目将试图将这两种方法联系起来，将它们的输出联系起来，汇集两种证明策略，并减少所需的条件，以便这项工作将适用于更广泛的模型类别。这个NSF EAPSI奖是与澳大利亚科学院合作资助的。