Théorie et applications des catégories

研究建议分为三个部分：(1)拟范畴理论。代数几何和同伦理论已经发展出相当数量的机制来解决它们的问题。下一个工具是更高堆栈理论。西格尔范畴是由Hirschowitz和Simpson引入的，作为发展高堆栈理论的框架。但是这个框架的复杂性使得取得进展变得困难。我建议使用准范畴。准范畴的概念在30多年前由博德曼和沃格特提出，但它的潜力被低估了。我已经证明了他们的表达能力比西格尔类别更强大。我的目标是证明范畴论可以推广到准范畴。我相信这将会产生将大量的知识和直觉从范畴论转移到同伦论、高范畴论和高拓扑论的效果。(2)同调运算的代数和几何。在Terrence Bisson的指导下，我们利用q环理论重组了mod2 Dyer-Lashof运算的代数。这包括环空间中的乘法同调运算。我们希望将这项工作推广到奇素数的同调模。(3)自由双补类。我们计划发表我们关于自由双完成分类的工作，并探索其在博弈论和计算机科学中的应用。

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.