Applications of higher topos theory to homotopy theory

我的研究计划包括三个相互关联的主题：***(1)高等拓扑理论和古德威利微积分；***(2)高准范畴理论；***(3)同伦类型理论。*********主题1：我们的长期目标是发展更高拓扑和更高拓扑之间的态射的微分学。一个关键的工具是我们在论文[ABFJ2]中证明的Goodwillie微积分的Blakers-Massey定理。我们计划进一步发展高等拓扑理论与古德威利微积分之间的联系。我们将在更高的拓扑的子拓扑之间引入一种新的操作。通过对固定子拓扑的迭代运算，我们得到了子拓扑的递增序列，从而得到了广义的Goodwillie塔的左精确定位塔。我们将证明塔的各层是稳定物体上的主束。*********主题2：我们的目标是在拟范畴理论的模型上发展（无穷，n）范畴理论。n-拟范畴定义为Ara在-n集范畴上构造的模型结构中的一个纤维对象[Ar]。模型结构等价于Rezk构造的完全Segal theta -n空间的模型结构。但是在n bbbb1的情况下，没有已知的n个拟范畴的组合描述。我正在和Ara一起确定一个特殊的外部角的列表，这些外部角与内部角相连，应该表征n个准类别。我们期望n类范畴的理论将比现有的（无穷，n）类范畴的理论更简单。我们期望将其应用于协数和高等代数。******主题3：没有什么比Awodey-Warren和Voevodsky发现的Martin-Löf类型理论的同伦解释更能说明数学的统一性了。同伦类型理论（HoTT）是由这些发现而产生的数学新领域。有证据表明，HoTT可以有效地促进同伦理论，但两者之间的距离不可低估。我们正在发展部落理论，作为这两个领域之间的桥梁。粗略地说，部落是一个具有称为纤维化的内部家庭概念的类别。任何能用部落语言证明的东西在类型论和同伦论中都应该是可证明的。******我们计划在部落语言中发展一个（无限，n）范畴的公理化理论。我们还计划用部落的语言来表达更高的拓扑理论和古德威利的微积分。*****************

My research proposal consists of three interconnected themes:***(1) Higher topos theory and Goodwillie's calculus; ***(2) Theory of higher quasi-categories; ***(3) Homotopy type theory. *********Theme 1: Our long term goal is to develop a differential calculus for higher toposes and morphisms between higher toposes. A key tool is the Blakers-Massey theorem for Goodwillie's Calculus proved in our paper [ABFJ2]. We are planning to develop further the connection between higher topos theory and Goodwillie's calculus. We shall introduce a new operation between the sub-toposes of a higher topos. By iterating the operation with a fixed sub-topos, we obtain increasing sequence of sub-toposes and hence a tower of left exact localizations which generalizes the Goodwillie tower. We shall prove that the layers of the tower are principal bundles over stable objects. *********Theme 2: Our goal is to develop the theory of (infinity, n)-categories on the model of the theory of quasi-categories. A n-quasi-category is defined to be a fibrant object in the model structure constructed by Ara on the category of Theta-n-sets [Ar]. The model structure is equivalent to the model structure for complete Segal Theta-n-spaces constructed by Rezk. But there is no known combinatorial description of n-quasi-categories in the case n>1. I am working with Ara to determine a list of special outer horns which, joined to inner horns, should caracterise n-quasi-categories. We expect that the theory of n-quasi-categories will be simpler than existing theory of (infinity,n)-categories. We expect applications to cobordism and to higher algebras.******Theme 3: Few things can better illustrate the unity of mathematics than the homotopy interpretation of Martin-Löf type theory discovered by Awodey-Warren and by Voevodsky. Homotopy type theory (HoTT) is the new field of mathematics arising from these discoveries. There are evidences that HoTT can contribute effectively to homotopy theory, but the distance between the two fields should not be underestimated. We are developing the theory of tribes as a bridge between the two fields. Roughly speaking, a tribe is a category equipped with an internal notion of family called fibration. Every thing provable in the language of tribes should be provable in type theory and in homotopy theory.******We are planning to develop an axiomatic theory of (infinity, n)-categories in the language of tribes. We are also planning to express higher topos theory and Goodwillie's Calculus in the language of tribes.*****************