Théorie et applications des catégories

研究内容包括三个部分：（1）拟范畴理论：代数几何和同伦理论已经发展了相当数量的解决问题的机制。下一个工具是更高层次的理论。Segal范畴是由Hirschowitz和Simpson引入的，作为发展高阶堆栈理论的框架。但这一框架的复杂性使进展变得困难。我建议使用准范畴。准范畴的概念是由博德曼和沃格特在30多年前提出的，但它的潜力被低估了。我已经证明了它们的表达能力比西格尔范畴更强。我的目标是证明范畴论可以扩展到拟范畴。我相信，这将产生这样的效果，即把大量的知识和直觉从范畴论转移到同伦论、更高的范畴论和更高的拓扑论。(2)同调运算的代数和几何.我们与Terrence Bisson一起利用Q-环理论重新组织了模2 Dyer-Lashof运算的代数.这包括环空间中的乘法同调运算。我们希望将这一工作推广到模奇素数的同调。(3)自由双完备的范畴。我们计划发表我们的工作自由双完备的范畴，并探讨其应用到博弈论和计算机科学。

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.