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

本地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。在范畴理论这一研究对象及其关系的数学理论的背景下，局部化是一种将两个不同的对象视为等同的方式。作为人类，我们一直都在这么做。例如，如果两条不同的行驶路线花费的时间相同，我们可能会认为它们是相等的。研究具有附加代数结构的对象，例如具有乘法的对象，通常是有利的。这项研究利用同伦理论中的工具来研究局部化和代数结构之间的相互作用。同伦理论认为，如果一个对象可以变形为另一个对象，则两个对象是等价的。同伦理论是代数拓扑学的一个子领域，是数学的一个基本领域，它融合了代数和拓扑学，并使用两者的工具来研究对方。这项工作的应用将加强同伦理论和范畴理论之间的发展联系，将允许来自两个领域的工具在另一个领域应用，并将导致对这两个领域的更好理解。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。一元论模型类别的设置保证了上述本地化过程的有效性。在这个背景下，还有一种进一步的局部化，称为布斯菲尔德局部化，而代数结构是通过被称为操作符的通用代数的载体编码的。首席研究员S最近的工作允许在Bousfield局部化应用于基础范畴之后研究算子代数。迈克尔·巴坦宁最近的工作--他将与麦格理大学共同进行这项研究--允许将布斯菲尔德本地化应用于算子代数范畴。该项目将试图将这两种方法联系起来，将它们的产出联系起来，将两种证明战略结合在一起，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。该NSF EAPSI奖是与澳大利亚科学院合作资助的。

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.