Théorie et applications des catégories

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The research proposal has three parts:(1) The theory of quasi-categories.Algebraic geometry and homotopy theory have developed a considerable amount of machinery for solving their problems. The theory of higher stacks is the next tool. Segal categories were introduced by Hirschowitz and Simpson as a framework for developing the theory of higher stacks. But the complexity of this framework makes progress difficult. I propose using quasi-categories instead. The notion of quasi-category was introduced  by Boardman and Vogt more than 30 years ago but its potential was underestimated. I have proved that their expressive power is greater than Segal categories. My goal is to show that category theory can be extended to quasi-categories. I believe that this will have the effect of transfering a huge body of knowledge and intuition from category theory to homotopy theory, higher category theory and higher topos theory.(2) The algebra and geometry of homology operations.With Terrence Bisson we have reorganised the algebra of mod 2 Dyer-Lashof operation by using the theory of Q-rings. This includes the multiplicative homology operations in ring spaces. We hope the extend this work to homology modulo an odd prime.(3) Free bicompletion of categories.We plan to publish our work on free bicompletion of categories and to explore its applications to game theory and computer science.