Applications of higher topos theory to homotopy theory

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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.*****************