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

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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.