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演算;（2）高等拟范畴理论;（3）同伦类型理论。主题1：我们的长期目标是发展一个微分学的高toposes和高toposes之间的态射。一个关键的工具是Blakers-Massey定理古德威利的演算证明在我们的文件[ABFJ 2]。我们正计划进一步发展之间的联系更高的拓扑理论和古德威利的演算。我们将在一个更高拓扑的子拓扑之间引入一个新的运算。通过迭代的操作与一个固定的子拓扑，我们得到增加的子拓扑序列，从而塔的左精确局部化推广的古德威利塔。我们将证明塔的层是稳定对象上的主丛。主题二：我们的目标是在拟范畴理论的基础上发展（无穷，n）-范畴理论。在Ara关于Theta-n-sets范畴[Ar]所构造的模型结构中，n-拟范畴被定义为一个不规则对象.该模型结构等价于Rezk构造的完备Segal Theta-n-空间的模型结构。但是对于n-拟范畴，在n>1的情况下，还没有已知的组合描述。 我正在与阿拉一起确定一个特殊的外角的列表，这些外角与内角相连，应该表征n-准范畴。我们期望n-拟范畴理论比现有的（无穷，n）-范畴理论更简单。我们希望应用到配边和高等代数。主题3：很少有东西可以更好地说明统一的数学比同伦解释马丁-洛夫型理论发现的Awodey沃伦和Voevodsky。同伦型理论（Homotopy Type Theory，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，但两个领域之间的距离不应被低估。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，部落是一个配备了称为纤维化的家庭内部概念的范畴。每一个在部落语言中可证明的东西在类型论和同伦论中也应该是可证明的。我们计划在部落语言中发展一个（无穷，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.