We define and develop two-level type theory, a version of MartinLöf type theory which is able to combine two type theories. In our case of interest, the first of these two theories is homotopy type theory (HoTT) which may include univalent universes and higher inductive types. The second is a traditional form of type theory validating uniqueness of identity proofs (UIP) and may be understood as...

We study the homotopy types of spaces of algebraic (rational) maps from real projective spaces into complex projective spaces. It was already shown in [1] that the inclusion of the first space into the second one is a homotopy equivalence. In this paper we prove that the homotopy types of the terms of the natural ‘degree’ filtration approximate closer and closer the homotopy type of the space o...

This note is a response to Bas Spitter's email of 28 February 2014 about ML4PG:"We (Jason actually) are adding dependency graphs to our HoTT library: https://github.com/HoTT/HoTT/wiki I seem to recall that finding the dependency graph was a main obstacle preventing machine learning to be used in Coq. However, you seem to have made progress on this. What tool are you using? https://anne.pacalet....

Let fa,b(z, z̄) = z a1+b1 1 z̄ b1 1 + · · ·+z an+bn n z̄ bn n be a polar weighted homogeneous polynomial with aj > 0, bj ≥ 0, j = 1, . . . , n and let fa(z) = z1 1 + · · ·+z an n be the associated weighted homogeneous polynomial. Consider the corresponding link variety Ka,b = f −1 a,b (0) ∩ S2n−1 and Ka = f−1 a (0) ∩ S2n−1. Ruas-Seade-Verjovsky [4] proved that the Milnor fibrations of fa,b and fa ...

space is said to be rational if its homotopy groups are rational vector spaces. Quillen has shown that up to homotopy there is a one-one correspondence between rational spaces and differential graded Lie algebras over 0. Call a two-connected space tame if the divisibility of its homotopy groups increases with dimension just quickly enough to prevent stable k-invariants from appearing. We will s...

In homotopy type theory, the truncation operator ∥−∥n (for a number n ≥ −1) is often useful if one does not care about the higher structure of a type and wants to avoid coherence problems. However, its elimination principle only allows to eliminate into n-types, which makes it hard to construct functions ∥A∥n → B if B is not an n-type. This makes it desirable to derive more powerful elimination...

In the present paper, based on the previous work (Part I), we present a game semantics for the intensional variant of intuitionistic type theory that refutes the principle of uniqueness of identity proofs and validates the univalence axiom, though we do not interpret non-trivial higher propositional equalities. Specifically, following the historic groupoid interpretation by Hofmann and Streiche...

Homotopy type theory is an extension of Martin-Löf type theory, based on a correspondence with homotopy theory and higher category theory. In homotopy type theory, the propositional equality type becomes proof-relevant, and corresponds to paths in a space. This allows for a new class of datatypes, called higher inductive types, which are specified by constructors not only for points but also fo...

We present a new game semantics for Martin-Löf type theory (MLTT); our aim is to give a mathematical and intensional explanation of MLTT. Specifically, we propose a category with families (a categorical model of MLTT) of a novel variant of games, which induces an injective (when Id-types are excluded) and surjective interpretation of the intensional variant of MLTT equipped with unit-, empty-, ...

Recall from previous lectures the definitions of functionality and transport. Functionality states that functions preserve identity; that is, domain elements equal in their type map to equal elements in the codomain. Transportation states the same for type families. Traditionally, this means that if a =A a′, then B[a] true iff B[a′] true. In proof-relevant mathematics, this logical equivalence ...