Joyal's Conjecture in Homotopy Type Theory

Joyal's Conjecture asserts, in a mathematically precise way, that Martin--Lof dependent type theory gives rise to locally cartesian closed quasicategories. We prove this conjecture

Threshold Properties of Prime Power Subgroups with Application to Secure Integer Comparisons

We present a semantically secure somewhat homomorphic public-key cryptosystem working in sub-groups of $\mathbb{Z}_{n}^{*}$ of prime power order. Our scheme introduces a novel threshold homomorphic property, which we use to build a two-party protocol for secure integer comparison. In contrast to related work which encrypts and acts on each bit of the input separately, our protocol compares multiple input bits simultaneously within a single ciphertext. Compared to the related protocol of Damgård et al.~we present results showing this approach to be both several times faster in computation and lower in communication complexity

Univalent categories and the Rezk completion

We develop category theory within Univalent Foundations, which is a foundational system for mathematics based on a homotopical interpretation of dependent type theory. In this system, we propose a definition of ‘category’ for which equality and equivalence of categories agree. Such categories satisfy a version of the univalence axiom, saying that the type of isomorphisms between any two objects is equivalent to the identity type between these objects; we call them ‘saturated’ or ‘univalent’ categories. Moreover, we show that any category is weakly equivalent to a univalent one in a universal way. In homotopical and higher-categorical semantics, this construction corresponds to a truncated version of the Rezk completion for Segal spaces, and also to the stack completion of a prestack.</jats:p

Homotopy limits in type theory

<p>Working in homotopy type theory, we provide a systematic study of homotopy limits of diagrams over graphs, formalized in the Coq proof assistant. We discuss some of the challenges posed by this approach to the formalizing homotopy-theoretic material. We also compare our constructions with the more classical approach to homotopy limits via fibration categories.</p