Applications of higher topos theory to homotopy theory

我的研究方案包括三个相互关联的主题： (1)高等拓扑论和古德威利演算； (2)高次拟范畴理论； (3)同伦型理论。 主题1：我们的长期目标是为更高的拓扑和更高的拓扑之间的态射开发一种微积分。本文证明的古德威利微积分的Blakers-Massey定理是一个重要的工具[ABFJ2]。我们计划进一步发展高等拓朴理论和古德威利微积分之间的联系。我们将在更高的TOPO的子地形之间引入一种新的操作。通过使用固定的子拓扑点迭代操作，我们得到了递增的子拓扑点序列，从而得到了一个推广了Goodwillie塔的左精确定域化塔。我们将证明塔的各层是稳定物体上的主丛。 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-拟范畴被定义为Ara在Theta-n-Sets[Ar]范畴上构造的模型结构中的Fibrant对象。该模型结构等价于Rezk构造的完备SegalTheta-n-空间的模型结构。但在n&gt；1的情况下，还没有已知的n-拟范畴的组合描述。我正在与Ara合作确定一个特殊的外角列表，这些外角与内角相连，应该刻画n-拟范畴。我们期望n-拟范畴的理论比现有的(无穷大，n)-范畴理论更简单。我们期待着它在余边论和更高代数中的应用。 主题3：没有什么能比阿沃迪-沃伦和沃沃茨基发现的马丁-Löf类型理论的同伦解释更能说明数学的统一性了。同伦型理论(Hott)是从这些发现中产生的新的数学领域。有证据表明，Hott可以有效地对同伦理论做出贡献，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，部落是一个带有家庭的内在概念的类别，称为纤维。凡是可以用部落语言证明的东西，都应该在类型论和同伦论中得到证明。 我们正在计划发展部落语言中的(无穷大，n)范畴的公理理论。我们还计划用部落语言来表达高等拓扑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.