We propose a new sheaf-theoretical method for the calculation of the monodromy zeta functions of Milnor fibrations. As an application, classical formulas of Kushnirenko [11] and Varchenko [23] etc. concerning polynomials on C will be generalized to polynomial functions on any toric variety.

We define in the setting of homotopy type theory an H-space structure on S3. Hence we obtain a description of the quaternionic Hopf fibration S3 ↪→ S7 S4, using only homotopy invariant tools.

We show “free theorems” in the style of Wadler on homotopy type theory as consequences of relational parametricity. It follows that, for each space, its higher inductive definition and impredicative definition have the same homotopy groups. An impredicative definition of a space can help us to find generators of homotopy groups of the space.

We propose a notion of groupoid homotopy for generalized maps. This notion of groupoid homotopy generalizes the notions of natural transformation and strict homotopy for functors. The groupoid homotopy type of a Lie groupoid is shown to be invariant under Morita equivalence. As an application we consider orbifolds as groupoids and study the orbifold homotopy between orbifold maps induced by the...

We will give a detailed account of why the simplicial sets model of the univalence axiom due to Voevodsky also models W-types. In addition, we will discuss W-types in categories of simplicial presheaves and an application to models of set theory.

The notion of a natural model of type theory is defined in terms of that of a representable natural transfomation of presheaves. It is shown that such models agree exactly with the concept of a category with families in the sense of Dybjer, which can be regarded as an algebraic formulation of type theory. We determine conditions for such models to satisfy the inference rules for dependent sums ...

∗This research was sponsored in part by the National Science Foundation under grant numbers CCF-1116703 and CCF-1445995 (REU). The views and conclusions contained in this document are those of the author and should not be interpreted as representing the official policies, either expressed or implied, of any sponsoring institution, the U.S. government or any other entity. See http://homotopytype...

The goal of this thesis is to prove that π4(S) ' Z/2Z in homotopy type theory. In particular it is a constructive and purely homotopy-theoretic proof. We first recall the basic concepts of homotopy type theory, and we prove some well-known results about the homotopy groups of spheres: the computation of the homotopy groups of the circle, the triviality of those of the form πk(S) with k < n, and...