| The aim of this project is to investigate the relation between three different concepts which have each played an important role in topology and in other parts of mathematics, and which in recent years have come together in new and intriguing ways. These three concepts are that of category, of operad and of spectrum. Operads arose in topology with the specific aim of describing the algebraic structure of iterated loop spaces, and have become an indispensable tool for describing more general homotopy invariant structures in geometry. Categories were originally designed to organise all structures within a certain domain of mathematics, but later turned out to be important in themselves. Spectra arose in topology as embodiments of cohomology theories, and now form the central object of study in stable homotopy theory. Many of the structures arising in areas of modern homotopy theory, such as elliptic cohomology, topological quantum field theory and moduli stacks, involve higher categories, where composition is only associative up to higher isomorphisms. Much work is being done at present on finding suitable models for different kinds of such higher categories, and many of these involve operads. The Joyal-Lurie theory of quasi-categories is a case in point, where various problems from cohomology theory and higher category theory come together. This project aims to join topological expertise from Sheffield and categorical expertise from Utrecht, and investigate in this exciting new area of mathematics some of the specific questions described below. |
